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**Unformatted text preview: **Digital Logic and
Microprocessor Design
With VHDL Enoch O. Hwang
La Sierra University, Riverside To my wife and children, Windy, Jonathan and Michelle Contents
Contents ....................................................................................................................................................................
Preface ................................................................................................................................................................... Chapter 1 Designing Microprocessors......................................................................................
1.1 Overview of a Microprocessor .......................................................................................................................
1.2 Design Abstraction Levels..............................................................................................................................
1.3 Examples of a 2-to-1 Multiplexer ...................................................................................................................
1.3.1
Behavioral Level....................................................................................................................................
1.3.2
Gate Level..............................................................................................................................................
1.3.3
Transistor Level .....................................................................................................................................
1.4 Introduction to VHDL ....................................................................................................................................
1.5 Synthesis.......................................................................................................................................................
1.6 Going Forward..............................................................................................................................................
1.7 Summary Checklist.......................................................................................................................................
1.8 Problems ....................................................................................................................................................... Chapter 2 Digital Circuits.......................................................................................................... 2
2.1 Binary Numbers.............................................................................................................................................. 3
2.2 Binary Switch .................................................................................................................................................
2.3 Basic Logic Operators and Logic Expressions ...............................................................................................
2.4 Truth Tables....................................................................................................................................................
2.5 Boolean Algebra and Boolean Function .........................................................................................................
2.5.1
Boolean Algebra ....................................................................................................................................
2.5.2
* Duality Principle ...............................................................................................................................
2.5.3
Boolean Function and the Inverse........................................................................................................
2.6 Minterms and Maxterms...............................................................................................................................
2.6.1
Minterms..............................................................................................................................................
2.6.2
* Maxterms ..........................................................................................................................................
2.7 Canonical, Standard, and non-Standard Forms.............................................................................................
2.8 Logic Gates and Circuit Diagrams................................................................................................................
2.9 Example: Designing a Car Security System .................................................................................................
2.10 VHDL for Digital Circuits............................................................................................................................
2.10.1 VHDL code for a 2-input NAND gate.................................................................................................
2.10.2 VHDL code for a 3-input NOR gate....................................................................................................
2.10.3 VHDL code for a function ...................................................................................................................
2.11 Summary Checklist.......................................................................................................................................
2.12 Problems ....................................................................................................................................................... Chapter 3 Combinational Circuits............................................................................................
3.1 Analysis of Combinational Circuits................................................................................................................
3.1.1
Using a Truth Table ...............................................................................................................................
3.1.2
Using a Boolean Function......................................................................................................................
3.2 Synthesis of Combinational Circuits ..............................................................................................................
3.3 * Technology Mapping...................................................................................................................................
3.4 Minimization of Combinational Circuits ......................................................................................................
3.4.1
Karnaugh Maps....................................................................................................................................
3.4.2
Don’t-cares ..........................................................................................................................................
3.4.3
* Tabulation Method............................................................................................................................
3.5 * Timing Hazards and Glitches .................................................................................................................... 5 3.5.1
Using Glitches .....................................................................................................................................
3.6 BCD to 7-Segment Decoder .........................................................................................................................
3.7 VHDL for Combinational Circuits ...............................................................................................................
3.7.1
Structural BCD to 7-Segment Decoder................................................................................................
3.7.2
Dataflow BCD to 7-Segment Decoder ................................................................................................
3.7.3
Behavioral BCD to 7-Segment Decoder..............................................................................................
3.8 Summary Checklist.......................................................................................................................................
3.9 Problems ....................................................................................................................................................... Chapter 4 Standard Combinational Components ...................................................................
4.1 Signal Naming Conventions ...........................................................................................................................
4.2 Adder ..............................................................................................................................................................
4.2.1
Full Adder..............................................................................................................................................
4.2.2
Ripple-carry Adder ................................................................................................................................
4.2.3
* Carry-lookahead Adder.......................................................................................................................
4.3 Two’s Complement Binary Numbers .............................................................................................................
4.4 Subtractor........................................................................................................................................................
4.5 Adder-Subtractor Combination.....................................................................................................................
4.6 Arithmetic Logic Unit...................................................................................................................................
4.7 Decoder.........................................................................................................................................................
4.8 Encoder.........................................................................................................................................................
4.8.1
* Priority Encoder................................................................................................................................
4.9 Multiplexer ...................................................................................................................................................
4.9.1
* Using Multiplexers to Implement a Function ...................................................................................
4.10 Tri-state Buffer .............................................................................................................................................
4.11 Comparator ...................................................................................................................................................
4.12 Shifter ...........................................................................................................................................................
4.12.1 * Barrel Shifter ....................................................................................................................................
4.13 * Multiplier ...................................................................................................................................................
4.14 Summary Checklist.......................................................................................................................................
4.15 Problems ....................................................................................................................................................... Chapter 5 * Implementation Technologies .............................................................................
5.1 Physical Abstraction .......................................................................................................................................
5.2 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET)..................................................................
5.3 CMOS Logic...................................................................................................................................................
5.4 CMOS Circuits ...............................................................................................................................................
5.4.1
CMOS Inverter ......................................................................................................................................
5.4.2
CMOS NAND gate................................................................................................................................
5.4.3
CMOS AND gate...................................................................................................................................
5.4.4
CMOS NOR and OR Gates ................................................................................................................ 1
5.4.5
Transmission Gate ...............................................................................................................................
5.4.6
2-input Multiplexer CMOS Circuit..................................................................................................... 1
5.4.7
CMOS XOR and XNOR Gates............................................................................................................ 1
5.5 Analysis of CMOS Circuits ......................................................................................................................... . 1
5.6 Using ROMs to Implement a Function ........................................................................................................ . 15
5.7 Using PLAs to Implement a Function .......................................................................................................... 1
5.8 Using PALs to Implement a Function .......................................................................................................... 1
5.9 Complex Programmable Logic Device (CPLD) ...........................................................................................
5.10 Field Programmable Gate Array (FPGA) .....................................................................................................
5.11 Summary Checklist.......................................................................................................................................
5.12 Problems ....................................................................................................................................................... 6 Chapter 6 Latches and Flip-Flops ............................................................................................
6.1 Bistable Element.............................................................................................................................................
6.2 SR Latch .........................................................................................................................................................
6.3 SR Latch with Enable .....................................................................................................................................
6.4 D Latch ...........................................................................................................................................................
6.5 D Latch with Enable .......................................................................................................................................
6.6 Clock...............................................................................................................................................................
6.7 D Flip-Flop .................................................................................................................................................. . 1
6.7.1
* Alternative Smaller Circuit ............................................................................................................... 1
6.8 D Flip-Flop with Enable ............................................................................................................................... 1
6.9 Asynchronous Inputs .................................................................................................................................... 1
6.10 Description of a Flip-Flop ............................................................................................................................
6.10.1 Characteristic Table ............................................................................................................................. 1
6.10.2 Characteristic Equation........................................................................................................................ 1
6.10.3 State Diagram ...................................................................................................................................... 1
6.10.4 Excitation Table................................................................................................................................... 1
6.11 Timing Issues................................................................................................................................................ 1
6.12 Example: Car Security System – Version 2..................................................................................................
6.13 VHDL for Latches and Flip-Flops................................................................................................................ 1
6.13.1 Implied Memory Element.................................................................................................................... 1
6.13.2 VHDL Code for a D Latch with Enable ..............................................................................................
6.13.3 VHDL Code for a D Flip-Flop ........................................................................................................... . 19
6.13.4 VHDL Code for a D Flip-Flop with Enable and Asynchronous Set and Clear ...................................
6.14 * Flip-Flop Types .........................................................................................................................................
6.14.1 SR Flip-Flop ........................................................................................................................................
6.14.2 JK Flip-Flop.........................................................................................................................................
6.14.3 T Flip-Flop...........................................................................................................................................
6.15 Summary Checklist.......................................................................................................................................
6.16 Problems ....................................................................................................................................................... Chapter 7 Sequential Circuits ................................................................................................... 2
7.1 Finite-State-Machine (FSM) Models..............................................................................................................
7.2 State Diagrams................................................................................................................................................
7.3 Analysis of Sequential Circuits.......................................................................................................................
7.3.1
Excitation Equation ...............................................................................................................................
7.3.2
Next-state Equation ...............................................................................................................................
7.3.3
Next-state Table.....................................................................................................................................
7.3.4
Output Equation...................................................................................................................................
7.3.5
Output Table ........................................................................................................................................
7.3.6
State Diagram ..................................................................................................................................... . 0
7.3.7
Example: Analysis of a Moore FSM ...................................................................................................
7.3.8
Example: Analysis of a Mealy FSM....................................................................................................
7.4 Synthesis of Sequential Circuits ...................................................................................................................
7.4.1
State Diagram ......................................................................................................................................
7.4.2
Next-state Table...................................................................................................................................
7.4.3
Implementation Table..........................................................................................................................
7.4.4
Excitation Equation and Next-state Circuit .........................................................................................
7.4.5
Output Table and Equation ................................................................................................................. . 1
7.4.6
FSM Circuit .........................................................................................................................................
7.4.7
Examples: Synthesis of Moore FSMs..................................................................................................
7.4.8
Example: Synthesis of a Mealy FSM...................................................................................................
7.5 Unused State Encodings and the Encoding of States....................................................................................
7.6 Example: Car Security System – Version 3..................................................................................................
7.7 VHDL for Sequential Circuits ...................................................................................................................... 7 7.8 * Optimization for Sequential Circuits .........................................................................................................
7.8.1
State Reduction................................................................................................................................... . 3
7.8.2
State Encoding .....................................................................................................................................
7.8.3
Choice of Flip-Flops ............................................................................................................................
7.9 Summary Checklist.......................................................................................................................................
7.10 Problems ....................................................................................................................................................... Chapter 8 Standard Sequential Components .......................................................................... 2
8.1 Registers .........................................................................................................................................................
8.2 Shift Registers.................................................................................................................................................
8.2.1
Serial-to-Parallel Shift Register .............................................................................................................
8.2.2
Serial-to-Parallel and Parallel-to-Serial Shift Register ..........................................................................
8.3 Counters..........................................................................................................................................................
8.3.1
Binary Up Counter.................................................................................................................................
8.3.2
Binary Up-Down Counter....................................................................................................................
8.3.3
Binary Up-Down Counter with Parallel Load .....................................................................................
8.3.4
BCD Up Counter .................................................................................................................................
8.3.5
BCD Up-Down Counter ......................................................................................................................
8.4 Register Files ................................................................................................................................................
8.5 Static Random Access Memory.................................................................................................................... 2
8.6 * Larger Memories .......................................................................................................................................
8.6.1
More Memory Locations ..................................................................................................................... 2
8.6.2
Wider Bit Width .................................................................................................................................. 2
8.7 Summary Checklist....................................................................................................................................... 2
8.8 Problems ....................................................................................................................................................... 2 Chapter 9 Datapaths .................................................................................................................. 2
9.1 Designing Dedicated Datapaths......................................................................................................................
9.1.1
Selecting Registers.................................................................................................................................
9.1.2
Selecting Functional Units.....................................................................................................................
9.1.3
Data Transfer Methods ..........................................................................................................................
9.1.4
Generating Status Signals ....................................................................................................................
9.2 Using Dedicated Datapaths...........................................................................................................................
9.3 Examples of Dedicated Datapaths ................................................................................................................
9.3.1
Simple IF-THEN-ELSE.......................................................................................................................
9.3.2
Counting 1 to 10 ..................................................................................................................................
9.3.3
Summation of n down to 1...................................................................................................................
9.3.4
Factorial ...............................................................................................................................................
9.3.5
Count Zero-One ...................................................................................................................................
9.4 General Datapaths.........................................................................................................................................
9.5 Using General Datapaths ..............................................................................................................................
9.6 A More Complex General Datapath .............................................................................................................
9.7 Timing Issues................................................................................................................................................
9.8 VHDL for Datapaths.....................................................................................................................................
9.8.1
Dedicated Datapath..............................................................................................................................
9.8.2
General Datapath .................................................................................................................................
9.9 Summary Checklist.......................................................................................................................................
9.10 Problems ....................................................................................................................................................... Chapter 10 Control Units ............................................................................................................
10.1 Constructing the Control Unit.........................................................................................................................
10.2 Examples ........................................................................................................................................................ 8 10.2.1 Count 1 to 10 .........................................................................................................................................
10.2.2 Summation of 1 to n ..............................................................................................................................
10.3 Generating Status Signals .............................................................................................................................
10.4 Timing Issues................................................................................................................................................
10.5 Standalone Controllers..................................................................................................................................
10.5.1 Rotating Lights ....................................................................................................................................
10.5.2 PS/2 Keyboard Controller....................................................................................................................
10.5.3 VGA Monitor Controller .....................................................................................................................
10.6 * ASM Charts and State Action Tables ....................................................................................................... . 3
10.6.1 ASM Charts ........................................................................................................................................ 3
10.6.2 State Action Tables..............................................................................................................................
10.7 VHDL for Control Units...............................................................................................................................
10.8 Summary Checklist.......................................................................................................................................
10.9 Problems ....................................................................................................................................................... 6
7
7
0
1
2
4 Chapter 11 Dedicated Microprocessors .....................................................................................
11.1 Manual Construction of a Dedicated Microprocessor ....................................................................................
11.2 Examples ........................................................................................................................................................
11.2.1 Greatest Common Divisor .....................................................................................................................
11.2.2 Summing Input Numbers.....................................................................................................................
11.2.3 High-Low Guessing Game ..................................................................................................................
11.2.4 Finding Largest Number ......................................................................................................................
11.3 VHDL for Dedicated Microprocessors.........................................................................................................
11.3.1 FSM + D Model...................................................................................................................................
11.3.2 FSMD Model .......................................................................................................................................
11.3.3 Behavioral Model ................................................................................................................................
11.4 Summary Checklist.......................................................................................................................................
11.5 Problems ....................................................................................................................................................... Chapter 12 General-Purpose Microprocessors .........................................................................
12.1 Overview of the CPU Design .........................................................................................................................
12.2 The EC-1 General-Purpose Microprocessor ...................................................................................................
12.2.1 Instruction Set........................................................................................................................................
12.2.2 Datapath.................................................................................................................................................
12.2.3 Control Unit ...........................................................................................................................................
12.2.4 Complete Circuit....................................................................................................................................
12.2.5 Sample Program...................................................................................................................................
12.2.6 Simulation............................................................................................................................................
12.2.7 Hardware Implementation ...................................................................................................................
12.3 The EC-2 General-Purpose Microprocessor .................................................................................................
12.3.1 Instruction Set......................................................................................................................................
12.3.2 Datapath...............................................................................................................................................
12.3.3 Control Unit .........................................................................................................................................
12.3.4 Complete Circuit..................................................................................................................................
12.3.5 Sample Program...................................................................................................................................
12.3.6 Hardware Implementation ...................................................................................................................
12.4 VHDL for General-Purpose Microprocessors ..............................................................................................
12.4.1 Structural FSM+D ...............................................................................................................................
12.4.2 Behavioral FSMD ................................................................................................................................
12.5 Summary Checklist.......................................................................................................................................
12.6 Problems ....................................................................................................................................................... 9 5 0
2
3
4
5
8
9
1
2
2
9
2
2 Appendix A Schematic Entry Tutorial 1 ....................................................................................
A.1 Getting Started ................................................................................................................................................
A.1.1
Preparing a Folder for the Project..........................................................................................................
A.1.2
Starting MAX+plus II............................................................................................................................
A.1.3
Starting the Graphic Editor ....................................................................................................................
A.2 Using the Graphic Editor ................................................................................................................................ 4
A.2.1
Drawing Tools ....................................................................................................................................... 4
A.2.2
Inserting Logic Symbols........................................................................................................................ 4
A.2.3
Selecting, Moving, Copying, and Deleting Logic Symbols...................................................................
A.2.4
Making and Naming Connections ......................................................................................................... 6
A.2.5
Selecting, Moving and Deleting Connection Lines ...............................................................................
A.3 Specifying the Top-Level File and Project .....................................................................................................
A.3.1
Saving the Schematic Drawing..............................................................................................................
A.3.2
Specifying the Project............................................................................................................................
A.4 Synthesis for Functional Simulation...............................................................................................................
A.5 Circuit Simulation...........................................................................................................................................
A.5.1
Selecting Input Test Signals ..................................................................................................................
A.5.2
Customizing the Waveform Editor ......................................................................................................
A.5.3
Assigning Values to the Input Signals .................................................................................................
A.5.4
Saving the Waveform File ...................................................................................................................
A.5.5
Starting the Simulator ..........................................................................................................................
A.6 Creating and Using the Logic Symbol.......................................................................................................... Appendix B VHDL Entry Tutorial 2 ...........................................................................................
B.1 Getting Started ................................................................................................................................................
B.1.1
Preparing a Folder for the Project..........................................................................................................
B.1.2
Starting MAX+plus II............................................................................................................................
B.1.3
Creating a Project ..................................................................................................................................
B.1.4
Editing the VHDL Source Code ............................................................................................................
B.2 Synthesis for Functional Simulation...............................................................................................................
B.3 Circuit Simulation...........................................................................................................................................
B.3.1
Selecting Input Test Signals ..................................................................................................................
B.3.2
Customizing the Waveform Editor ........................................................................................................
B.3.3
Assigning Values to the Input Signals ...................................................................................................
B.3.4
Saving the Waveform File .....................................................................................................................
B.3.5
Starting the Simulator ............................................................................................................................ Appendix C UP2 Programming Tutorial 3 .................................................................................
C.1 Getting Started ................................................................................................................................................
C.1.1
Preparing a Folder for the Project..........................................................................................................
C.1.2
Creating a Project ..................................................................................................................................
C.1.3
Viewing the Source File ........................................................................................................................
C.2 Synthesis for Programming the PLD ..............................................................................................................
C.3 Circuit Simulation...........................................................................................................................................
C.4 Using the Floorplan Editor .............................................................................................................................
C.4.1
Selecting the Target Device ...................................................................................................................
C.4.2
Maping the I/O Pins with the Floorplan Editor......................................................................................
C.5 Fitting the Netlist and Pins to the PLD ...........................................................................................................
C.6 Hardware Setup ............................................................................................................................................
C.6.1
Installing the ByteBlaster Driver .........................................................................................................
C.6.2
Jumper Settings....................................................................................................................................
C.6.3
Hardware Connections........................................................................................................................
C.7 Programming the PLD ..................................................................................................................................
C.8 Testing the Hardware....................................................................................................................................
C.9 MAX7000S EPM7128SLC84-7 Summary................................................................................................... 10 C.9.1
JTAG Jumper Settings .........................................................................................................................
C.9.2
Prototyping Resources for Use ............................................................................................................
C.9.3
General Pin Assignments.....................................................................................................................
C.9.4
Two Pushbutton Switches....................................................................................................................
C.9.5
16 DIP Switches ..................................................................................................................................
C.9.6
16 LEDs...............................................................................................................................................
C.9.7
7-Segment LEDs..................................................................................................................................
C.9.8
Clock....................................................................................................................................................
C.10 FLEX10K EPF10K70RC240-4 Summary....................................................................................................
C.10.1 JTAG Jumper Settings .........................................................................................................................
C.10.2 Prototyping Resources for Use ............................................................................................................
C.10.3 Two Pushbutton Switches....................................................................................................................
C.10.4 8 DIP Switches ....................................................................................................................................
C.10.5 7-Segment LEDs..................................................................................................................................
C.10.6 Clock....................................................................................................................................................
C.10.7 PS/2 Port ..............................................................................................................................................
C.10.8 VGA Port............................................................................................................................................. Appendix D VHDL Summary ......................................................................................................
D.1 Basic Language Elements...............................................................................................................................
D.1.1
Comments ..............................................................................................................................................
D.1.2
Identifiers...............................................................................................................................................
D.1.3
Data Objects ..........................................................................................................................................
D.1.4
Data Types .............................................................................................................................................
D.1.5
Data Operators .......................................................................................................................................
D.1.6
ENTITY.................................................................................................................................................
D.1.7
ARCHITECTURE.................................................................................................................................
D.1.8
GENERIC ..............................................................................................................................................
D.1.9
PACKAGE ............................................................................................................................................
D.2 Dataflow Model Concurrent Statements.......................................................................................................
D.2.1
Concurrent Signal Assignment ............................................................................................................ 0
D.2.2
Conditional Signal Assignment ...........................................................................................................
D.2.3
Selected Signal Assignment.................................................................................................................
D.2.4
Dataflow Model Example....................................................................................................................
2
D.3 Behavioral Model Sequential Statements .....................................................................................................
D.3.1
PROCESS ............................................................................................................................................
D.3.2
Sequential Signal Assignment .............................................................................................................
D.3.3
Variable Assignment ...........................................................................................................................
D.3.4
WAIT...................................................................................................................................................
D.3.5
IF THEN ELSE....................................................................................................................................
D.3.6
CASE ...................................................................................................................................................
4
D.3.7
NULL...................................................................................................................................................
D.3.8
FOR .....................................................................................................................................................
D.3.9
WHILE ................................................................................................................................................
D.3.10 LOOP...................................................................................................................................................
D.3.11 EXIT ....................................................................................................................................................
D.3.12 NEXT...................................................................................................................................................
D.3.13 FUNCTION .........................................................................................................................................
D.3.14 PROCEDURE......................................................................................................................................
D.3.15 Behavioral Model Example .................................................................................................................
D.4 Structural Model Statements.........................................................................................................................
D.4.1
COMPONENT Declaration.................................................................................................................
D.4.2
PORT MAP .........................................................................................................................................
D.4.3
OPEN...................................................................................................................................................
D.4.4
GENERATE ........................................................................................................................................
D.4.5
Structural Model Example ................................................................................................................... 11 D.5 Conversion Routines.....................................................................................................................................
D.5.1
CONV_INTEGER() ............................................................................................................................
D.5.2
CONV_STD_LOGIC_VECTOR(,)..................................................................................................... 12 1
1 Digital Logic and Microprocessor Design with VHDL Preface Preface
This book is about the digital logic design of microprocessors. It is intended to provide both an understanding of
the basic principles of digital logic design, and how these fundamental principles are applied in the building of
complex microprocessor circuits using current technologies. Although the basic principles of digital logic design
have not changed, the design process, and the implementation of the circuits have changed. With the advances in
fully integrated modern computer aided design (CAD) tools for logic synthesis, simulation, and the implementation
of circuits in programmable logic devices (PLDs) such as field programmable gate arrays (FPGAs), it is now
possible to design and implement complex digital circuits very easily and quickly.
Many excellent books on digital logic design have followed the traditional approach of introducing the basic
principles and theories of logic design, and the building of separate combinational and sequential components.
However, students are left to wonder about the purpose of these individual components, and how they are used in
the building of microprocessors – the ultimate in digital circuits. One primary goal of this book is to fill in this gap
by going beyond the logic principles, and the building of individual components. The use of these principles and the
individual components are combined together to create datapaths and control units, and finally the building of real
dedicated custom microprocessors and general-purpose microprocessors.
Previous logic design and implementation techniques mainly focus on the logic gate level. At this low level, it is
difficult to discuss larger and more complex circuits beyond the standard combinational and sequential circuits.
However, with the introduction of the register-transfer technique for designing datapaths, and the concept of a finitestate machine for control units, we can easily implement an arbitrary algorithm as a dedicated microprocessor in
hardware. The construction of a general-purpose microprocessor then comes naturally as a generalization of a
dedicated microprocessor.
With the provided CAD tool, and the optional FPGA hardware development kit, students can actually
implement these microprocessor circuits, and see them execute, both in software simulation, and in hardware. The
book contains many interesting examples with complete circuit schematic diagrams, and VHDL codes for both
simulation and implementation in hardware. With the hands-on exercises, the student will learn not only the
principles of digital logic design, but also in practice, how circuits are implemented using current technologies.
To actually see your own microprocessor comes to life in real hardware is an exciting experience. Hopefully,
this will help the students to not only remember what they have learned, but will also get them interested in the
world of digital circuit design. Advanced and Historical Topics
Sections that are designated with an asterisk ( * ) are either advanced topics, or topics for a historical
perspective. These sections may be skipped without any loss of continuity in learning how to design a
microprocessor. Summary Checklist
There is a chapter summary checklist at the end of each chapter. These checklists provide a quick way for
students to evaluate whether they have understood the materials presented in the chapter. The items in the checklists
are divided into two categories. The first set of items deal with new concepts, ideas, and definitions, while the
second set deals with practical how to do something types. Design of Circuits Using VHDL
Although this book provides coverage on VHDL for all the circuits, it can be omitted entirely for the
understanding and designing of digital circuits. For an introductory course in digital logic design, learning the basic
principles is more important than learning how to use a hardware description language. In fact, instructors may find
that students may get lost in learning the principles while trying to learn the language at the same time. With this in
mind, the VHDL code in the text is totally independent of the presentation of each topic, and may be skipped
without any loss of continuity. 13 Digital Logic and Microprocessor Design with VHDL Preface On the other hand, by studying the VHDL codes, the student can not only learn the use of a hardware
description language, but also learn how digital circuits can be designed automatically using a synthesizer. This
book provides a basic introduction to VHDL, and uses the learn-by-examples approach. In writing VHDL code at
the dataflow and behavioral levels, the student will see the power and usefulness of a state-of-the-art CAD synthesis
tool. Using this Book
This book can be used in either an introductory, or a more advanced course in digital logic design. For an
introductory course with no previous background in logic, Chapters 1 to 4 are intended to provide the fundamental
concepts in designing combinational circuits, and Chapters 6 to 8 cover the basic sequential circuits. Chapters 9 to
12 on microprocessor design can be introduced and covered lightly. For an advanced course where students already
have an exposure to logic gates and simple digital circuits, Chapters 1 to 4 will serve as a review. The focus should
be on the register-transfer design of datapaths and control units, and the building of dedicated and general-purpose
microprocessors as covered in Chapters 9 to 12. A lab component should complement the course where students can
have a hands-on experience in implementing the circuits presented using the included CAD software, and the
optional development kit. A brief summary of the topics covered in each chapter follows.
Chapter 1 – Designing a Microprocessor gives an overview of the various components of a microprocessor
circuit, and the different abstraction levels in which a circuit can be designed.
Chapter 2 – Digital Circuits provides the basic principles and theories for designing digital logic circuits by
introducing the use of truth tables and Boolean algebra, and how the theories get translated into logic gates, and
circuit diagrams. A brief introduction to VHDL is also given.
Chapter 3 – Combinational Circuits shows how combinational circuits are analyzed, synthesized and
reduced.
Chapter 4 – Combinational Components discusses the standard combinational components that are used as
building blocks for larger digital circuits. These components include adder, subtractor, arithmetic logic unit,
decoder, encoder, multiplexer, tri-state buffer, comparator, shifter, and multiplier. In a hierarchical design, these
components will be used to build larger circuits such as the microprocessor.
Chapter 5 Implementation Technologies digresses a little by looking at how logic gates are implemented at
the transistor level, and the various programmable logic devices available for implementing digital circuits.
Chapter 6 – Latches and Flip-Flops introduces the basic storage elements, specifically, the latch and the flipflop.
Chapter 7 – Sequential Circuits shows how sequential circuits in the form of finite-state machines, are
analyzed, and synthesized. This chapter also shows how the operation of sequential circuits can be precisely
described using state diagrams.
Chapter 8 – Sequential Components discusses the standard sequential components that are used as building
blocks for larger digital circuits. These components include register, shift register, counter, register file, and
memory. Similar to the combinational components, these sequential components will be used in a hierarchical
fashion to build larger circuits.
Chapter 9 – Datapaths introduces the register-transfer design methodology, and shows how an arbitrary
algorithm can be performed by a datapath.
Chapter 10 – Control Units shows how a finite-state machine (introduced in Chapter 7) is used to control the
operations of a datapath so that the algorithm can be executed automatically.
Chapter 11 – Dedicated Microprocessors ties the separate datapath and control unit together to form one
coherent circuit – the custom dedicated microprocessor. Several complete dedicated microprocessor examples are
provided.
Chapter 12 – General-Purpose Microprocessors continues on from Chapter 11 to suggest that a generalpurpose microprocessor is really a dedicated microprocessor that is dedicated to only read, decode, and execute
instructions. A simple general-purpose microprocessor is designed and implemented, and programs written in
machine language can be executed on it. 14 Digital Logic and Microprocessor Design with VHDL Preface Software and Hardware Packages
The newest student edition of Altera’s MAX+Plus II CAD software is included with this book on the
accompanying CD-ROM. The optional UP2 hardware development kit is available from Altera at a special student
price. An order form for the kit can be obtained from Altera’s website at www.altera.com.
Source files for all the circuit drawings and VHDL codes presented in this book can also be found on the
accompanying CD-ROM. Website for the Book
The website for this book is located at the following URL:
www.cs.lasierra.edu/~ehwang
The website provides many resources for both faculty and students. Enoch O. Hwang
Riverside, California 15 Chapter 1 Designing Microprocessors Control
Inputs Data
Inputs Microprocessor
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors Being a computer science or electrical engineering student, you probably have assembled a PC before. You may
have gone out to purchase the motherboard, CPU (central processing unit), memory, disk drive, video card, sound
card, and other necessary parts, assembled them together, and have made yourself a state-of-the-art working
computer. But have you ever wondered how the circuits inside those IC (integrated circuit) chips are designed? You
know how the PC works at the system level by installing the operating system and seeing your machine come to life.
But have you thought about how your PC works at the circuit level, how the memory is designed, or how the CPU
circuit is designed?
In this book, I will show you from the ground up, how to design the digital circuits for microprocessors, also
known as CPUs. When we hear the word “microprocessor,” the first thing that probably comes to many of our
minds is the Intel Pentium® CPU, which is found in most PCs. However, there are many more microprocessors that
are not Pentiums, and many more microprocessors that are used in areas other than the PCs.
Microprocessors are the heart of all “smart” devices, whether they be electronic devices or otherwise. Their
smartness comes as a direct result of the decisions and controls that microprocessors make. For example, we usually
do not consider a car to be an electronic device. However, it certainly has many complex, smart electronic systems,
such as the anti-lock brakes and the fuel-injection system. Each of these systems is controlled by a microprocessor.
Yes, even the black, hardened blob that looks like a dried-up and pressed-down piece of gum inside a musical
greeting card is a microprocessor.
There are generally two types of microprocessors: general-purpose microprocessors and dedicated
microprocessors. General-purpose microprocessors, such as the Pentium CPU, can perform different tasks under
the control of software instructions. General-purpose microprocessors are used in all personal computers.
Dedicated microprocessors, also known as application-specific integrated circuits (ASICs), on the other
hand, are designed to perform just one specific task. For example, inside your cell phone, there is a dedicated
microprocessor that controls its entire operation. The embedded microprocessor inside the cell phone does nothing
else but control the operation of the phone. Dedicated microprocessors are, therefore, usually much smaller and not
as complex as general-purpose microprocessors. However, they are used in every smart electronic device, such as
the musical greeting cards, electronic toys, TVs, cell phones, microwave ovens, and anti-lock break systems in your
car. From this short list, I’m sure that you can think of many more devices that have a dedicated microprocessor
inside them. Although the small dedicated microprocessors are not as powerful as the general-purpose
microprocessors, they are being sold and used in a lot more places than the powerful general-purpose
microprocessors that are used in personal computers.
Designing and building microprocessors may sound very complicated, but don’t let that scare you, because it is
not really all that difficult to understand the basic principles of how microprocessors are designed. We are not trying
to design a Pentium microprocessor here, but after you have learned the material presented in this book, you will
have the basic knowledge to understand how it is designed.
This book will show you in an easily understandable approach, starting with the basics and leading you through
to the building of larger components, such as the arithmetic logic unit (ALU), register, datapath, control unit, and
finally to the building of the microprocessor — first dedicated microprocessors, and then general-purpose
microprocessors. Along the way, there will be many sample circuits that you can try out and actually implement in
hardware using the optional Altera UP2 development board. These circuits, forming the various components found
inside a microprocessor, will be combined together at the end to produce real, working microprocessors. Yes, the
exciting part is that at the end, you actually can implement your microprocessor in a real IC, and see that it really
can execute software programs or make lights flash! 1.1 Overview of a Microprocessor The Von Neumann model of a computer, shown in Figure 1.1, consists of four main components: the input, the
output, the memory, and the microprocessor (or CPU). The parts that you purchased for your computer can all be
categorized into one of these four groups. The keyboard and mouse are examples of input devices. The CRT
(cathode ray tube) and speakers are examples of output devices. The different types of memory (cache, read-only
memory (ROM), random-access memory (RAM), and the disk drive) are all considered part of the memory box in
the model. In this book, the focus is not on the mechanical aspects of the input, output, and storage devices. Rather, 17 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors the focus is on the design of the digital circuitry of the microprocessor, the memory, and other supporting digital
logic circuits.
The logic circuit for the microprocessor can be divided into two parts: the datapath and the control unit, as
shown in Figure 1.1. Figure 1.2 shows the details inside the control unit and the datapath. The datapath is
responsible for the actual execution of all data operations performed by the microprocessor, such as the addition of
two numbers inside the arithmetic logic unit (ALU). The datapath also includes registers for the temporary storage
of your data. The functional units inside the datapath, which in our example includes the ALU and the register, are
connected together with multiplexers and data signal lines. The data signal lines are for transferring data between
two functional units. Data signal lines in the circuit diagram are represented by lines connecting two functional
units. Sometimes, several data signal lines are grouped together to form a bus. The width of the bus (that is, the
number of data signal lines in the group) is annotated next to the bus line. In the example, the bus lines are thicker
and are 8-bits wide. Multiplexers, also known as MUXes, are for selecting data from two or more sources to go to
one destination. In the sample circuit, a 2-to-1 multiplexer is used to select between the input data and the constant
‘0’ to go to the left operand of the ALU. The output of the ALU is connected to the input of the register. The output
of the register is connected to three different destinations: (1) the right operand of the ALU, (2) an OR gate used as a
comparator for the test “not equal to 0,” and (3) a tri-state buffer. The tri-state buffer is used to control the output of
the data from the register.
Memory Control
Unit Input Datapath Output Microprocessor Figure 1.1. Von Neumann model of a computer.
Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Figure 1.2. Internal parts of a microprocessor.
Even though the datapath is capable of performing all of the data operations of the microprocessor, it cannot,
however, do it on its own. In order for the datapath to execute the operations automatically, the control unit is
required. The control unit, also known as the controller, controls all of the operations of the datapath, and therefore,
the operations of the entire microprocessor. The control unit is a finite state machine (FSM) because it is a machine
that executes by going from one state to another and that there are only a finite number of states for the machine to
go to. The control unit is made up of three parts: the next-state logic, the state memory, and the output logic. The
purpose of the state memory is to remember the current state that the FSM is in. The next-state logic is the circuit for 18 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors determining what the next state should be for the machine. And the output logic is the circuit for generating the
actual control signals for controlling the datapath.
Every digital logic circuit, regardless of whether it is part of the control unit or the datapath, is categorized as
either a combinational circuit or a sequential circuit. A combinational circuit is one where the output of the circuit
is dependent only on the current inputs to the circuit. For example, an adder circuit is a combinational circuit. It
takes two numbers as inputs. The adder evaluates the sum of these two numbers and outputs the result.
A sequential circuit, on the other hand, is dependent not only on the current inputs, but also on all the previous
inputs. In other words, a sequential circuit has to remember its past history. For example, the up-channel button on a
TV remote is part of a sequential circuit. Pressing the up-channel button is the input to the circuit. However, just
having this input is not enough for the circuit to determine what TV channel to display next. In addition to the upchannel button input, the circuit must also know the current channel that is being displayed, which is the history. If
the current channel is channel 3, then pressing the up-channel button will change the channel to channel 4.
Since sequential circuits are dependent on the history, they must therefore contain memory elements for
remembering the history; whereas combinational circuits do not have memory elements. Examples of combinational
circuits inside the microprocessor include the next-state logic and output logic in the control unit, and the ALU,
multiplexers, tri-state buffers, and comparators in the datapath. Examples of sequential circuits include the register
for the state memory in the controller and the registers in the datapath. The memory in the Von Neuman computer
model is also a sequential circuit.
Irregardless of whether a circuit is combinational or sequential, they are all made up of the three basic logic
gates: AND, OR, and NOT gates. From these three basic gates, the most powerful computer can be made.
Furthermore, these basic gates are built using transistors — the fundamental building blocks for all digital logic
circuits. Transistors are just electronic binary switches that can be turned on or off. The on and off states of a
transistor are used to represent the two binary values: 1 and 0.
Figure 1.3 summarizes how the different parts and components fit together to form the microprocessor. From
transistors, the basic logic gates are built. Logic gates are combined together to form either combinational circuits or
sequential circuits. The difference between these two types of circuits is only in the way the logic gates are
connected together. Latches and flip-flops are the simplest forms of sequential circuits, and they provide the basic
building blocks for more complex sequential circuits. Certain combinational circuits and sequential circuits are used
as standard building blocks for larger circuits, such as the microprocessor. These standard combinational and
sequential components usually are found in standard libraries and serve as larger building blocks for the
microprocessor. Different combinational components and sequential components are connected together to form
either the datapath or the control unit of a microprocessor. Finally, combining the datapath and the control unit
together will produce the circuit for either a dedicated or a general microprocessor. 19 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors Transistors Gates 5 2 Combinational
Circuits 3 Flip-flops 6 Sequential
Circuits 7
Combinational
Components 4 Datapath + Sequential
Components 8 Control Unit 9 10 Dedicated
Microprocessor
11
General
Microprocessor
12 Figure 1.3. Summary of how the parts of a microprocessor fit together. The numbers in each box denote the chapter
number in which the topic is discussed. 1.2 Design Abstraction Levels Digital circuits can be designed at any one of several abstraction levels. When designing a circuit at the
transistor level, which is the lowest level, you are dealing with discrete transistors and connecting them together to
form the circuit. The next level up in the abstraction is the gate level. At this level, you are working with logic gates
to build the circuit. At the gate level, you also can specify the circuit using either a truth table or a Boolean equation.
In using logic gates, a designer usually creates standard combinational and sequential components for building
larger circuits. In this way, a very large circuit, such as a microprocessor, can be built in a hierarchical fashion.
Design methodologies have shown that solving a problem hierarchically is always easier than trying to solve the
entire problem as a whole from the ground up. These combinational and sequential components are used at the
register-transfer level in building the datapath and the control unit in the microprocessor. At the register-transfer
level, we are concerned with how the data is transferred between the various registers and functional units to realize
or solve the problem at hand. Finally, at the highest level, which is the behavioral level, we construct the circuit by
describing the behavior or operation of the circuit using a hardware description language. This is very similar to
writing a computer program using a programming language. 1.3 Examples of a 2-to-1 Multiplexer As an example, let us look at the design of the 2-to-1 multiplexer from the different abstraction levels. At this
point, don’t worry too much if you don’t understand the details of how all of these circuits are built. This is intended
just to give you an idea of what the description of the circuits look like at the different abstraction levels. We will get
to the details in the rest of the book.
An important point to gain from these examples is to see that there are many different ways to create the same
functional circuit. Although they are all functionally equivalent, they are different in other respects such as size (how
big the circuit is or how many transistors it uses), speed (how long it takes for the output result to be valid), cost 20 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors (how much it costs to manufacture), and power usage (how much power it uses). Hence, when designing a circuit,
besides being functionally correct, there will always be economic versus performance tradeoffs that we need to
consider.
The multiplexer is a component that is used a lot in the datapath. An analogy for the operation of the 2-to-1
multiplexer is similar in principle to a railroad switch in which two railroad tracks are to be merged onto one track.
The switch controls which one of the two trains on the two separate tracks will move onto the one track. Similarly,
the 2-to-1 multiplexer has two data inputs, d0 and d1, and a select input, s. The select input determines which data
from the two data inputs will pass to the output, y.
Figure 1.4 shows the graphical symbol also referred to as the logic symbol for the 2-to-1 multiplexer. From
looking at the logic symbol, you can tell how many signal lines the 2-to-1 multiplexer has, and the name or function
designated for each line. For the 2-to-1 multiplexer, there are two data input signals, d1 and d0, a select input signal,
s, and an output signal, y.
d1
s d0 1 0 y Figure 1.4. Logic symbol for the 2-to-1 multiplexer. 1.3.1 Behavioral Level
We can describe the operation of the 2-to-1 multiplexer simply, using the same names as in the logic symbol, by
saying that
d0 passes to y when s = 0, and
d1 passes to y when s = 1
Or more precisely, the value that is at d0 passes to y when s = 0, and the value that is at d1 passes to y when s = 1.
We use a hardware description language (HDL) to describe a circuit at the behavioral level. When describing a
circuit at this level, you would write basically the same thing as in the description, except that you have to use the
correct syntax required by the hardware description language. Figure 1.5 shows the description of the 2-to-1
multiplexer using the hardware description language called VHDL.
LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY multiplexer IS PORT (
d0, d1, s: IN STD_LOGIC;
y: OUT STD_LOGIC);
END multiplexer;
ARCHITECTURE Behavioral OF multiplexer IS
BEGIN
PROCESS(s, d0, d1)
BEGIN
y <= d0 WHEN s = '0' ELSE d1;
END PROCESS;
END Behavioral;
Figure 1.5. Behavioral level VHDL description of the 2-to-1 multiplexer.
The LIBRARY and USE statements are similar to the “#include” preprocessor command in C. The IEEE library
contains the definition for the STD_LOGIC type used in the declaration of signals. The ENTITY section declares the 21 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors interface for the circuit by specifying the input and output signals of the circuit. In this example, there are three input
signals of type STD_LOGIC, and one output signal also of type STD_LOGIC. The ARCHITECTURE section defines the
actual operation of the circuit. The operation of the multiplexer is defined in the one conditional signal assignment
statement
y <= d0 WHEN s = '0' ELSE d1;
The statement, which uses the symbol <= to denote the signal assignment, says that the signal y gets the value of d0
when s is equal to 0, otherwise, y gets the value of d1.
As you can see, when designing circuits at the behavioral level, we do not need to know what logic gates are
needed or how they are connected together. We only need to know their interface and operation. 1.3.2 Gate Level
At the gate level, you can draw a schematic diagram, which is a diagram showing how the logic gates are
connected together. Two schematic diagrams of a circuit are shown in Figure 1.6(a) and (b). In Figure 1.6(a), the
circuit uses three inverters (
), four 3-input AND gates (
), and one 4-input OR gate (
). In Figure 1.6(b),
only one inverter, two 2-input AND gates, and one 2-input OR gate are needed. Although one circuit is larger (in
terms of the number of gates needed) than the other, both of these circuits realize the same 2-to-1 multiplexer
function. Therefore, when we want to actually implement a 2-to-1 multiplexer circuit, we will want to use the
second, smaller circuit rather than the first.
s d 1 d0 d0
s
y y d1 (b) (a) Figure 1.6. Gate level circuit diagram for the 2-to-1 multiplexer: (a) circuit using eight gates; (b) circuit using four
gates.
At the gate level, you can also describe the 2-to-1 multiplexer using a truth table or with a Boolean equation as
shown in Figure 1.7(a) and (b) respectively. For the truth table, we list all possible combinations of the binary values
for the three inputs s, d0 and d1, and then determine what the output value y should be based on the functional
description of the circuit. We see that for the first four rows of the table when s = 0, y has the same values as d0,
whereas in the last four rows when s = 1, y has the same values as d1.
The Boolean equation in (b) can be derived from either the schematic diagram or the truth table. The first
equality in (b) matches the truth table in (a), and also the schematic diagram in Figure 1.6(a). The second equality in
(b) matches the schematic diagram in Figure 1.6(b). To derive the equation from the truth table, we look at all the
rows where the output y is a 1. Each of these rows results in a term in the equation. For each term, the variable is
primed (' ) when the value of the variable is a 0, and unprimed when the value of the variable is a 1.
s
0
0
0
0
1 d1
0
0
1
1
0 d0
0
1
0
1
0 y
0
1
0
1
0 y = s' d1' d0 + s' d1 d0 + s d1 d0' + s d1 d0
= s' d0 + s d1 22 Digital Logic and Microprocessor Design with VHDL 1
1
1 0
1
1 1
0
1 Chapter 1 - Designing Microprocessors 0
1
1 (b) (a)
Figure 1.7. Gate level description of the 2-to-1 multiplexer: (a) using a truth table; (b) using a Boolean equation. 1.3.3 Transistor Level
The 2-to-1 multiplexer circuit at the transistor level is shown in Figure 1.8. It contains six transistors, three of
which are PMOS (
), and three are NMOS (
). The pair of transistors on the left forms an inverter for the
signal s, while the two pairs of transistors on the right form two transmission gates. The transmission gate allows or
disallows the data signal d0 or d1 to pass through, depending on the control signal s. The top transmission gate is
turned on when s is a 0, and the bottom transmission gate is turned on when s is a 1. Hence, when s is 0, the value at
d0 is passed to y, and when s is 1, the value at d1 is passed to y. d0
Vcc s y d1 Figure 1.8. Transistor circuit for the 2-to-1 multiplexer. 1.4 Introduction to VHDL The popularity of using hardware description languages (HDL) for designing digital circuits began in the mid1990s when commercial synthesis tools became available. Two popular HDLs used by many engineers today are
VHDL and Verilog. VHDL, which stands for VHSIC Hardware Description Language, and VHSIC, in turn, stands
for Very High Speed Integrated Circuit, was jointly sponsored and developed by the U.S. Department of Defense
and the IEEE in the mid-1980s. It was standardized by the IEEE in 1987 (VHDL-87), and later extended in 1993
(VHDL-93). Verilog, on the other hand, was first introduced in 1984, and later in 1988, as a proprietary hardware
description language by the two companies Synopsys and Cadence Design Systems. In this book, we will use
VHDL.
VHDL, in many respects, is similar to a regular computer programming language, such as C++. For example, it
has constructs for variable assignments, conditional statements, loops, and functions, just to name a few. In a
computer programming language, a compiler is used to translate the high-level source code to machine code. In
VHDL, however, a synthesizer is used to translate the source code to a description of the actual hardware circuit that
implements the code. From this description, which we call a netlist, the actual physical digital device that realizes
the source code can be made automatically. Accurate functional and timing simulation of the code is also possible in
order to test the correctness of the circuit.
You saw in Section 1.3.1 how we used VHDL to describe the 2-to-1 multiplexer at the behavioral level. VHDL
can also be used to describe a circuit at other levels. Figure 1.9 shows the VHDL code for the multiplexer written at
the dataflow level. The main difference between the behavioral VHDL code shown in Figure 1.5 and the dataflow
VHDL code is that in the behavioral code there is a PROCESS block statement, whereas in the dataflow code, there is 23 Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors no PROCESS statement. Statements within a PROCESS block are executed sequentially like in a computer program,
while statements outside a PROCESS block (including the PROCESS block itself) are executed concurrently or in
parallel. The signal assignment statement, using the symbol <=, is derived directly from the Boolean equation for the
multiplexer as shown in Figure 1.7(b) using the built-in VHDL operators AND, OR, and NOT.
LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY multiplexer IS PORT(
d0, d1, s: IN STD_LOGIC;
y: OUT STD_LOGIC);
END multiplexer;
ARCHITECTURE Dataflow OF multiplexer IS
BEGIN
y <= ((NOT s) AND d0) OR (s AND d1);
END Dataflow;
Figure 1.9. Dataflow level VHDL description of the 2-to-1 multiplexer.
In addition to the behavioral and dataflow levels, we can also write VHDL code at the structural level. Figure
1.11 shows the VHDL code for the multiplexer written at the structural level. The code is based on the circuit shown
in Figure 1.10. The three different gates (and2gate, or2gate, and notgate) used in the circuit are first declared and
defined using the ENTITY and ARCHITECTURE statements respectively. After this, the multiplexer is declared, also
with the ENTITY statement. The actual structural definition of the multiplexer is in the ARCHITECTURE section for
multiplexer2. First of all, the COMPONENT statements specify what components are used in the circuit. The SIGNAL
statement declares three internal signals that will be used in the connection of the circuit. Finally, the PORT MAP
statements declare the instances of the gates used in the circuit, and also specify how they are connected using the
external and internal signals.
d0 u1 u2 snd0 sn
u4 s
d1 u3 sd1 Figure 1.10. 2-to-1 multiplexer circuit.
----------------- NOT gate ----------------------LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY notgate IS PORT(
i: IN STD_LOGIC;
o: OUT STD_LOGIC);
END notgate;
ARCHITECTURE Dataflow OF notgate IS
BEGIN
o <= not i;
END Dataflow;
----------------- 2-input AND gate --------------LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY and2gate IS PORT(
i1, i2: IN STD_LOGIC; 24 y Digital Logic and Microprocessor Design with VHDL Chapter 1 - Designing Microprocessors o: OUT STD_LOGIC);
END and2gate;
ARCHITECTURE Dataflow OF and2gate IS
BEGIN
o <= i1 AND i2;
END Dataflow;
----------------- 2-input OR gate ---------------LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY or2gate IS PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END or2gate;
ARCHITECTURE Dataflow OF or2gate IS
BEGIN
o <= i1 OR i2;
END Dataflow;
----------------- 2-to-1 multiplexer -----------LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
ENTITY multiplexer IS PORT(
d0, d1, s: IN STD_LOGIC;
y: OUT STD_LOGIC);
END multiplexer;
ARCHITECTURE Structural OF multiplexer IS
COMPONENT notgate PORT(
i: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT and2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT and3gate PORT(
i1, i2, i3: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT or2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
SIGNAL sn, snd0, sd1: STD_LOGIC;
BEGIN
U1: notgate PORT MAP(s,sn);
U2: and2gate PORT MAP(d0, sn, snd0);
U3: and2gate PORT MAP(d1, s, sd1);
U4: or2gate PORT MAP(snd0, sd1, y);
END Structural;
Figure 1.11. Structural level VHDL description of the 2-to-1 multiplexer. 25 Digital Logic and Microprocessor Design with VHDL 1.5 Chapter 1 - Designing Microprocessors Synthesis Given a gate level circuit diagram, such as the one shown in Figure 1.6, you can actually get some discrete logic
gates, and manually connect them together with wires on a breadboard. Traditionally, this is how electronic
engineers actually designed and implemented digital logic circuits. However, this is not how electronic engineers
design circuits anymore. They write programs, such as the one in Figure 1.5, just like what computer programmers
do. The question then is how does the program that describes the operation of the circuit actually get converted to
the physical circuit?
The problem here is similar to translating a computer program written in a high-level language to machine
language for a particular computer to execute. For a computer program, we use a compiler to do the translation. For
translating a digital logic circuit, we use a synthesizer. Instead of using a high-level computer language to describe
a computer program, we use a hardware description language (HDL) to describe the operations of a digital logic
circuit. Writing a description of a digital logic circuit is similar to writing a computer program; the only difference is
that a different language is used. A synthesizer is then used to translate the HDL program into the circuit netlist. A
netlist is a description of how a circuit is actually realized or connected using basic gates. This translation process
from a HDL description of a circuit to its netlist is referred to as synthesis.
Furthermore, the netlist from the output of the synthesizer can be used directly to implement the actual circuit in
a programmable logic device (PLD) chip such as a field programmable gate array (FPGA). With this final step, the
creation of a digital circuit that is fully implemented in an integrated circuit (IC) chip can be easily done. The
Appendix gives a tutorial of the complete process from writing the VHDL code to synthesizing the circuit and
uploading the netlist to the FPGA chip using Altera’s development system. 1.6 Going Forward We will now embark upon a journey that will take you from a simple transistor to the building of a
microprocessor. Figure 1.2 will serve as our guide and map. If you get lost on the way, and do not know where a
particular component fits in the overall picture, just refer to this map. At the beginning of each chapter, I will refresh
your memory with this map by highlighting the components in the map that the chapter will cover.
Figure 1.12 is an actual picture of the circuitry inside an Intel Pentium 4 CPU. When you reach the end of this
book, you still may not be able to design the circuit for the P4, but you will certainly have the knowledge of how a
microprocessor is designed because you will actually have designed and implemented a working microprocessor
yourself. Figure 1.12. The internal circuitry of the Intel P4 CPU. 26 Digital Logic and Microprocessor Design with VHDL 1.7 Chapter 1 - Designing Microprocessors Summary Checklist
Microprocessor
General-purpose microprocessor
Dedicated microprocessor, ASIC
Datapath
Control unit
Finite state machine (FSM)
Next-state logic
State memory
Output logic
Combinational circuit
Sequential circuit
Transistor level design
Gate level design
Register-transfer level design
Behavioral level design
Logic symbol
VHDL
Synthesis
Netlist 1.8 Problems 1.1. Find out the approximate number of general-purpose microprocessors sold in the US in a year versus the
number of dedicated microprocessors sold. 1.2. Compile a list of devices that you use during one regular day that are controlled by a microprocessor. 1.3. Describe what your regular daily routine will be like if there is no electrical power, including battery power,
available. 1.4. Apply the Von Neumann model of a computer system as shown in Figure 1.1 to the following systems.
Determine what parts of the system correspond to the different parts of the model.
a)
b)
c)
d)
e) Traffic light
Heart pace maker
Microwave oven
Musical greeting card
Hard disk drive (not the entire personal computer) 1.5. The speed of a microprocessor is often measured by its clock frequency. What is the clock frequency of the
fastest general-purpose microprocessor available? 1.6. Compare some typical clock speeds between general-purpose microprocessors versus dedicated
microprocessors. 1.7. Summarize the mainstream generations of the Intel general-purpose microprocessors used in personal
computers starting with the 8086 CPU. List the year introduced, the clock speed, and the number of
transistors in each.
Answer 27 Digital Logic and Microprocessor Design with VHDL CPU
8086
80286
80386
80486
Pentium
Pentium Pro
Pentium II
Celeron
Pentium III
Pentium 4 Year Introduced
1978
1982
1985
1989
1993
1995
1997
1998
1999
2000 Chapter 1 - Designing Microprocessors Clock Speed
4.7 – 10 MHz
6 – 12 MHz
16 – 33 MHz
25 – 100 MHz
60 – 200 MHz
150 – 200 MHz
234 – 450 MHz
266 – 800 MHz
400 MHz – 1.2 GHz
1.4 – 3 GHz Number of Transistors
29,000
134,000
275,000
1.2 million
3.3 million
5.5 million
7.5 million
19 million
28 million
42 million 1.8. Using Figure 1.9 as a template, write the dataflow VHDL code for the 2-to-1 multiplexer circuit shown in
Figure 1.6(a). 1.9. Using Figure 1.11 as a template, write the structural VHDL code for the 2-to-1 multiplexer circuit shown in
Figure 1.6(a). 1.10. Do Tutorial 1 in Appendix A.
1.11. Do Tutorial 2 in Appendix B.
1.12. Do Tutorial 3 in Appendix C. 28 Chapter 2 Digital Circuits Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Our world is an analog world. Measurements that we make of the physical objects around us are never in
discrete units, but rather in a continuous range. We talk about physical constants such as 2.718281828… or
3.141592…. To build analog devices that can process these values accurately is next to impossible. Even building a
simple analog radio requires very accurate adjustments of frequencies, voltages, and currents at each part of the
circuit. If we were to use voltages to represent the constant 3.14, we would have to build a component that will give
us exactly 3.14 volts every time. This is again impossible; due to the imperfect manufacturing process, each
component produced is slightly different from the others. Even if the manufacturing process can be made as perfect
as perfect can get, we still would not be able to get 3.14 volts from this component every time we use it. The reason
being that the physical elements used in producing the component behave differently in different environments, such
as temperature, pressure, and gravitational force, just to name a few. Therefore, even if the manufacturing process is
perfect, using this component in different environments will not give us exactly 3.14 volts every time.
To make things simpler, we work with a digital abstraction of our analog world. Instead of working with an
infinite continuous range of values, we use just two values! Yes, just two values: 1 and 0, on and off, high and low,
true and false, black and white, or however you want to call it. It is certainly much easier to control and work with
two values rather than an infinite range. We call these two values a binary value for the reason that there are only
two of them. A single 0 or a single 1 is then a binary digit or bit. This sounds great, but we have to remember that
the underlining building block for our digital circuits is still based on an analog world.
This chapter provides the theoretical foundations for building digital logic circuits using logic gates, the basic
building blocks for all digital circuits. In order to understand how logic gates are used to implement digital circuits,
we need to have a good understanding of the basic theory of Boolean algebra, Boolean functions, and how to use
and manipulate them. Most people may find Sections 2.5 and 2.6 on these theories to be boring, but let me
encourage you to grind through it patiently, because if you do not understand it now, you will quickly get lost in the
later chapters. The good news is that these two sections are the only sections in this book on theory, and I will try to
keep it as short and simple as possible. You will also find that many of the Boolean Theorems are very familiar,
because they are similar to the Algebra Theorems that you have learned from your high school math class. As you
can see from the microprocessor road map, this chapter affects all the parts for building a microprocessor. 2.1 Binary Numbers Since digital circuits deal with binary values, we will begin with a quick introduction to binary numbers. A bit,
having either the value of 0 or 1, can represent only two things or two pieces of information. It is, therefore,
necessary to group many bits together to represent more pieces of information. A string of n bits can represent 2n
different pieces of information. For example, a string of two bits results in the four combinations 00, 01, 10, and 11.
By using different encoding techniques, a group of bits can be used to represent different information, such as a
number, a letter of the alphabet, a character symbol, or a command for the microprocessor to execute.
The use of decimal numbers is quite familiar to us. However, since the binary digit is used to represent
information within the computer, we also need to be familiar with binary numbers. Note that the use of binary
numbers is just a form of representation for a string of bits. We can just as well use octal, decimal, or hexadecimal
numbers to represent the string of bits. In fact, you will find that hexadecimal numbers are often used as a shorthand
notation for binary numbers.
The decimal number system is a positional system. In other words, the value of the digit is dependent on the
position of the digit within the number. For example, in the decimal number 48, the decimal digit 4 has a greater
value than the decimal digit 8 because it is in the tenth position, whereas the digit 8 is in the unit position. The value
of the number is calculated as 4×101 + 8×100.
Like the decimal number system, the binary number system is also a positional system. The only difference
between the two is that the binary system is a base-2 system, and so it uses only two digits, 0 and 1, instead of ten.
The binary numbers from 0 to 15 (decimal) are shown in Figure 2.1. The range from 0 to 15 has 16 different
combinations. Since 24 = 16, therefore, we need a 4-bit binary number, i.e., a string of four bits, to represent this
range.
When we count in decimal, we count from 0 to 9. After 9, we go back to 0, and have a carry of a 1 to the next
digit. When we count in binary, we do the same thing except that we only count from 0 to 1. After 1, we go back to
0, and have a carry of a 1 to the next bit. 30 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits The decimal value of a binary number can be found just like for a decimal number except that we raise the base
number 2 to a power rather than the base number 10 to a power. For example, the value for the decimal number 658
is
65810 = 6×102 + 5×101 + 8×100 = 600 + 50 + 8 = 65810
Similaly, the decimal value for the binary number 10110112 is
10110112 = 1×26 + 0×25 + 1×24 + 1×23 + 0×22 + 1×21 + 1×20 = 64 + 16 + 8 + 2 + 1 = 9110
To get the decimal value, the least significant bit (in this case, the rightmost 1) is multiplied with 20. The next
bit to the left is multiplied with 21, and so on. Finally, they are all added together to give the value 9110.
Notice the subscript 10 in the decimal number 65810, and the 2 in the binary number 10110112. This subscript is
used to denote the base of the number whenever there might be confusion as to what base the number is in.
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111 Octal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17 Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F Figure 2.1 Numbers from 0 to 15 in binary, octal, and hexadecimal.
Converting a decimal number to its binary equivalent can be done by successively dividing the decimal number
by 2 and keeping track of the remainder at each step. Combining the remainders together (starting with the last one)
forms the equivalent binary number. For example, using the decimal number 91, we divide it by 2 to get 45 with a
remainder of 1. Then we divide 45 by 2 to get 22 with a remainder of 1. We continue in this fashion until the end as
shown below.
2
2
2
2
2
2 91
45
22
11
5
2
1 1
1
0
1
1
0 least significant bit = 1011011 most significant bit Concatenating the remainders together starting with the last one results in the binary number 10110112.
Binary numbers usually consist of a long string of bits. A shorthand notation for writing out this lengthy string
of bits is to use either the octal or hexadecimal numbers. Since octal is base-8 and hexadecimal is base-16, both of
which are a power of 2, a binary number can be easily converted to an octal or hexadecimal number, or vice versa. 31 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Octal numbers only use the digits from 0 to 7 for the eight different combinations. When counting in octal, the
number after 7 is 10 as shown in Figure 2.1. To convert a binary number to octal, we simply group the bits into
groups of threes starting from the right. The reason for this is because 8 = 23. For each group of three bits, we write
the equivalent octal digit for it. For example, the conversion of the binary number 1 110 0112 to the octal number
1638 is shown below.
001 110 011
1
6
3
Since the original binary number has seven bits, we need to extend it with two leading zeros to get three bits for
the leftmost group. Note that when we are dealing with negative numbers, we may require extending the number
with leading ones instead of zeros.
Converting an octal number to its binary equivalent is just as easy. For each octal number, we write down the
equivalent three bits. These groups of three bits are concatenated together to form the final binary number. For
example, the conversion of the octal number 57248 to the binary number 101 111 010 1002 is shown below.
5
7
2
4
101 111 010 100
The decimal value of an octal number can be found just like for a binary or decimal number except that we raise
the base number 8 to a power instead. For example, the octal number 57248 has the value
57248 = 5×83 + 7×82 + 2×81 + 4×80 = 2560 + 448 + 16 + 4 = 302810
Hexadecimal numbers are treated basically the same way as octal numbers except with the appropriate changes
to the base. Hexadecimal (or hex for short) numbers use base-16, and thus require 16 different digit symbols as
shown in Figure 2.1. Converting binary numbers to hexadecimal numbers involve grouping the bits into groups of
fours since 16 = 24. For example, the conversion of the binary number 110 1101 10112 to the hexadecimal number
6DB16 is shown below. Again, we need to extend it with a leading zero to get four bits for the leftmost group.
0110
6 1101
D 1011
B To convert a hex number to a binary number, we write down the equivalent four bits for each hex digit, and
then concatenate them together to form the final binary number. For example, the conversion of the hexadecimal
number 5C4A16 to the binary number 0101 1100 0100 10102 is shown below.
5
0101 C
1100 4
0100 A
1010 The following example shows how the decimal value of the hexadecimal number C4A16 is evaluated.
C4A16 = C×162 + 4×161 + A×160 = 12×162 + 4×161 + 10×160 = 3072 + 64 + 10 = 314610 2.2 Binary Switch Besides the fact that we are working only with binary values, digital circuits are easy to understand because
they are based on one simple idea of turning a switch on or off to obtain either one of the two binary values. Since
the switch can be in either one of two states (on or off), we call it a binary switch, or just a switch for short. The
switch has three connections: an input, an output, and a control for turning the switch on or off as shown in Figure
2.2. When the switch is opened as in (a), it is turned off and nothing gets through from the input to the output. When
the switch is closed as in (b), it is turned on, and whatever is presented at the input is allowed to pass through to the
output. 32 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits control
in out in (a) out
(b) Figure 2.2 Binary switch: (a) opened or off; (b) closed or on.
Uses of the binary switch idea can be found in many real world devices. For example, the switch can be an
electrical switch with the input connected to a power source and the output connected to a siren S as shown in Figure
2.3.
Switch Battery Siren Figure 2.3 A siren controlled by a switch.
When the switch is closed, the siren turns on. The usual convention is to use a 1 to mean “on” and a 0 to mean
“off.” Therefore, when the switch is closed, the output is a 1 and the siren will turn on. We can also use a variable, x,
to denote the state of the switch. We can let x = 1 to mean the switch is closed and x = 0 to mean the switch is
opened. Using this convention, we can describe the state of the siren S in terms of the variable x using a simple logic
expression. Since S = 1 if x = 1 and S = 0 if x = 0, we can write
S=x
This logic expression describes the output S in terms of the input variable x. 2.3 Basic Logic Operators and Logic Expressions
Two binary switches can be connected together either in series or in parallel as shown in Figure 2.4. x
x y
F F
y (a) (b) Figure 2.4 Connection of two binary switches: (a) in series; (b) in parallel.
If two switches are connected in series as in (a), then both switches have to be on in order for the output F to be
a 1. In other words, F = 1 if x = 1 AND y = 1. If either x or y is off, or both are off, then F = 0. Translating this into a
logic expression, we get
F = x AND y
Hence, two switches connected in series give rise to the logical AND operator. In a Boolean function (which we
will explain in more detail in section 2.5) the AND operator is either denoted with a dot ( • ) or no symbol at all. Thus
we can rewrite the above expression as
F=x•y
or simply 33 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits F = xy
If we connect two switches in parallel as in (b), then only one switch needs to be on in order for the output F to
be a 1. In other words, F = 1 if either x = 1, or y = 1, or both x and y are 1’s. This means that F = 0 only if both x
and y are 0’s. Translating this into a logic expression, we get
F = x OR y
and this gives rise to the logical OR operator. In a Boolean function, the OR operator is denoted with a plus symbol
( + ). Thus we can rewrite the above expression as
F=x+y
In addition to the AND and OR operators, there is another basic logic operator – the NOT operator, also known as
the INVERTER. Whereas, the AND and OR operators have multiple inputs, the NOT operator has only one input and
one output. The NOT operator simply inverts its input, so a 0 input will produce a 1 output, and a 1 becomes a 0. In a
Boolean function, the NOT operator is either denoted with an apostrophe symbol ( ' ) or a bar on top ( ) as in
F = x' or F=x When several operators are used in the same expression, the precedence given to the operators are, from highest
to lowest, NOT, AND, and OR. The order of evaluation can be changed by means of using parenthesis. For example,
the expression
F = xy + z' means (x and y) or (not z), and the expression
F = x(y + z)' means x and (not (y or z)). 2.4 Truth Tables The operation of the AND, OR, and NOT logic operators can be formally described by using a truth table as
shown in Figure 2.5. A truth table is a two-dimensional array where there is one column for each input and one
column for each output (a circuit may have more than one output). Since we are dealing with binary values, each
input can be either a 0 or a 1. We simply enumerate all possible combinations of 0’s and 1’s for all the inputs.
Usually, we want to write these input values in the normal binary counting order. With two inputs, there are 22
combinations giving us the four rows in the table. The values in the output column are determined from applying the
corresponding input values to the functional operator. For the AND truth table in Figure 2.5(a), F = 1 only when x
and y are both 1, otherwise, F = 0. For the OR truth table (b), F = 1 when either x or y or both is a 1, otherwise F = 0.
For the NOT truth table, the output F is just the inverted value of the input x.
x
0
0
1
1 y
0
1
0
1 (a) F
0
0
0
1 x
0
0
1
1 y
0
1
0
1 F
0
1
1
1 (b) x
0
1 F
1
0 (c) Figure 2.5 Truth tables for the three basic logical operators: (a) AND; (b) OR; (c) NOT. Using a truth table is one method to formally describe the operation of a circuit or function. The truth table for
any given logic expression (no matter how complex it is) can always be derived. Examples on the use of truth tables 34 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits to describe digital circuits are given in the following sections. Another method to formally describe the operation of
a circuit is by using Boolean expressions or Boolean functions. 2.5 Boolean Algebra and Boolean Function 2.5.1 Boolean Algebra
George Boole, in 1854, developed a system of mathematical logic, which we now call Boolean algebra. Based
on Boole’s idea, Claude Shannon, in 1938, showed that circuits built with binary switches can easily be described
using Boolean algebra. The abstraction from switches being on and off to the use of Boolean algebra is as follows.
Let B = {0, 1} be the Boolean algebra whose elements are one of the two values, 0 and 1. We define the operations
AND (•), OR (+), and NOT (' ) for the elements of B by the axioms in Figure 2.6(a). These axioms are simply the
definitions for the AND, OR, and NOT operators.
A variable x is called a Boolean variable if x takes on only values in B, i.e. either 0 or 1. Consequently, we
obtain the theorems in Figure 2.6(b) for single variable and Figure 2.6(c) for two and three variables.
Theorems in Figure 2.6(b) can be proved easily by substituting the binary values into the expressions and using
the axioms. For example, to show that Theorem 6a is true, we substitute 0 into x to get axiom 3a, and substitute 1
into x to get axiom 2a.
To prove the theorems in Figure 2.6(c), we can use either one of two methods: 1) use a truth table, or 2) use
axioms and theorems that have already been proven. We show these two methods in the following two examples.
1a.
2a.
3a.
4a. 0•0=0
1•1=1
0•1=1•0=0
0' = 1 1b.
2b.
3b.
4b. 1+1=1
0+0=0
1+0=0+1=1
1' = 0 5b.
6b.
7b. x+1=1
x+0=0+x=x
x+x=x 9b. x + x' = 1 (a)
5a.
6a.
7a.
8a.
9a. x•0=0
x•1=1•x=x
x•x=x
(x' )' = x
x • x' = 0 Null element
Identity
Idempotent
Double complement
Inverse (b)
10a.
11a.
12a.
13a.
14a.
15a. x•y=y•x
(x • y) • z = x • (y • z)
x • (y + z) = (x • y) + (x • z)
x • (x + y) = x
(x • y) + (x • y ' ) = x
(x • y)' = x' + y' 10b.
11b.
12b.
13b.
14b.
15b. x+y=y+x
(x + y) + z = x + (y + z)
x + (y • z) = (x + y) • (x + z)
x + (x • y) = x
(x + y) • (x + y' ) = x
(x + y)' = x' • y' Commutative
Associative
Distributive
Absorption
Combining
DeMorgan’s (c)
Figure 2.6 Boolean algebra axioms and theorems: (a) Axioms; (b) Single variable theorems; (c) two and three
variable theorems.
Example 2.1: Proof of theorem using a truth table. Theorem 12a states that x • (y + z) = (x • y) + (x • z). To prove that Theorem 12a is true using a truth table, we
need to show that for every combination of values for the three variables x, y, and z, the left-hand side of the
expression is equal to the right-hand side. The truth table below is constructed as follows: 35 Digital Logic and Microprocessor Design with VHDL x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 (y + z)
0
1
1
1
0
1
1
1 (x • y)
0
0
0
0
0
0
1
1 Chapter 2 - Digital Circuits (x • z)
0
0
0
0
0
1
0
1 x • (y + z)
0
0
0
0
0
1
1
1 (x • y) + (x • z)
0
0
0
0
0
1
1
1 We start with the first three columns labeled x, y, and z, and enumerate all possible combinations of values for
these three variables. For each combination (row), we evaluate the intermediate expressions y+z, x•y, and x•z by
substituting the values of x, y, and z into the expression. Finally, we obtain the values for the last two columns,
which correspond to the left-hand side and right-hand side of Theorem 12a. The values in these two columns are
♦
identical for every combination of x, y, and z, therefore, we can say that Theorem 12a is true.
Example 2.2: Proof of theorem using axioms and theorems. Theorem 13b states that x + (x • y) = x. To prove that Theorem 13b is true using axioms and theorems, we can
argue as follows:
x + ( x • y ) = ( x • 1) + ( x • y )
= x • (1 + y )
= x • (1)
=x by Identity Theorem 6a
by Distributive Theorem 12a
by Null element Theorem 5b
by Identity Theorem 6a ♦ Example 2.2 shows that some theorems can be derived from others that have already been proven with the truth
table. Full treatment of Boolean algebra is beyond the scope of this book and can be found in the references. For our
purposes, we simply assume that all the theorems are true and will just use them to show that two circuits are
equivalent as depicted in the next two examples.
Example 2.3: Use Boolean algebra to reduce the equation F(x,y,z) = (x' + y' + x'y' + xy) (x' + yz) as much as possible.
F = (x' + y' + x'y' + xy) (x' + yz)
= (x' • 1 + y' • 1 + x'y' + xy) (x' + yz)
= (x' (y + y' ) + y' (x + x' ) + x'y' + xy) (x' + yz)
= (x'y + x'y' + y'x + y'x' + x'y' + xy) (x' + yz)
= (x'y + x'y' + y'x + y'x' + x'y' + xy) (x' + yz)
= (x' (y + y') + x (y + y')) (x' + yz)
= (x' • 1 + x • 1) (x' + yz)
= (x' + x) (x' + yz)
= 1 (x' + yz)
= (x' + yz) by Identity Theorem 6a
by Inverse Theorem 9b
by Distributive Theorem 12a
by Idempotent Theorem 7b
by Distributive Theorem 12a
by Inverse Theorem 9b
by Identity Theorem 6a
by Inverse Theorem 9b
by Identity Theorem 6a Since the expression (x' + y' + x'y' + xy) (x' + yz) reduces down to (x' + yz), therefore, we do want to implement
♦
the circuit for the latter expression rather then the former because the circuit size for the latter is much smaller.
Example 2.4: Show, using Boolean algebra, that the two equations F1 = (xy' + x'y + x' + y' + z' ) (x + y' + z) and
F2 = y' + x'z + xz' are equivalent.
F1 = (xy' + x'y + x' + y' + z' ) (x + y' + z)
= xy'x + xy'y' + xy'z + x'yx + x'yy' + x'yz + x'x + x'y' + x'z + y'x + y'y' + y'z + z'x + z'y' + z'z
= xy' + xy' + xy'z + 0 + 0 + x'yz + 0 + x'y' + x'z + xy' + y' + y'z + xz' + y'z' + 0
= xy' + xy'z + x'yz + x'y' + x'z + y' + y'z + xz' + y'z'
= y'(x + xz + x' + 1 + z + z') + x'z(y + 1) + xz' 36 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits = y' + x'z + xz'
= F2 ♦ 2.5.2 * Duality Principle
Notice in Figure 2.6 that we have listed the axioms and theorems in pairs. Specifically, we define the dual of a
logic expression as one that is obtained by changing all + operators with • operators, and vice versa, and by
changing all 0’s with 1’s, and vice versa. For example, the dual of the logic expression
(x•y'•z) + (x•y•z' ) + (y•z) + 0
is
(x+y'+z) • (x+y+z' ) • (y+z) • 1
The duality principle states that if a Boolean expression is true, then its dual is also true. Be careful in that it
does not say that a Boolean expression is equivalent to its dual. For example, Theorem 5a in Figure 2.6 says that
x • 0 = 0 is true, thus by the duality principle, its dual, x + 1 = 1 is also true. However, x • 0 = 0 is not equal to x + 1
= 1, since 0 is definitely not equal to 1.
We will see in Section 2.5.3 that the inverse of a Boolean expression can be obtained by first taking the dual of
that expression, and then complementing each Boolean variable in the resulting dual expression. In this respect, the
duality principle is often used in digital logic design. Whereas an expression might be complex to implement, its
inverse might be simpler, thus resulting in a smaller circuit, and inverting the final output of this circuit will produce
the same result as from the original expression. 2.5.3 Boolean Function and the Inverse
As we have seen, any digital circuit can be described by a logical expression, also known as a Boolean function.
Any Boolean functions can be formed from binary variables and the Boolean operators •, +, and ' (for AND, OR, and
NOT respectively). For example, the following Boolean function uses the three variables or literals x, y, and z. It has
three AND terms (also referred to as product terms), and these AND terms are ORed (summed) together. The first
two AND terms contain all three variables each, while the last AND term contains only two variables. By definition,
an AND (or product) term is either a single variable, or two or more variables ANDed together. Quite often, we refer
to functions that are in this format as a sum-of-products or or-of-ands.
3 AND terms F(x,y,z) = x y' z + x y z' + y z
3 variables 2 variables The value of a function evaluates to either a 0 or a 1 depending on the given set of values for the variables. For
example, the function above evaluates to a 1 when any one of the three AND terms evaluate to a 1, since 1 OR x is 1.
The first AND term, xy'z, equals to a 1 if
x = 1, y = 0, and z = 1 because if we substitute these values for x, y, and z into the first AND term xy'z, we get a 1. Similarly, the second AND
term, xyz', equals to a 1 if
x = 1, y = 1, and z = 0. The last AND term, yz, has only two variables. What this means is that the value of this term is not dependent on the
missing variable x. In other words x can be either a 0 or a 1, but as long as y = 1 and z = 1, this term will equal to a 1.
Thus, we can summarize by saying that F evaluates to a 1 if
x = 1, y = 0, and z = 1 or 37 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits x = 1, y = 1, and z = 0 or
x = 0, y = 1, and z = 1 or
x = 1, y = 1, and z = 1. Otherwise, F evaluates to a 0.
It is often more convenient to summarize the above verbal description of a function with a truth table as shown
in Figure 2.7 under the column labeled F. Notice that the four rows in the table where F = 1 match the four cases in
the description above.
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 F
0
0
0
1
0
1
1
1 F'
1
1
1
0
1
0
0
0 Figure 2.7 Truth table for the function F = xy'z + xyz' + yz The inverse of a function, denoted by F', can be easily obtained from the truth table for F by simply changing
all the 0’s to 1’s and 1’s to 0’s as shown in the truth table in Figure 2.7 under the column labeled F'. Therefore, we
can write the Boolean function for F' in the sum-of-products format, where the AND terms are obtained from those
rows where F' = 1. Thus, we get
F' = x'y'z' + x'y'z + x'yz' + xy'z' To deduce F' algebraically from F requires the use of DeMorgan’s Theorem (Theorem 15a) twice. For example,
using the same function
F = xy'z + xyz' + yz we obtain F' as follows
F' = (xy'z + xyz' + yz)'
= (xy'z)' • (xyz')' • (yz)'
= (x'+y+z' ) • (x'+y'+z) • (y'+z' ) There are three things to notice about this equation for F'. First, F' is just the dual of F (as defined in
Section 2.5.2) and then having all the variables inverted. Second, instead of being in a sum-of-products format, it is
in a product-of-sums (and-of-ors) format where three OR terms (also referred to as sum terms) are ANDed together.
Third, from the same original function F, we obtained two different equations for F'. From the truth table, we
obtained
F' = x'y'z' + x'y'z + x'yz' + xy'z' and from applying DeMorgan’s Theorem to F, we obtained
F' = (x'+y+z' ) • (x'+y'+z) • (y'+z' ) Hence, we must conclude that these two expressions for F', where one is in the sum-of-products format, and the
other is in the product-of-sums format, are equivalent. In general, all functions can be expressed in either the sumof-products or product-of-sums format. 38 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Thus, we should also be able to express the same function F = xy'z + xyz' + yz in the product-of-sums format.
We can derive it using one of two methods. For method one, we can start with F' and apply DeMorgan’s Theorem to
it just like how we obtained F' from F.
F = F' '
= (x'y'z' + x'y'z + x'yz' + xy'z' )'
= (x'y'z' )' • (x'y'z)' • (x'yz' )' • (xy'z' )'
= (x+y+z) • (x+y+z' ) • (x+y'+z) • (x'+y+z) For the second method, we start with the original F and convert it to the product-of-sums format using the
Boolean theorems.
F = xy'z + xyz' + yz
= (x+x+y) • (x+x+z) • (x+y+y) • (x+y+z) • (x+z'+y) • (x+z'+z) •
(y'+x+y) • (y'+x+z) • (y'+y+y) • (y'+y+z) • (y'+z'+y) • (y'+z'+z) •
(z+x+y) • (z+x+z) • (z+y+y) • (z+y+z) • (z+z'+y) • (z+z'+z)
= (x+y) • (x+z) • (x+y) • (x+y+z) • (x+z'+y) • (y'+x+z) • (z+x+y) • (z+x) • (z+y) • (z+y)
= (x+y) • (x+z) • (x+y+z) • (x+y+z' ) • (x+y'+z) • (z+y)
= (x+y+zz' ) • (x+yy'+z) • (x+y+z) • (x+y+z' ) • (x+y'+z) • (xx'+y+z)
= (x+y+z) • (x+y+z') • (x+y+z) • (x+y'+z) • (x+y+z) • (x+y+z') • (x+y'+z) • (x+y+z) • (x'+y+z)
= (x+y+z) • (x+y+z' ) • (x+y'+z) • (x'+y+z) (step 1) (step 2)
(step 3)
(step 4)
(step 5) In the first step, we apply Theorem 12b (Distributive) to get every possible combination of sum terms. For
example, the first sum term (x+x+y) is obtained from getting the first x from xy'z, the second x from xyz', and the y
from yz. The second sum term (x+x+z) is obtained from getting the first x from xy'z, the second x from xyz', and the z
from yz. This is repeated for all combinations. In this step, the sum terms, such as (x+z'+z), where it contains
variables of the form v + v' can be eliminated since v + v' = 1, and 1 • x = x.
In the second and third steps, duplicate variables and terms are eliminated. For example, the term (x+x+y) is
equal to just (x+y+y), which is just (x+y). The term (x+z'+z) is equal to (x+1), which is equal to just 1, and therefore,
can be eliminated completely from the expression.
In the fourth step, every sum term with a missing variable will have that variable added back in by using
Theorems 6b and 9a, which says that x + 0 = x and yy' = 0, therefore, x + yy' = x.
Step five uses the Distributive Theorem, and the resulting duplicate terms are again eliminated to give us the
format that we want.
Functions that are in the product-of-sums format (such as the one shown below) are more difficult to deduce
when they evaluate to a 1. For example, using
F' = (x'+y+z' ) • (x'+y'+z) • (y'+z' )
F' evaluates to a 1 when all three terms evaluate to a 1. For the first term to evaluate to a 1, x can be 0, or y can be 1,
or z can be 0. For the second term to evaluate to a 1, x can be 0, or y can be 0, or z can be 1. Finally, for the last term,
y can be 0, or z can be 0, or x can be either a 0 or a 1. As a result, we end up with many more combinations to
consider, even though many of the combinations are duplicates. However, it is easier to determine when a product-of-sums format expression evaluates to a 0. For example,
using the same expression
F' = (x'+y+z' ) • (x'+y'+z) • (y'+z' )
F' evaluates to 0 when any one of the three OR terms is 0, since 0 AND x is 0; and this happens when
x = 1, y = 0, and z = 1 for the first OR term, or
x = 1, y = 1, and z = 0 for the second OR term, or
y = 1, z = 1, and x can be either 0 or 1 for the last or term. 39 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Similarly, for a sum-of-products format expression, it is easy to evaluate when it is a 1, but difficult to evaluate
when it is a 0.
These four conditions in which F' evaluates to a 0 match exactly those rows in the table shown in Figure 2.7
where F' = 0. Therefore, we see that in general, the unique algebraic expression for any Boolean function can be
specified by either (1) selecting the rows from the truth table where the function is a 1 and use the sum-of-products
format, or (2) selecting the rows from the truth table where the function is a 0 and use the product-of-sums format.
Whatever format we decide to use, the one thing to remember is that we are always interested in only when the
function (or its inverse) is equal to a 1.
Figure 2.8 summarizes these two formats for the function F = xy'z + xyz' + yz and its inverse F'. Notice that the
sum-of-products format for F is the dual with its variables inverted of the product-of-sums format for F'. Similarly,
the product-of-sums format for F is the dual with its variables inverted of the sum-of-products format for F'.
Sum-of-products
x'yz + xy'z + xyz' + xyz x'y'z' + x'y'z + x'yz' + xy'z' (x+y+z) • (x+y+z') • (x+y'+z) • (x'+y+z) d
rte
ve a l
in du inverse F' equal in
v
du ert
al ed F Product-of-sums equal (x+y'+z') • (x'+y+z') • (x'+y'+z) • (x'+y'+z') Figure 2.8 Relationships between the function F = xy'z + xyz' + yz and its inverse F', and the sum-of-products and
product-of-sums formats. The label “inverted dual” means applying the duality principle and then inverting the
variables. 2.6 Minterms and Maxterms As you recall, a product term is a term with either a single variable, or two or more variables ANDed together,
and a sum term is a term with either a single variable, or two or more variables ORed together. To differentiate
between a term that contains any number of variables with a term that contains all the variables used in the function,
we use the words minterm and maxterm. We are not introducing new ideas here, rather, we are just introducing two
new words and notations for defining what we have already learned. 2.6.1 Minterms
A minterm is a product term that contains all the variables used in a function. For a function with n variables,
the notation mi where 0 ≤ i < 2n, is used to denote the minterm whose index i is the binary value of the n variables
such that the variable is complemented if the value assigned to it is a 0, and uncomplemented if it is a 1.
For example, for a function with three variables x, y, and z, the notation m3 is used to represent the term in
which the values for the variables xyz are 011 (for the subscript 3). Since we want to complement the variable whose
value is a 0, and uncomplement it if it is a 1. Hence m3 is for the minterm x'yz. Figure 2.9(a) shows the eight
minterms and their notations for n = 3 using the three variables x, y, and z.
When specifying a function, we usually start with product terms that contain all the variables used in the
function. In other words, we want the sum of minterms, and more specifically the sum of the one-minterms, that is
the minterms for which the function is a 1 (as opposed to the zero-minterms, that is the minterms for which the
function is a 0). We use the notation 1-minterm to denote one-minterm, and 0-minterm to denote zero-minterm. 40 Digital Logic and Microprocessor Design with VHDL x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 Minterm
x' y' z'
x' y' z
x' y z'
x' y z
x y' z'
x y' z
x y z'
xyz Chapter 2 - Digital Circuits Notation
m0
m1
m2
m3
m4
m5
m6
m7 x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 Maxterm
x+y+z
x + y + z'
x + y' + z
x + y' + z'
x' + y + z
x' + y + z'
x' + y' + z
X' + y' + z' (a) Notation
M0
M1
M2
M3
M4
M5
M6
M7 (b) Figure 2.9 (a) Minterms for three variables. (b) Maxterms for three variables. The function from the previous section
F = xy'z + xyz' + yz
= x'yz + xy'z + xyz' + xyz and repeated in the following truth table has the 1-minterms m3, m5, m6, and m7.
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 F
0
0
0
1
0
1
1
1 F'
1
1
1
0
1
0
0
0 Minterm
x' y' z'
x' y' z
x' y z'
x' y z
x y' z'
x y' z
x y z'
xyz Notation
m0
m1
m2
m3
m4
m5
m6
m7 Thus, a shorthand notation for the function is
F(x, y, z) = m3 + m5 + m6 + m7 By just using the minterm notations, we do not know how many variables are in the original function.
Consequently, we need to explicitly specify the variables used by the function as in F(x, y, z). We can further
simplify the notation by using the standard algebraic symbol Σ for summation. Therefore, we have
F(x, y, z) = Σ(3, 5, 6, 7) These are just different ways of expressing the same function.
Since a function is obtained from the sum of the 1-minterms, the inverse of the function, therefore, must be the
sum of the 0-minterms. This can be easily obtained by replacing the set of indices with those that were excluded
from the original set.
Example 2.5: Given the Boolean function F(x, y, z) = y + x'z, use Boolean algebra to convert the function to the
sum-of-minterms format. This function has three variables. In a sum-of-minterms format, all product terms must have all variables. To do
so, we need to expand each product term by ANDing it with (v + v' ) for every missing variable v in that term. Since
(v + v' ) = 1, therefore, ANDing a product term with (v + v' ) does not change the value of the term.
F = y + x'z
= y(x+x' )(z+z' ) + x'z(y+y' )
expand 1st term by ANDing it with (x+x' )(z+z' ), and 2nd term with (y+y' )
= xyz + xyz' + x'yz + x'yz' + x'yz + x'y'z
= m7 + m6 + m3 + m2 + m1 41 Digital Logic and Microprocessor Design with VHDL = Σ(1, 2, 3, 6, 7) Chapter 2 - Digital Circuits
♦ sum of 1-minterms Example 2.6: Given the Boolean function F(x, y, z) = y + x'z, use Boolean algebra to convert the inverse of the
function to the sum-of-minterms format.
F' = (y + x'z)'
= y' • (x'z)'
= y' • (x+z' )
= y'x + y'z'
= y'x(z+z' ) + y'z' (x+x' )
= xy'z + xy'z' + xy'z' + x'y'z'
= m5 + m4 + m0
= Σ(0, 4, 5) inverse
use DeMorgan
use DeMorgan
use Distributive Theorem to change to sum of products format
expand 1st term by ANDing it with (z+z' ), and 2nd term with (x+x' )
♦ sum of 0-minterms 2.6.2 * Maxterms
Analogous to a minterm, a maxterm is a sum term that contains all the variables used in the function. For a
function with n variables, the notation Mi where 0 ≤ i < 2n, is used to denote the maxterm whose index i is the binary
value of the n variables such that the variable is complemented if the value assigned to it is a 1, and
uncomplemented if it is a 0.
For example, for a function with three variables x, y, and z, the notation M3 is used to represent the term in
which the values for the variables xyz are 011. For maxterms, we want to complement the variable whose value is a
1, and uncomplement it if it is a 0. Hence M3 is for the maxterm x + y' + z'. Figure 2.9(b) shows the eight maxterms
and their notations for n = 3 using the three variables x, y, and z.
We have seen that a function can also be specified as a product of sums, or more specifically, a product of 0maxterms, that is, the maxterms for which the function is a 0. Just like the minterms, we use the notation
1-maxterm to denote one-maxterm, and 0-maxterm to denote zero-maxterm. Thus, the function
F(x, y, z) = xy'z + xyz' + yz
= (x + y + z) • (x + y + z') • (x + y' + z) • (x' + y + z) which is shown in the following table
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 F
0
0
0
1
0
1
1
1 F'
1
1
1
0
1
0
0
0 Maxterm
x+y+z
x + y + z'
x + y' + z
x + y' + z'
x' + y + z
x' + y + z'
x' + y' + z
x' + y' + z' Notation
M0
M1
M2
M3
M4
M5
M6
M7 can be specified as the product of the 0-maxterms M0, M1, M2, and M4. The shorthand notation for the function is
F(x, y, z) = M0 • M1 • M2 • M4 Again, by using the standard algebraic symbol Π for product, the notation is further simplified to
F(x, y, z) = Π (0, 1, 2, 4) The following summarizes these relationships for the function F = xy'z + xyz' + yz and its inverse. Comparing
these equations with those in Figure 2.8, we see that they are identical. 42 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits F(x, y, z) = x' y z + x y' z + x y z' + x y z
= m3 + m5 + m6 + m7
= Σ(3, 5, 6, 7) Σ 1-minterms = (x+y+z) • (x+y+z' ) • (x+y'+z) • (x'+y+z)
= M0 • M1 • M2 • M4
= Π(0, 1, 2, 4)
F'(x, y, z) = x' y' z' + x' y' z + x' y z' + x y' z'
= m0 + m1 + m2 + m4
= Σ(0, 1, 2, 4) equivalent
Π 0-maxterms inverted
duals inverse Σ 0-minterms = (x+y'+z' ) • (x'+y+z' ) • (x'+y'+z) • (x'+y'+z' )
= M3 • M5 • M6 • M7
Π 1-maxterms
= Π(3, 5, 6, 7) equivalent Notice that it is always the Σ of minterms and Π of maxterms; you never have Σ of maxterms or Π of minterms.
Example 2.7: Given the Boolean function F(x, y, z) = y + x'z, use Boolean algebra to convert the function to the
product-of-maxterms format. To change a sum term to a maxterm, we expand each term by ORing it with (vv' ) for every missing variable v in
that term. Since (vv' ) = 0, therefore, ORing a sum term with (vv' ) does not change the value of the term.
F = y + x'z
= y + (x'z)
use Distributive Theorem to change to product of sums format
= (y+x' )(y+z)
expand 1st term by ORing it with zz', and 2nd term with xx'
= (y+x' +zz' )(y+z+xx' )
= (x' +y+z) (x' +y+z' ) (x+y+z) (x' +y+z)
= M4 • M5 • M0
product of 0-maxterms
= Π(0, 4, 5) ♦ Example 2.8: Given the Boolean function F(x, y, z) = y + x'z, use Boolean algebra to convert the inverse of the
function to the product-of-maxterms format.
F' = (y + x' z)'
inverse
use DeMorgan
= y' • (x' z)'
= y' • (x+z' )
use DeMorgan
= (y' +xx' +zz' ) • (x+z' +yy' )
expand 1st term by ORing it with xx' +zz', and 2nd term with yy'
= (x+y' +z) (x+y' +z' ) (x' +y' +z) (x' +y' +z' ) (x+y+z' ) (x+y' +z' )
= M2 • M3 • M6 • M7 • M1
= Π(1, 2, 3, 6, 7)
product of 1-maxterms 2.7 ♦ Canonical, Standard, and non-Standard Forms Any Boolean function that is expressed as a sum of minterms, or as a product of maxterms is said to be in its
canonical form. For example, the following two expressions are in their canonical forms
F = x' y z + x y' z + x y z' + x y z
F' = (x+y'+z' ) • (x'+y+z' ) • (x'+y'+z) • (x'+y'+z' ) As noted from the previous section, to convert a Boolean function from one canonical form to its other
equivalent canonical form, simply interchange the symbols Σ with Π, and list the index numbers that were excluded
from the original form. For example, the following two expressions are equivalent 43 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits F1(x, y, z) = Σ(3, 5, 6, 7)
F2(x, y, z) = Π(0, 1, 2, 4) To convert a Boolean function from one canonical form to its inverse, simply interchange the symbols Σ with
Π, and list the same index numbers from the original form. For example, the following two expressions are inverses
F1(x, y, z) = Σ(3, 5, 6, 7)
F2(x, y, z) = Π(3, 5, 6, 7) A Boolean function is said to be in a standard form if a sum-of-products expression or a product-of-sums
expression has at least one term that is not a minterm or a maxterms respectively. In other words, at least one term in
the expression is missing at least one variable. For example, the following expression is in a standard form because
the last term is missing the variable x.
F = xy'z + xyz' + yz Sometimes, common variables in a standard form expression can be factored out. The resulting expression is no
longer in a sum-of-products or product-of-sums format. These expressions are in a non-standard form. For
example, starting with the previous expression, if we factor out the common variable x from the first two terms, we
get the following expression, which is in a non-standard form.
F = x(y'z + yz') + yz 2.8 Logic Gates and Circuit Diagrams Logic gates are the actual physical implementations of the logical operators discussed in the previous sections.
Transistors, acting as tiny electronic binary switches are connected together to form these gates. Thus, we have the
AND gate, the OR gate, and the NOT gate (also called the INVERTER ) for the corresponding AND, OR, and NOT logical
operators. These gates form the basic building blocks for all digital logic circuits. The name “gate” comes from the
fact that these devices operate like a door or gate to let or not to let things (in our case, current) through. In drawing digital circuit diagrams, also called schematic diagrams, or just schematics, we use special logic
symbols to denote these gates as shown in Figure 2.10. The AND gate, or specifically, the 2-input AND gate, in
Figure 2.10(a) has two input connections coming in from the left and one output connection going out on the right.
Similarly, the 2-input OR gate in Figure 2.10(b) has two input connections and one output connection. The INVERTER
in Figure 2.10(c) has one input coming from the left and one output going to the right. The outputs from these gates,
of course, are dependent on their inputs, and are defined by their logical functions. (a) (b) (c) Figure 2.10 Logic symbols for the three basic logic gates: (a) 2-input AND; (b) 2-input OR; (c) NOT. Sometimes, an AND gate or an OR gate with more than two inputs are needed. Hence, in addition to the 2-input
and OR gates, there are 3-input, 4-input, or as many inputs as are needed of the AND and OR gates. In practice,
however, the number of inputs is limited to a small number, such as five. The logic symbols for some of these gates
are shown in Figure 2.11(a) to (d).
AND There are several other gates that are variants of the three basic gates that are also often used in digital circuits.
They are the NAND gate, the NOR gate, the XOR gate, and the XNOR gate. The NAND gate is derived from an AND gate
and the INVERTER connected in series so that the output of the AND gate is inverted. The name “NAND” comes from
the description “Not AND.” Similarly, the NOR gate is the OR gate with its output inverted. The XOR, or eXclusive OR
gate is like the OR gate except that when both inputs are 1, the output is a 0 instead. The XNOR, or eXclusive NOR
gate is just the inverse of the XOR gate for when there are an even number of inputs (like 2 inputs). When there are
an odd number of inputs (like 3 inputs), the XOR is the same as the XNOR. The logic symbols and their truth tables
for some of these gates are shown in Figure 2.11 and Figure 2.12 respectively. 44 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Notice, in Figure 2.11, the use of the little circle or bubble at the output of some of the logic symbols. This
bubble is used to denote the inverted value of a signal. For example, the NAND gate is the inverse of the AND gate,
thus the NAND gate logic symbol is the same as the AND gate logic symbol except that it has the extra bubble at the
output. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 2.11 Logic symbols for: (a) 3-input AND; (b) 4-input AND; (c) 3-input OR; (d) 4-input OR; (e) 2-input NAND;
(f) 2-input NOR; (g) 3-input NAND; (h) 3-input NOR; (i) 2-input XOR; (j) 2-input XNOR. x
0
0
1
1 y
0
1
0
1 3-AND
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 x•y•z
0
0
0
0
0
0
0
1 2-NAND
(x•y)'
1
1
1
0 2-NOR
(x+y)'
1
0
0
0 3-OR
x+y+z
0
1
1
1
1
1
1
1 (x • y • z)'
1
1
1
1
1
1
1
0 3-NAND 2-XOR
x⊕y
0
1
1
0 2-XNOR
xy
1
0
0
1 3-NOR
(x + y + z)'
1
0
0
0
0
0
0
0 3-XOR
x⊕y⊕z
0
1
1
0
1
0
0
1 3-XNOR
xyz
0
1
1
0
1
0
0
1 Figure 2.12 Truth tables for: 2-input NAND; 2-input NOR; 2-input XOR; 2-input XNOR; 3-input AND; 3-input
OR; 3-input NAND; 3-input NOR; 3-input XOR; 3-input XNOR. The notations used for these gates in a logical expression are (xy)' for the 2-input NAND gate, (x+y)' for the 2input NOR gate, x ⊕ y for the XOR gate, and x y for the XNOR gate.
Looking at the truth table for the 2-input XOR gate, we can derive the equation for the 2-XOR gate as
x ⊕ y = x'y + xy' Similarly, the equation for the 2-input XNOR gate as derived from the 2-XNOR truth table is
x y = x'y' + xy The equation for the 3-input XOR gate is derived as follows
x⊕y⊕z
= (x ⊕ y) ⊕ z
= (x'y + xy' ) ⊕ z
= (x'y + xy' )z' + (x'y + xy' )'z
= x'yz' + xy'z' + (x'y)' (xy' )'z 45 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits = x'yz' + xy'z' + (x+y' ) (x'+y) z
= x'yz' + xy'z' + xx'z + xyz + x'y'z + y'yz
= x'y'z + x'yz' + xy'z' + xyz
The last four product terms in the above derivation are the four 1-minterms in the 3-input XOR truth table. For 3
or more inputs, the XOR gate has a value of 1when there is an odd number of 1’s in the inputs, otherwise, it is a 0.
Notice also that the truth tables for the 3-input XOR and XNOR gates are identical. It turns out that for an even
number of inputs, XOR is the inverse of XNOR, but for an odd number of inputs, XOR is equal to XNOR.
All these gates can be interconnected together to form large complex circuits which we call networks. These
networks can be described graphically using circuit diagrams, with Boolean expressions or with truth tables.
Example 2.9: Draw the circuit diagram for the equation F(x, y, z) = y + x'z. In the equation, we need to first invert x, and then AND it with z. Finally, we need to OR y with the output of the
The resulting circuit is shown below. For easy reference, the internal nodes in the circuit are annotated with the
two intermediate values x' and x'z.
AND. x' x x'z z
F y ♦ Example 2.10: Draw the circuit diagram for the equation F(x, y, z) = xyz + xyz' + x'yz + x'yz' + x'y'z. The equation consists of five AND terms that are ORed together. Each AND term requires three inputs for the
three variables. Hence, the circuit shown below has five 3-input AND gates, whose outputs are connected to a 5-input
OR gate. The inputs to the AND gates come directly from the three variables x, y, and z, or their inverted values.
Notice that in the equation, there are six inverted variables. However, in the circuit, we do not need six inverters,
rather, only three inverters are used; one for each variable.
x y z F ♦ 2.9 Designing a Car Security System In a car security system, we usually want to connect the siren in such a way that the siren will activate when it is
triggered by one or more sensors. In addition, there will be a master switch to turn the system on or off. Let us
assume that there is a car door switch D, a vibration detector switch V, and the master switch M. We will use the
convention that when the door is opened D = 1, otherwise, D = 0. Similarly, when the car is being shaken, V = 1,
otherwise, V = 0. Thus, we want the siren S to turn on, that is, set S = 1, when either D = 1 or V = 1, or when both
D = 1and V = 1, but only for when the system is turned on, that is, when M = 1. However, when we turn off the
system, and either enter or drive the car, we do not want the siren to turn on. Hence, when M = 0, it does not matter
what values D and V have, the siren should remain off.
Given the above description of a car security system, we can build a digital circuit that meets our specifications.
We start by constructing a truth table, which is basically a precise way of stating the operations for the device. The
table will have three input columns M, D, and V, and an output column S as shown below 46 Digital Logic and Microprocessor Design with VHDL M
0
0
0
0
1
1
1
1 Chapter 2 - Digital Circuits D
0
0
1
1
0
0
1
1 V
0
1
0
1
0
1
0
1 S
0
0
0
0
0
1
1
1 The values under the S column are obtained from interpreting the description of when we want the siren to turn
on. When M = 0, we don’t want the siren to come on, regardless of what the values for D and V are. When M = 1,
we want the siren to come on when either or both D and V is a 1.
The truth table can be described formally with a logic expression written in words as
S = (M AND (NOT D) AND V) OR (M AND D AND (NOT V)) OR (M AND D AND V) or preferably using the simpler notation of a Boolean function
S = (M D' V) + (M D V') + (M D V) Again, what this equation is saying is that we want the siren to activate, S = 1, when the master switch is on and
the door is not opened and the vibration switch is on, or the master switch is on and the door is opened and the
vibration switch is not on, or the master switch is on and the door is opened and the vibration switch is on. Notice
that we are only interested in the situations when S = 1. We ignore the rows when S = 0. When we construct circuits
from truth tables, we always use only the rows where the output is a 1.
Finally, we can translate this equation into a circuit diagram. The translation is a simple one-to-one mapping of
changing the AND operator into the AND gate, the OR operator into the OR gate, and the NOT operator into the
INVERTER. Thus, we get the following circuit diagram for our car security system
M D V S A careful reader might notice that the Boolean equation shown above and the circuit for specifying when the
siren is to be turned on can be simplified to
D
V
M S = M (D + V ) S This simplified equation says that the siren is to be turned on only when the master switch is on and either the door
switch or vibration switch is on. Just by using simple reasoning, we can see that this simplified circuit will do
exactly what the previous circuit does. In other words, both circuits are functionally equivalent.
More formally, we can use the Boolean Theorems from Section 2.5.1 to show that these two equations are
indeed equivalent as follows
S = ( M D' V ) + ( M D V ' ) + ( M D V ) 47 Digital Logic and Microprocessor Design with VHDL = M (D' V + D V' + D V)
= M (D' V + D V' + D V + DV)
= M ( D(V' + V) + V(D' + D) )
= M ( D(1) + V(1))
= M (D + V ) Chapter 2 - Digital Circuits by Distributive Theorem 12a
by Idempotent Theorem 7b
by Distributive Theorem 12a
by Inverse Theorem 9b
by Identity Theorem 6a Figure 2.13(a) shows a sample simulation trace of the car security system circuit. Between times 0 and 200ns,
the master switch M is a 0, so regardless of the values of D and V, the siren is off (Siren=0). Between times 200ns
and 600ns, M = 1. During this time, whenever either D = 1 or V = 1, the siren is on. This is a functional trace of the
circuit, and so all the signal edges line up exactly, i.e., the output signal edge changes at exactly the same time (with
no delay) as the input edge that caused it to change. For a timing trace, on the other hand, the output signal edge will
be delayed slightly after the causing input edge as shown in Figure 2.13(b). (a) (b)
Figure 2.13 Sample simulation trace of the car security system circuit: (a) functional trace; (b) timing trace. When building a circuit, besides having a functionally correct circuit, we also want to optimize it in terms of its
size, speed, heat dissipation, and power consumption. We will see in later sections how circuits are optimized. 2.10 VHDL for Digital Circuits
A digital circuit that is described with a Boolean function can easily be converted to VHDL code using the
dataflow model. At the dataflow level, a circuit is defined using built-in VHDL logic operators such as AND, OR, and
NOT. These operators are applied to signals using concurrent signal assignment statements. 2.10.1 VHDL code for a 2-input NAND gate
Figure 2.14 shows the VHDL code for a 2-input NAND gate. It also serves as a basic template for all VHDL
codes. Lines starting with two hyphens are comments. The LIBRARY and USE statements specify that the IEEE
library is needed and that all the components in that library package can be used. These two statements are
equivalent to the “#include” preprocessor line in C++.
Every component defined in VHDL, whether it is a simple NAND gate or a complex microprocessor, has two
parts: an ENTITY section and an ARCHITECTURE section. The entity section is similar to a function declaration in C++
and serves as the interface between the component and the outside. It declares all the input and output signals for a
circuit. Every entity must have a unique name; in the example, the name NAND2gate is used. The entity contains a
PORT list, which, like a parameter list, specifies the data to be passed in and out of the component. In the example,
there are two input signals called x and y of type STD_LOGIC and an output signal called f of the same type. The
STD_LOGIC type is like the BIT type, except that it contains additional values besides just 0 and 1. 48 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits The architecture is the definition of the component, and contains the code that realizes the operation of the
component. For every architecture, you need to specify its name, and which entity it is for. In the example, the name
is Dataflow, and it is for the entity NAND2 gate. It is possible for one entity to have more then one architecture,
since an entity can be implemented in more than one way. Within the body of the architecture, we can have one or
more concurrent statements. Unlike statements in C++ where they are executed in sequential order, concurrent
statements in the architecture body are executed in parallel. Thus, the ordering of these statements is irrelevant. The
symbol “<=” is used for a signal assignment statement. The expression on the right-hand side of the <= symbol is
evaluated when either x or y changes values (either from 0 to 1, or from 1 to 0), and the result is assigned to the
signal on the left-hand side. The NAND operator is a built-in VHDL operator.
-- this is a dataflow model of a 2-input NAND gate
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.all;
ENTITY NAND2gate IS PORT (
x: IN STD_LOGIC;
y: IN STD_LOGIC;
f: OUT STD_LOGIC);
END NAND2gate;
ARCHITECTURE Dataflow OF NAND2gate IS
BEGIN
f <= x NAND y;
-- signal assignment
END Dataflow;
Figure 2.14 VHDL code for a 2-input NAND gate. 2.10.2 VHDL code for a 3-input NOR gate
Figure 2.15(a) shows the VHDL code for a 3-input NOR gate. In addition to the three input signals x, y, and z,
and one output signal f declared in the entity section, this example has two internal signals, xory and xoryorz, both of
which are of type STD_LOGIC. The keyword SIGNAL in the architecture section is used to declare these two internal
signals. Internal signals are used for naming connection points (or nodes) within a circuit. Three concurrent signal
assignment statements are used. All the signal assignment statements are executed concurrently, so the ordering of
the statements is irrelevant. The coding of these three signal assignment statements is based on the 3-input NOR gate
circuit shown in Figure 2.15(b).
Figure 2.15(c) shows a sample simulation trace of the circuit. In the trace, we see that the output signal f is a 1
only when all three inputs are 0’s. This occurs twice, first time between 0 and 100ns, and second time between
800ns and 900ns. For all the other times, f is a 0, since not all three inputs are 0’s. Hence, the simulation trace shows
the correct operation of this circuit for the 3-input NOR gate.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.all;
ENTITY NOR3gate IS PORT (
x: IN STD_LOGIC;
y: IN STD_LOGIC;
z: IN STD_LOGIC;
f: OUT STD_LOGIC);
END NOR3gate;
ARCHITECTURE Dataflow OF NOR3gate IS
SIGNAL xory, xoryorz : STD_LOGIC;
BEGIN
xory <= x OR y;
-- three concurrent signal assignments 49 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits xoryorz <= xory OR z;
f <= NOT xoryorz;
END Dataflow; (a)
x
y
z xory
xoryorz z (b) (c)
Figure 2.15 3-input NOR gate: (a) VHDL code; (b) circuit; (c) simulation trace. 2.10.3 VHDL code for a function
Figure 2.16 shows the VHDL code, and the simulation trace for the car security system circuit of Section 2.9.
The function implemented is S = (M D' V) + (M D V') + (M D V).
This VHDL code (as well as the ones from the two previous sections) is written at the dataflow level, not
because the name of the architecture is “Dataflow.” Dataflow level coding uses logic equations to describe a circuit,
and this is done by using the built-in VHDL operators such as AND, OR, and NOT in concurrent signal assignment
statements.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.all;
ENTITY Siren IS PORT (
M: IN STD_LOGIC;
D: IN STD_LOGIC;
V: IN STD_LOGIC;
S: OUT STD_LOGIC);
END Siren;
ARCHITECTURE Dataflow OF Siren IS
SIGNAL term_1, term_2, term_3: STD_LOGIC;
BEGIN
term_1 <= M AND (NOT D) AND V;
term_2 <= M AND D AND (NOT V);
term_3 <= M AND D AND V;
S <= term_1 OR term_2 OR term_3;
END Dataflow; (a) 50 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits (b)
Figure 2.16 The car security circuit of Section 2.9: (a) dataflow VHDL code; (b) simulation trace. 2.11 Summary Checklist
Binary number
Hexadecimal number
Binary switch
AND, OR, and NOT
Truth table
Boolean algebra axioms and theorems
Duality principle
Boolean function and the inverse
Product term
Sum term
Sum-of-products, or-of-ands
Product of sums, and-of-ors
Minterm and maxterm
Sum-of minterms
Product-of-maxterms
Canonical, standard, and non-standard form
Logic gate, logic symbol
Circuit diagram
NAND, NOR, XOR, XNOR
Network
VHDL
Be able to derive the Boolean equation from a truth table, or vice versa
Be able to derive the circuit diagram from a Boolean equation, or vice versa
Be able to derive the circuit diagram from a truth table, or vice versa
Be able to use Boolean algebra to reduce a Boolean equation 51 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits 2.12 Problems
2.1
a)
b)
c)
d)
e)
f) Convert the following decimal numbers to binary:
66
49
513
864
1897
2004 a)
b)
c)
d)
e)
f) Convert the following unsigned binary numbers to decimal, hexadecimal and octal:
11110
11010
100100011
1011011
1101101110
101111010100 2.2 Answer:
a) 3010, 1E16, 368
e) 87810, 36E16, 15568
2.3
a)
b)
c)
d)
e)
f) Convert the following hexadecimal numbers to binary:
66
E3
2FE8
7C2
5A2D
E08B a)
b)
c)
d)
e)
f)
g)
h) Derive the truth table for the following Boolean functions.
F(x,y,z) = x'y'z' + x'yz + xy'z' + xyz
F(x,y,z) = xy'z + x'yz' + xyz + xyz'
F(w,x,y,z) = w'xy'z + w'xyz + wxy'z + wxyz
F(w,x,y,z) = wxy'z + w'yz' + wxz + xyz'
F(x,y,z) = xy' + x'y'z + xyz'
F(w,x,y,z) = w'z' + w'xy + wx'z + wxyz
F(x,y,z) = [(x+y' ) (yz)' ] (xy' + x'y)
F(N3,N2,N1,N0) = N3'N2'N1N0' + N3'N2'N1N0 + N3N2'N1N0' + N3N2'N1N0 + N3N2N1'N0' + N3N2N1N0 2.4 Answer:
g)
xy
00
00
01
01
10
10
11
11 2.5 z
0
1
0
1
0
1
0
1 x'
1
1
1
1
0
0
0
0 y'
1
1
0
0
1
1
0
0 x+y'
1
1
0
0
1
1
1
1 yz
0
0
0
1
0
0
0
1 (yz)'
1
1
1
0
1
1
1
0 [(x+y' ) (yz)' ]
1
1
0
0
1
1
1
0 xy'
0
0
0
0
1
1
0
0 Derive the Boolean function for the following truth tables: 52 x'y
0
0
1
1
0
0
0
0 (xy' + x'y)
0
0
1
1
1
1
0
0 [(x+y' ) (yz)' ] (xy' + x'y)
0
0
0
0
1
1
0
0 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits a) b)
a
0
0
0
0
1
1
1
1 b
0
0
1
1
0
0
1
1 c
0
1
0
1
0
1
0
1 F
0
0
1
1
0
0
1
0 w
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 c) a)
b)
c)
d) y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F
0
0
1
0
1
1
0
1
0
1
1
0
1
1
0
1 d)
w
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 2.6 x
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 x
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F1
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1 F2
1
1
1
1
0
1
0
0
1
1
0
0
1
1
1
1 N3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 N2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 Use a truth table to show that the following expressions are true:
w'z' + w'xy + wx'z + wxyz = w'z' + xyz + wx'y'z + wyz
z + y' + yz' = 1
xy'z' + x' + xyz' = x' + z'
xy + x'z + yz = xy + x'z Answer:
d)
x
0
0
0
0
1
1 y
0
0
1
1
0
0 z
0
1
0
1
0
1 xy
0
0
0
0
0
0 x'z
0
1
0
1
0
0 yz
0
0
0
1
0
0 53 xy+x'z+yz
0
1
0
1
0
0 xy+x'z
0
1
0
1
0
0 N1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 N0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F
0
0
1
1
0
0
1
0
0
0
1
1
1
0
0
1 Digital Logic and Microprocessor Design with VHDL 1
1
e)
f)
g)
h)
2.7
a) 1
1 0
1 1
1 Chapter 2 - Digital Circuits 0
0 0
1 1
1 1
1 w'x'yz' + w'x'yz + wx'yz' + wx'yz + wxyz = y(x' + wz)
w'xy'z + w'xyz + wxy'z + wxyz = xz
xiyi + ci(xi + yi) = xiyici + xiyici' + xiyi'ci + xi'yici
xiyi + ci(xi + yi) = xiyi + ci(xi ⊕ yi) Use Boolean algebra to show that the expressions in Problem 2.6 are true.
Answer w'z' + w'xy + wx'z + wxyz
= w'x'y'z' + w'x'yz' + w'xy'z' + w'xyz' + w'xyz' + w'xyz + wx'y'z + wx'yz + wxyz
= w'x'y'z' + w'x'yz' + w'xy'z' + w'xyz' + w'xyz + wx'y'z + wx'yz + wxyz
= w'z' + w'xyz + wx'y'z + wx'yz + wxyz
= w'z' + (w'+w)xyz + wx'y'z + w(x'+x)yz
= w'z' + xyz + wx'y'z + wyz c)
Answer:
x y' z' + x' + x y z' = x z' (y' + y) + x'
= x z' + x'
= x z' + 1 x'
= (x + 1)(x + x' )(z' + 1)(z' + x' )
= 1 • 1 • 1 (z' + x' )
= x' + z' f)
Answer:
w'xy'z + w'xyz + wxy'z + wxyz
= xy'z(w' + w) + xyz(w' + w)
= xy'z + xyz
= xz(y + y')
= xz
g)
Answer:
xiyi + ci(xi + yi)
= xiyi + xici + yici
= xiyi(ci + ci' ) + xi(yi + yi' )ci + (xi + xi' )yici
= xiyici + xiyici' + xiyici + xiyi'ci + xiyici + xi'yici
= xiyici + xiyici' + xiyi'ci + xi'yici 2.8 Use Boolean algebra to reduce the functions in Problem 2.4 as much as possible.
Answer:
a)
F(x,y,z) = x'y'z' + x'yz + xy'z' + xyz
= y'z' (x + x') + yz(x + x' )
= y'z' + yz
= (y + z' ) (y' + z)
=y z 2.9 Any function can be implemented directly as specified or as its inverted form with a not gate added at the 54 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits final output. Assume that the circuit size is proportional to only the number of AND gates and OR gates, i.e.
ignore the number of NOT gates in determining the circuit size. Determine which form of the function (the
inverted or un-inverted) will result in a smaller circuit size for the following function. Give your reason and
specify how many AND and OR gates are needed to implement the smaller circuit.
F(x,y,z) = x'y'z' + x'y'z + xy'z + xy'z' + xyz 2.10 Derive the truth table for the following logic gates:
a) A 4-input AND gate.
b) A 4-input NAND gate.
c) A 4-input NOR gate.
d) A 4-input XOR gate.
e) A 4-input XNOR gate.
f) A 5-input XOR gate.
g) A 5-input XNOR gate.
2.11 Derive the truth table for the following Boolean functions.
a) F(w,x,y,z) = [(x y)' + (xyz)'] (w' + x + z)
b) F(x,y,z) = x ⊕ y ⊕ z
c) F(w,x,y,z) = [w'xy'z + w'z (y ⊕ x)]'
2.12 Use Boolean algebra to convert the functions in Problem 2.11 to:
a) The sum-of-minterms format.
b) The product-of-maxterms format.
2.13 Use Boolean algebra to reduce the functions in Problem 2.11 as much as possible.
2.14 Use a truth table to show that the following expressions are true:
a) (x ⊕ y) = (x y)'
b) x ⊕ y' = x y
c) (w ⊕ x) (y ⊕ z) = (w x) (y z) = (((w x) y) z).
Answer
w x y Z w⊕x y⊕z (w⊕x) (y⊕z) w x y z (w x) (y z)
0000
0
0
1
1
1
1
0001
0
1
0
1
0
0
0010
0
1
0
1
0
0
0011
0
0
1
1
1
1
0100
1
0
0
0
1
0
0101
1
1
1
0
0
1
0110
1
1
1
0
0
1
0111
1
0
0
0
1
0
1000
1
0
0
0
1
0
1001
1
1
1
0
0
1
1010
1
1
1
0
0
1
1011
1
0
0
0
1
0
1100
0
0
1
1
1
1
1101
0
1
0
1
0
0
1110
0
1
0
1
0
0
1111
0
0
1
1
1
1
d) [((xy)'x)' ((xy)'y)' ]' = x ⊕ y
2.15 Use Boolean algebra to show that the expressions in Question 2.14 are true.
d) 55 (((w x) y) z)
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits Answer:
F = [((xy)'x)' ((xy)'y)' ]'
= ((xy)'x) + ((xy)'y)
= (x' + y' )x + (x' + y' )y
= xx' + xy' + x'y + y'y
= xy' + x'y
=x⊕y
2.16 Use Boolean algebra to show that XOR = XNOR for three inputs.
Answer:
x ⊕ y ⊕ z = (x ⊕ y ) ⊕ z
= (x'y + xy') ⊕ z
= (x'y + xy')' z + (x'y + xy') z'
= (x'y)' · (xy')' z + x'yz' + xy'z'
= (x + y') · (x' + y) z + x'yz' + xy'z'
= xx'z + xyz + x'y'z + y'yz + x'yz' + xy'z'
= (xy + x'y') z + (x'y + xy') z'
= (xy + x'y') z + (xy + x'y')' z'
= (x y) z + (x y)' z'
=x y z
2.17 Express the Boolean functions in Question 2.4 using:
a) The Σ notation.
b) The Π notation.
2.18 Write the following expression as a Boolean function in the canonical form:
c) F(x, y, z) = Σ(1, 3, 7)
d) F(w, x, y, z) = Σ(1, 3, 7)
e) F(x, y, z) = Π(1, 3, 7)
f) F(w, x, y, z) = Π(1, 3, 7)
g) F' (x, y, z) = Σ(1, 3, 7)
h) F' (x, y, z) = Π(1, 3, 7)
2.19 Given F' (x, y, z) = Σ(1, 3, 7), express the function F using a truth table.
Answer:
F' is expressed as a sum of its 0-minterms. Therefore, F is the sum of its 1-minterms = Σ(0, 2, 4, 5, 6). Using
three variables, the truth table is as follows:
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 Minterms
m0=x' y' z'
m1=x' y' z
m2=x' y z'
m3=x' y z
m4=x y' z'
m5=x y' z
m6=x y z'
m7=x y z F
1
0
1
0
1
1
1
0 2.20 Use Boolean algebra to convert the function F(x, y, z) = Σ(3, 4, 5) to its equivalent product-of-sum canonical
form.
Answer:
F = Σ(3, 4, 5) = m3 + m4 + m5
= x’yz + xy’z’ + xy’z
= (x’ + x + x)(x’ + x + y’)(x’ + x + z) 56 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits (x’ + y’ + x)(x’ + y’ + y’)(x’ + y’ + z)
(x’ + z’ + x)(x’ + z’ + y’)(x’ + z’ + z)
(y + x + x)(y + x + y’)(y + x + z)
(y + y’ + x)(y + y’ + y’)(y + y’ + z)
(y + z’ + x)(y + z’ + y’)(y + z’ + z)
(z + x + x)(z + x + y’)(z + x + z)
(z + y’ + x)(z + y’ + y’)(z + y’ + z)
(z + z’ + x)(z + z’ + y’)(z + z’ + z)
= (x’ + y’ + z) (x’ + y’ + z’) (x + y + z) (x + y + z’) (x + y’ + z)
2.21 Given F = xy'z' + xy'z + xyz' + xyz, write the expression for F ' using:
a) the AND-of-OR format
Answer:
AND-of-OR format is obtained by using the duality principle or De Morgan’s Theorem:
F' = (x'+y+z) • (x'+y+z') • (x'+y'+z) • (x'+y'+z')
b) the OR-of-AND formats
Answer:
OR-of-AND format is obtained by first constructing the truth table for F and then inverting the 0’s and 1’s
to get F '. Then we simply use the AND terms where F' = 1.
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 F
0
0
0
0
1
1
1
1 F'
1
1
1
1
0
0
0
0 F ' = x'y'z' + x'y'z + x'yz' + x'yz 2.22 Use Boolean algebra to convert the equation F = w x y z to:
a) The sum-of-minterms format.
Answer
F=w x y z
= (wx + w'x' ) y z
= [(wx + w'x' )y + (wx + w'x' )' y' ] z + [(wx + w'x' )y + (wx + w'x' )' y' ]' z'
= wxyz + w'x'yz + (wx)' (w'x' )'y'z + [(wx + w'x' )y + (wx + w'x' )' y' ]' z'
= m15 + m3 + (w'+x' )(w+x)y'z + [(wx + w'x' )y + (wx + w'x' )' y' ]' z'
= m15 + m3 + w'xy'z + wx'y'z + [(wx + w'x' )y + (wx + w'x' )' y' ]' z'
= m15 + m3 + m5 + m9 + [(wx + w'x' ) y]' [(wx + w'x' )' y' ]' z'
= m15 + m3 + m5 + m9 + [(wx + w'x' )' + y' ] [(wx + w'x' )+ y] z'
= m15 + m3 + m5 + m9 + [(wx)' (w'x' )' + y' ] [wxz' + w'x' z' + yz' ]
= m15 + m3 + m5 + m9 + [(w'+x' )(w+x) + y' ] [wxz' + w'x' z' + yz' ]
= m15 + m3 + m5 + m9 + [w'x + wx' + y' ] [wxz' + w'x' z' + yz' ]
= m15 + m3 + m5 + m9 + w'xyz' + wx'yz' + wxy'z' + w'x'y'z'
= m15 + m3 + m5 + m9 + m6 + m10 + m12 + m0
b) The product-of-maxterms format.
2.23 Write the complete dataflow VHDL code for the Boolean functions in Problem 2.4.
2.24 Write the complete dataflow VHDL code for the logic gates in Problem 2.10. 57 Digital Logic and Microprocessor Design with VHDL Chapter 2 - Digital Circuits 2.25 Write the complete dataflow VHDL code for the Boolean functions in Problem 2.11. 58 Chapter 3 Combinational Circuits Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Digital circuits, regardless of whether they are part of the control unit or the datapath, are classified as either
one of two types: combinational or sequential. Combinational circuits are the class of digital circuits where the
outputs of the circuit are dependent only on the current inputs. In other words, a combinational circuit is able to
produce an output simply from knowing what the current input values are. Sequential circuits, on the other hand,
are circuits whose outputs are dependent on not only the current inputs, but also on all of the past inputs. Therefore,
in order for a sequential circuit to produce an output, it must know the current input and all past inputs. Because of
their dependency on past inputs, sequential circuits must contain memory elements in order to remember the history
of past input values. Combinational circuits do not need to know the history of past inputs, and therefore, do not
require any memory elements. A “large” digital circuit may contain both combinational circuits and sequential
circuits. However, regardless of whether it is a combinational circuit or a sequential circuit, it is nevertheless a
digital circuit, and so they use the same basic building blocks – the AND, OR, and NOT gates. What makes them
different is in the way the gates are connected.
The car security system from Section 2.9 is an example of a combinational circuit. In the example, the siren is
turned on when the master switch is on and someone opens the door. If you close the door then the siren will turn off
immediately. With this setup, the output, which is the siren, is dependent only on the inputs, which are the master
and door switches. For the security system to be more useful, the siren should remain on even after closing the door
after it is first triggered. In order to add this new feature to the security system, we need to modify it so that the
output is not only dependent on the master and door switches, but also dependent on whether the door has been
previously opened or not. A memory element is needed in order to remember whether the door was previously
opened or not, and this results in a sequential circuit.
In this and the next chapter, we will look at the design of combinational circuits. In this chapter, we will look at
the analysis and design of general combinational circuits. Chapter 4 will look at the design of specific combinational
components. Some sample combinational circuits in our microprocessor road map include the next-state logic and
output logic in the control unit, and the multiplexer, ALU, comparator, and tri-state buffer in the datapath. We will
leave the design of sequential circuits for a later chapter.
In addition to being able to design a functionally correct circuit, we would also like to be able to optimize the
circuit in terms of size, speed, and power consumption. Usually, reducing the circuit size will also increase the speed
and reduce the power usage. In this chapter, we will look only at reducing the circuit size. Optimizing the circuit for
speed and power usage is beyond the scope of this book. 3.1 Analysis of Combinational Circuits Very often, we are given a digital logic circuit, and we would like to know the operation of the circuit. The
analysis of combinational circuits is the process in which we are given a combinational circuit, and we want to
derive a precise description of the operation of the circuit. In general, a combinational circuit can be described
precisely either with a truth table or with a Boolean function. 3.1.1 Using a Truth Table
For example, given the combinational circuit of Figure 3.1, we want to derive the truth table that describes the
circuit. We create the truth table by first listing all of the inputs found in the circuit, one input per column, followed
by all of the outputs found in the circuit. Hence, we start with a table with four columns: three columns (x, y, z) for
the inputs, and one column (f) for the output, as shown in Figure 3.2(a). 60 Digital Logic and Microprocessor Design with VHDL x y Chapter 3 - Combinational Circuits z f Figure 3.1 Sample combinational circuit. x y z x
0
0
0
0
1
1
1
1 f y
0 1 z
0 x
0 1 z
0
1
0
1
0
1
0
1 f (b) (a)
x
0 y
0
0
1
1
0
0
1
1 y
0 1 0 z
1 1 1 0 0
0 0 1 f 0 0 f 0 (c) (d)
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 f
0
1
0
1
0
1
0
1 (e)
Figure 3.2 Deriving the truth table for the sample circuit in Figure 3.1: (a) listing the input and output columns; (b)
enumerating all possible combinations of the three input values; (c) circuit annotated with the input values xyz =
000; (d) circuit annotated with the input values xyz = 001; (e) complete truth table for the circuit. 61 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits The next step is to enumerate all possible combinations of 0’s and 1’s for all of the input variables. In general,
for a circuit with n inputs, there are 2n combinations, from 0 to 2n – 1. Continuing on with the example, the table in
Figure 3.2(b) lists the eight combinations for the three variables in order.
Now, for each row in the table (that is, for each combination of input values) we need to determine what the
output value is. This is done by substituting the values for the input variables and tracing through the circuit to the
output. For example, using xyz = 000, the outputs for all of the AND gates are 0, and ORing all the zeros gives a zero,
therefore, f = 0 for this set of values for x, y, and z. This is shown in the annotated circuit in Figure 3.2(c).
For xyz = 001, the output of the top AND gate gives a 1, and 1 OR with anything gives a 1, therefore, f = 1, as
shown in the annotated circuit in Figure 3.2(d).
Continuing in this fashion for all of the input combinations, we can complete the final truth table for the circuit,
as shown in Figure 3.2(e).
A faster method for evaluating the values for the output signals is to work backwards, that is, to trace the circuit
from the output back to the inputs. You want to ask the question: When is the output a 1 (or a 0)? Then trace back to
the inputs to see what the input values ought to be in order to get the 1 output. For example, using the circuit in
Figure 3.1, f is a 1 when any one of the four OR gate inputs is a 1. For the first input of the OR gate to be a 1, the
inputs to the top AND gate must be all 1’s. This means that the values for x, y, and z must be 0, 0, and 1, respectively.
Repeat this analysis with the remaining three inputs to the OR gate. What you will end up with are the four input
combinations for which f is a 1. The remaining input combinations, of course, will produce a 0 for f.
Example 3.1: Deriving a truth table from a circuit diagram
Derive the truth table for the following circuit with three inputs, A, B and C, and two outputs, P and Q:
A B C P Q The truth table will have three columns for the three inputs and two columns for the two outputs. Enumerating
all possible combinations of the three input values gives eight rows in the table. For each combination of input
values, we need to evaluate the output values for both P and Q. For P to be a 1, either of the OR gate inputs must be
a 1. The first input to this OR gate is a 1 if ABC = 001. The second input to this OR gate is a 1 if AB = 11. Since C is
not specified in this case, it means that C can be either a 0 or a 1. Hence, we get the three input combinations for
which P is a 1, as shown in the following truth table under the P column. The rest of the input combinations will
produce a 0 for P. For Q to be a 1, both inputs of the AND gate must be a 1. Hence, A must be a 0, and either B is a 0
or C is a 1. This gives three input combinations for which Q is a 1, as shown in the truth table below under the Q
column.
A
0
0
0
0
1
1
1
1 B
0
0
1
1
0
0
1
1 C
0
1
0
1
0
1
0
1 62 P
0
1
0
0
0
0
1
1 Q
1
1
0
1
0
0
0
0 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits
♦ 3.1.2 Using a Boolean Function
To derive a Boolean function that describes a combinational circuit, we simply write down the Boolean logical
expressions at the output of each gate (instead of substituting actual values of 0’s and 1’s for the inputs) as we trace
through the circuit from the primary input to the primary output. Using the sample combinational circuit of Figure
3.1, we note that the logical expression for the output of the top AND gate is x'y'z. The logical expressions for the
following AND gates are, respectively x'yz, xy'z, and xyz. Finally, the outputs from these AND gates are all ORed
together. Hence, we get the final expression
f = x'y'z + x'yz + xy'z + xyz
To help keep track of the expressions at the output of each logic gate, we can annotate the outputs of each logic
gate with the resulting logical expression as shown here.
x y x' z y' x'y'z
x'yz x'y'z + x'yz + xy'z + xyz
f xy'z
xyz If we substitute all possible combinations of values for all of the variables in the final equation, we should
obtain the same truth table as before.
Example 3.2: Deriving a Boolean function from a circuit diagram
Derive the Boolean function for the following circuit with three inputs, x, y, and z, and one output, f.
x y z f Starting from the primary inputs x, y, and z, we annotate the outputs of each logic gate with the resulting logical
expression. Hence, we obtain the annotated circuit:
x y z y' x' xy'
y⊕z xy' + (y ⊕ z) f = x' (xy' + (y ⊕ z)) The Boolean function for the circuit is the final equation, f = x' (xy' + (y ⊕ z)), at the output of the circuit. ♦ If a circuit has two or more outputs, then there must be one equation for each of the outputs. All the equations
are then derived totally independent of each other. 63 Digital Logic and Microprocessor Design with VHDL 3.2 Chapter 3 - Combinational Circuits Synthesis of Combinational Circuits Synthesis of combinational circuits is just the reverse procedure of the analysis of combinational circuits. In
synthesis, we start with a description of the operation of the circuit. From this description, we derive either the truth
table or the Boolean logical function that precisely describes the operation of the circuit. Once we have either the
truth table or the logical function, we can easily translate that into a circuit diagram.
For example, let us construct a 3-bit comparator circuit that outputs a 1 if the number is greater than or equal to
5 and outputs a 0 otherwise. In other words, construct a circuit that outputs a 0 if the input is a number between 0
and 4 inclusive and outputs a 1 if the input is a number between 5 and 7 inclusive. The reason why the maximum
number is 7 is because the range for an unsigned 3-bit binary number is from 0 to 7. Hence, we can use the three
bits, x2, x1, and x0, to represent the 3-bit input value to the comparator. From the description, we obtain the following
truth table:
Decimal
number
0
1
2
3
4
5
6
7 Binary number
x2
x1
x0
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1 Output
f
0
0
0
0
0
1
1
1 In constructing the circuit, we are interested only in when the output is a 1 (i.e., when the function f is a 1).
Thus, we only need to consider the rows where the output function f = 1. From the previous truth table, we see that
there are three rows where f = 1, which give the three AND terms x2x1'x0, x2x1x0', and x2x1x0. Notice that the variables
in the AND terms are such that it is inverted if its value is a 0, and not inverted if its value is a 1. In the case of the
first AND term, we want f = 1 when x2 = 1 and x1 = 0 and x0 = 1, and this is satisfied in the expression x2x1'x0.
Similarly, the second and third AND terms are satisfied in the expressions x2x1x0' and x2x1x0 respectively. Finally, we
want f = 1 when either one of these three AND terms is equal to 1. So we ORed the three AND terms together giving us
our final expression:
f = x2x1'x0 + x2x1x0' + x2x1x0 (3.1) In drawing the schematic diagram, we simply convert the AND operators to AND gates and OR operators to OR
gates. The equation is in the sum-of-products format, meaning that it is summing (ORing) the product (AND) terms.
A sum-of-products equation translates to a two-level circuit with the first level being made up of AND gates and the
second level made up of OR gates. Each of the three AND terms contains three variables, so we use a 3-input AND
gate for each of the three AND terms. The three AND terms are ORed together, so we use a 3-input OR gate to connect
the output of the three AND gates. For each inverted variable, we need an inverter. The schematic diagram derived
from Equation (3.1) is shown here.
x2 x1 x0 f From this discussion, we see that any combinational circuit can be constructed using only AND, OR, and NOT
gates from either a truth table or a Boolean equation.
Example 3.3: Synthesizing a combinational circuit from a truth table 64 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Synthesize a combinational circuit from the following truth table. The three variables, a, b, and c, are input
signals, and the two variables, x, and y, are output signals.
a
0
0
0
0
1
1
1
1 b
0
0
1
1
0
0
1
1 c
0
1
0
1
0
1
0
1 x
1
0
1
1
0
1
1
0 y
0
0
1
0
1
1
0
0 We can either first derive the Boolean equation from the truth table, and then derive the circuit from the
equation, or we can derive the circuit directly from the truth table. For this example, we will first derive the Boolean
equation. Since there are two output signals, there will be two equations; one for each output signal.
From Section 2.6, we saw that a function is formed by summing its 1-minterms. For output x, there are five 1minterms: m0, m2, m3, m5, and m6. These five minterms represent the five AND terms, a'b'c', a'bc', a'bc, ab'c, and
abc'. Hence, the equation for x is
x = a'b'c' + a'bc' + a'bc + ab'c + abc'
Similarly, the output signal y has three 1-minterms, and they are a'bc', ab'c', and ab'c. Hence, the equation for y
is
y = a'bc' + ab'c' + ab'c
The combinational circuit constructed from these two equations is shown in Figure 3.3(a). Each 3-variable AND
term is replaced by a 3-input AND gate. The three inputs to these AND gates are connected to the three input variables
a, b, and c, either directly if the variable is not primed or through a NOT gate if the variable is primed. For output x, a
5-input OR gate is used to connect the outputs of the five AND gates for the corresponding five AND terms. For output
y, a 3-input OR gate is used to connect the outputs of the three AND gates.
Notice that the two AND terms, a'bc', and ab'c, appear in both the x and the y equations. As a result, we do not
need to generate these two signals twice. Hence, we can reduce the size of the circuit by not duplicating these two
AND gates, as shown in Figure 3.3(b).
♦ a b c a b c x x y y (a) (b) Figure 3.3 Combinational circuit for Example 3.2: (a) no reduction; (b) with reduction. 65 Digital Logic and Microprocessor Design with VHDL 3.3 Chapter 3 - Combinational Circuits * Technology Mapping To reduce implementation cost and turnaround time to produce a digital circuit on an IC, designers often make
use of off-the-shelf semi-custom gate arrays. Many gate arrays are ICs that have only NAND gates or NOR gates built
in them, but their input and output connections are not yet connected. To use these gate arrays, a designer simply has
to specify where to make these connections between the gates. The problem here is that, when we use these gate
arrays to implement a circuit, we need to convert all AND, OR, and NOT gates in the circuit to use only NAND or NOR
gates, depending on what is available in the gate array. In addition, these NAND and NOR gates usually have the same
number of fixed inputs, for example, only three inputs.
In Section 3.2, we saw that any combinational circuit can be constructed with only AND, OR, and NOT gates. It
turns out that any combinational circuit can also be constructed with either only NAND gates or only NOR gates. The
reason why we want to use only NAND or NOR gates will be made clear when we look at how these gates are built at
the transistor level in Chapter 5. We will now look at how a circuit with AND, OR, and NOT gates is converted to one
with only NAND or only NOR gates.
The conversion of any given circuit to use only 2-input NAND or 2-input NOR gates is possible by observing the
following equalities. These equalities, in fact, are obtained from the Boolean algebra theorems from Chapter 2.
Rule 1: x' ' = x (double NOT) Rule 2: x' = (x • x)' = (x • 1)' (NOT to NAND) Rule 3: x' = (x + x)' = (x + 0)' (NOT to NOR) Rule 4: xy = ((xy)')' (AND to NAND) Rule 5: x + y = ((x + y)')' = (x' y' )' (OR to NAND) Rule 6: xy = ((xy)')' = (x' + y')' (AND to NOR) Rule 7: x + y = ((x + y)')' (OR to NOR) Rule 1 simply says that a double inverter can be eliminated altogether. Rules 2 and 3 convert a NOT gate to a
gate or a NOR gate, respectively. For both Rules 2 and 3, there are two ways to convert a NOT gate to either a
gate or a NOR gate. For the first method, the two inputs are connected in common. For the second method, one
input is connected to the logic 1 for the NAND gate and to 0 for the NOR gate. Rule 4 applies Rule 1 to the AND gate.
The resulting expression gives us a NAND gate followed by a NOT gate. We can then use Rule 2 to change the NOT
gate to a NAND gate. Rule 5 changes an OR gate to use two NOT gates and a NAND gate by first applying Rule 1 and
then De Morgan’s theorem. Again, the two NOT gates can be changed to two NAND gates using Rule 2. Similarly,
Rule 6 converts an AND gate to use two NOT gates and a NOR gate, and Rule 7 converts an OR gate to a NOR gate
followed by a NOT gate.
NAND
NAND In a circuit diagram, these rules are translated to the equivalent circuits, as shown in Figure 3.4. Rules 2, 4, and
5 are used if we want to convert a circuit to use only 2-input NAND gates; whereas, Rules 3, 6, and 7 are used if we
want to use only 2-input NOR gates. 66 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Rule 1: = Rule 2: = = Rule 3: = = Rule 4: = = Rule 5: = = Rule 6: = = Rule 7: = = 1 0 Figure 3.4 Circuits for converting from AND, OR, or NOT gates to NAND or NOR gates.
Another thing that we might want is to get the functionality of a 2-input NAND or 2-input NOR gate from a 3input NAND or 3-input NOR gate, respectively. In other words, we want to use a 3-input NAND or NOR gate to work
like a 2-input NAND or NOR gate, respectively. On the other hand, we might also want to get the reverse of that (that
is, to get the functionality of a 3-input NAND or 3-input NOR gate from a 2-input NAND or 2-input NOR gate,
respectively). These equalities are shown in the following rules and their corresponding circuits in Figure 3.5.
Rule 8: (x • y)' = (x • y • y)' (2-input to 3-input NAND) Rule 9: (x + y)' = (x + y + y)' (2-input to 3-input NOR) Rule 10: (abc)' = ((ab) c)' = ((ab)'' c)' (3-input to 2-input NAND) Rule 11: (a+b+c)' = ((a+b) + c)' = ((a+b)'' + c)' (3-input to 2-input NOR) Rule 8 converts from a 2-input NAND gate to a 3-input NAND gate. Rule 9 converts from a 2-input NOR gate to a
3-input NOR gate. Rule 10 converts from a 3-input NAND gate to using only 2-input NAND gates. Rule 11 converts
from a 3-input NOR gate to using only 2-input NOR gates. Notice that, for Rules 10 and 11, an extra NOT gate is
needed in between the two gates. 67 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Rule 8: = Rule 9: = Rule 10: = = Rule 11: = = Figure 3.5 Circuits for converting 2-input to 3-input NAND or NOR gate and vice versa.
Example 3.4: Converting a circuit to 3-input NAND gates
Convert the following circuit to use only 3-input NAND gates.
x y z f First, we need to change the 4-input OR gate to a 3- and 2-input OR gates.
x y z f Then we will use Rule 4 to change all of the AND gates to 3-input NAND gates with inverters and Rule 5 to
change all of the OR gates to 3-input NAND gates with inverters. The 2-input NAND gates are replaced with 3-input
NAND gates with two of its inputs connected together.
x y z f 68 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Finally, we eliminate all the double inverters and replace the remaining inverters with NAND gates with their
inputs connected together.
x y z f ♦ 3.4 Minimization of Combinational Circuits When constructing digital circuits, in addition to obtaining a functionally correct circuit, we like to optimize it
in terms of circuit size, speed, and power consumption. In this section, we will focus on the reduction of circuit size.
Usually, by reducing the circuit size, we will also improve on speed and power consumption. We have seen in the
previous sections that any combinational circuit can be represented using a Boolean function. The size of the circuit
is directly proportional to the size or complexity of the functional expression. In fact, it is a one-to-one
correspondence between the functional expression and the circuit size. In Section 2.5.1, we saw how we can
transform a Boolean function to another equivalent function by using the Boolean algebra theorems. If the resulting
function is simpler than the original, then we want to implement the circuit based on the simpler function, since that
will give us a smaller circuit size.
Using Boolean algebra to transform a function to one that is simpler is not an easy task, especially for the
computer. There is no formula that says which is the next theorem to use. Luckily, there are easier methods for
reducing Boolean functions. The Karnaugh map method is an easy way for reducing an equation manually and is
discussed in Section 3.4.1. The Quine-McCluskey or tabulation method for reducing an equation is ideal for
programming the computer and is discussed in Section 3.4.3. 3.4.1 Karnaugh Maps
To minimize a Boolean equation in the sum-of-products form, we need to reduce the number of product terms
by applying the Combining Boolean theorem (Theorem 14) from Section 2.5.1. In so doing, we will also have
reduced the number of variables used in the product terms. For example, given the following 3-variable function:
F = xy'z' + xyz'
we can factor out the two common variables xz' and reduce it to
F = xz' (y' + y)
= xz' 1
= xz'
In other words, two product terms that differ by only one variable whose value is a 0 (primed) in one term and a
1 (unprimed) in the other term, can be combined together to form just one term with that variable omitted as shown
in the previous equations. Thus, we have reduced the number of product terms, and the resulting product term has
one less variable. By reducing the number of product terms, we reduce the number of OR operators required, and by
reducing the number of variables in a product term, we reduce the number of AND operators required.
Looking at a logic function’s truth table, sometimes it is difficult to see how the product terms can be combined
and minimized. A Karnaugh map (K-map for short) provides a simple and straightforward procedure for
combining these product terms. A K-map is just a graphical representation of a logic function’s truth table, where
the minterms are grouped in such a way that it allows one to easily see which of the minterms can be combined. The 69 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits K-map is a 2-dimensional array of squares, each of which represents one minterm in the Boolean function. Thus, the
map for an n-variable function is an array with 2n squares.
Figure 3.6 shows the K-maps for functions with 2, 3, 4, and 5 variables. Notice the labeling of the columns and
rows are such that any two adjacent columns or rows differ in only one bit change. This condition is required
because we want minterms in adjacent squares to differ in the value of only one variable or one bit, and so these
minterms can be combined together. This is why the labeling for the third and fourth columns and for the third and
fourth rows are always interchanged. When we read K-maps, we need to visualize them as such that the two end
columns or rows wrap around, so that the first and last columns and the first and last rows are really adjacent to each
other, because they also differ in only one bit.
In Figure 3.6, the K-map squares are annotated with their minterms and minterm numbers for easy reference
only. For example, in Figure 3.6(a) for a 2-variable K-map, the entry in the first row and second column is labeled
x'y and annotated with the number 1. This is because the first row is when the variable x is a 0, and the second
column is when the variable y is a 1. Since, for minterms, we need to prime a variable whose value is a 0 and not
prime it if its value is a 1, therefore, this entry represents the minterm x'y, which is minterm number 1. Be careful
that, if we label the rows and columns differently, the minterms and the minterm numbers will be in different
locations. When we use K-maps to minimize an equation, we will not write these in the squares. Instead, we will be
putting 0’s and 1’s in the squares.
For a 5-variable K-map, as shown in Figure 3.6(d), we need to visualize the right half of the array (where v = 1)
to be on top of the left half (where v = 0). In other words, we need to view the map as three-dimensional. Hence,
although the squares for minterms 2 and 16 are located next to each other, they are not considered to be adjacent to
each other. On the other hand, minterms 0 and 16 are adjacent to each other, because one is on top of the other.
yz
wx 00 01 11 10 y 0 1
0 0 x'y' 1 xy' x yz 00 1 0 0 xy'z' 3 x'y'z
4 xy 11
1 x'y'z' 1 x'y
2 01 3
x'yz 5
xy'z 00 w'x'y'z w'x'yz w'x'yz' 4
w'xy'z' 5
w'xy'z 7
w'xyz 6
w'xyz' 11 wxy'z' wx'y'z' 2
x'yz' 7
xyz w'x'y'z' 01 10 1 10 x 0 6 12 xyz' 8 (a) (b) wx 00 01
0 10
3 00
2 16 00 v'w'x'y'z' v'w'x'y'z v'w'x'yz v'w'x'yz' 01 v'w'xy'z' v'w'xy'z v'w'xyz v'w'xyz'
13 15 14 28 11 v'wxy'z' v'wxy'z v'wxyz v'wxyz' vwxy'z' 11 10 10 v'wx'y'z' v'wx'y'z v'wx'yz v'wx'yz' 4 12 8 5 9 11
wx'yz 14
wxyz'
10
wx'yz' v=1 11
1 9
wx'y'z 15
wxyz 2 (c) v=0 yz 13
wxy'z 3 7 6 01
17 11
19 10
18 vw'x'y'z' vw'x'y'z vw'x'yz vw'x'yz'
21 23 22 vw'xy'z' vw'xy'z vw'xyz vw'xyz' 20 24 29
vwxy'z 31
vwxyz 30
vwxyz' 25 27 26 vwx'y'z' vwx'y'z vwx'yz vwx'yz' (d)
Figure 3.6 Karnaugh maps for: (a) 2 variables; (b) 3 variables; (c) 4 variables; (d) 5 variables.
Given a Boolean function, we set the value for each K-map square to either a 0 or a 1, depending on whether
that minterm for the function is a 0-minterm or a 1-minterm, respectively. However, since we are only interested in
using the 1-minterms for a function, the 0’s are sometimes not written in the 0-minterm squares.
For example, the K-map for the 2-variable function: 70 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits F = x'y' + x'y + xy
is
F x' y x 0
0 1
0 1 1
1 2 1 3
1 y The 1-minterms, m0 (x'y') and m1 (x'y), are adjacent to each other, which means that they differ in the value of
only one variable. In this case, x is 0 for both minterms, but for y, it is a 0 for one minterm and a 1 for the other
minterm. Thus, variable y can be dropped, and the two terms are combined together giving just x'. The prime in x' is
because x is 0 for both minterms. This reasoning corresponds with the expression:
x'y' + x'y = x' (y' + y) = x' (1) = x'
Similarly, the 1-minterms m1 (x'y) and m3 (xy) are also adjacent and y is the variable having the same value for
both minterms, and so they can be combined to give
x'y + xy = (x' + x) y = (1) y = y
We use the term subcube to refer to a rectangle of adjacent 1-minterms. These subcubes must be rectangular in
shape and can only have sizes that are powers of two. Formally, for an n-variable K-map, an m-subcube is defined as
that set of 2m minterms in which n – m of the variables will have the same value in every minterm, while the
remaining variables will take on the 2m possible combinations of 0’s and 1’s. Thus, a 1-minterm all by itself is called
a 0-subcube, two adjacent 1-minterms is called a 1-subcube, and so on. In the previous 2-variable K-map, there are
two 1-subcubes: one labeled with x' and one labeled with y.
A 2-subcube will have four adjacent 1-minterms and can be in the shape of any one of those shown in Figure
3.7(a) through (e). Notice that Figure 3.7(d) and (e) also form 2-subcubes, even though the four 1-minterms are not
physically adjacent to each other. They are considered to be adjacent because the first and last rows and the first and
last columns wrap around in a K-map. In Figure 3.7(f), the four 1-minterms cannot form a 2-subcube, because even
though they are physically adjacent to each other, they do not form a rectangle. However, they can form three 1subcubes – y'z, x'y' and x'z.
We say that a subcube is characterized by the variables having the same values for all of the minterms in that
subcube. In general, an m-subcube for an n-variable K-map will be characterized by n – m variables. If the value that
is similar for all of the variables is a 1, that variable is unprimed; whereas, if the value that is similar for all of the
variables is a 0, that variable is primed. In an expression, this is equivalent to the resulting smaller product term
when the minterms are combined together. For example, the 2-subcube in Figure 3.7(d) is characterized by z', since
the value of z is 0 for all of the minterms, whereas the values for x and y are not all the same for all of the minterms.
Similarly, the 2-subcube in Figure 3.7(e) is characterized by x'z'.
F
F F x' yz
00 x 01 11 10 1 1 1 1 z yz
00 01 11 0 1 1 1 00 wx 01 1
1 x 10 y'z yz (a) (b) 1 11 1 1 01 0 00 1 10 1 (c) 71 11 10 Digital Logic and Microprocessor Design with VHDL F
F z' yz
00 x
0 01 1 11 Chapter 3 - Combinational Circuits x'z' yz
00 wx
10 00 01 11 F 10 1 1 01
1 1 x'z 00
0 1 x'y' yz x 01 11 1 1 1 10 y'z
1 1 1 11
10 (d) 1 1 (f) (e)
Figure 3.7 Examples of K-maps with 2-subcubes: (a) and (b) 3-variable; (c) 4-variable; (d) 3-variable with wrap
around subcube; (e) 4-variable with wrap around subcube; (f) four adjacent minterms that cannot form one 2subcube.
For a 5-variable K-map, as shown in Figure 3.8, we need to visualize the right half of the array (where v = 1) to
be on top of the left half (where v = 0). Thus, for example, minterm 20 is adjacent to minterm 4 since one is on top
of the other, and they form the 1-subcube w'xy'z'. Even though minterm 6 is physically adjacent to minterm 20 on
the map, they cannot be combined together, because when you visualize the right half as being on top of the left
half, then they really are not on top of each other. Instead, minterm 6 is adjacent to minterm 4 because the columns
wrap around, and they form the subcube v'w'xz'. Minterms 9, 11, 13, 15, 25, 27, 29, and 31 are all adjacent, and
together they form the subcube wz. Now that we are viewing this 5-variable K-map in three dimensions, we also
need to change the condition of the subcube shape to be a three-dimensional rectangle.
You can see that this visualization becomes almost impossible to work with very quickly as we increase the
number of variables. In more realistic designs with many more variables, tabular methods (instead of K-maps) are
used for reducing the size of equations. F v'w'xz' w'xy'z'
v=0 yz
00 wx 11
10 11 v=1
10 0 1 3 4 00
01 01
5 7 1 00
2 12 13
1 8 15 19 18 20 21 23 22 31 30 27 26 1
14 28 29
1 11 10 17 1
9 1 11 16 6
1 01 10 24 1 1
25 1 1 wz Figure 3.8 A 5-variable K-map with wrap-around subcubes.
The K-map method reduces a Boolean function from its canonical form to its standard form. The goal for the Kmap method is to find as few subcubes as possible to cover all of the 1-minterms in the given function. This
naturally implies that the size of the subcube should be as big as possible. The reasoning for this is that each subcube
corresponds to a product term, and all of the subcubes (or product terms) must be ORed together to get the function.
Larger subcubes require fewer AND gates because of fewer variables in the product term, and fewer subcubes will
require fewer inputs to the OR gate.
The procedure for using the K-map method is as follows:
1. Draw the appropriate K-map for the given function and place a 1 in the squares that correspond to the function’s
1-minterms. 2. For each 1-minterm, find the largest subcube that covers this 1-minterm. This largest subcube is known as a 72 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits prime implicant (PI). By definition, a prime implicant is a subcube that is not contained within any other
subcube. If there is more than one subcube that is of the same size as the largest subcube, then they are all prime
implicants.
3. Look for 1-minterms that are covered by only one prime implicant. Since this prime implicant is the only
subcube that covers this particular 1-minterm, this prime implicant must be in the final solution. This prime
implicant is referred to as an essential prime implicant (EPI). By definition, an essential prime implicant is a
prime implicant that includes a 1-minterm that is not included in any other prime implicant. 4. Create a minimal cover list by selecting the smallest possible number of prime implicants such that every 1minterm is contained in at least one prime implicant. This cover list must include all of the essential prime
implicants plus zero or more of the remaining prime implicants. It is acceptable that a particular 1-minterm is
covered in more than one prime implicant, but all 1-minterms must be covered. 5. The final minimized function is obtained by ORing all of the prime implicants from the minimal cover list. Note that the final minimized function obtained by the K-map method may not be in its most reduced form. It is
only in its most reduced standard form. Sometimes, it is possible to reduce the standard form further into a nonstandard form.
Example 3.5: Using K-map to minimize a 4-variable function
Use the K-map method to minimize a 4-variable (w, x, y, and z) function F with the 1-minterms: m0, m2, m5, m7,
m10, m13, m14, and m15.
We start with the following 4-variable K-map with a 1 placed in each of the eight minterm squares:
F yz
00 wx
00
01
11
10 01
0 11
1 10
3 1 2
1 4 5
1 12 13
1 8 7 6 1
15
1
9 14
1 11 10
1 The prime implicants for each of the 1-minterms are shown in the following K-map and table:
F 00 wx
00
01
11
10 1-minterm 01
0 11
1 10
3 1 2
1 4 5
1 12 6 15 13
1 8 7 14 1
1
9 1
11 10 x'yz'
wyz' 1 xz wxy Prime Implicant m0
m2
m5
m7
m10
m13
m14
m15 w'x'z' yz w'x'z'
w'x'z', x'yz'
xz
xz
x'yz', wyz'
xz
wyz', wxy
xz For minterm m0, there is only one prime implicant w'x'z'. For minterm m2, there are two 1-subcubes that cover
it, and they are the largest. Therefore, m2 has two prime implicants, w'x'z' and x'yz'. When we consider m14, again
there are two 1-subcubes that cover it, and they are the largest. So m14 also has two prime implicants. Minterm m15,
however, has only one prime implicant xz. Although the 1-subcube wxy also covers m15, it is not a prime implicant
for m15 because it is smaller than the 2-subcube xz. 73 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits From the K-map, we see that there are five prime implicants: w'x'z', x'yz', xz, wyz', and wxy. Of these five prime
implicants, w'x'z' and xz are essential prime implicants, since m0 is covered only by w'x'z', and m5, m7, and m13 are
covered only by xz.
We start the cover list by including the two essential prime implicants w'x'z' and xz. These two subcubes will
have covered the minterms m0, m2, m5, m7, m13, and m15. To cover the remaining two uncovered minterms, m10 and
m14, we want to use as few prime implicants as possible. Hence, we select the prime implicant wyz', which covers
both of them.
Finally, our reduced standard form equation is obtained by ORing the two essential prime implicants and one
prime implicant in the cover list:
F = w'x'z' + xz + wyz'
Notice that we can reduce this standard form equation even further by factoring out the z' from the first and last
term to get the non-standard form equation
♦ F = z' (w'x' + wy) + xz
Example 3.6: Using K-map to minimize a 5-variable function Use the K-map method to minimize a 5-variable function F (v, w, x, y and z) with the 1-minterms: v'w'x'yz',
v'w'x'yz, v'w'xy'z, v'w'xyz, vw'x'yz', vw'x'yz, vw'xyz', vw'xyz, vwx'y'z, vwx'yz, vwxy'z, and vwxyz.
w'x'y F yz
00 wx 01 v=1 1 11 10 1 00
01 w'yz v=0
00 01 1 10 1 1 1 1 11 1 v'w'xz
11 1 10 1 vw'y 1
1 vwz vyz The list of prime implicants is: v'w'xz, w'x'y, w'yz, vw'y, vyz, and vwz. From this list of prime implicants, w'yz
and vyz are not essential. The four remaining essential prime implicants are able to cover all of the 1-minterms.
Hence, the solution in standard form is
F = v'w'xz + w'x'y + vw'y + vwz ♦ 3.4.2 Don’t-cares
There are times when a function is not specified fully. In other words, there are some minterms for the function
where we do not care whether their values are a 0 or a 1. When drawing the K-map for these “don’t-care”
minterms, we assign an “×” in that square instead of a 0 or a 1. Usually, a function can be reduced even further if we
remember that these ×’s can be either a 0 or a 1. As you recall when drawing K-maps, enlarging a subcube reduces
the number of variables for that term. Thus, in drawing subcubes, some of them may be enlarged if we treat some of
these ×’s as 1’s. On the other hand, if some of these ×’s will not enlarge a subcube, then we want to treat them as 0’s
so that we do not need to cover them. It is not necessary to treat all ×’s to be all 1’s or all 0’s. We can assign some
×’s to be 0’s and some to be 1’s.
For example, given a function having the following 1-minterms and don’t-care minterms:
1-minterms: m0, m1, m2, m3, m4, m7, m8, and m9
×-minterms: m10, m11, m12, m13, m14, and m15
we obtain the following K-map with the prime implicants x', yz, and y'z'. 74 Digital Logic and Microprocessor Design with VHDL F yz
00 wx 01
0 00 1 01 × 10 1 11
1 1 11 y'z' Chapter 3 - Combinational Circuits 1
4 10
3 1 2
1 5 6 7 x' 14 yz 1
12 × 8 13
9 1 ×
× 15
11 ×
× 10 Notice that, in order to get the 4-subcube characterized by x', the two don’t-care minterms, m10 and m11, are
taken to have the value 1. Similarly, the don’t-care minterms, m12 and m15, are assigned a 1 for the subcubes y'z' and
yz, respectively. On the other hand, the don’t-care minterms, m13 and m14, are taken to have the value 0, so that they
do not need to be covered in the solution. The reduced standard form function as obtained from the K-map is,
therefore,
F = x' + yz + y'z'
Again, this equation can be reduced further by recognizing that yz + y'z' = y
F = x' + ( y z. Thus, z) 3.4.3 * Tabulation Method
K-maps are useful for manually obtaining the minimized standard form Boolean function for maybe up to, at
most, five variables. However, for functions with more than five variables, it becomes very difficult to visualize how
the minterms should be combined into subcubes. Moreover, the K-map algorithm is not as straightforward for
converting to a computer program. There are tabulation methods that are better suited for programming the
computer, and thus, can solve any function given in canonical form having any number of variables. One tabulation
method is known as the Quine-McCluskey method.
Example 3.7: Illustrating the Qui e-McCluskey algorithm
We now illustrate the Quine-McCluskey algorithm using the same four-variable function from Example 3.5 and
repeated here
F(w,x,y,z) = Σ(0,2,5,7,10,13,14,15)
To construct the initial table, the minterms are grouped according to the number of 1’s in that minterm
number’s binary representation. For example, m0 (0000) has no 1’s; m2 (0010) has one 1; m5 (0101) has two 1’s; etc.
Thus, the initial table of 0-subcubes (i.e. subcubes having only one minterm) as obtained from the function stated
above is
Group
G0
G1
G2
G3 G4 Subcube
Minterms
m0
m2
m5
m10
m7
m13
m14
m15 Subcube Value
wxyz
0000
0010
0101
1010
0111
1101
1110
1111 Subcube
Covered The “Subcube Covered” column is filled in from the next step.
In the second step, we construct a second table by combining those minterms in adjacent groups from the first
table that differ in only one bit position, as shown next. For example, m0 and m2 differ in only the y bit. Therefore, in
the second table, we have an entry for the 1-subcube containing the two minterms, m0 and m2. A hyphen (–) is used 75 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits in the bit position that is different in the two minterms. Since this 1-subcube covers the two individual minterms, m0
and m2, we make a note of it by checking these two minterms in the “Subcube Covered” column in the previous
table. This process is equivalent to saying that the two minterms, m0 (w'x'y'z' ) and m2 (w'x'yz' ), can be combined
together and is reduced to the one term, w'x'z'. The hypen under the y column simply means that y can be either a 0
or a 1, and therefore, y can be discarded. Thus, this second table simply lists all of the 1-subcubes. Again, the
“Subcube Covered” column in this second table will be filled in from the third step.
Group
G0
G1
G2 G3 Subcube
Minterms
m 0, m 2
m2,m10
m 5, m 7
m5,m13
m10,m14
m7,m15
m13,m15
m14,m15 Subcube Value
wxyz
00–0
–010
01–1
–101
1–10
–111
11–1
111– Subcube
Covered In step three, we perform the same matching process as before. We look for subcubes in adjacent groups that
differ in only one bit position. In the matching, the hyphen must also match. These subcubes are combined to create
the next subcube table. The resulting table, however, is a table containing 2-subcubes. From the above 1-subcube
table, we get the following 2-subcube table:
Group
G2 Subcube
Minterms
m5,m7,m13,m15 Subcube Value
wxyz
–1–1 Subcube
Covered From the 1-subcube table, subcubes m5m7 and m13m15 can be combined together to form the subcube
m5m7m13m15 in the 2-subcube table, since they differ in only the w bit. Similarly, subcubes m5m13 and m7m15 from the
1-subcube table can also be combined together to form the subcube m5m7m13m15, because they differ in only the y
bit. From both of these combinations, the resulting subcube is the same. Therefore, we have the four checks in the 1subcube table, but only one resulting subcube in the 2-subcube table. Notice that in the subcube m5m7m13m15, there
are two hypens; one that is carried over from the previous step, and one for where the bit is different from the
current step.
We continue to repeat the matching as long as there are adjacent subcubes that differ in only one bit position.
We stop when there are no more subcubes that can be combined. The prime implicants are those subcubes that are
not covered, (i.e. those without a check mark in the “Subcube Covered” column). The only subcube in the 2-subcube
table does not have a check mark, and it has the value x = 1 and z = 1; thus we get the prime implicant xz. The 1subcube table has four subcubes that do not have a check mark; they are the four prime implicants: w'x'z', x'yz', wyz',
and wxy. Note that these prime implicants may not necessarily be all in the last table. These five prime implicants
(xz, w'x'z', x'yz', wyz', and wxy) are exactly the same as those obtained in Example 3.5.
♦ 3.5 * Timing Hazards and Glitches As you probably know, things in practice don’t always work according to what you learn in school. Hazards and
glitches in circuits are such examples of things that may go awry. In our analysis of combinational circuits, we have
been performing only functional analysis. A functional analysis assumes that there is no delay for signals to pass
from the input to the output of a gate. In other words, we look at a circuit only with respect to its logical operation as
defined by the Boolean theorems. We have not considered the timing of the circuit. When a circuit is actually
implemented, the timing of the circuit (that is, the time for the signals to pass from the input of a logic gate to the
output) is very critical and must be treated with care. Otherwise, an actual implementation of the circuit may not
work according to the functional analysis of the same circuit. Timing hazards are problems in a circuit as a result of
timing issues. These problems can be observed only from a timing analysis of the circuit or from an actual
implementation of the circuit. A functional analysis of the circuit will not reveal timing hazard problems. 76 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits A glitch is when a signal is expected to be stable (from a functional analysis), but it changes value for a brief
moment and then goes back to what it is expected to be. For example, if a signal is expected to be at a stable 0, but
instead, it goes up to a 1 and then drops back to a 0 very quickly. This sudden, unexpected transition of the signal is
a glitch, and the circuit having this behavior contains a hazard.
Take, for example, the simple 2-to-1 multiplexer circuit shown in Figure 3.9(a). Let us assume that both d0 and
d1 are at a constant 1, and that s goes from a 1 to a 0. For a functional analysis of the circuit, the output y should
remain at a constant 1. However, if we perform a timing analysis of the circuit, we will see something different in
the timing diagram. Let us assume that all of the logic gates in the circuit have a delay of one time unit. The
resulting timing trace is shown in Figure 3.9(b). At time t0, s drops to a 0. Since it takes one time unit for s to be
inverted through the inverter, s' changes to a 1 after one time unit at time t1. At the same time, it takes the bottom
AND gate one time unit for the output sd1 to change to a 0 at time t1. However, the top AND gate will not see any
input change until time t1, and when it does, it takes another one time unit for its output s'd0 to rise to a 1 at time t2.
Starting at time t1, both inputs of the OR gate are 0, so after one time unit, the OR gate outputs a 0 at time t2. At time
t2, when the top AND gate outputs a 1, the OR gate will take this 1 input, and outputs a 1 after one time unit at t3. So
between times t2 and t3, output y unexpectedly drops to a 0 for one time unit, and then rises back to a 1. Hence, the
output signal y has a glitch, and the circuit has a hazard.
As you may have noticed, glitches in a signal are caused by multiple sources having paths of different delays
driving that signal. These types of simple glitches can be easily solved using K-maps. A glitch generally occurs if,
by simply changing one input, we have to go out of one prime implicant in a K-map and into an adjacent one (i.e.
moving from one subcube to another). The glitch can be eliminated by adding an extra prime implicant, so that when
going from one prime implicant to the adjacent one, we remain inside the third prime implicant.
Figure 3.9(c) shows the K-map with the two original prime implicants, s'd0 and sd1, that correspond to the
circuit in (a). When we change s from a 1 to a 0, we have to go out of the prime implicant sd1 and into the prime
implicant s'd0. Figure 3.9(d) shows the addition of the extra prime implicant d1d0. This time, when moving from the
prime implicant sd1 to the prime implicant s'd0, we remain inside the prime implicant d1d0. The 2-to-1 multiplexer
circuit with the extra prime implicant d1d0 added as shown in Figure 3.9(e) will prevent the glitch from happening.
d0
d1
d0 s'd0
s' s
d1 s 00 s' sd1 0 s'd0 y s'd0 y dd
10 s 01 11 1 1 1 1 sd1 10 1 sd1 y
t0
(a) 00
0 t2 t3
(c) (b)
s'd0 y dd
10
s t1 01 11 1 1 1 1 d1d0 (d) s'd0 d0 10 s sd1 d1 1 sd1 y d 1d 0
(e) Figure 3.9 Example of a glitch: (a) 2-to-1 multiplexer circuit with glitches; (b) timing trace; (c) K-map with
glitches; (d) K-map without glitches; (e) 2-to-1 multiplexer circuit without the glitches. 77 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits 3.5.1 Using Glitches
Sometimes, we can use glitches to our advantage, as shown in the following example.
Example 3.8: A one-shot circuit using glitches
A circuit that outputs a single short pulse when given an input of arbitrary time length is known as a one-shot. A
one-shot circuit is used, for example, for generating a single short 1 pulse when a key is pressed. Sometimes, when a
key is pressed, we do not want to generate a continuous 1 signal for as long as the key is pressed. Instead, we want
the output signal to be just a single short 1 pulse, even if the key is still being pressed.
Since logic gates have an inherent signal delay, we can use this delay to determine the duration of the short
pulse that we want. This short pulse, of course, is really just a glitch in the circuit. Figure 3.10(a) shows a sample
one-shot circuit using signal delays through three inverters. Figure 3.10(b) shows a sample timing trace for it.
Delay through
the Inverters
Input
A Input Output Output A Delay through
the AND gate
(b) (a) Figure 3.10 A one-shot circuit: (a) using signal delay through three inverters; (b) timing trace.
Initially, assume that the value for Input is a 0, and point A is a 1, therefore, the output of the AND gate is a 0.
When we set Input to a 1 momentarily, both inputs to the AND gate will be 1’s, and so after a delay through the AND
gate, Output will be a 1. After a delay through the three inverters, with Input still at a 1, point A will go to a 0, and
Output will change back to a 0. When we set Input back to a 0, Output will continue to be a 0. After the delay
through the inverters when point A goes back to a 1, Output remains at a 0.
As a result, a glitch is created by the signal delay through the three inverters. This glitch, however, is the short 1
pulse that we want, and the length of this pulse is determined by the delay through the inverters. With this one-shot
circuit, it does not matter how long the input key is being pressed, the output signal will always be the same 1 pulse
each time that the key is pressed.
♦ 3.6 BCD to 7-Segment Decoder We will now synthesize the circuit for a BCD to 7-segment decoder for driving a 7-segment LED display. The
decoder converts a 4-bit binary coded decimal (BCD) input to seven output signals for turning on the seven lights in
a 7-segment LED display. The 4-bit input encodes the binary representation of a decimal digit. Given the decimal
digit input, the seven output lines are turned on in such a way so that the LED displays the corresponding digit. The
7-segment LED display schematic with the names of each segment labeled is shown here
a
f g e b
c d 78 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits The operation of the BCD to 7-segment decoder is specified in the truth table in Figure 3.11. The four inputs to
the decoder are i3, i2, i1, and i0, and the seven outputs for each of the seven LEDs are labeled a, b, c, d, e, f, and g.
For each input combination, the corresponding digit to display on the 7-segment LED is shown in the “Display”
column. The segments that need to be turned on for that digit will have a 1 while the segments that need to be turned
off for that digit will have a 0. For example, for the 4-bit input 0000, which corresponds to the digit 0, segments a, b,
c, d, e, and f need to be turned on, while segment g needs to be turned off.
Notice that the input combinations 1010 to 1111 are not used, and so don’t-care values are assigned to all of the
segments for these six combinations.
a b c d e f g 0 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 2 1 1 0 1 1 0 1 0 1 1 3 1 1 1 1 0 0 1 0 1 0 0 4 0 1 1 0 0 1 1 0 1 0 1 5 1 0 1 1 0 1 1 0 1 1 0 6 1 0 1 1 1 1 1 0 1 1 1 7 1 1 1 0 0 0 0 1 0 0 0 8 1 1 1 1 1 1 1 1 0 0 1 9 1 1 1 0 0 1 1 × × × × × × × i3 Inputs
i2
i1 i0 Decimal
Digit 0 0 0 0 0 0 0 0 0 0 Display rest of the combinations Figure 3.11 Truth table for the BCD to 7-segment decoder.
From the truth table in Figure 3.11, we are able to specify seven equations that are dependent on the four inputs
for each of the seven segments. For example, the canonical form equation for segment a is
a = i3'i2'i1'i0' + i3'i2'i1i0' + i3'i2'i1i0 + i3'i2i1'i0 + i3'i2i1i0' + i3'i2i1i0 + i3i2'i1'i0' + i3i2'i1'i0
Before implementing this equation directly in a circuit, we want to simplify it first using the K-map method.
The K-map for the equation for segment a is
a i1 i0 i3 i2 00 01
0 00 11
1 1
4 01 5
1 11 × 10 1 10 12 × 8 9
1 6 7 ×
× i2'i0'
i2i0 1 1
13 2 3
1 1
15
11 ×
× 14 i1 10 i3 From evaluating the K-map, we derive the simpler equation for segment a as
a = i3 + i1 + i2'i0' + i2i0 = i3 + i1 + (i2 i0) Proceeding in a similar manner, we get the following remaining six equations
b = i2' + (i1 i0)
c = i2 + i1' + i0 79 Digital Logic and Microprocessor Design with VHDL d
e
f
g Chapter 3 - Combinational Circuits = i1i0' + i2'i0' + i2'i1 + i2i1'i0
= i1i0' + i2'i0'
= i3 + i2i1' + i2i0' + i1'i0'
= i3 + (i2 ⊕ i1) + i1i0' From these seven simplified equations, we can now implement the circuit, as shown in Figure 3.12. The
labeling of the nodes and gates in the drawing will be explained and used in Section 3.7.1.
i3 i2 i1
U2
ip2 i0
U1 U0 ip1 ip0 U5 U8
U9
U10
U11 a U6 a1 U4
b U7 U3 c U12 d U15 e U19 f U22 g b1 d1
d2
d3
d4 U13 e
1
U14 e
2
U16
U17
U18
U20 f1
f2
f3
g1 U21 g
2 Figure 3.12 Circuit for the BCD to 7-segment decoder. 3.7 VHDL for Combinational Circuits Writing VHDL code to describe a digital circuit can be done using any one of three models or levels of
abstraction: structural, dataflow, or behavioral. The choice of which model to use usually depends on what is
known about the circuit. At the structural level, which is the lowest level, you first have to manually design the
circuit. Having drawn the circuit, you use VHDL to specify the components and gates that are needed by the circuit
and how they are connected together by following your circuit exactly. Synthesizing a structural VHDL description
of a circuit will produce a netlist that is like your original circuit. The advantage of working at the structural level is
that you have full control as to what components are used and how they are connected together. The disadvantage,
of course, is that you need to manually come up with the circuit, and so the full capabilities of the synthesizer are not
utilized. A simple example of a structural VHDL code for a 2-input multiplexer was shown in Figure 1.11.
At the dataflow level, the circuit is defined using built-in VHDL logic operators (such as the AND, OR, and NOT)
that are applied to input signals. In order to work at this level, you need to have the Boolean equations for the circuit.
Hence, the dataflow level is best suited for describing a circuit that is already expressed as a Boolean function. The 80 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits equations are easily converted to the required VHDL syntax using signal assignment statements. A simple example
of a dataflow VHDL code for a 2-input multiplexer was shown in Figure 1.9.
All of the statements used in the structural and dataflow levels are executed concurrently, as opposed to
statements in a computer program, which are usually executed in a sequential manner. In other words, the ordering
of the VHDL statements written in the structural or dataflow level does not matter – the results would be exactly the
same.
Describing a circuit at the behavioral level is very similar to writing a computer program. You have all of the
standard high-level programming constructs—such as the FOR LOOP, WHILE LOOP, IF THEN ELSE, CASE, and variable
assignments. The statements are enclosed in a PROCESS block and are executed sequentially. A simple example of a
behavioral VHDL code for a 2-input multiplexer was shown in Figure 1.5. 3.7.1 Structural BCD to 7-Segment Decoder
Figure 3.13 shows the structural VHDL code for the BCD to 7-segment decoder based on the circuit shown in
Figure 3.12. The code starts with declaring and defining all of the components needed in the circuit. For this decoder
circuit, only basic gates (such as the NOT gate, 2-input AND, 3-input AND, etc.) are used. The ENTITY statement is
used to declare all of these components, and the ARCHITECTURE statement is used to define the operation of these
components. Since we are using only simple gates, defining these components using the dataflow model is the
simplest. For more complex components (as we will see in later chapters) we want to choose the model that is best
suited for the information that we have available for the circuit. The reason why the code shown in Figure 3.13 is
structural is not because of how these components are defined, but rather on how these components are connected
together to form the enclosing entity; in this case, the bcd entity. Notice that the LIBRARY and USE statements need to
be repeated for every ENTITY declaration.
The actual structural code begins with the bcd ENTITY declaration. The bcd circuit shown in Figure 3.12 has four
input signals, i3, i2, i1, and i0, and seven output signals, a, b, c, d, e, f, and g. These signals are declared in the PORT
list using the keyword IN for the input signals, and OUT for the output signals; both of which are of type STD_LOGIC.
The ARCHITECTURE section begins by specifying the components needed in the circuit using the COMPONENT
statement. The port list in the COMPONENT statements must match exactly the port list in the entity declarations of
the components. They must match not only in the number, direction, and type of the signals, but also in the names
given to the signals. Note also that names in the component port list can be the same as the names in the bcd entity
port list, but they are not the same signals. For example, the and2gate component port list and the bcd entity port list
both have two signals called i1 and i2. References to these two signals in the body of the bcd architecture are for the
signals declared in the bcd entity.
After the COMPONENT statements, the internal node signals are declared using the SIGNAL statement. The names
listed are the same as the internal node names used in the circuit in Figure 3.12 for easy reference.
Following all of the declarations, the body of the architecture starts with the keyword BEGIN. For each gate used
in the circuit, there is a corresponding PORT MAP statement. Each PORT MAP statement begins with an optional label
(such as U1, U2, and so on) followed by the name of the component (as previously declared with the COMPONENT
statements) to use. Again, the labels used in the PORT MAP statements correspond to the labels on the gates in the
circuit in Figure 3.12. The parameter list in the PORT MAP statement matches the port list in the component
declaration. For example, U0 is instantiated with the component notgate. The first parameter in the PORT MAP
statement is the input signal i0, and the second parameter is the output signal ip0. U4 is instantiated with the 3-input
OR gate. The three inputs are i3, i1, and a1, and the output is a. Here, a1 is the output from the 2-input XNOR gate of
U3. The rest of the PORT MAP statements in the program are obtained in a similar manner.
All the PORT MAP statements are executed concurrently, and therefore, the ordering of these statements is
irrelevant. In other words, changing the ordering of these statements will still produce the same result. Any time
when a signal in a PORT MAP statement changes value, i.e., from a 0 to a 1, or vice versa, that PORT MAP statement is
executed. 81 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits ----------------- NOT gate ----------------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY notgate IS PORT(
i: IN STD_LOGIC;
o: OUT STD_LOGIC);
END notgate;
ARCHITECTURE Dataflow OF notgate IS
BEGIN
o <= NOT i;
END Dataflow;
----------------- 2-input AND gate --------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY and2gate IS PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END and2gate;
ARCHITECTURE Dataflow OF and2gate IS
BEGIN
o <= i1 AND i2;
END Dataflow;
----------------- 3-input AND gate --------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY and3gate IS PORT(
i1, i2, i3: IN STD_LOGIC;
o: OUT STD_LOGIC);
END and3gate;
ARCHITECTURE Dataflow OF and3gate IS
BEGIN
o <= (i1 AND i2 AND i3);
END Dataflow;
----------------- 2-input OR gate ---------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY or2gate IS PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END or2gate;
ARCHITECTURE Dataflow OF or2gate IS
BEGIN
o <= i1 OR i2;
END Dataflow;
----------------- 3-input OR gate ---------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY or3gate IS PORT(
i1, i2, i3: IN STD_LOGIC;
o: OUT STD_LOGIC);
END or3gate;
ARCHITECTURE Dataflow OF or3gate IS 82 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits BEGIN
o <= i1 OR i2 OR i3;
END Dataflow;
----------------- 4-input OR gate ---------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY or4gate IS PORT(
i1, i2, i3, i4: IN STD_LOGIC;
o: OUT STD_LOGIC);
END or4gate;
ARCHITECTURE Dataflow OF or4gate IS
BEGIN
o <= i1 OR i2 OR i3 OR i4;
END Dataflow;
----------------- 2-input XOR gate --------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY xor2gate IS PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END xor2gate;
ARCHITECTURE Dataflow OF xor2gate IS
BEGIN
o <= i1 XOR i2;
END Dataflow;
----------------- 2-input XNOR gate -------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY xnor2gate IS PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END xnor2gate;
ARCHITECTURE Dataflow OF xnor2gate IS
BEGIN
o <= NOT(i1 XOR i2);
END Dataflow;
----------------- bcd entity --------------------LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY bcd IS PORT(
i0, i1, i2, i3: IN STD_LOGIC;
a, b, c, d, e, f, g: OUT STD_LOGIC);
END bcd;
ARCHITECTURE Structural OF bcd IS
COMPONENT notgate PORT(
i: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT and2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC); 83 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits END COMPONENT;
COMPONENT and3gate PORT(
i1, i2, i3: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT or2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT or3gate PORT(
i1, i2, i3: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT or4gate PORT(
i1, i2, i3, i4: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT xor2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
COMPONENT xnor2gate PORT(
i1, i2: IN STD_LOGIC;
o: OUT STD_LOGIC);
END COMPONENT;
SIGNAL ip0,ip1,ip2,a1,b1,d1,d2,d3,d4,e1,e2,f1,f2,f3,g1,g2: STD_LOGIC;
BEGIN
U0: notgate PORT MAP(i0,ip0);
U1: notgate PORT MAP(i1,ip1);
U2: notgate PORT MAP(i2,ip2);
U3: xnor2gate PORT MAP(i2, i0, a1);
U4: or3gate PORT MAP(i3, i1, a1, a);
U5: xnor2gate PORT MAP(i1, i0, b1);
U6: or2gate PORT MAP(ip2, b1, b);
U7: or3gate PORT MAP(i2, ip1, i0, c);
U8: and2gate PORT MAP(i1, ip0, d1);
U9: and2gate PORT MAP(ip2, ip0, d2);
U10: and2gate PORT MAP(ip2, i1, d3);
U11: and3gate PORT MAP(i2, ip1, i0, d4);
U12: or4gate PORT MAP(d1, d2, d3, d4, d);
U13: and2gate PORT MAP(i1, ip0, e1);
U14: and2gate PORT MAP(ip2, ip0, e2);
U15: or2gate PORT MAP(e1, e2, e);
U16: and2gate PORT MAP(i2, ip1, f1);
U17: and2gate PORT MAP(i2, ip0, f2);
U18: and2gate PORT MAP(ip1, ip0, f3);
U19: or4gate PORT MAP(i3, f1, f2, f3, f);
U20: xor2gate PORT MAP(i2, i1, g1);
U21: and2gate PORT MAP(i1, ip0, g2);
U22: or3gate PORT MAP(i3, g1, g2, g);
END Structural;
Figure 3.13 Structural VHDL code of the BCD to 7-segment decoder. 84 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits 3.7.2 Dataflow BCD to 7-Segment Decoder
Figure 3.14 shows the dataflow VHDL code for the BCD to 7-segment decoder based on the Boolean equations
derived in Section 3.6. The ENTITY declaration for this dataflow code is exactly the same as that for the structural
code, since the interface for the decoder remains the same.
In the ARCHITECTURE section, seven concurrent signal assignment statements are used: one for each of the seven
Boolean equations, which corresponds to the seven LED segments. For example, the equation for segment a is
a = i3 + i1 + (i2 i0) This is converted to the signal assignment statement:
a <= i3 OR i1 OR (i2 XNOR i0);
Proceeding in a similar manner, we obtain the signal assignment statements in the dataflow code for the
remaining six equations.
All of the signal assignment statements are executed concurrently, and therefore, the ordering of these
statements is irrelevant. In other words, changing the ordering of these statements will still produce the same result.
Any time when a signal on the right-hand side of an assignment statement changes value (i.e., from a 0 to a 1, or
vice versa) that assignment statement is executed.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY bcd IS PORT(
i0, i1, i2, i3: IN STD_LOGIC;
a, b, c, d, e, f, g: OUT STD_LOGIC);
END bcd;
ARCHITECTURE Dataflow OF bcd IS
BEGIN
a <= i3 OR i1 OR (i2 XNOR i0);
b <= (NOT i2) OR NOT (i1 XOR i0);
c <= i2 OR (NOT i1) OR i0;
d <= (i1 AND NOT i0) OR (NOT i2 AND NOT i0)
OR (NOT i2 AND i1) OR (i2 AND NOT i1 AND i0);
e <= (i1 AND NOT i0) OR (NOT i2 AND NOT i0);
f <= i3 OR (i2 AND NOT i1)
OR (i2 AND NOT i0) OR (NOT i1 AND NOT i0);
g <= i3 OR (i2 XOR i1) OR (i1 AND NOT i0);
END Dataflow; ----- seg
seg
seg
seg a
b
c
d -- seg e
-- seg f
-- seg g Figure 3.14 Dataflow VHDL code of the BCD to 7-segment decoder. 3.7.3 Behavioral BCD to 7-Segment Decoder
The behavioral VHDL code for the BCD to 7-segment decoder is shown in Figure 3.15. The port list for this
entity is slightly different from the two entities in the previous sections. Instead of having the four separate input
signals, i0, i1, i2, and i3, we have declared a vector, I, of length four. This vector, I, is declared with the type keyword
STD_LOGIC_VECTOR, that is, a vector of type STD_LOGIC. The length of the vector is specified by the range (3
DOWNTO 0). The first number (3) in the range denotes the index of the most significant bit of the vector, and the
second number (0) in the range denotes the index of the least significant bit of the vector. Likewise, the seven output
signals, a to g, is replaced with the STD_LOGIC_VECTOR Segs of length 7. This time, however, the keyword TO is
used in the range to mean that the most significant bit in the vector is index 1, and the least significant bit in the
vector is index 7.
In the architecture section, a PROCESS statement is used. All of the statements inside the process block are
executed sequentially. The process block itself, however, is treated as a single concurrent statement. Thus, the 85 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits architecture section can have two or more process blocks together with other concurrent statements, and these will
all execute concurrently.
The parenthesized list of signals after the PROCESS keyword is referred to as the sensitivity list. The purpose of
the sensitivity list is that, when a value for any of the listed signals changes, the entire process block is executed
from the beginning to the end.
In the code, there is a CASE statement inside the process block. Depending on the value of I, one of the WHEN
parts will be executed. A WHEN part consists of the keyword WHEN followed by a constant value for the variable I to
match, followed by the symbol “=>.” The statement or statements after the symbol “=>” is executed when I matches
that corresponding constant. In the code, all of the WHEN parts contain one signal assignment statement. All of the
signal assignment statements assign a string of seven bits to the output signal Segs. This string of seven bits
corresponds to the on-off values of the seven segments, a to g, as shown in the 7-segment decoder truth table of
Figure 3.11. For example, looking at the truth table, we see that when I = “0000” (that is, for the decimal digit 0) we
want all of the segments to be on except for segment g. Recall that in the declaration of the Segs vector, the most
significant bit, which is the leftmost bit in the bit string, is index 1, and the least significant bit, which is the
rightmost bit, is index 7. In VHDL, the notation Segs(n) is used to denote the index n of the Segs vector. In the code,
we have designated Segs(1) for segment a, Segs(2) for segment b, and so on to Segs(7) for segment g. So, in order to
display the decimal digit 0, we need to assign the bit string “1111110” to Segs.
If the value of I does not match any of the WHEN parts, then the WHEN OTHERS part will be chosen. In this case,
all of the segments will be turned off. Notice that for both the structural and the dataflow code, the segments are not
all turned off when I is one of these values. Instead, a certain combination of LEDs are turned on because the Kmaps assigned some of the don’t-cares to 1’s. If we assign all the don’t-cares to 0, then all the LEDs will be turned
off. An alternative to turning all of the segments off for the remaining six cases is to display the six alphabets, A, b,
C, d, E, and F, for the six hexadecimal digits. The two letters, b, and d, have to be displayed in lower case, because
otherwise, it will be the same as the numbers 8 and 0, respectively.
A sample simulation trace of the behavioral 7-segment decoder code is shown in Figure 3.16.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY bcd IS PORT (
I: IN STD_LOGIC_VECTOR (3 DOWNTO 0);
Segs: OUT STD_LOGIC_VECTOR (1 TO 7));
END bcd;
ARCHITECTURE Behavioral
BEGIN
PROCESS(I)
BEGIN
CASE I IS
WHEN "0000" => Segs
WHEN "0001" => Segs
WHEN "0010" => Segs
WHEN "0011" => Segs
WHEN "0100" => Segs
WHEN "0101" => Segs
WHEN "0110" => Segs
WHEN "0111" => Segs
WHEN "1000" => Segs
WHEN "1001" => Segs
WHEN OTHERS => Segs
END CASE;
END PROCESS;
END Behavioral; OF bcd IS <=
<=
<=
<=
<=
<=
<=
<=
<=
<=
<= "1111110";
"0110000";
"1101101";
"1111001";
"0110011";
"1011011";
"1011111";
"1110000";
"1111111";
"1110011";
"0000000"; 86 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits Figure 3.15 Behavioral VHDL code of the BCD to 7-segment decoder. Figure 3.16 A sample simulation trace of the behavioral 7-segment decoder code. 3.8 Summary Checklist
Combinational circuit
Analysis of combinational circuit
Synthesis of combinational circuit
Technology mapping
Using K-maps to minimize a Boolean function
The use of don’t-cares
Using don’t-cares in a K-map
Using the Quine-McCluskey method to minimize a Boolean function
Timing hazards and glitches
How to eliminate simple glitches
Writing structural, dataflow, and behavioral VHDL code
Be able to analyze any combinational circuit by deriving its truth table, or Boolean function
Be able to synthesize a combinational circuit from a given description, truth table, or Boolean function
Be able to reduce any combinational circuit to its smallest size 87 Digital Logic and Microprocessor Design with VHDL 3.9 Chapter 3 - Combinational Circuits Problems 3.1 Derive the truth table for the following circuits:
a)
xyz F b)
w
x
y F z c)
x y z F d)
a b c F Answer:
a)
x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 F
0
0
0
0
1
1
1
1 b) 88 Digital Logic and Microprocessor Design with VHDL w
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 x
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 w(x+y)
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1 x
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1 ( x + y)
0
0
1
1
1
1
1
1 a
0
0
0
0
1
1
1
1 b
0
0
1
1
0
0
1
1 c
0
1
0
1
0
1
0
1 a’
1
1
1
1
0
0
0
0 (w(x+y))'
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0 (y' + z)
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1 Chapter 3 - Combinational Circuits [(w(x+y))' (y' + z)]
1
1
0
1
1
1
0
1
1
1
0
0
0
0
0
0 F
0
0
1
0
0
0
1
0
0
0
1
1
1
1
1
1 c)
(x + y + z' )
1
0
1
1
1
1
1
1 (x + y + z' )'
0
1
0
0
0
0
0
0 (x + y) 1 (x + y + z' )'
0
0
0
0
0
0
0
0 d)
(a' ⊕ c)
1
0
1
0
0
1
0
1 (c' b' a)
0
0
0
0
1
0
0
0 (c' b' a)'
1
1
1
1
0
1
1
1 (a' ⊕ c) + (c' b' a)'
1
1
1
1
0
1
1
1 3.2 Derive the Boolean function directly from the circuits in Problem 3.1.
Answer:
a) F = xy + (xy' (x+z' ))
b) F = [(w(x+y))' (y' + z)]'
c) F = [(x + y) (y' + z' + x' + y) (x + y + z' )' ]'
d) F = [(a' ⊕ c) + (c' b' a)' ] b
3.3 Draw the circuit diagram that implements the following truth tables: 89 F
0
0
1
1
1
0
1
1 F
1
1
1
1
1
1
1
1 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits a) b)
a
0
0
0
0
1
1
1
1 b
0
0
1
1
0
0
1
1 c
0
1
0
1
0
1
0
1 F
0
0
1
1
0
0
1
0 w
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 c) x
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F
0
0
1
0
1
1
0
1
0
1
1
0
1
1
0
1 d)
w
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 x
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 y
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 z
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F1
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1 N3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 F2
1
1
1
1
0
1
0
0
1
1
0
0
1
1
1
1 3.4 Draw the circuit diagram that implements the following expressions:
a) F (x, y, z) = Σ(0, 1, 6)
Answer:
F (x, y, z) = Σ(0, 1, 6)
= m0 + m1 + m6
= x' y' z' + x' y' z + x y z' 90 N2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 N1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 N0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 F
0
0
1
1
0
0
1
0
0
0
1
1
1
0
0
1 Digital Logic and Microprocessor Design with VHDL x y Chapter 3 - Combinational Circuits z F b)
c)
d)
e)
f) F (w,x, y, z) = Σ(0, 1, 6)
F (w,x, y, z) = Σ(2, 6, 10, 11, 14, 15)
F (x, y, z) = Π(0, 1, 6)
F (w,x, y, z) = Π(0, 1, 6)
F (w,x, y, z) = Π(2, 6, 10, 11, 14, 15) 3.5 Draw the circuit diagram that implements the following Boolean functions using as few basic gates as possible,
but without modifying the equation.
a)
b)
c)
d)
e)
f)
g) F = xy' + x'y'z + xyz'
F = w'z' + w'xy + wx'z + wxyz
F = w'xy'z + w'xyz + wxy'z + wxyz
F = N3'N2'N1N0' + N3'N2'N1N0 + N3N2'N1N0' + N3N2'N1N0 + N3N2N1'N0' + N3N2N1N0
F = [(x y)' + (xyz)'] (w' + x + z)
F=x⊕y⊕z
F = [w'xy'z + w'z (y ⊕ x)]' 3.6 Draw the circuit diagram that implements the Boolean functions in Problem 3.5 using only 2-input AND, 2-input
OR, and NOT gates.
3.7 Design a circuit that inputs a 4-bit number. The circuit outputs a 1 if the input number is any one of the
following numbers: 2, 3, 10, 11, 12, and 15. Otherwise, it outputs a 0.
3.8 Design a circuit that inputs a 4-bit number. The circuit outputs a 1 if the input number is greater than or equal to
5. Otherwise, it outputs a 0.
3.9 Design a circuit that inputs a 4-bit number. The circuit outputs a 1 if the input number has an even number of
zeros. Otherwise, it outputs a 0.
3.10 Construct the following circuit. The circuit has five input signals and one output signal. The five input lines are
labeled W, X, Y, Z, and E, and the output line is labeled F. E is used to enable (turn on) or disable (turn off) the
circuit; thus, when E = 0, the circuit is disabled, and F is always 0. When E = 1, the circuit is enabled, and F is
determined by the value of the four input signals, W, X, Y, and Z, where W is the most significant bit. If the
value is odd, then F = 1, otherwise F = 0.
Answer: 91 Digital Logic and Microprocessor Design with VHDL
E W X Y Chapter 3 - Combinational Circuits Z F or E
Z F 3.11 Draw the smallest circuit that inputs two 2-bit numbers. The circuit outputs a 2-bit number that represents the
count of the number of even numbers in the inputs. The number 0 is taken as an even number. For example, if
the two input numbers are 0 and 3, then the circuit outputs the number 1 in binary. If the two input numbers are
0 and 2, then the circuit outputs the number 2 in binary. Show your work by deriving the truth table, the
equation, and finally the circuit. You need to minimize all of the equations to standard forms.
Answer:
x1 x0 y1 y0 out1 out0
0000
1
0
0001
0
1
0010
1
0
0011
0
1
0100
0
1
0101
0
0
0110
0
1
0111
0
0
1000
1
0
1001
0
1
1010
1
0
1011
0
1
1100
0
1
1101
0
0
1110
0
1
1111
0
0 out1 = x1'x0'y1'y0' + x1'x0'y1y0' + x1x0'y1'y0' + x1x0'y1y0'
= x 0 ' y 0'
out0 = x1'x0'y1'y0 + x1'x0'y1y0 + x1'x0y1'y0' + x1'x0y1y0' + x1x0'y1'y0 + x1x0'y1y0 + x1x0y1'y0' + x1x0y1y0'
= x0 ⊕ y0 92 Digital Logic and Microprocessor Design with VHDL
x0 Chapter 3 - Combinational Circuits y0 out1
out0 3.12 Derive and draw the circuit that inputs two 2-bit unsigned numbers. The circuit outputs a 3-bit signed number
that represents the difference between the two input numbers (i.e. it is the result of the first number minus the
second number). Derive the truth table and equations in canonical form.
Answer:
x1 x0
00
00
00
00
01
01
01
01
10
10
10
10
11
11
11
11 y1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 y0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1 d2
0
1
1
1
0
0
1
1
0
0
0
1
0
0
0
0 d1
0
1
1
0
0
0
1
1
1
0
0
1
1
1
0
0 d0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0 d2 = x1'x0'y1'y0 + x1'x0'y1y0' + x1'x0'y1y0 + x1'x0y1y0' + x1'x0y1y0 + x1x0'y1y0
d1 = x1'x0'y1'y0 + x1'x0'y1y0' + x1'x0y1y0' + x1'x0y1y0 + x1x0'y1'y0' + x1x0'y1y0 + x1x0y1'y0' + x1x0y1'y0
d0 = x1'x0'y1'y0 + x1'x0'y1y0 + x1'x0y1'y0' + x1'x0y1y0' + x1x0'y1'y0 + x1x0'y1y0 + x1x0y1'y0' + x1x0y1y0' 93 Digital Logic and Microprocessor Design with VHDL
x1 x0 y1 Chapter 3 - Combinational Circuits y0 m1 m1 m2 m3 m4 m6 m7 m8 m9m11 12 13 14
mmm m2
m3 d2 m4
m6 d1 m7
m8 d0 m9
m11
m12
m13
m14 3.13 Use Boolean algebra to show that the following circuit is equivalent to the NOT gate. 3.14 Construct a 4-input NAND gate circuit using only 2-input NAND gates.
Answer: 3.15 Implement the following circuit using as few NAND gates (with any number of inputs) as possible.
N E T A B Out Answer:
A xnor B = (A xor B)'
= (AB' + A'B)'
= (AB')' (A'B)'
= [(AB' )' (A'B)']'' 94 Digital Logic and Microprocessor Design with VHDL = A
B N Chapter 3 - Combinational Circuits E T A B A
B Out N E T A B Out 3.16 Draw the circuit diagram that implements the Boolean functions in Problem 3.5 using only 2-input NAND gates.
3.17 Draw the circuit diagram that implements the Boolean functions in Problem 3.5 using only 3-input NAND gates.
3.18 Draw the circuit diagram that implements the Boolean functions in Problem 3.5 using only 3-input NOR gates.
3.19 Convert the following circuit as is (i.e., do not reduce it first) to use only 2-input NOR gates. Answer: 3.20 Convert the following full adder circuit to use only eleven 2-input NAND gates. 95 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits x y c out
cin s Answer:
Recall that wx + yz = ((wx)' (yz)')'. Furthermore, x ⊕ y ⊕ z = x
x y z and x y = (x'y' + xy). y cin cout s 3.21 Perform a timing analysis of the circuit shown in Figure 3.9(c) to see that the circuit does not produce any
glitches.
3.22 Derive a circuit for the 2-input XOR gate that uses only 2-input NAND gates.
Answer
x
y F F=x⊕y
= xy' + x'y
= xx' + xy' + x'y + y'y
= (x' + y' )x + (x' + y' )y
= ((xy)'x) + ((xy)'y)
= [((xy)'x)' ((xy)'y)' ]'
3.23 Use K-maps to reduce the Boolean functions represented by the truth tables in Problem 3.3 to standard form. 96 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits 3.24 Use K-maps to reduce the Boolean functions in Problem 3.4 to standard form.
3.25 Use K-maps to reduce the Boolean functions in Problem 3.5 to standard form.
3.26 List all of the PIs, EPIs, and all of the minimized standard form solutions for the following equation.
F(v,w,x,y,z) = Π(2,3,4,5,6,7,8,9,11,13,15,18,19,20,21,22,29,30,31)
Answer
F(v,w,x,y,z) = Π(2,3,4,5,6,7,8,9,11,13,15,18,19,20,21,22,29,30,31)
= Σ(0,1,10,12,14,16,17,23,24,25,26,27,28)
F v=0 yz
00 wx
00 01 1 11 v=1
10 00 1 01 1 11 1 01
11 10 1
1 1
1 10 1
1 1 1 1 w'x'y' F 00
00 wxy'z' v=0 yz wx 01 1 11 v=1
10 00 01 1 1 1 01
11 11 10 vw'xyz 1
1 1 vwx' 1 1 10 1 vx'y' 1 v'wxz' v'wyz' vwy'z' 1 1 wx'yz' PI’s: wxy'z', w'x'y', vx'y', vw'xyz, vwx', wx'yz', vwy'z', v'wyz', v'wxz'
PI’s
wxy'z'
w'x'y'
EPI
vx'y'
vw'xyz
EPI
vwx'
EPI
wx'yz'
vwy'z'
v'wyz'
v'wxz'
EPI covered minterms 0 1 0 10 12 1 Solution:
F = w'x'y' + vw'xyz + vwx' + v'wyz' + wxy'z' 97 16 17 23 24 25 26 27 16 EPI’s: w'x'y', vw'xyz, vwx' 14 17 23 24 25 26 27 28 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits 3.27 Use K-maps to reduce the following 4-variable Boolean functions F(w, x, y, z) to standard form:
a) 1-minterms: m2, m3, m4, m5
Don’t-care minterms: m10, m11, m12, m13, m14, m15
b) 1-minterms: 1, 3, 4, 7, 9
Don’t-care minterms: 0, 2, 13, 14, 15
c) 1-minterms: 2, 3, 8, 9
Don’t-care minterms: 1, 5, 6, 7, 8, 13, 15
Answer:
a)
F yz wx 00 01
0 00 10
3 1
4 01 1 11 × 10 11
1
5 2
1 7 6 1
12
8 × 13
9 ×
× 15
11 ×
× 14
10 3.28 Use K-maps to reduce the following 5-variable Boolean functions F(v, w, x, y, z) to standard form:
a) 1-minterms: 1, 3, 4, 7, 9
Don’t-care minterms: 0, 2, 13, 14, 15
b) 1-minterms:
3.29 Use the Quine-McCluskey method to simplify the function f(w,x,y,z) = Σ(0,2,5,7,13,15). List all the PI’s, EPI’s,
cover lists, and solutions.
Answer:
Group Subcube
ID
Minterms
G0
m0
G1
m2
G2
m5
G3
m7
m13
G4
m15
Group
ID
G0
G2
G3 Group
ID
G2 Subcube Value
wxyz
0000
0010
0101
0111
1101
1111 Subcube
Covered Subcube
Minterms
m0,2
m5,7
m5,13
m7,15
m13,15 Subcube Value
wxyz
00-0
01-1
-101
-111
11-1 Subcube
Covered Subcube
Minterms
m5,7,13,15 Subcube Value
wxyz
-1-1 Subcube
Covered Prime
Implicant
Name Prime
Implicant
Expression Implicant
Minterms Function Minterms
0 2 5 7 13 98 15 Digital Logic and Microprocessor Design with VHDL ⊗ P1 w'x'z' (0,2) P4 xz Chapter 3 - Combinational Circuits ⊗ (5,7,13,15) ⊗ EPI Covered Minterms:
Not Covered Minterms: 0 2 ⊗ ⊗ ⊗ 5 7 13 15 PI list: w'x'z', xz
EPI list: w'x'z', xz
Cover list: w'x'z', xz
f = w'x'z' + xz
3.30 Use the Quine-McCluskey method to reduce the Boolean functions in Problem 3.4 to standard form.
3.31 Write the function that eliminates the static hazard(s) in the function F = w'z + xyz' + wx'y.
Answer:
yz 00 wx 11 01
0 00 3 1
1 4
01 2 13 14
1 9 11
1 2 1
7 5 10 6 11 10
1 15 10 4
01 10
3 1
1 6 11
8 0 1 1 11 01 00
7 5 00 wx 1 1
12 yz 10 1
12 13 8 Original function 1 1 15 11
1 9 14
1
10
1 After adding the overlaps F = w'z + xyz' + wx'y + w'xy + wyz' + x'yz
3.32 Write the function that eliminates the static hazard(s) in the function F = y'z' + wz + w'x'y.
Answer:
F F
yz wx 00 01 11 yz wx 10 00
00
01 01 11 10 1 1 1 1 00 1 01 1 11 1 1 1 11 1 1 1 10 1 1 1 10 1 1 1 1 1 With hazard Without hazard F = y'z' + wz + w'x'y + wy' + x'yz + w'x'z'
3.33 Write the complete structural VHDL code for the Boolean functions in Problem 3.4.
3.34 Write the complete dataflow VHDL code for the Boolean functions in Problem 3.4.
3.35 Write the complete behavioral VHDL code for the Boolean functions in Problem 3.4. 99 Digital Logic and Microprocessor Design with VHDL Chapter 3 - Combinational Circuits 3.36 Write the complete dataflow VHDL code for the Boolean functions in Problem 3.5.
3.37 Write the behavioral VHDL code for converting an 8-bit unsigned binary number to two 4-bit BCD numbers.
These two BCD numbers represent the tenth and unit digits of a decimal number. Also, turn on the decimal
point LED for the unit digit if the 8-bit binary number is in the one hundreds range, and turn on the decimal
point LED for the tenth digit if the 8-bit binary number is in the two hundreds range. This circuit is used as the
output circuit for many designs in later chapters. 100 Chapter 4 Standard Combinational Components Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components As with many construction projects, it is often easier to build in a hierarchical fashion. Initially, we use the very
basic building blocks to build slightly larger building blocks, and then from these larger building blocks, we build
yet larger building blocks, and so on. Similarly, in constructing large digital circuits, instead of starting with the
basic logic gates as building blocks each time, we often start with larger building blocks. Many of these larger
building blocks are often used over and over again in different digital circuits, and therefore, are considered as
standard components for large digital circuits. In order to reduce the design time, these standard components are
often made available in standard libraries so that they do not have to be redesigned each time that they are needed.
For example, many digital circuits require the addition of two numbers; therefore, an adder circuit is considered a
standard component and is available in most standard libraries.
Standard combinational components are combinational circuits that are available in standard libraries. These
combinational components are used mainly in the construction of datapaths. For example, in our microprocessor
road map, the standard combinational components are the multiplexer, ALU, comparator, and tri-state buffer. Other
standard combinational components include adders, subtractors, decoders, encoders, shifters, rotators, and
multipliers. Although the next-state logic and output logic circuits in the control unit are combinational circuits, they
are not considered as standard combinational components because they are designed uniquely for a particular control
unit to solve a specific problem, and usually are never reused in another design.
In this chapter, we will design some standard combinational components. These components will be used in
later chapters for the building of the datapath in the microprocessor. When we use these components to build the
datapath, we do not need to know the detailed construction of these components. Instead, we only need to know how
these components operate, and how they connect to other components. Nevertheless, in order to see the whole
picture, we should understand how these individual components are designed. 4.1 Signal Naming Conventions So far in our discussion, we have always used the words “high” and “low” to mean 1 or 0, or “on” or “off”,
respectively. However, this is somewhat arbitrary, and there is no reason why we can’t say a 0 is a high or a 1 is off.
In fact, many standard off-the-shelf components use what we call negative logic where 0 is for on and 1 is for off.
Using negative logic is usually more difficult to understand because we are used to positive logic where 1 is for on,
and 0 is for off. In all of our discussions, we will use the more natural, positive logic that we are familiar with.
Nevertheless, in order to prevent any confusion as to whether we are using positive logic or negative logic, we
often use the words “assert,” “de-assert,” “active-high,” and “active-low.” Regardless of whether we are using
positive or negative logic, active-high always means that a 1 (i.e., a high) will cause the signal to be active or
enabled and that a 0 will cause the signal to be inactive or disabled. For example, if there is an active-high signal
called add and we want to enable it (i.e. to make it do what it is intended for, which in this case is to add something)
then we need to set this signal line to a 1. Setting this signal to a 0 will cause this signal to be disabled or inactive.
An active-low signal, on the other hand, means that a 0 will cause the signal to be active or enabled, and that a 1 will
cause the signal to be inactive or disabled. So if the signal add is an active-low signal, then we need to set it to a 0 to
make it add something.
We also use the word “assert” to mean: to make a signal active or to enable the signal. To de-assert a signal is
to disable the signal or to make it inactive. For example, to assert the active-high add signal line means to set the
add signal to a 1. To de-assert an active-low line also means to set the line to a 1—since a 0 will enable the line
(active-low)—and we want to disable (de-assert) it. 4.2 Adder 4.2.1 Full Adder
To construct an adder for adding two n-bit binary numbers, X = xn-1 … x0 and Y = yn-1 … y0, we need to first
consider the addition of a single bit slice, xi with yi, together with the carry-in bit, ci, from the previous bit position
on the right. The result from this addition is a sum bit, si, and a carry-out bit, ci+1, for the next bit position. In other
words, si = xi + yi + ci, and ci+1 = 1 if there is a carry from the addition to the next bit on the left. Note that the +
operator in this equation is addition and not the logical OR. 102 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components For example, consider the following addition of the two 4-bit binary numbers, X = 1001 and Y = 0011. c2 c1 1
+ 0 0 01 11 1
100 1 0 1 The result of the addition is 1100. The addition is performed just like that for decimal numbers, except that there is a
carry whenever the sum is either a 2 or a 3 in decimals, since 2 is 10 in binary and 3 is 11. The most significant bit in
the 10 or the 11 is the carry-out bit. Looking at the bit slice that is highlighted in blue where x1 = 0, y1 = 1, and c1 =
1, the addition for this bit slice is x1 + y1 + c1 = 0 + 1 + 1 = 10. Therefore, the sum bit is s1 = 0, and the carry-out bit
is c2 = 1.
The circuit for the addition of a single bit slice is known as a full adder (FA), and its truth table is shown in
Figure 4.1(a). The derivation of the equations for si and ci+1 are shown in Figure 4.1(b). From these two equations,
we get the circuit for the full adder, as shown in Figure 4.1(c). Figure 4.1(d) shows the logic symbol for it. The
dataflow VHDL code for the full adder is shown in Figure 4.2.
xi
0
0
0
0
1
1
1
1 yi
0
0
1
1
0
0
1
1 ci
0
1
0
1
0
1
0
1 ci+1
0
0
0
1
0
1
1
1 si
0
1
1
0
1
0
0
1 si = xi'yi'ci + xi'yici' + xiyi'ci' + xiyici
= (xi'yi + xiyi')ci' + (xi'yi' + xiyi)ci
= (xi ⊕ yi)ci' + (xi ⊕ yi)'ci
= xi ⊕ yi ⊕ ci ci+1 = xi'yici + xiyi'ci + xiyici' + xiyici
= xiyi(ci' + ci) + ci(xi'yi + xiyi')
= xiyi + ci(xi ⊕ yi) (b) (a)
xi yi xi yi ci+1FA ci ci+1
ci si si (d) (c) Figure 4.1 Full adder: (a) truth table; (b) equations for si and ci+1; (c) circuit; (d) logic symbol.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY fa IS PORT (
Ci, Xi, Yi: IN STD_LOGIC;
Ci1, Si: OUT STD_LOGIC);
END fa; 103 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components ARCHITECTURE Dataflow OF fa IS
BEGIN
Ci1 <= (Xi AND Yi) OR (Ci AND (Xi XOR Yi));
Si <= Xi XOR Yi XOR Ci;
END Dataflow;
Figure 4.2 Dataflow VHDL code for a 1-bit full adder. 4.2.2 Ripple-carry Adder
The full adder is for adding two operands that are only one bit wide. To add two operands that are, say four bits
wide, we connect four full adders together in series. The resulting circuit (shown in Figure 4.3) is called a ripplecarry adder for adding two 4-bit operands.
Since a full adder adds the three bits, xi, yi and ci, together, we need to set the first carry-in bit, c0, to 0 in order
to perform the addition correctly. Moreover, the output signal, cout, is a 1 whenever there is an overflow in the
addition.
The structural VHDL code for the 4-bit ripple-carry adder is shown in Figure 4.4. Since we need to duplicate
the full adder component four times, we can use either the PORT MAP statement four times or by using the FORGENERATE statement as shown in the code to automatically generate the four components. The statement FOR k
IN 3 DOWNTO 0 GENERATE determines how many times to repeat the PORT MAP statement that is in the body
of the GENERATE statement and the values used for k. The vector signal Carryv is used to propagate the carry bit
from one FA to the next. x3 y3
cout FA3
s3 x2 y2
c3 FA2 x1 y1
c2 s2 FA1
s1 x0 y0
c1 FA0 c0 = 0 s0 Figure 4.3 Ripple-carry adder.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY Adder4 IS PORT (
A, B: IN STD_LOGIC_VECTOR(3 DOWNTO 0);
Cout: OUT STD_LOGIC;
SUM: OUT STD_LOGIC_VECTOR(3 DOWNTO 0));
END Adder4;
ARCHITECTURE Structural OF Adder4 IS
COMPONENT FA PORT (
ci, xi, yi: IN STD_LOGIC;
co, si: OUT STD_LOGIC);
END COMPONENT;
SIGNAL Carryv: STD_LOGIC_VECTOR(4 DOWNTO 0);
BEGIN
Carryv(0) <= '0';
Adder: FOR k IN 3 DOWNTO 0 GENERATE
FullAdder: FA PORT MAP (Carryv(k), A(k), B(k), Carryv(k+1), SUM(k)); 104 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components END GENERATE Adder;
Cout <= Carryv(4);
END Structural;
Figure 4.4 VHDL code for a 4-bit ripple-carry adder using a FOR-GENERATE statement. 4.2.3 * Carry-lookahead Adder
The ripple-carry adder is slow because the carry-in for each full adder is dependent on the carry-out signal from
the previous FA. So before FAi can output valid data, it must wait for FAi–1 to have valid data. In the carrylookahead adder, each bit slice eliminates this dependency on the previous carry-out signal and instead uses the
values of the two input operands, X and Y, directly to deduce the needed signals. This is possible from the following
observations regarding the carry-out signal. For each FAi, the carry-out signal, ci+1, is set to a 1 if either one of the
following two conditions is true:
xi = 1 and yi = 1
or
(xi = 1 or yi = 1) and ci = 1
In other words,
ci+1 = xiyi + ci(xi + yi) (4.1) At first glance, this carry-out equation looks completely different from the carry-out equation deduced in Figure
4.1(b). However, they are equivalent (see Problem 2.6(g)).
If we let
g i = xi yi
and
p i = xi + yi
then Equation (4.1) can be rewritten as
ci+1 = gi + pici (4.2) Using Equation (4.2) for ci+1, we can recursively expand it to get the carry-out equations for any bit slice, ci, that
is dependent only on the two input operands, X and Y, and the initial carry-in bit, c0. Using this technique, we get the
following carry-out equations for the first four bit slices
c1 = g 0 + p 0c 0 (4.3) c2 = g 1 + p 1c 1
= g1 + p1(g0 + p0c0)
= g1 + p1g0 + p1p0c0 (4.4) = g 2 + p 2c 2
= g2 + p2 (g1 + p1g0 + p1p0c0)
= g2 + p2 g1 + p2p1g0 + p2p1 p0c0 (4.5) = g 3 + p 3c 3
= g3 + p3(g2 + p2g1 + p2p1g0 + p2p1p0c0)
= g3 + p3g2 + p3p2g1 + p3p2p1g0 + p3p2p1p0c0 (4.6) c3 c4 Using Equations (4.3) to (4.6), we obtain the circuit for generating the carry-lookahead signals for c1 to c4, as
shown in Figure 4.5(a). Note that each equation is translated to a three-level combinational logic—one level for
generating the gi and pi, and two levels (for the sum-of-products format) for generating the ci expression. This carry- 105 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components lookahead circuit can be reduced even further because we want c0 to be a 0 when performing additions, and this 0
will cancel the rightmost product term in each equation.
The full adder for the carry-lookahead adder can also be made simpler since it is no longer required to generate
the carry-out signal for the next bit slice. In other words, the carry-in signal for the full adder now comes from the
new carry-lookahead circuit rather than from the carry-out signal of the previous bit slice. Thus, this full adder only
needs to generate the sumi signal. Figure 4.5(b) shows one bit slice of the carry-lookahead adder. For an n-bit carrylookahead adder, we use n bit slices. These n bit slices are not connected in series as with the ripple-carry adder,
otherwise, it defeats the purpose of having the more complicated carry-out circuit.
x3 y3 x2 y2 x1 y1 x0 y0
xi yi g3 p3 g2 p2 g1 p1 g0 p0 c0 x0...xi-1 y0...yi-1
Carrylookahead
Circuit
ci sumi
c4 c3 c2 c1
(b) (a) Figure 4.5 (a) Circuit for generating the carry-lookahead signals, c1 to c4; (b) one bit slice of the carry-lookahead
adder. 4.3 Two’s Complement Binary Numbers Before introducing subtraction circuits, we need to review how negative numbers are encoded using two’s
complement representation. Binary encoding of numbers can be interpreted as either signed or unsigned. Unsigned
numbers include only positive numbers and zero, whereas signed numbers include positive, negative, and zero. For
signed numbers, the most significant bit (MSB) tells whether the number is positive or negative. If the most
significant bit is a 1, then the number is negative; otherwise, the number is positive. The value of a positive signed
number is obtained exactly as for unsigned numbers described in Section 2.1. For example, the value for the positive
signed number 011010012 is just 1 × 26 + 1 × 25 + 1 × 23 + 1 × 20 = 105 in decimal.
However, to determine the value of a negative signed number, we need to perform a two-step process: 1) flip all
the 1 bits to 0’s and all the 0 bits to 1’s, and 2) add a 1 to the result obtained from step 1). The number obtained from
applying this two-step process is evaluated as an unsigned number for its value. The negative of this resulting value
is the value of the original negative signed number.
Example 4.1: Finding the value for a signed number
Given the 8-bit signed number 111010012, we know that it is a negative number because of the leading 1. To
find out the value of this negative number, we perform the two-step process as follows:
11101001
00010110
00010111 (original number)
(flip bits)
(add a 1 to the previous number) The value for the resulting number 00010111 is 1 × 24 + 1 × 22 + 1 × 21 + 1 × 20 = 23. Therefore, the value of the
original number 11101001 is negative 23 (–23).
♦ 106 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components Example 4.2: Finding the value for a signed number
To find the value for the 4-bit signed number 1000, we apply the two-step process to the number as follows:
1000
0111
1000 (original number)
(flip bits)
(add a 1 to the previous number) The resulting number 1000 is exactly the same as the original number! This, however, should not confuse us if
we follow exactly the instructions for the conversion process. We need to interpret the resulting number as an
unsigned number to determine the value. Interpreting the resulting number 1000 as an unsigned number gives us the
value of 8. Therefore, the original number, which is also 1000, is negative 8 (–8).
♦
Figure 4.6 shows the two’s complement numbers for four bits. The range goes from –8 to 7. In general, for an
n-bit two’s complement number, the range is from –2n-1 to 2n-1 – 1.
4-bit Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111 Two’s Complement
0
1
2
3
4
5
6
7
–8
–7
–6
–5
–4
–3
–2
–1 Figure 4.6 4-bit two’s complement numbers.
The nice thing about using two’s complement to represent negative numbers is that when we add a number with
the negative of the same number, the result is zero as expected. This is shown in the next example.
Example 4.3: Adding 4-bit signed numbers
Use 4-bit signed arithmetic to perform the following addition.
3
+ (–3)
0 =
=
= 0011
+ 1101
10000 The result 10000 has five bits. But since we are using 4-bit arithmetic (that is, the two operands are 4-bits wide)
the result must also be in 4-bits. The leading 1 in the result is, therefore, an overflow bit. By dropping the leading
one, the remaining result 0000 is the correct answer for the problem. Although this addition resulted in an overflow
bit, but by dropping this extra bit, we obtained the correct answer.
♦
Example 4.4: Adding 4-bit signed numbers
Use 4-bit signed arithmetic to perform the following addition. 107 Digital Logic and Microprocessor Design with VHDL 6
+3
9 =
=
≠ Chapter 4 - Standard Combinational Components 0110
+ 0011
1001 The result 1001 is a 9 if we interpret it as an unsigned number. However, since we are using signed numbers,
we need to interpret the result as a signed number. Interpreting 1001 as a signed number gives – 7, which of course
is incorrect. The problem here is that the range for a 4-bit signed number is from – 8 to + 7, and + 9 is outside of this
range.
♦
Although the addition in this example did not resulted in an overflow bit, but the final answer is incorrect. In
order to correct this problem, we need to add (at least) one extra bit by sign extending the number. The corrected
arithmetic is shown in Example 4.5.
Example 4.5: Adding 5-bit signed numbers
Use 5-bit signed arithmetic to perform the following addition.
6
+3
9 =
=
= 00110
+ 00011
01001
♦ The result 01001, when interpreted as a signed number, is 9. To extend a signed number, we need to add leading 0’s or 1’s depending on whether the original most
significant bit is a 0 or a 1. If the most significant bit is a 0, we sign extend the number by adding leading 0’s. If the
most significant bit is a 1, we sign extend the number by adding leading 1’s. By performing this sign extension, the
value of the number is not changed, as shown in Example 4.6.
Example 4.6: Performing sign extensions
Sign extend the numbers 10010 and 0101 to 8-bits wide.
For the number 10010, since the most significant bit is a 1, therefore, we need to add leading 1’s to make the
number 8-bits long. The resulting number is 11110010. For the number 0101, since the most significant bit is a 0,
therefore, we need to add leading 0’s to make the number 8-bits long. The resulting number is 00000101. The
following shows that the two resulting numbers have the same value as the two original numbers. Since the first
number is negative (because of the leading 1 bit) we need to perform the two-step process to evaluate its value. The
second number is positive, so we can evaluate its value directly. Flip bits
Add 1
Value Original
Number
10010
01101
01110
– 14 Sign
Extended
11110010
00001101
00001110
– 14 Original
Number
0101 Sign
Extended
00000101 5 5
♦ 4.4 Subtractor We can construct a one-bit subtractor circuit similar to the method used for constructing the full adder.
However, instead of the sum bit, si, for the addition, we have a difference bit, di, for the subtraction, and instead of
having carry-in and carry-out signals, we have borrow-in (bi) and borrow-out (bi+1) signals. So, when we subtract the
ith bit of the two operands, xi and yi, we get the difference di = xi − yi. If, however, the previous bit on the right has to
borrow from this ith bit, then input bi will be set to a 1, and the equation for the difference will be di = xi − bi − yi. On
the other hand, if the ith bit has to borrow from the next bit on the left for the subtraction, then the output bi+1 will be
set to a 1. The value borrowed is a 2, and so the resulting equation for the difference will be di = xi − bi + 2bi+1 − yi.
Note that the symbols + and − used in this equation are for addition and subtraction, and not for logical operations.
The term 2bi+1 is “2 multiply by bi+1.” Since bi+1 is a 1 when we have to borrow, and we borrow a 2 each time, 108 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components therefore, the equation just adds a 2 when there is a borrow. When there is no borrow, bi+1 is 0, and so the term bi+1
cancels out to 0.
For example, consider the following subtraction of the two 4-bit binary numbers, X = 0100 and Y = 0011:
bi+1
1 0 bi
1 0 1 0 0 0 1 1 0 0 0 1 Consider the bit position that is highlighted in blue. Since the subtraction for the previous bit on the right has to
borrow, therefore, bi is a 1. Moreover, bi+1 is also a 1 because the current bit has to borrow from the next bit on the
left. When it borrows, it gets a 2. Therefore, di = xi − bi + 2bi+1 − yi = 0 – 1 + 2(1) – 1 = 0.
The truth table for the 1-bit subtractor is shown in Figure 4.7(a), from which the equations for di and bi+1, as
shown in Figure 4.7(b), are derived. From these two equations, we get the circuit for the subtractor as shown in
Figure 4.7(c). Figure 4.7(d) shows the logic symbol for the subtractor.
Building a subtractor circuit for subtracting an n-bit operand can be done by daisy-chaining n 1-bit subtractor
circuits together, similar to the adder circuit shown in Figure 4.3. However, there is a much better subtractor circuit,
as shown in the next section.
xi
0
0
0
0
1
1
1
1 yi
0
0
1
1
0
0
1
1 bi
0
1
0
1
0
1
0
1 bi+1
0
1
1
1
0
0
0
1 di
0
1
1
0
1
0
0
1 di = xi'yi'bi + xi'yibi' + xiyi'bi' + xiyibi
= (xi'yi + xiyi' )bi' + (xi'yi' + xiyi)bi
= (xi ⊕ yi)bi' + (xi ⊕ yi)'bi
= xi ⊕ yi ⊕ b i bi+1 = xi'yi'bi + xi'yibi' + xi'yibi + xiyibi
= xi'bi(yi' + yi) + xi'yi(bi' + bi) + yibi(xi' + xi)
= xi'bi + xi'yi + yibi (a) (b)
xi yi bi
bi+1 xi yi bi+1FS bi
di di (d) (c) Figure 4.7 1-bit subtractor: (a) truth table; (b) equations for di and bi+1; (c) circuit; (d) logic symbol. 109 Digital Logic and Microprocessor Design with VHDL 4.5 Chapter 4 - Standard Combinational Components Adder-Subtractor Combination It turns out that instead of having to build a separate adder and subtractor units, we can modify the ripple-carry
adder (or the carry-lookahead adder) slightly to perform both operations. The modified circuit performs subtraction
by adding the negated value of the second operand. In other words, instead of performing the subtraction A – B, the
addition operation A + (– B) is performed.
Recall that in two’s complement representation, to negate a value involves inverting all 0’s to 1’s and 1’s to 0’s,
and then adding a 1. Hence, we need to modify the adder circuit so that we can selectively do either one of two
things: 1) flip the bits of the B operand and then add an extra 1 for the subtraction operation, or 2) not flip the bits
and not add an extra 1 for the addition operation.
For this adder-subtractor combination circuit, in addition to the two input operands A and B, a select signal, s, is
needed to select which operation to perform. The assignment of the two operations to the select signal s is shown in
Figure 4.8(a). When s = 0, we want to perform an addition, and when s = 1, we want to perform a subtraction. When
s = 0, B does not need to be modified, and like the adder circuit from Section 4.2.2, the initial carry-in signal c0
needs to be set to a 0. On the other hand, when s = 1, we need to invert the bits in B and add a 1. The addition of a 1
is accomplished by setting the initial carry-in signal c0 to a 1. Two circuits are needed for handling the above
situations: one for inverting the bits in B and one for setting c0. Both of these circuits are dependent on s.
The truth table for these two circuits is shown in Figure 4.8(b). The input variable bi is the ith bit of the B
operand. The output variable yi is the output from the circuit that either inverts or does not invert the bits in B. From
this truth table, we can conclude that the circuit for yi is just a 2-input XOR gate, while the circuit for c0 is just a
direct connection from s. Putting everything together, we obtain the adder-subtractor combination circuit (for four
bits) as shown in Figure 4.8(c). The logic symbol for the circuit is shown in Figure 4.8(d). s
0
1 Function
Add
Subtract s
0
0
1
1 Operation
F=A+B
F = A + B' + 1 bi
0
1
0
1 (a)
a3 b3 yi
0
1
1
0 c0
0
0
1
1 (b) a2 b2 a1 b1 a0 b0 s 4
y3
Unsigned_
Overflow cout FA y2
c3 FA y1
c2 FA sA y0
c1 FA 4 c0 Signed_
Overflow B Unsigned_ AdderOverflow
Signed_ Subtractor
Overflow 4
f3 f2 f1 F f0 (c) (d) Figure 4.8 Adder-subtractor combination: (a) operation table; (b) truth table for yi and c0; (c) circuit; (d) logic
symbol.
Notice the adder-subtractor circuit in Figure 4.8(c) has two different overflow signals, Unsigned_Overflow and
Signed_Overflow. This is because the circuit can deal with both signed and unsigned numbers. When working with
unsigned numbers only, the output signal Unsigned_Overflow is sufficient to determine whether there is an overflow
or not. However, for signed numbers, we need to perform the XOR of Unsigned_Overflow with c3, producing the
Signed_Overflow signal in order to determine whether there is an overflow or not. 110 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components For example, the valid range for a 4-bit signed number goes from –23 to 23–1 (i.e., from –8 to 7). Adding the
two signed numbers, 4 + 5 = 9 should result in a signed number overflow since 9 is outside the range. However, the
valid range for a 4-bit unsigned number goes from 0 to 24–1 (i.e., 0 to 15). If we treat the two numbers 4 and 5 as
unsigned numbers, then the result of adding these two unsigned numbers, 9, is inside the range. So when adding the
two numbers 4 and 5, the Unsigned_Overflow signal should be de-asserted, while the Signed_Overflow signal
should be asserted. Performing the addition of 4 + 5 in binary as shown here: c3
0
Unsigned
Overflow Signed
Overflow 1 0 0 + 01 1 0 1 0 1 0 1 0 0 XOR 1 = 1 we get 0100 + 0101 = 1001, which produces a 0 for the Unsigned_Overflow signal. However, the addition produces
a 1 for c3, and XORing these two values, 0 for Unsigned_Overflow and 1 for c3, results in a 1 for the
Signed_Overflow signal.
In another example, adding the two 4-bit signed numbers, –4 + (–3) = –7 should not result in a signed overflow.
Performing the arithmetic in binary, –4 = 1100 and –3 = 1101, as shown here: c3
1
Unsigned
Overflow Signed
Overflow 1 0 0 + 11 1 0 1 1 1 0 1 0 1 XOR 1 = 0 we get 1100 + 1101 = 11001, which produces a 1 for both Unsigned_Overflow and c3. XORing these two values
together gives a 0 for the Signed_Overflow signal. On the other hand, if we treat the two binary numbers, 1100 and
1101, as unsigned numbers, then we are adding 12 + 13 = 25. 25 is outside the unsigned number range, and so the
Unsigned_Overflow signal should be asserted.
The behavioral VHDL code for the 4-bit adder-subtractor combination circuit is shown in Figure 4.9. The
keyword declares a read-only constant identifier, n, of type INTEGER having a default value of 4. This
constant identifier is then used in the declaration of the STD_LOGIC_VECTOR size for the three vectors: A, B, and F.
GENERIC The Unsigned_Overflow bit is obtained by performing the addition or subtraction operation using n + 1 bits.
The two operands are zero extended using the & symbol for concatenation before the operation is performed. The
result of the operation is stored in the n + 1 bit vector, result. The most significant bit of this vector, result(n), is the
Unsigned_Overflow bit.
To get the Signed_Overflow bit, we need to XOR the Unsigned_Overflow bit with the carry bit, c3, from the
second-to-last bit slice. The c3 bit is obtained just like how the Unsigned_Overflow bit is obtained, except that the
operation is performed on only the first n – 1 bits of the two operands. The vector c3 of length n is used for storing
the result of the operation. The Signed_Overflow signal is the XOR of signed_result(n) with c3(n–1).
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL; 111 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components ENTITY AddSub IS
GENERIC(n: INTEGER :=4); -- default number of bits = 4
PORT(S: IN STD_LOGIC; -- select subtract signal
A: IN STD_LOGIC_VECTOR(n-1 DOWNTO 0);
B: IN STD_LOGIC_VECTOR(n-1 DOWNTO 0);
F: OUT STD_LOGIC_VECTOR(n-1 DOWNTO 0);
unsigned_overflow: OUT STD_LOGIC;
signed_overflow: OUT STD_LOGIC);
END AddSub;
ARCHITECTURE Behavioral OF AddSub IS
-- temporary result for extracting the unsigned overflow bit
SIGNAL result: STD_LOGIC_VECTOR(n DOWNTO 0);
-- temporary result for extracting the c3 bit
SIGNAL c3: STD_LOGIC_VECTOR(n-1 DOWNTO 0);
BEGIN
PROCESS(S, A, B)
BEGIN
IF (S = '0') THEN
-- addition
-- the two operands are zero extended one extra bit before adding
-- the & is for string concatination
result <= ('0' & A) + ('0' & B);
c3 <= ('0' & A(n-2 DOWNTO 0)) + ('0' & B(n-2 DOWNTO 0));
F <= result(n-1 DOWNTO 0);
-- extract the n-bit result
unsigned_overflow <= result(n);
-- get the unsigned overflow bit
signed_overflow <= result(n) XOR c3(n-1); -- get signed overflow bit
ELSE
-- subtraction
-- the two operands are zero extended one extra bit before subtracting
-- the & is for string concatination
result <= ('0' & A) - ('0' & B);
c3 <= ('0' & A(n-2 DOWNTO 0)) - ('0' & B(n-2 DOWNTO 0));
F <= result(n-1 DOWNTO 0);
-- extract the n-bit result
unsigned_overflow <= result(n);
-- get the unsigned overflow bit
signed_overflow <= result(n) XOR c3(n-1); -- get signed overflow bit
END IF;
END PROCESS;
END Behavioral;
Figure 4.9 Behavioral VHDL code for a 4-bit adder-subtractor combination component. 4.6 Arithmetic Logic Unit The arithmetic logic unit (ALU) is one of the main components inside a microprocessor. It is responsible for
performing arithmetic and logic operations, such as addition, subtraction, logical AND, and logical OR. The ALU,
however, is not used to perform multiplications or divisions. It turns out that, in constructing the circuit for the ALU,
we can use the same idea as for constructing the adder-subtractor combination circuit, as discussed in the previous
section. Again, we will use the ripple-carry adder as the building block and then insert some combinational logic
circuitry in front of the two input operands to each full adder. This way, the primary inputs will be modified
accordingly, depending on the operations being performed before being passed to the full adder. The general, overall
circuit for a 4-bit ALU is shown in Figure 4.10(a), and its logic symbol in (b).
As we can see in the figure, the two combinational circuits in front of the full adder (FA) are labeled LE and
AE. The logic extender (LE) is for manipulating all logical operations; whereas, the arithmetic extender (AE) is for
manipulating all arithmetic operations. The LE performs the actual logical operations on the two primary operands,
ai and bi, before passing the result to the first operand, xi, of the FA. On the other hand, the AE only modifies the
second operand, bi, and passes it to the second operand, yi, of the FA where the actual arithmetic operation is
performed. 112 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components a3 b3 a2 b2 a1 b1 a0 b0 LE AE LE AE LE AE LE AE s2
s1
s0 x3
Unsigned_
Overflow c4 y3
FA x2
c3 y2 x1 y1 c2 FA FA x0
c1 y0
FA c0 CE Signed_
Overflow
f3 f2 f1 f0 (a)
4
3 4 SA B Unsigned_
Overflow
Signed_
Overflow ALU 4
F
(b)
Figure 4.10 4-bit ALU: (a) circuit; (b) logic symbol.
We saw from the adder-subtractor circuit that, to perform additions and subtractions, we only need to modify yi
(the second operand to the FA) so that all operations can be done with additions. Thus, the AE only takes the second
operand of the primary input, bi, as its input and modifies the value depending on the operation being performed. Its
output is yi, and it is connected to the second operand input of the FA. As in the adder-subtractor circuit, the addition
is performed in the FA. When arithmetic operations are being performed, the LE must pass the first operand
unchanged from the primary input ai to the output xi for the FA.
Unlike the AE (where it only modifies the operand) the LE performs the actual logical operations. Thus, for
example, if we want to perform the operation A OR B, the LE for each bit slice will take the corresponding bits, ai
and bi, and OR them together. Hence, one bit from both operands, ai and bi, are inputs to the LE. The output of the
LE is passed to the first operand, xi, of the FA. Since this value is already the result of the logical operation, we do
not want the FA to modify it but to simply pass it on to the primary output, fi. This is accomplished by setting both
the second operand, yi, of the FA, and c0 to 0 since adding a 0 will not change the resulting value.
The combinational circuit labeled CE (for carry extender) is for modifying the primary carry-in signal, c0, so
that arithmetic operations are performed correctly. Logical operations do not use the carry signal, so c0 is set to 0 for
all logical operations. 113 Digital Logic and Microprocessor Design with VHDL s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1 Operation Name
Pass
AND
OR
NOT
Addition
Subtraction
Increment
Decrement Chapter 4 - Standard Combinational Components Operation
Pass A to output
A AND B
A OR B
A'
A+B
A–B
A+1
A–1 xi (LE)
ai
ai AND bi
ai OR bi
ai'
ai
ai
ai
ai yi (AE)
0
0
0
0
bi
bi'
0
1 c0 (CE)
0
0
0
0
0
1
1
0 (a) s2
0
0
0
0
1 s1
0
0
1
1
× s0
0
1
0
1
× (b) xi
ai
ai bi
ai + bi
ai'
ai s2
0
1
1
1
1
1
1
1
1 s1
×
0
0
0
0
1
1
1
1 s0
×
0
0
1
1
0
0
1
1
(c) bi
×
0
1
0
1
0
1
0
1 yi
0
0
1
1
0
0
0
1
1 s2
0
1
1
1
1 s1
×
0
0
1
1 s0
×
0
1
0
1 c0
0
0
1
1
0 (d) Figure 4.11 ALU operations: (a) function table; (b) LE truth table; (c) AE truth table; (d) CE truth table.
In the circuit shown in Figure 4.10, three select lines, s2, s1, and s0, are used to select the operations of the ALU.
With these three select lines, the ALU circuit can implement up to eight different operations. Suppose that the
operations that we want to implement in our ALU are as defined in Figure 4.11(a). The xi column shows the values
that the LE must generate for the different operations. The yi column shows the values that the AE must generate.
The c0 column shows the carry signals that the CE must generate.
For example, for the pass-through operation, the value of ai is passed through without any modifications to xi.
For the AND operation, xi gets the result of ai AND bi. As mentioned before, both yi and c0 are set to 0 for all of the
logical operations, because we do not want the FA to change the results. The FA is used only to pass the results from
the LE straight through to the output F. For the subtraction operation, instead of subtracting B, we want to add –B.
Changing B to –B in two’s complement format requires flipping the bits of B and then adding a 1. Thus, yi gets the
inverse of bi, and the 1 is added through the carry-in c0. To increment A, we set yi to all 0’s, and add the 1 through
the carry-in c0. To decrement A, we add a –1 instead. Negative one in two’s complement format is a bit string with
all 1’s. Hence, we set yi to all 1’s and the carry-in c0 to 0. For all the arithmetic operations, we need the first operand,
A, unchanged for the FA. Thus, xi gets the value of ai for all arithmetic operations.
Figure 4.11(b), (c) and (d) show the truth tables for the LE, AE, and CE respectively. The LE circuit is derived
from the xi column of Figure 4.11(b); the AE circuit is derived from the yi column of Figure 4.11(c); and the CE
circuit is derived from the c0 column of Figure 4.11(d). Notice that xi is dependent on five variables, s2, s1, s0, ai, and
bi; whereas yi is dependent on only four variables, s2, s1, s0, and bi; and c0 is dependent on only the three select lines,
s2, s1, and s0. The K-maps, equations, and schematics for these three circuits are shown in Figure 4.12.
The behavioral VHDL code for the ALU is shown in Figure 4.13, and a sample simulation trace for all the
operations using the two inputs 5 and 3 is shown in Figure 4.14. 114 Digital Logic and Microprocessor Design with VHDL xi
s1s0 aibi s2 = 0
01 s2 = 1 11 10 00 1 01 01 1 1 11 00 1 00 10 1 1 1 1 s2ai 1 1 s2's1s0ai' ai 11 1 1 1 1 1 s2's1ai'bi bi s0'ai 1 10 Chapter 4 - Standard Combinational Components s2
s1
s0 s1'aibi LE
xi xi = s2ai + s0'ai + s1'aibi + s2's1ai'bi + s2's1s0ai'
= s2ai + s0'ai + s1'aibi + s2's1ai'(bi + s0)
(a)
yi
s2s1 s0bi 01 00 11 bi 10 00 s2
s1
s0 s2s1s0 01
1 11 1
s2s0bi' 1 10 1 s2s1's0'bi AE yi yi = s2s1s0 + s2s0bi' + s2s1's0'bi
= s2s0(s1 + bi') + s2s1's0'bi
(b)
c0
s2 s1s0 00 01 11 s0 s1 s2 10 0
1 CE
1 1
c0 c0 = s2s1's0 + s2s1s0'
= s2(s1 ⊕ s0)
(c)
Figure 4.12 K-maps, equations, and schematics for: (a) LE; (b) AE; and (c) CE. 115 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
-- The following package is needed so that the STD_LOGIC_VECTOR signals
-- A and B can be used in unsigned arithmetic operations.
USE IEEE.STD_LOGIC_UNSIGNED.ALL;
ENTITY alu IS PORT (
S: IN STD_LOGIC_VECTOR(2 DOWNTO 0);
-- select for operations
A, B: IN STD_LOGIC_VECTOR(3 DOWNTO 0); -- input operands
F: OUT STD_LOGIC_VECTOR(3 DOWNTO 0)); -- output
END alu;
ARCHITECTURE Behavior OF alu IS
BEGIN
PROCESS(S, A, B)
BEGIN
CASE S IS
WHEN "000" => -- pass A through
F <= A;
WHEN "001" => -- AND
F <= A AND B;
WHEN "010" => -- OR
F <= A OR B;
WHEN "011" => -- NOT A
F <= NOT A;
WHEN "100" => -- add
F <= A + B;
WHEN "101" => -- subtract
F <= A - B;
WHEN "110" => -- increment
F <= A + 1;
WHEN OTHERS => -- decrement
F <= A - 1;
END CASE;
END PROCESS;
END Behavior;
Figure 4.13 Behavioral VHDL code for an ALU. Pass A AND OR NOT A Add Subtract Increment Decrement Figure 4.14 Sample simulation trace with the two input operands 5 and 3 for all of the eight operations. 116 Digital Logic and Microprocessor Design with VHDL 4.7 Chapter 4 - Standard Combinational Components Decoder A decoder, also known as a demultiplexer, asserts one out of n output lines, depending on the value of an mbit binary input data. In general, an m-to-n decoder has m input lines, Am-1, …, A0, and n output lines, Yn-1, …, Y0,
where n = 2m. In addition, it has an enable line, E, for enabling the decoder. When the decoder is disabled with E set
to 0, all the output lines are de-asserted. When the decoder is enabled, then the output line whose index is equal to
the value of the input binary data is asserted. For example, for a 3-to-8 decoder, if the input address is 101, then the
output line Y5 is asserted (set to 1 for active-high) while the rest of the output lines are de-asserted (set to 0 for
active-high).
A decoder is used in a system having multiple components, and we want only one component to be selected or
enabled at any one time. For example, in a large memory system with multiple memory chips, only one memory
chip is enabled at a time. One output line from the decoder is connected to the enable input on each memory chip.
Thus, an address presented to the decoder will enable that corresponding memory chip. The truth table, circuit, and
logic symbol for a 3-to-8 decoder are shown in Figure 4.15.
A larger size decoder can be implemented using several smaller decoders. For example, Figure 4.16 uses seven
1-to-2 decoders to implement a 3-to-8 decoder. The correct operation of this circuit is left as an exercise for the
reader.
The behavioral VHDL code for the 3-to-8 decoder is shown in Figure 4.17.
E
0
1
1
1
1
1
1
1
1 A2
×
0
0
0
0
1
1
1
1 A1
×
0
0
1
1
0
0
1
1 A0
×
0
1
0
1
0
1
0
1 Y7
0
0
0
0
0
0
0
0
1 Y6
0
0
0
0
0
0
0
1
0 Y5
0
0
0
0
0
0
1
0
0 Y4
0
0
0
0
0
1
0
0
0 Y3
0
0
0
0
1
0
0
0
0 Y2
0
0
0
1
0
0
0
0
0 Y1
0
0
1
0
0
0
0
0
0 Y0
0
1
0
0
0
0
0
0
0 (a)
E
A2
A1
A0 A2 A1 A0 E
Y7 Y6 Y5 Y4 Y3 Y2 Y1 Y0 Y7 Y6 Y5 Y4 Y3 Y2 Y1 Y0
(b) (c) Figure 4.15 A 3-to-8 decoder: (a) truth table; (b) circuit; (c) logic symbol. 117 Digital Logic and Microprocessor Design with VHDL E Chapter 4 - Standard Combinational Components A2
E
1 A1 A0 0 E
1 E
1 0 E
1 E
1 0 Y7 Y6 0 Y5 Y4 0 E 1 0 Y3 Y2 E 1 Y1 Y0 Figure 4.16 A 3-to-8 decoder implemented with seven 1-to-2 decoders
-- A 3-to-8 decoder
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY Decoder IS PORT(
E: IN STD_LOGIC;
-- enable
A: IN STD_LOGIC_VECTOR(2 DOWNTO 0); -- 3 bit address
Y: OUT STD_LOGIC_VECTOR(7 DOWNTO 0)); -- data bus output
END Decoder;
ARCHITECTURE Behavioral OF Decoder IS
BEGIN
PROCESS (E, A)
BEGIN
IF (E = '0') THEN
-- disabled
Y <= (OTHERS => '0');
-- 8-bit vector of 0
ELSE
CASE A IS
-- enabled
WHEN "000" => Y <= "00000001";
WHEN "001" => Y <= "00000010";
WHEN "010" => Y <= "00000100";
WHEN "011" => Y <= "00001000";
WHEN "100" => Y <= "00010000";
WHEN "101" => Y <= "00100000";
WHEN "110" => Y <= "01000000";
WHEN "111" => Y <= "10000000";
WHEN OTHERS => NULL;
END CASE;
END IF;
END PROCESS;
END Behavioral;
Figure 4.17 Behavioral VHDL code for a 3-to-8 decoder. 118 0 Digital Logic and Microprocessor Design with VHDL 4.8 Chapter 4 - Standard Combinational Components Encoder An encoder is almost like the inverse of a decoder where it encodes a 2n-bit input data into an n-bit code. The
encoder has 2n input lines and n output lines, as shown by the logic symbol in Figure 4.18(c) for n = 3. The
operation of the encoder is such that exactly one of the input lines should have a 1 while the remaining input lines
should have 0’s. The output is the binary value of the index of the input line that has the 1. The truth table for an 8to-3 encoder is shown in Figure 4.18(a). For example, when input I3 is a 1, the three output bits Y2, Y1, and Y0, are set
to 011, which is the binary number for the index 3. Entries having multiple 1’s in the truth table inputs are ignored,
since we are assuming that only one input line can be a 1.
Looking at the three output columns in the truth table, we obtain the three equations shown in Figure 4.18(b),
and the resulting circuit in (c). The logic symbol is shown in Figure 4.18(d).
Encoders are used to reduce the number of bits needed to represent some given data either in data storage or in
data transmission. Encoders are also used in a system with 2n input devices, each of which may need to request for
service. One input line is connected to one input device. The input device requesting for service will assert the input
line that is connected to it. The corresponding n-bit output value will indicate to the system which of the 2n devices
is requesting for service. For example, if device 5 requests for service, it will assert the I5 input line. The system will
know that device 5 is requesting for service, since the output will be 101 = 5. However, this only works correctly if
it is guaranteed that only one of the 2n devices will request for service at any one time.
If two or more devices request for service at the same time, then the output will be incorrect. For example, if
devices 1 and 4 of the 8-to-3 encoder request for service at the same time, then the output will also be 101, because
I4 will assert the Y2 signal, and I1 will assert the Y0 signal. To resolve this problem, a priority is assigned to each of
the input lines so that when multiple requests are made, the encoder outputs the index value of the input line with the
highest priority. This modified encoder is known as a priority encoder.
I7
0
0
0
0
0
0
0
1 I6
0
0
0
0
0
0
1
0 I5
0
0
0
0
0
1
0
0 I4
0
0
0
0
1
0
0
0 I3
0
0
0
1
0
0
0
0 I2
0
0
1
0
0
0
0
0 I1
0
1
0
0
0
0
0
0 I0
1
0
0
0
0
0
0
0 Y2
0
0
0
0
1
1
1
1 Y1
0
0
1
1
0
0
1
1 Y0
0
1
0
1
0
1
0
1 Y0 = I1 + I3 + I5 + I7
Y1 = I2 + I3 + I6 + I7
Y2 = I4 + I5 + I6 + I7 (b) (a)
I0
I1 Y0 I2
I3 I7 I6 I5 I4 I3 I2 I1 I0 Y1 I4
I5 Y2 Y2 Y1 I6
I7 (d) (c)
Figure 4.18 An 8-to-3 encoder: (a) truth table; (b) equations; (c) circuit; (d) logic symbol. 119 Y0 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components 4.8.1 * Priority Encoder
The truth table for an active-high 8-to-3 priority encoder is shown in Figure 4.19. The table assumes that input
I7 has the highest priority, and I0 has the lowest priority. For example, if the highest priority input asserted is I3, then
it doesn’t matter whether the lower priority input lines, I2, I1 and I0, are asserted or not; the output will be for that of
I3, which is 011. Since it is possible that no inputs are asserted, there is an extra output, Z, that is needed to
differentiate between when no inputs are asserted and when one or more inputs are asserted. Z is set to a 1 when one
or more inputs are asserted; otherwise, Z is set to 0. When Z is 0, all of the Y outputs are meaningless.
I7
0
0
0
0
0
0
0
0
1 I6
0
0
0
0
0
0
0
1
× I5
0
0
0
0
0
0
1
×
× I4
0
0
0
0
0
1
×
×
× I3
0
0
0
0
1
×
×
×
× I2
0
0
0
1
×
×
×
×
× I1
0
0
1
×
×
×
×
×
× I0
0
1
×
×
×
×
×
×
× Y2
×
0
0
0
0
1
1
1
1 Y1
×
0
0
1
1
0
0
1
1 Y0
×
0
1
0
1
0
1
0
1 Z
0
1
1
1
1
1
1
1
1 Figure 4.19 An 8-to-3 priority encoder truth table.
An easy way to derive the equations for the 8-to-3 priority encoder is to define a set of eight intermediate
variables, v0, …, v7, such that vk is a 1 if Ik is the highest priority 1 input. Thus, the equations for v0 to v7 are:
v0 = I7' I6' I5' I4' I3' I2' I1' I0
v1 = I7' I6' I5' I4' I3' I2' I1
v2 = I7' I6' I5' I4' I3' I2
v3 = I7' I6' I5' I4' I3
v4 = I7' I6' I5' I4
v5 = I7' I6' I5
v 6 = I 7' I 6
v7 = I 7
Using these eight intermediate variables, the final equations for the priority encoder are similar to the ones for
the regular encoder, namely:
Y0 = v1 + v3 + v5 + v7
Y1 = v2 + v3 + v6 + v7
Y2 = v4 + v5 + v6 + v7
Finally, the equation for Z is simply
Z = I7 + I6 + I5 + I4 + I3 + I2 + I1 + I0 4.9 Multiplexer The multiplexer, or MUX for short, allows the selection of one input signal among n signals, where n > 1, and
is a power of two. Select lines connected to the multiplexer determine which input signal is selected and passed to
the output of the multiplexer. In general, an n-to-1 multiplexer has n data input lines, m select lines where m = log2
n, i.e. 2m = n, and one output line. For a 2-to-1 multiplexer, there is one select line, s, to select between the two
inputs, d0 and d1. When s = 0, the input line d0 is selected, and the data present on d0 is passed to the output y. When
s = 1, the input line d1 is selected and the data on d1 is passed to y. The truth table, equation, circuit, and logic
symbol for a 2-to-1 multiplexer are shown in Figure 4.20. 120 Digital Logic and Microprocessor Design with VHDL s
0
0
0
0
1
1
1
1 d1
0
0
1
1
0
0
1
1 d0
0
1
0
1
0
1
0
1 Chapter 4 - Standard Combinational Components y
0
1
0
1
0
0
1
1 y = s'd1'd0 + s'd1d0 + sd1d0' + sd1d0
= s'd0(d1' + d1) + sd1(d0' + d0)
= s'd0 + sd1 (b) (a)
d0
s s y d1 d0
y d1 (c) (b) Figure 4.20 A 2-to-1 multiplexer: (a) truth table; (b) equation; (c) circuit; (d) logic symbol.
Constructing a larger size multiplexer, such as the 8-to-1 multiplexer, can be done similarly. In addition to
having the eight data input lines, d0 to d7, the 8-to-1 multiplexer has three (23 = 8) select lines, s0, s1, and s2.
Depending on the value of the three select lines, one of the eight input lines will be selected and the data on that
input line will be passed to the output. For example, if the value of the select lines is 101, then the input line d5 is
selected, and the data that is present on d5 will be passed to the output.
The truth table, circuit, and logic symbol for the 8-to-1 multiplexer are shown in Figure 4.21. The truth table is
written in a slightly different format. Instead of including the d’s in the input columns and enumerating all 211 =
2048 rows (the eleven variables come from the eight d’s and the three s’s), the d’s are written in the entry under the
output column. For example, when the select line value is 101, the entry under the output column is d5, which means
that y takes on the value of the input line d5.
To understand the circuit in Figure 4.21(b), notice that each AND gate acts as a switch and is turned on by one
combination of the three select lines. When a particular AND gate is turned on, the data at the corresponding d input
is passed through that AND gate. The outputs of the remaining AND gates are all 0’s.
d7 d6 d5 d4 d3 d2 d1 d0 s2 s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1
(a) y
d0
d1
d2
d3
d4
d5
d6
d7 s1
s0 d7 d6 d5 d4 d3 d2 d1 d0
s2
s1
s0
y y (b) (c) Figure 4.21 An 8-to-1 multiplexer: (a) truth table; (b) circuit; (c) logic symbol.
Instead of using 4-input AND gates (where three of its inputs are used by the three select lines to turn it on) we
can use 2-input AND gates, as shown in Figure 4.22(a). This way the AND gate is turned on with just one line. The 121 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components eight 2-input AND gates can be turned on individually from the eight outputs of a 3-to-8 decoder. Recall from
Section 4.7 that the decoder asserts only one output line at any time.
Larger multiplexers can also be constructed from smaller multiplexers. For example, an 8-to-1 multiplexer can
be constructed using seven 2-to-1 multiplexers as shown in Figure 4.22(b). The four top-level 2-to-1 multiplexers
provide the eight data inputs and all are switched by the same least significant select line s0. This top level selects
one from each group of two data inputs. The middle level then groups the four outputs from the top level again into
groups of two and selects one from each group using the select line s1. Finally, the multiplexer at the bottom level
uses the most significant select line s2 to select one of the two outputs from the middle level multiplexers.
The VHDL code for an 8-bit wide 4-to-1 multiplexer is shown in Figure 4.23. Two different implementations of
the same multiplexer are shown. Figure 4.23(a) shows the architecture code written at the behavioral level, since it
uses a PROCESS statement. Inside the PROCESS block, a CASE statement is used to select between the four choices for
S. Figure 4.23(b) shows a dataflow level architecture code using a concurrent selected signal assignment statement
using the keyword WITH … SELECT. In the first choice, if S is equal to 00, then the value D0 is assigned to Y. If S
does not match any one of the four choices, 00, 01, 10, and 11, then the WHEN OTHERS clause is selected. The syntax
(OTHERS => 'U') means to fill the entire vector with the value “U”. s0
s1
s2 Decoder d7 d6 d5 d4 d3 d2 d1 d0 0
1
2
3
4
5
6
7 d7 d6
1
s d5 d4 0 1
s y d3 d2 0 1
s y d1 d0 0 1
s y 0
y s0
1
s 0 1
s y 0
y s1
1
s s2 0
y y y (a) (b) Figure 4.22 An 8-to-1 multiplexer implemented using: (a) a 3-to-8 decoder; (b) seven 2-to-1 multiplexers.
-- A 4-to-1 8-bit wide multiplexer
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY Multiplexer IS
PORT(S: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
-- select lines
D0, D1, D2, D3: IN STD_LOGIC_VECTOR(7 DOWNTO 0); -- data bus input
Y: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
-- data bus output
END Multiplexer;
-- Behavioral level code
ARCHITECTURE Behavioral OF Multiplexer IS
BEGIN
PROCESS (S,D0,D1,D2,D3)
BEGIN
CASE S IS
WHEN "00" => Y <= D0;
WHEN "01" => Y <= D1;
WHEN "10" => Y <= D2;
WHEN "11" => Y <= D3;
WHEN OTHERS => Y <= (OTHERS => 'U'); 122 -- 8-bit vector of U Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components END CASE;
END PROCESS;
END Behavioral;
(a)
-- Dataflow level code
ARCHITECTURE Dataflow OF Multiplexer IS
BEGIN
WITH S SELECT Y <=
D0 WHEN "00",
D1 WHEN "01",
D2 WHEN "10",
D3 WHEN "11",
(OTHERS => 'U') WHEN OTHERS;
END Dataflow; -- 8-bit vector of U (b)
Figure 4.23 VHDL code for an 8-bit wide 4-to-1 multiplexer: (a) behavioral level; (b) dataflow level. 4.9.1 * Using Multiplexers to Implement a Function
Multiplexers can be used to implement a Boolean function very easily. In general, for an n-variable function, a
2n-to-1 multiplexer (that is, a multiplexer with n select lines) is needed. An n-variable function has 2n minterms, and
each minterm corresponds to one of the 2n multiplexer inputs. The n input variables are connected to the n select
lines of the multiplexer. Depending on the values of the n variables, one data input line will be selected, and the
value on that input line is passed to the output. Therefore, all we need to do is to connect all the data input lines to
either a 1 or a 0, depending on whether we want that corresponding minterm to be a 1-minterm or a 0-minterm,
respectively.
Figure 4.24 shows the implementation of the 3-variable function, F (x, y, z) = x'y'z' + x'yz' + xy'z + xyz' + xyz.
The 1-minterms for this function are m0, m2, m5, m6, and m7, so the corresponding data input lines, d0, d2, d5, d6, and
d7 are connected to a 1, while the remaining data input lines are connected to a 0. For example, the 0-minterm x'yz
has the value 011, which will select the d3 input, so a 0 passes to the output. On the other hand, the 1-minterm xy'z
has the value 101, which will select the d5 input, so a 1 passes to the output.
11100100
x
y
z d7 d6 d5 d4 d3 d2 d1 d0
s2
s1
s0
y
F Figure 4.24 Using an 8-to-1 multiplexer to implement the function F (x, y, z) = x'y'z' + x'yz' + xy'z + xyz' + xyz. 4.10 Tri-state Buffer
A tri-state buffer, as the name suggests, has three states: 0, 1, and a third state denoted by Z. The value Z
represents a high-impedance state, which for all practical purposes acts like a switch that is opened or a wire that is
cut. Tri-state buffers are used to connect several devices to the same bus. A bus is one or more wire for transferring
signals. If two or more devices are connected directly to a bus without using tri-state buffers, signals will get
corrupted on the bus because the devices are always outputting either a 0 or a 1. However, with a tri-state buffer in
between, devices that are not using the bus can disable the tri-state buffer so that it acts as if those devices are 123 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components physically disconnected from the bus. At any one time, only one active device will have its tri-state buffers enabled,
and thus, use the bus.
The truth table and symbol for the tri-state buffer is shown in Figure 4.25(a) and (b). The active high enable line
E turns the buffer on or off. When E is de-asserted with a 0, the tri-state buffer is disabled, and the output y is in its
high-impedance Z state. When E is asserted with a 1, the buffer is enabled, and the output y follows the input d.
A circuit consisting of only logic gates cannot produce the high impedance state required by the tri-state buffer,
since logic gates can only output a 0 or a 1. To provide the high impedance state, the tri-state buffer circuit uses two
transistors in conjunction with logic gates, as shown in Figure 4.25(c). Section 5.3 will discuss the operations of
these two transistors in detail. For now, we will keep it simple. The top PMOS transistor is enabled with a 0 at the
node labeled A, and when it is enabled, a 1 signal from Vcc passes down through the transistor to y. The bottom
NMOS transistor is enabled with a 1 at the node labeled B, and when it is enabled, a 0 signal from ground passes up
through the transistor to y. When the two transistors are disabled (with A = 1 and B = 0) they will both output a high
impedance Z value; so y will have a Z value.
Having the two transistors, we need a circuit that will control these two transistors so that together they realize
the tri-state buffer function. The truth table for this control circuit is shown in Figure 4.25(d). The truth table is
derived as follows. When E = 0, it does not matter what the input d is, we want both transistors to be disabled so that
the output y has the Z value. The PMOS transistor is disabled when the input A = 1; whereas, the NMOS transistor is
disabled when the input B = 0. When E = 1 and d = 0, we want the output y to be a 0. To get a 0 on y, we need to
enable the bottom NMOS transistor and disable the top PMOS transistor so that a 0 will pass through the NMOS
transistor to y. To get a 1 on y for when E = 1 and d = 1, we need to do the reverse by enabling the top PMOS
transistor and disabling the bottom NMOS transistor.
The resulting circuit is shown in Figure 4.25(c). When E = 0, the output of the NAND gate is a 1, regardless of
what the other input is, and so the top PMOS transistor is turned off. Similarly, the output of the AND gate is a 0, so
the bottom NMOS transistor is also turned off. Thus, when E = 0, both transistors are off, so the output y is in the Z
state.
When E = 1, the outputs of both the NAND and AND gates are equal to d'. So if d = 0, the output of the two gates
are both 1, so the bottom transistor is turned on while the top transistor is turned off. Thus, y will have the value 0,
which is equal to d. On the other hand, if d = 1, the top transistor is turned on while the bottom transistor is turned
off, and y will have the value 1.
The behavioral VHDL code for an 8-bit wide tri-state buffer is shown in Figure 4.26.
E Vcc
A E
0
1 (a) y
Z
d E d d PMOS y y
B (b) (c) NMOS E
0
0
1
1 d
0
1
0
1 A
1
1
1
0 B
0
0
1
0 y
Z
Z
0
1 (d) Figure 4.25 Tri-state buffer: (a) truth table; (b) logic symbol; (c) circuit; (d) truth table for the control portion of the
tri-state buffer circuit.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY TriState_Buffer IS PORT (
E: IN STD_LOGIC;
d: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
y: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END TriState_Buffer; 124 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components ARCHITECTURE Behavioral OF TriState_Buffer IS
BEGIN
PROCESS (E, d)
BEGIN
IF (E = '1') THEN
y <= d;
ELSE
y <= (OTHERS => 'Z');
-- to get 8 Z values
END IF;
END PROCESS;
END Behavioral;
Figure 4.26 VHDL code for an 8-bit wide tri-state buffer. 4.11 Comparator
Quite often, we need to compare two values for their arithmetic relationship (equal, greater, less than, etc.). A
comparator is a circuit that compares two binary values and indicates whether the relationship is true or not. To
compare whether a value is equal or not equal to a constant value, a simple AND gate can be used. For example, to
compare a 4-bit variable x with the constant 3, the circuit in Figure 4.27(a) can be used. The AND gate outputs a 1
when the input is equal to the value 3. Since 3 is 0011 in binary, therefore, x3 and x2 must be inverted.
The XOR and XNOR gates can be used for comparing inequality and equality, respectively, between two values.
The XOR gate outputs a 1 when its two input values are different. Hence, we can use one XOR gate for comparing
each bit pair of the two operands. A 4-bit inequality comparator is shown in Figure 4.27(b). Four XOR gates are used,
with each one comparing the same bit from the two operands. The outputs of the XOR gates are ORed together so that
if any bit pair is different then the two operands are different, and the resulting output is a 1. Similarly, an equality
comparator can be constructed using XNOR gates instead, since the XNOR gate outputs a 1 when its two input values
are the same.
To compare the greater-than or less-than relationships, we can construct a truth table and build the circuit from
it. For example, to compare whether a 4-bit value X is less than five, we get the truth table, equation, and circuit
shown in Figure 4.27(c). 125 Digital Logic and Microprocessor Design with VHDL x3
x2
x1
x0 Chapter 4 - Standard Combinational Components x3
y3
x2
y2
x1
y1
x0
y0 F F (a)
x3
0
0
0
0
0
0
0
0
1 x2
0
0
0
0
1
1
1
1
× x1
0
0
1
1
0
0
1
1
× (b)
x0
0
1
0
1
0
1
0
1
× X<5
1
1
1
1
1
0
0
0
0 x3 x2 x1 x0 (X < 5) (X < 5) = x3'x2' + x3'x2x1'x0'
(c)
Figure 4.27 Simple 4-bit comparators for: (a) X = 3; (b) X ≠ Y; (c) X < 5.
Instead of constructing a comparator for a fixed number of bits for the input values, we often prefer to construct
an iterative circuit by constructing a 1-bit slice comparator and then daisy chaining n of them together to make an
n-bit comparator. The 1-bit slice comparator will have, in addition to the two input operand bits, xi and yi, a pi bit
that keeps track of whether all the previous bit pairs compared so far are true or not for that particular relationship.
The circuit outputs a 1 if pi = 1, and the relationship is true for the current bit pair xi and yi. Figure 4.28(a) shows a 1bit slice comparator for the equal relationship. If the current bit pair, xi and yi, is equal, the XNOR gate will output a 1.
Hence, pi+1 = 1 if the current bit pair is equal and the previous bit pair, pi, is a 1. To obtain a 4-bit iterative equality
comparator, we connect four 1-bit equality comparators in series, as shown in Figure 4.28(b). The initial p0 bit must
be set to a 1. Thus, if all four bit pairs are equal, then the last bit, p4, will be a 1; otherwise, p4 will be a 0. xi yi x3 y3 x2 y2 x1 y1 x0 y0 EQ pi+1 pi p4 EQ p3 EQ (a) p2 EQ p1 EQ p0 '1' (b) Figure 4.28 Iterative comparators: (a) 1-bit slice for xi = yi; (b) 4-bit X = Y.
Building an iterative comparator circuit for the greater-than relationship is slightly more difficult. The 1-bit slice
comparator circuit for the condition xi > yi is constructed as follows. In addition to the two operand input bits, xi and
yi, there are also two status input bits, gin and ein. Here, gin is a 1 if the condition xi > yi is true for the previous bit
slice; otherwise, gin is a 0. Furthermore, ein is a 1 if the condition xi = yi is true; otherwise ein is a 0. The circuit also
has two status output bits, gout and eout, having the same meaning as the gin and ein signals. These two input and two
output status bits allow the bit slices to be daisy-chained together. Following the above description of the 1-bit slice,
we obtain the truth table shown in Figure 4.29(a). The equations for eout and gout are shown in Figure 4.29(b), and the
1-bit slice circuit in (c). 126 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components In order for the bit slices to operate correctly, we need to perform the comparisons from the most significant bit
to the least significant bit. The complete 4-bit iterative comparator circuit for the condition xi > yi is shown in Figure
4.29(d). The initial values for gin and ein must be set to gin = 0 and ein = 1.
If x = y, then the last eout is a 1, otherwise, eout is a 0. If the last eout is a 0, then the last gout can be either a 1 or a
0. If x > y then gout is a 1; otherwise, gout is a 0. Notice that both eout and gout cannot be both 1’s. The operation of this
comparator circuit is summarized in Figure 4.29(e).
gin
0
0
0
0
0
1
1 ein
0
1
1
1
1
0
1 xi
×
0
0
1
1
×
× yi
×
0
1
0
1
×
× Meaning
<
=
<
>
=
>
Invalid gout
0
0
0
1
0
1
1 eout
0
1
0
0
1
0
1 (a)
gout eout
xiyi ginein xiyi
00 01 11 10 ginein 00 00 01 11 10 00 01 1 01 1
1 11 1 1 1 1 11 10 1 1 1 1 1 10 gout = gin + einxiyi' 1 xi yi gin gout eout ein
(c)
x3 y3 1 gin
ein x2 y2 gout > 1 eout = ginein + einxi'yi' + einxiyi
(b) 0 1 eout x1 y1 gout > x0 y0 gout > eout (d) 127 eout gout > eout x>y
x=y Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components Condition
Invalid
x=y
x>y
x<y eout
1
1
0
0 gout
1
0
1
0 (e)
Figure 4.29 Comparator for x > y: (a) truth table for 1-bit slice; (b) K-maps and equations for gout and eout; (c) circuit
for 1-bit slice; (d) 4-bit x > y comparator circuit; (e) operational table. 4.12 Shifter
The shifter is used for shifting bits in a binary word one position either to the left or to the right. The operations
for the shifter are referred to either as shifting or rotating, depending on how the end bits are shifted in or out. For a
shift operation, the two end bits do not wrap around; whereas for a rotate operation, the two end bits wrap around.
Figure 4.30 shows six different shift and rotate operations.
For example, for the “Shift left with 0” operation, all the bits are shifted one position to the left. The original
leftmost bit is shifted out (i.e. discarded) and the rightmost bit is filled with a 0. For the “Rotate left” operation, all
the bits are shifted one position to the left. However, instead of discarding the leftmost bit, it is shifted in as the
rightmost bit (i.e. it rotates around).
For each bit position, a multiplexer is used to move a bit from either the left or right to the current bit position. The
size of the multiplexer will determine the number of operations that can be implemented. For example, we can use a
4-to-1 multiplexer to implement the four operations, as specified by the table in Figure 4.31(a). Two select lines, s1
and s0, are needed to select between the four different operations. For a 4-bit operand, we will need to use four 4-to1 multiplexers as shown in Figure 4.31(b). How the inputs to the multiplexers are connected will depend on the
given operations.
Operation
Shift left with 0 Shift left with 1 Shift right with 0 Shift right with 1 Rotate left Rotate right Comment Example Shift bits to the left one position. The
leftmost bit is discarded and the rightmost
bit is filled with a 0. 1 0110100 Same as above, except that the rightmost bit
is filled with a 1. 101101000
1 0110100
101101001 Shift bits to the right one position. The
rightmost bit is discarded and the leftmost
bit is filled with a 0. 1 0110100 Same as above, except that the leftmost bit is
filled with a 1. 1 0110100 Shift bits to the left one position. The
leftmost bit is moved to the rightmost bit
position. 1 0110100 Shift bits to the right one position. The
rightmost bit is moved to the leftmost bit
position. 1 0110100 Figure 4.30 Shifter and rotator operations. 128 0 1 0110100 1 1 0110100 01101001 0 1 011010 Digital Logic and Microprocessor Design with VHDL s1
0
0
1
1 s0
0
1
0
1 Chapter 4 - Standard Combinational Components Operation
Pass through
Shift left and fill with 0
Shift right and fill with 0
Rotate right
(a) in3 in2 in1 '0' in0 '0' 3210
s1 MUX3
s0 y 3210
s1 MUX2
s0 y 3210
s1 MUX1
s0 y 3210
s1 MUX0
s0 y out3 out2 out1 s1
s0 in3 in2 in1 in0
4-bit Shifter
out3 out2 out1 out0 out0 s1
s0 (c) (b)
Figure 4.31 A 4-bit shifter: (a) operation table; (b) circuit; (c) logic symbol. In this example, when s1 = s0 = 0, we want to pass the bit straight through without shifting, i.e. we want the
value for ini to pass to outi. Given s1 = s0 = 0, d0 of the multiplexer is selected, hence, ini is connected to d0 of muxi
which outputs to outi. For s1 = 0 and s0 = 1, we want to shift left, i.e. we want the value for ini to pass to outi+1. With
s1 = 0 and s0 = 1, d1 of the multiplexer is selected, hence, ini is connected to d1 of MUXi+1 which outputs to outi+1.
For this selection, we also want to shift in a 0 bit, so d1 of MUX0 is connected directly to a 0.
The behavioral VHDL code for an 8-bit shifter having the functions as defined in Figure 4.31(a) is shown in
Figure 4.32.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL;
ENTITY shifter IS PORT (
S: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
-- select for operations
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- input
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
-- output
END shifter;
ARCHITECTURE Behavior OF shifter IS
BEGIN
PROCESS(S, input)
BEGIN
CASE S IS
WHEN "00" =>
-- pass through
output <= input;
WHEN "01" =>
-- shift left with 0
output <= input(6 downto 0) & '0';
WHEN "10" =>
-- shift right with 0
output <= '0' & input(7 downto 1);
WHEN OTHERS =>
-- rotate right
output <= input(0) & input(7 DOWNTO 1); 129 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components END CASE;
END PROCESS;
END Behavior;
Figure 4.32 Behavioral VHDL code for an 8-bit shifter having the operations as defined in Figure 4.31(a). 4.12.1 * Barrel Shifter
A barrel shifter is a shifter that can shift or rotate the data by any number of bits in a single operation. The
select lines for a barrel shifter are used—not to determine what kind of operations (shift or rotate) to perform as for
the general shifter—but rather to determine how many bits to move. Hence, only one particular operation can be
implemented in a barrel shifter circuit. In general, an n-bit barrel shifter can shift the data bits by as much as n – 1
bit distance away in one operation.
Figure 4.33(a) shows the operation table of a 4-bit barrel shifter implementing the rotate left operation. When
s1s0 = 00, no rotation is performed (i.e. a pass through). When s1s0 = 01, the data bits are rotated one position to the
left. When s1s0 = 10, the data bits are rotated two positions to the left. The corresponding circuit is shown in Figure
4.33(b).
Select
s1 s0
00
01
10
11 Operation
No rotation
Rotate left by 1 bit position
Rotate left by 2 bit positions
Rotate left by 3 bit positions Output
out3 out2 out1 out0
in3 in2 in1 in0
in2 in1 in0 in3
in1 in0 in3 in2
in0 in3 in2 in1 (a)
in3 in2 in1 in0 3210
s1 MUX3
s0 y 3210
s1 MUX2
s0 y 3210
s1 MUX1
s0 y 3210
s1 MUX0
s0 y out3 out2 out1 out0 s1
s0 (b)
Figure 4.33 A 4-bit barrel shifter for the rotate left operation: (a) operation table; (b) circuit. 4.13 * Multiplier
In grade school, we were taught to multiply two numbers using a shift-and-add procedure. Regardless of
whether the two numbers are in decimal or binary, we use the same shift-and-add procedure for multiplying them. In
fact, multiplying with binary numbers is even easier because you are always multiplying with either a 0 or a 1.
Figure 4.34(a) shows the multiplication of two 4-bit unsigned binary numbers, the multiplicand M (m3m2m1m0) with
the multiplier Q (q3q2q1q0) to produce the resulting product P (p7p6p5p4p3p2p1p0). Notice that the intermediate
products are always either the same as the multiplicand (if the multiplier bit is a 1) or it is zero (if the multiplier bit
is a 0).
We can derive a combinational multiplication circuit based on this shift-and-add procedure as shown in Figure
4.34(b). Each intermediate product is obtained by ANDing the multiplicand M with one bit of the multiplier qi. Since 130 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components qi is always a 1 or a 0, therefore, the output of the AND gates is always either mi or 0. For example, bit zero of the
first intermediate product is obtained by ANDing m0 with q0; bit one is obtained by ANDing m1 with q0, and so on.
Hence, the four bits for the first intermediate product are m3q0, m2q0, m1q0, and m0q0; the four bits for the second
intermediate product are m3q1, m2q1, m1q1, and m0q1; and so on.
Multiple adders are used to sum all the intermediate products together to give the final product. Each
intermediate product is shifted over to the correct bit position for the addition. For example, p0 is just m0q0; p1 is the
sum of m1q0 and m0q1; p2 is the sum of m2q0, m1q1 and m0q2; and so on. The four full adders (1-bit adders) in each
row are connected, as in the ripple-carry adder with each carry-out signal connected to the carry-in of the next full
adder. The carry-out of the last full adder is connected to the input of the last full adder in the row below. The last
carry-out from the last row of adders is the value for p7 of the final product. As in the ripple-carry adder, all the
initial carry-ins, c0, are set to a 0.
Multiplicand (M)
Multiplier (Q)
Intermediate products Product (P) 1101
× 1011
1101
1101
0000
+ 1101
10001111 m3q1
m3q2 m2q2
+ m3q3 m2q3 m1q3
p7 p6 p5 p4 m3 m2 m1 m0
× q3 q2 q1 q0
m3q0 m2q0 m1q0 m0q0
m2q1 m1q1 m0q1
m1q2 m0q2
m0q3
p3 p2 p1 p0 (a)
m3 q0 m3 q1 m2 q1 m2 q0 m1 q1 m1 q0 m0 q0 m0 q1 0
+
m3 q2 m2 q2 +
m3 q3 m2 q3 +
m1 q2 +
m1 q3 + 0 m0 q2 + + 0 m0 q3 +
p7 + + + + p6 p5 p4 p3 (b)
Figure 4.34 Multiplication: (a) method; (b) circuit. 131 0
p2 p1 p0 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components 4.14 Summary Checklist
Full adder
Ripple-carry adder
Carry-lookahead adder
Two’s complement
Sign extension
Subtractor
Arithmetic logic unit (ALU)
Arithmetic extender (AE)
Logic extender (LE)
Carry extender (CE)
Decoder
Encoder
Priority encoder
Multiplexer (MUX)
Tri-state buffer
Z value
Comparator
Shifter
Barrel Shifter
Multiplier 132 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components 4.15 Problems
4.1. Convert the following numbers to 12-bit binary numbers using two’s complement representation.
a) 23410
b) –23410
c) 2348
d) BC416
e) –47210
Answer:
a) 000011101010
b) 111100010110
c) 000 010 011100
d) 101111000100
e) 111000101000
4.2. Convert the following two’s complement binary numbers to decimal, octal, & hexadecimal.
a) 1001011
b) 011110
c) 101101
d) 1101011001
e) 0110101100
Answer:
a)
b)
c)
d)
e) Decimal
–53
30
–19
–167
428 Octal
713
36
55
7531
654 Hexadecimal
CB
1E
ED
F59
1AC 4.3. Write the complete structural VHDL code for the full adder circuit shown in Figure 4.1(c).
Answer
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY Full_Adder IS PORT (
xi: IN STD_LOGIC;
yi: IN STD_LOGIC;
cin: IN STD_LOGIC;
si: OUT STD_LOGIC;
cout OUT STD_LOGIC);
END Full_Adder;
ARCHITECTURE Full_Adder_structural OF Full_Adder IS
SIGNAL xy, xxory, xxorycin: STD_LOGIC;
COMPONENT AND2 PORT (I0,I1: IN STD_LOGIC; O: OUT STD_LOGIC);
END COMPONENT;
COMPONENT XOR2 PORT (I0,I1: IN STD_LOGIC; O: OUT STD_LOGIC);
END COMPONENT;
COMPONENT OR2 PORT (I0,I1: IN STD_LOGIC; O: OUT STD_LOGIC);
END COMPONENT;
BEGIN 133 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components U1: AND2 PORT MAP (xi, yi, xy);
U2: XOR2 PORT MAP (xi, yi, xxory);
U3: XOR2 PORT MAP (xxory, cin, si);
U4: AND2 PORT MAP (xxory, cin, xxorycin);
U5: OR2 PORT MAP (xy, xxorycin, cout);
END Full_Adder_structural;
4.4. Draw the smallest possible complete circuit for a 2-bit carry-lookahead adder.
Answer:
x1 y1 x0 y0 xi yi
ci FA1 c1 FA0 c1 FAi = c0 x0
y0 c0 = 0
s1 si s0
x1 y1 x0 y0 c1 s1 s0 4.5. Draw the complete circuit for a 4-bit carry-lookahead adder.
4.6. Derive the carry-lookahead equation and circuit for c5.
4.7. Show that when adding two n-bit signed numbers, An-1…A0 and Bn-1…B0, producing the result, Sn-1…S0, the
Signed_Overflow flag can be deduced by the equation:
Signed_Overflow = An-1 XOR Bn-1 XOR Sn-1 XOR Sn
4.8. Draw the complete 4-bit ALU circuit having the following operations. Use K-maps to reduce all the equations
to standard form.
s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1 Operations
B–1
A NOR B
A–B
A XNOR B
1
A NAND B
A plus B
A' Answer:
s2 s1 s0 ALU Operations 134 LE AE CE Digital Logic and Microprocessor Design with VHDL 0
0
0
0
1
1
1
1 0
0
1
1
0
0
1
1 0
1
0
1
0
1
0
1 Chapter 4 - Standard Combinational Components xi
bi B–1
A NOR B
A–B
A XNOR B
1
A NAND B
A+B
A' ai NOR bi
ai
ai XNOR bi
0
ai NAND bi
ai
ai' s2's1's0'bi xi aibi s1s0 s2 = 0
00 01 1 00 01 1 1 10 1 1 1 1 11 s0ai'bi' 11 1 00 01 s2 = 1 1 10 1 1 11 10 1 s2s1's0bi'
s2s0ai' 1 1 s1s0'ai s2's1aibi xi = s0ai'bi' + s2's1's0'bi + s2s1's0bi' + s2s0ai' + s1s0'ai + s2's1aibi
yi
s1s0 s2bi s2's1's0'
01 1 00 00 11 10 1 01
11
s2's0'bi'
1 10 1 s2s1s0'bi yi = s2's0'bi' + s2's1's0' + s2s1s0'bi
c0 s1s0
s2 00 01 0
1 11 10
1 1 c0 = s2s1's0' + s2's1s0'
= (s2 ⊕ s1)s0' 135 yi
1
0
bi'
0
0
0
bi
0 c0
0
0
1
0
1
0
0
0 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components a3 b3 a2 b2 a1 b1 a0 b0 LE AE LE AE LE AE LE AE s2
s1
s0 x3 y3 c4 Unsigned
Overflow x2 y2 c3 FA x1
c2 FA y1
FA x0
c1 y0
FA c0 CE Signed
Overflow
f3 f2 f1 f0 4.9. Draw the complete 4-bit ALU circuit having the following operations. Don’t-care values are assigned to unused
select combinations. Use K-maps to reduce all the equations to standard form.
s2
0
0
0
0
1
1
1 s1
0
0
1
1
0
0
1 s0
0
1
0
1
0
1
0 Operations
Pass A through the LE
Pass B through the LE
NOT A
NOT B
A–B
B–A
B+1 s2 s1 s0 Operations 0
0
0
0
1
1
1
1 0
0
1
1
0
0
1
1 0
1
0
1
0
1
0
1 Pass A through the LE
Pass B through the LE
NOT A
NOT B
A–B
B–A
B+1
Unused Answer xi s2 = 0
00 01 CE
c0
0
0
0
0
1
1
1
× s2 = 1 01 1 11 1 10 1 11 10 1 00 s1s0bi' AE
yi
0
0
0
0
bi'
bi
0
× s1's0'ai
aibi s1s0
s2's1's0bi LE
xi
ai
bi
ai'
bi'
ai
ai'
bi
× 1 1 01 11 10 1 1 s2s0ai' 1
1 1 00 1 × × × 1 1 × s2s1bi s2's1s0'ai' 136 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components xi = s1s0bi' + s2's1's0bi + s1's0'ai + s2s0ai' + s2s1bi + s2's1s0'ai'
s2 s1 s0 ai bi xi yi
s1s0 s2bi 00 01 11 10
1 00
01 1 11 × s2s1's0'bi'
s2s0bi × 10 yi = s2s1's0'bi' + s2s0bi
s2 s1 s0 bi yi c0
s2
s1s0 0 1 00 1 01 1 11 × 10 1 s2 c0 = s2
s2 c0 137 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components a3 b3 a2 b2 a1 b1 a0 b0 LE AE LE AE LE AE LE AE s2
s1
s0 x3 y3 c4 Unsigned
Overflow x2 y2 c3 FA FA x1
c2 y1 x0
c1 FA y0
FA c0 CE Signed
Overflow
f3 f2 f1 f0 4.10.
Draw the complete 4-bit ALU circuit having the following operations. Use K-maps to reduce all the
equations to standard form.
s2
0
0
0
0
1
1
1 s1
0
0
1
1
0
0
1 s0
0
1
0
1
0
1
0 Operations
A plus B
Increment A
Increment B
Pass A
A–B
A XOR B
A AND B 4.11. Draw the complete 4-bit ALU circuit having the following operations. Use K-maps to reduce all the equations
to standard form.
s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1 Operations
Pass A
Pass B through the AE
A plus B
A'
A XOR B
A NAND B
A–1
A–B 4.12. Given the following K-maps for the LE, AE, and C0 of an ALU, determine the ALU operations assigned to
each of the select line combinations.
LE AE
aibi s1s0 s2 = 0
00 01 1 10 10 00 01 s2bi
10 00
00 1 1 1 1 1
1 11 s1s0 1 1 11 11 1 00 01 s2 = 1 01 1 1 11 CE
s1s0 01 1 1 s2 11
1 10 1 138 10 00
0 1 1 1 01 11 10
1 1 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components Answer:
s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1 LE
bi AE
1
0
bi'
0
0
0
bi
0 ai NOR bi
ai
ai XNOR bi
0
ai NAND bi
ai
ai' CE
0
0
1
0
1
0
0
0 ALU Operation
B–1
A NOR B
A–B
A XNOR B
1
A NAND B
A+B
A' 4.13. A four-function ALU has the following equations for its LE, AE, and CE:
xi = ai + s1's0bi
yi = s1's0' + s1s0bi'
c0 = s1s0
Determine the four functions in the correct order that are implemented in this ALU. Show all your work.
Answer:
From the equations, we get the following K-maps:
xi yi aibi
10 1 1 00 1 1 01 11 1 1 11 10 1 1 10 00 01 bi 11 s1s0
00
01 1 0 s1s0 1 1 1 1 From the K-maps, we deduce the functional table
s1 s0
00
01
10
11 LE
xi
A AE
yi
1
0
0
B' A OR B
A
A 4.14. Draw the circuit for the 2-to-4 decoder.
Answer:
E
A1
A0 Y3 Y2 Y1 Y0 139 CE
c0
0
0
0
1 Function
A–1
A OR B
Pass A
A–B Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components 4.15. Derive the truth table for a 3-to-8 decoder using negative logic.
4.16. Draw the circuit for the 4-to-16 decoder using only 2-to-4 decoders.
Answer:
A3 A2 A1
E E A1 A0 A0 2-to-4
decoder
C3 C2 C1 C0 A1
E A0 2-to-4
decoder A1
E C3 C2 C1 C0 A0 A1 2-to-4
decoder
C3 C2 C1 C0 C15 C14 C13 C12 A0 2-to-4
decoder E C3 C2 C1 C0 C11 C10 C9 C8 C7 C6 C5 C4 A1
E A0 2-to-4
decoder
C3 C2 C1 C0
C3 C2 C1 C0 4.17. Draw the circuit for the 4-to-2 priority encoder using only 2-input AND, 2-input OR, and NOT gates.
Answer:
Truth table:
D3
0
0
0
0
1 D2
0
0
0
1
× D1
0
0
1
×
× D0
0
1
×
×
× A1
0
0
0
1
1 A0
0
0
1
0
1 Z
0
1
1
1
1 Equations:
A 1 = D3 + D2
A0 = D3 + D2'D1
Z = D3 + D2 + D1 + D0
Circuit:
D3
D2
D1 A1
A0
Z D0
4.18. Draw the circuit for an 8-to-3 priority encoder.
4.19. Draw the circuit for the 8-to-3 priority encoder using only 4-to-2 priority encoders.
4.20. Write the behavioral VHDL code for the 8-to-3 priority encoder.
4.21. Draw the circuit for a 16-to-1 multiplexer using only 4-to-1 multiplexers.
Answer: 140 Digital Logic and Microprocessor Design with VHDL I15 Chapter 4 - Standard Combinational Components I0 3210
s0
s1 y 3210
s0
s1 y 3210
s0
s1 y 3210
s0
s1 y s0
s1
s2
s3 3210
s0
s1 y Y 4.22. Draw the circuit for a 16-to-1 multiplexer using only 2-to-1 multiplexers.
4.23. Use only 2-to-1 multiplexers to implement the function: f(w,x,y,z) = Σ(0,2,5,7,13,15).
Answer:
10 10 15 14
1
s 00 13 12 0 1
s y 00 11 10 0 1
s y 9 0 1
s y 10
8 7 0 1
s y 10
6 5 0 1
s y 01
4 3 0 1
s y 01
2 1 0 1
s y 0
0 y z
1
s 0 1
s y 0 1
s y 0 1
s y 0
y y
1
s 0 1
s y 0
y x
1
s w 0
y f 4.24. Use only 2-to-1 multiplexers (as many as you need) to implement the function: F(x, y, z) = Π(0, 3, 4, 5, 7).
Answer:
F(x, y, z) = Π(0, 3, 4, 5, 7)
= Σ(1, 2, 6)
01
d7 d6
1
s 00
d5 d4 0 1
s y 01
d3 d2 0 1
s y 10
d1 d0 0 1
s y 0
y z
1
s 0 1
s y 0
y y
x 1
s 0
y y 4.25. Use one 8-to-1 multiplexer to implement the function: F(x,y,z) = Σ(0,3,4,6,7).
4.26. Use 2-to-1 multiplexers to implement the function: F(x,y,z) = Σ(0,2,4,5).
4.27. Derive the truth table for comparing two 4-bit operands for the less-than-or-equal-to relationship. Derive the
equation and circuit from this truth table.
4.28. Construct the circuit for one bit slice of an n-bit magnitude comparator that compares xi ≥ yi.
Answer:
Truth table for xi ≥ yi:
xi yi 141 fi Digital Logic and Microprocessor Design with VHDL 0
0
1
1 Chapter 4 - Standard Combinational Components 0
1
0
1 1
0
1
1 When taking the previous bit-slice into consideration, fi can be a 1 only if the previous fi-1 is also a 1. Thus,
x y fi fi-1 4.29. Draw the circuit for a 4-bit iterative comparator that tests for the greater-than-or-equal-to relationship.
4.30. Draw the circuit for a 4-bit shifter that realizes the following operation table:
s2
0
0
0
0
1
1
1
1 s1
0
0
1
1
0
0
1
1 s0
0
1
0
1
0
1
0
1 Operation
Pass through
Rotate left
Shift right and fill with 1
Not used
Shift left and fill with 0
Pass through
Rotate right
Shift right and fill with 0 4.31. Draw a 4-bit shifter circuit for the following operational table. Use only the basic gates AND, OR, and NOT (i.e.
do not use multiplexers).
s1
0
0
1
1 s0
0
1
0
1 Operation
Shift left fill with 0
Shift right fill with 0
Rotate left
Rotate right s1
0
0
1
1 s0
0
1
0
1 Operation
Shift left fill with 0
Shift right fill with 1
Rotate left
Rotate right Answer:
O3
I2
1
I2
I0 O2
I1
I3
I1
I3 O1
I0
I2
I0
I2 O3 = s1's0'I2 + s1's0 + s1s0'I2 + s1s0I0
O2 = s1's0'I1 + s1's0I3 + s1s0'I1 + s1s0I3
O1 = s1's0'I0 + s1's0I2 + s1s0'I0 + s1s0I2
O0 = s1's0I1 + s1s0'I3 + s1s0I1 142 O0
0
I1
I3
I1 Digital Logic and Microprocessor Design with VHDL Chapter 4 - Standard Combinational Components I3 I2 I1 I0 O3 O2 O1 O0 s1
s0 4.32. Draw a 4-bit shifter circuit for the following operation table using only six 2-to-1 multiplexers.
Operation
Shift left fill with 0
Shift right fill with 0
Rotate left
Rotate right
Answer:
I3 I2 I1 I0 '0' '0'
1
s 0
y 1
s Shift 1
s 0
y 1
s 0
y 1
s 0
y 1
s 0
y 0
y Left
O3 O2 O1 O0 4.33. Derive the truth table for the following combinational circuit. Write also the operation name for each row in
the table. 143 Digital Logic and Microprocessor Design with VHDL d3 Chapter 4 - Standard Combinational Components d2 d1 d0 1
1
s 0 0 1
s y 0
y s2
3210
s1
s0 y 3210
s1
s0 y 3210
s1
s0 y 3210
s1
s0 y y3 y2 y1 y0 s1
s0 Answer:
s2 s1 s0
000
001
010
011
100
101
110
111 y3
1
d3
d2
d3
d0
d3
d2
d3 y2
d3
d2
d1
d2
d3
d2
d1
d2 y1
d2
d1
d0
d1
d2
d1
d0
d1 y0
d1
d0
0
d0
d1
d0
d3
d0 Operation
Shift right and pad with 1
Pass through
Shift left and pad with 0
Pass through
Rotate right
Pass through
Rotate left
Pass through 4.34. Draw a 4-bit barrel shifter circuit for the rotate right operation.
4.35. Draw the complete detail circuit diagram for the 4-bit multiplier based on the circuit shown in Figure 4.34(b). 144 Chapter 5 * Implementation Technologies Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies This chapter discusses how digital circuits are implemented at the physical level. As you know, transistors are
the fundamental building blocks for all digital circuits. They are the actual physical devices that implement the
binary switch and, therefore, also for the logic gates.
There are many different transistor technologies for creating a digital circuit. Some of these technologies are the
diode-transistor logic (DTL), transistor-transistor logic (TTL), bipolar logic, and complementary metal-oxidesemiconductor (CMOS) logic. Among them, the most widely used is the CMOS technology.
Figure 5.1(a) shows a single discrete transistor with its three connections for signal input, output, and control.
Above the transistor in the figure is a lump of silicon, which, of course, is the main ingredient for the transistor.
Figure 5.1(b) is a picture of transistors inside an IC taken with an electron microscope. Figure 5.1(c) is a higher
magnification of the rectangle area in (b). (a) (b) (c) Figure 5.1 Transistors: (a) a lump of silicon and a transistor; (b) transistors inside an EPROM as seen through an
electron microscope; (c) higher magnification of the rectangle area in (b).
In this chapter, we will look at how CMOS transistors work, and how they are used to build the basic logic
gates. Next, we will look at how digital circuits are actually implemented in various programmable logic devices
(PLDs), such as read-only memories (ROMs), programmable logic arrays (PLAs), programmable array logic
(PAL) devices, complex programmable logic devices (CPLDs), and field programmable gate arrays (FPGAs). The
optional Altera UP2 development board contains both a CPLD and a FPGA chip for implementing your circuits.
The information presented in this chapter, however, is not needed to understand how microprocessors are designed
at the logic circuit level. 5.1 Physical Abstraction Physical circuits deal with physical properties, such as voltages and currents. Digital circuits use the
abstractions of 0 and 1 to represent the presence or absence of these physical properties. In fact, a range of voltages
is interpreted as the logic 0, and another, non-overlapping range is interpreted as the logic 1. Traditionally, digital
circuits operate with a 5-volt power supply. In such a case, it is customary to interpret the voltages in the range 0–
1.5 V as logic 0, while voltages in the range 3.5–5 V as logic 1. This is shown in Figure 5.2. Voltages in the middle
range (from 1.5–3.5 V) are undefined and should not occur in the circuit except during transitions from one state to
the other. However, they may be interpreted as a “weak” logic 0 or a “weak” logic 1.
In our discussion of transistors, we will not get into their electrical characteristics of voltages and currents, but
we will simply use the abstraction of 0 and 1 to describe their operations. 146 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies 5V
Logic 1
3.5 V
Weak 1
Undefined
Weak 0
1.5 V
Logic 0
0V
Figure 5.2 Voltage levels for logic 0 and 1. 5.2 Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) The metal-oxide-semiconductor field-effect transistor (MOSFET) acts as a voltage-controlled switch with three
terminals: source, drain, and gate. The gate controls whether current can pass from the source to the drain or not.
When the gate is asserted or activated, the transistor is turned on and current flows from the source to the drain.
When looking at the transistor by itself, there is no physical difference between the source and the drain terminals.
They are distinguished only when connected with the rest of the circuit by the differences in the voltage levels.
There are two variations of the MOSFET: the n-channel and the p-channel. The physical structures of these two
transistors are shown in Figure 5.3(a) and (b), respectively. The name metal-oxide-semiconductor comes from the
three layers of material that make up the transistor. The “n” stands for negative and represents the electrons, while
“p” stands for positive and represents the holes that flow through a channel in the semiconductor material between
the source and the drain.
For the n-channel MOSFET shown in Figure 5.3(a), a p-type silicon semiconductor material (called the
substrate) is doped with n-type impurities at the two ends. These two n-type regions form the source and the drain of
the transistor. An insulating oxide layer is laid on top of the two n regions and the p substrate, except for two
openings leading to the two n regions. Finally, metal is laid in the two openings in the oxide to form connections to
the source and the drain. Another deposit of metal is laid on top of the oxide between the source and the drain to
form the connection to the gate.
The structure of the p-channel MOSFET shown in Figure 5.3(b) is similar to that in (a), except that the substrate
is of n-type material, and the doping for the source and drain is of p-type impurities.
Metal
Oxide layer
Doping of impurities
Silicon semiconductor
Source n Gate n-channel
p Drain Source n p Gate p-channel
n Substrate Substrate (a) (b) Figure 5.3 Physical structure of the MOSFET: (a) n-channel (NMOS); (b) p-channel (PMOS). 147 Drain p Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies The n-channel and p-channel MOSFETs work opposite of each other. For the n-channel MOSFET, only an nchannel between the source and the drain is created under the control of the gate. This n-channel (n for negative)
only allows negative charged electrons (0’s) to move from the source to the drain. On the other hand, the p-channel
MOSFET can only create a p-channel between the source and the drain under the control of the gate, and this pchannel (p for positive) only allows positive charged holes (1’s) to move from the source to the drain. 5.3 CMOS Logic In CMOS (complementary MOS) logic, only the two complementary MOSFET transistors, (n-channel also
known as NMOS, and p-channel also known as PMOS)1, are used to create the circuit. The logic symbols for the
NMOS and PMOS transistors are shown in Figure 5.4(a) and Figure 5.5(a), respectively. In designing CMOS
circuits, we are interested only in the three connections—source, drain, and gate—of the transistor. The substrate for
the NMOS is always connected to ground, while the substrate for the PMOS is always connected to VCC2, so it is
ignored in the diagrams for simplicity. Notice that the only difference between these two logic symbols is that one
has a circle at the gate input, while the other does not. Using the convention that the circle denotes active-low (i.e., a
0 activates the signal), the NMOS gate input (with no circle) is, therefore, active-high. The PMOS gate input (with a
circle) is active-low.
For the NMOS transistor, the source is the terminal with the lower voltage with respect to the drain. You can
intuitively think of the source as the terminal that is supplying the 0 value, while the drain consumes the 0 value.
The operation of the NMOS transistor is shown in Figure 5.4(b).When the gate is a 1 (asserted), the NMOS
transistor is turned on or enabled, and the source input that is supplying the 0 can pass through to the drain output
through the connecting n-channel. However, if the source has a 1, the 1 will not pass through to the drain even if the
transistor is turned on, because the NMOS does not create a p-channel. Instead, only a weak 1 will pass through to
the drain. On the other hand, when the gate is a 0 (or any value other than a 1), the transistor is turned off, and the
connection between the source and the drain is disconnected. In this case, the drain will always have a highimpedance Z value independent of the source value. The × (don’t-care) in the Input Signal column means that it
doesn’t matter what the input value is, the output will be Z. The high-impedance value, denoted by Z, means no
value or no output. This is like having an insulator with an infinite resistance or a break in a wire, therefore,
whatever the input is, it will not pass over to the output.
Drain Gate
1 Gate 0
(Any value other than a 1) Switch
On
(Closed)
Off
(Open) Input Signal
0
1 Output Signal
0
Weak 1 × Z Source
(a) (b) Figure 5.4 NMOS transistor: (a) logic symbol; (b) truth table.
The PMOS transistor works exactly the opposite of the NMOS transistor. For the PMOS transistor, the source is
the terminal with the higher voltage with respect to the drain. You can intuitively think of the source as the terminal
that is supplying the 1 value, while the drain consumes the 1 value. The operation of the PMOS transistor is shown
in Figure 5.5(b). When the gate is a 0 (asserted), the PMOS transistor is turned on or enabled, and the source input
that is supplying the 1 can pass through to the drain output through the connecting p-channel. However, if the source
has a 0, the 0 will not pass through to the drain even if the transistor is turned on, because the PMOS does not create
an n-channel. Instead, only a weak 0 will pass through to the drain. On the other hand, when the gate is a 1 (or any
value other than a 0), the transistor is turned off, and the connection between the source and the drain is
disconnected. In this case, the drain will always have a high-impedance Z value independent of the source value.
1
2 For bipolar transistors, these two transistors are referred to as NPN and PNP, respectively.
VCC is power or 5-volts in a 5 V circuit, while ground is 0 V. 148 Digital Logic and Microprocessor Design with VHDL Drain Chapter 5 - Implementation Technologies Gate
0 Gate 1
(Any value other than a 0) Switch
On
(Closed)
Off
(Open) Input Signal
0
1 Output Signal
Weak 0
1 × Z Source
(b) (a)
Figure 5.5 PMOS transistor: (a) logic symbol; (b) truth table. 5.4 CMOS Circuits CMOS circuits are built using only the NMOS and PMOS transistors. Because of the inherent properties of the
NMOS and PMOS transistors, CMOS circuits are always built with two halves. One half will use one transistor type
while the other half will use the other type, and when combined together to form the complete circuit, they will work
in complement of each other. The NMOS transistor is used to output the 0 half of the truth table, while the PMOS
transistor is used to output the 1 half of the truth table.
Furthermore, notice that the truth tables for these two transistors, shown in Figure 5.4(b) and Figure 5.5(b),
suggest that CMOS circuits essentially must deal with five logic values instead of two. These five logic values are 0,
1, Z (high-impedance), weak 0, and weak 1. Therefore, when the two halves of a CMOS circuit are combined
together, there is a possibility of mixing any combinations of these five logic values.
Figure 5.6 summarizes the result of combining these logic values. Here a 1 combined with another 1 does not
give you a 2, but rather just a 1! A short circuit results from connecting a 0 directly to a 1 (that is, connecting ground
directly to VCC). This is like sticking two ends of a wire into the two holes of an electrical outlet in the wall. You
know the result, and you don’t want to do it! Connecting a 0 with a weak 1, or a 1 with a weak 0 will also result in a
short, but it may take a longer time before you start to see smoke coming out. Any value combined with Z is just that
value, since Z is nothing.
A properly designed CMOS circuit should always output either a 0 or a 1. The other three values (weak 0, weak
1, and Z) should not occur in any part of the circuit. The construction of several basic gates using the CMOS
technology will now be shown. 0
1
Z
Weak 0
Weak 1 0
0
Short
0
0
Short 1
Short
1
1
Short
1 Z
0
1
Z
Weak 0
Weak 1 Weak 0
0
Short
Weak 0
Weak 0
Short Weak 1
Short
1
Weak 1
Short
Weak 1 Figure 5.6 Result of combining the five possible logic values. 5.4.1 CMOS Inverter
Half of the inverter truth table says that, given a 1, the circuit needs to output a 0. Therefore, the question to ask
is which CMOS transistor (NMOS or PMOS) when given a 1 will output a 0? Looking at the two truth tables for the
two transistors, we find that only the NMOS transistor outputs a 0. The PMOS transistor outputs either a 1 or a weak
0. A weak 0, as you recall from Section 5.1, is an undefined or an unwanted value. The next question to ask is how
do we connect the NMOS transistor so that, when we input a 1, the transistor outputs a 0? The answer is shown in
Figure 5.7(a) where the source of the NMOS transistor is connected to ground (to provide the 0 value), the gate is
the input, and the drain is the output. When the gate is a 1, the 0 from the source will pass through to the drain
output. 149 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies The complementary half of the inverter circuit is to output a 1 when given a 0. Again, from looking at the two
transistor truth tables, we find that the PMOS transistor will do the job. This is expected, since we have used the
NMOS for the first half, and the complementary second half of the circuit must use the other transistor. This time,
the source is connected to VCC to supply the 1 value, as shown in Figure 5.7(b). When the gate is a 0, the 1 from the
source will pass through to the drain output.
To form the complete inverter circuit, we simply combine these two complementary halves together, as shown
in Figure 5.7(c). When combining two halves of a CMOS circuit together, the one thing to be careful of is not to
create any possible shorts in the circuit. We need to make sure that, for all possible combinations of 0’s and 1’s to all
the inputs, there are no places in the circuit where both a 0 and a 1 can occur at the same node.
For our CMOS inverter circuit, when the gate input is a 1, the bottom NMOS transistor is turned on while the
top PMOS transistor is turned off. With this configuration, a 0 from ground will pass through the bottom NMOS
transistor to the output while the top PMOS transistor will output a high-impedance Z value. A Z combined with a 0
is still a 0, because a high-impedance is of no value. Alternatively, when the gate input is a 0, the bottom NMOS
transistor is turned off while the top PMOS transistor is turned on. In this case, a 1 from VCC will pass through the
top PMOS transistor to the output while the bottom NMOS transistor will output a Z. The resulting output value is a
1. Since the gate input can never be both a 0 and a 1 at the same time, therefore, the output can only have either a 0
or a 1, and so, no short can result.
Vcc 1 Source Vcc 1 Source Drain Gate Drain Input Output Gate
Gate
0 Source Drain (a) (b) 0 Source (c) Figure 5.7 CMOS inverter circuit: (a) NMOS half; (b) PMOS half; (c) complete circuit. 5.4.2 CMOS NAND Gate
Figure 5.8 shows the truth table for the NAND gate. Half of the truth table consists of the one 0 output while the
other half of the truth table consists of the three 1 outputs. For the 0 half of the truth table, we want the output to be a
0 when both A = 1 and B = 1. Again, we ask the question: Which CMOS transistor when given a 1 will output a 0?
Of course the answer is again the NMOS transistor. This time, however, since there are two inputs, A and B, we
need two NMOS transistors. We need to connect these two transistors so that a 0 is outputted only when both are
turned on with a 1. Recall from Section 2.3 that the AND operation results from two binary switches connected in
series. Figure 5.9(a) shows the two NMOS transistors connected in series, with the source of one connected to
ground to provide the 0 value, and the drain of the other providing the output 0. The two transistor gates are
connected to the two inputs, A and B, so that only when both inputs are a 1 will the circuit output a 0.
The complementary half of the NAND gate is to output a 1 when either A = 0 or B = 0. This time, two PMOS
transistors are used. To realize the OR operation, the two transistors are connected in parallel with both sources
connected to VCC and both drains to the output, as shown in Figure 5.9(b). This way, only one transistor needs to be
turned on for the circuit to output the 1 value.
The complete NAND gate circuit is obtained by combining these two halves together, as shown in Figure 5.9(c).
When both A and B are 1, the two bottom NMOS transistors are turned on while the two top PMOS transistors are
turned off. In this configuration, a 0 from ground will pass through the two bottom NMOS transistors to the output,
while the two top PMOS transistors will output a high-impedance Z value. Combining a 0 with a Z will result in a 0. 150 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies Alternatively, when either A = 0, or B = 0, or both equal to 0, at least one of the bottom NMOS transistors will be
turned off, thus outputting a Z. On the other hand, at least one of the top PMOS transistors will be turned on and a 1
from VCC will pass through that PMOS transistor. The resulting output value will be a 1. Again, we see that no short
circuit can occur.
B B 0
1 0 1
1 0
0 1 1 1 A 1 1 0 A
1 1 0 (a) (b) Figure 5.8 NAND gate truth table: (a) the 0 half; (b) the 1 half.
Vcc Vcc 1 Source B A 1 Source A B Output
Output Output
B B A A
0 Source
(a) 0 Source
(b) (c) Figure 5.9 CMOS NAND circuit: (a) the 0 half using two NMOS transistors; (b) the 1 half using two PMOS
transistors; (c) the complete NAND gate circuit. 5.4.3 CMOS AND Gate
Figure 5.10 shows the 0 and 1 halves of the truth table for the AND gate. We can proceed to derive this circuit in
the same manner as we did for the NAND gate. For the 0 half of the truth table, we want the output to be a 0 when
either A = 0 or B = 0. This means that we need a transistor that outputs a 0 when it is turned on also with a 0. This
being one of the main differences between the NAND gate and the AND gate, it causes a slight problem. Looking
again at the two transistor truth tables in Figure 5.4 and Figure 5.5, we see that neither transistor fits this criterion.
The NMOS transistor outputs a 0 when the gate is enabled with a 1, and the PMOS transistor outputs a 1 when the
gate is enabled with a 0. If we pick the NMOS transistor, then we need to invert its input. On the other hand, if we
pick the PMOS transistor, then we need to invert its output.
For this discussion, let us pick the PMOS transistor. To obtain the A or B operation, two PMOS transistors are
connected in parallel. The output from these two transistors is inverted with a single NMOS transistor, as shown in
Figure 5.11(a). When either A or B has a 0, that corresponding PMOS transistor is turned on, and a 1 from the VCC
source passes down to the gate of the NMOS transistor. With this NMOS transistor turned on, a 0 from ground is
passed through to the drain output of the circuit. 151 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies For the 1 half of the circuit, we want the output to be a 1 when both A = 1 and B = 1. Again, we have the
dilemma that neither transistor fits this criterion. To be complimentary with the 0 half, we will use two NMOS
transistors connected in series. When both transistors are enabled with a 1, the output 0 needs to be inverted with a
PMOS transistor, as shown in Figure 5.11(b).
Combining the two halves produces the complete AND gate CMOS circuit shown in Figure 5.11(c). Instead of
joining the two halves at the point of the output, the circuit connects together before inverting the signal to the
output. The resulting AND gate circuit is simply the circuit for the NAND gate followed by that of the INVERTER.
From this discussion, we understand why in practice that NAND gates are preferred over AND gates.
B B 0
0 0 0
0 0 0 1 0 1 0 1 1 A A
1 0 1 (a) (b) Figure 5.10 AND gate truth table: (a) the 0 half; (b) the 1 half.
Vcc 1 Source A Vcc B 1 Source A 0 B 0
Output (a) Output
Output
Vcc 1 B
Vcc 1 B A
A 0 Source
0 Source
(b) (c) Figure 5.11 CMOS AND circuit: (a) the 0 half using two PMOS transistors and an NMOS transistor; (b) the 1 half 152 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies using two NMOS transistors and a PMOS transistor; (c) the complete AND gate circuit. 5.4.4 CMOS NOR and OR Gates
The CMOS NOR gate and OR gate circuits can be derived similarly to that of the NAND and AND gate circuits.
Like the NAND gate, the NOR gate circuit uses four transistors, whereas the OR gate circuit uses six transistors. 5.4.5 Transmission Gate
The NMOS and PMOS transistors, when used alone as a control switch, can pass only a 0 or a 1, respectively.
Very often, we like a circuit that is able to pass both a 0 and a 1 under a control signal. The transmission gate is such
a circuit that allows both a 0 and a 1 to pass through when it is enabled. When it is disabled, it outputs the Z value.
The transmission gate uses the two complimentary transistors connected together, as shown in Figure 5.12. Both
ends of the two transistors are connected in common. The top PMOS transistor gate is connected to the inverted
control signal C', while the bottom NMOS transistor gate is connected directly to the control signal C. Hence, both
transistors are enabled when the control signal C = 1, and the circuit is disabled when C = 0.
When the circuit is enabled, if the input is a 1, the 1 signal will pass through the top PMOS transistor, while the
bottom NMOS transistor will pass through a weak 1. The final, combined output value will be a 1. On the other
hand, if the input is a 0, the 0 signal will pass through the bottom NMOS transistor, while the top PMOS transistor
will output a weak 0. The final, combined output value this time will be a 0. Therefore, in both cases, the output
value is the same as the input value.
When the circuit is disabled with C = 0, both transistors will output the Z value. Thus, regardless of the input,
there will be no output. C' Output Input C
Figure 5.12 CMOS transmission gate circuit. 5.4.6 2-input Multiplexer CMOS Circuit
CMOS circuits for larger components can be derived by replacing each gate in the circuit with the
corresponding CMOS circuit for that gate. Since we know the CMOS circuit for the three basic gates (AND, OR, and
NOT) this is a simple “copy and paste” operation.
For example, we can replace the gate level 2-input multiplexer circuit shown in Figure 5.13(a) with the CMOS
circuit shown in Figure 5.13(b). For this circuit, we simply replace the two AND gates with the two 6-transistor
circuits for the AND gate, another 6-transistor circuit for the OR gate, and the 2-transistor circuit for the INVERTER;
giving a total of 20 transistors for this version of the 2-input multiplexer.
However, since the NAND gate uses two fewer transistors than the AND gate, we can first convert the two level
or-of-ands circuit in Figure 5.13(a) to a two level NAND gate circuit shown in Figure 5.13(c). This conversion is
based on the technology mapping technique discussed in Section 3.3. Performing the same “cut-and-paste”
operation on this two level NAND gate circuit produces the CMOS circuit in Figure 5.13(d) that uses only 14
transistors. 153 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies We can do much better in terms of the number of transistors needed for the 2-input multiplexer circuit. From the
original gate level multiplexer circuit in Figure 5.13(a), we want to ask the question: What is the purpose of the two
AND gates? The answer is that each AND gate acts like a control switch. When it is turned on by the select signal s,
the input passes through to the output. Well, the operation of the transmission gate is just like this, and it uses only
two transistors. Hence, we can replace the two AND gates with two transmission gates. Furthermore, the AND gate
outputs a 0 when it is disabled. In order for this 0 from the output of the disabled AND gate not to corrupt the data
from the output of the other enabled AND gate, the OR gate is needed. If we connect the two outputs from the AND
gates directly without the OR gate, a short circuit will occur when the enabled AND gate outputs a 1, because the
disabled AND gate always outputs a 0. However, this problem disappears when we use two transmission gates
instead of the two AND gates, because when a transmission gate is disabled, it outputs a Z value and not a 0.
Thereby, we can connect the outputs of the two transmission gates directly without the need of the OR gate. The
resulting circuit is shown in Figure 5.13(e), using only six transistors. The two-transistor inverter is needed (just like
in the gate level circuit) for turning on only one switch while turning off the other switch at any one time.
Vcc Vcc d0 Vcc
Vcc
Vcc AND d0 s s Vcc y Vcc d1 y
OR d1 AND (a) (b)
Vcc
Vcc d0
Vcc y
NAND d0 s s Vcc y d1
d1 NAND (c) (d) 154 NAND Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies d0
Vcc s y d1
(e)
Figure 5.13 2-input multiplexer circuits: (a) gate level circuit using AND and OR gates; (b) transistor level circuit for
part (a); (c) gate level circuit using NAND gates; (d) transistor level circuit for part (c); (e) transistor level circuit
using transmission gates. 5.4.7 CMOS XOR and XNOR Gates
The XOR circuit can be constructed using the same reasoning as for the 2-input multiplexer discussed in
Section 5.4.6. First, we recall that the equation for the XOR gate is AB' + A'B. For the first AND term, we want to use
a transmission gate to pass the A value. This transmission gate is enabled with the value B'. The resulting circuit for
this first term is shown in Figure 5.14(a). For the second AND term, we want to use another transmission gate to pass
the A' value and have the transmission gate enabled with the value B, resulting in the circuit shown in Figure
5.14(b). Combining the two partial circuits together gives us the complete XOR circuit shown in Figure 5.14 (c).
Again, as with the 2-input multiplexer circuit, it is not necessary to use an OR gate to connect the outputs of the two
transmission gates together. A
A Output A Output
B B Output B (a) (b) (c) Figure 5.14 CMOS XOR gate circuit: (a) partial circuit for the term AB'; (b) partial circuit for the term A'B; (c)
complete circuit.
The CMOS XOR circuit shown in Figure 5.14(c) uses eight transistors: four transistors for the two transmission
gates and another four transistors for the two inverters. However, with some ingenuity, we can construct the XOR
circuit with only six transistors, as shown in Figure 5.15(a). Similarly, the XNOR circuit is shown in Figure 5.15(b).
In the next section, we will perform an analysis of this XOR circuit to see that it indeed has the same functionality as
the XOR gate. 155 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies B A A Output Output B
(a) (b) Figure 5.15 CMOS circuits using only six transistors for: (a) XOR gate; (b) XNOR gate. 5.5 Analysis of CMOS Circuits The analysis of a CMOS circuit follows the same procedure as with the analysis of a combinational circuit, as
discussed in Section 3.1. First, we must assume that the inputs to the circuit must have either a logic 0 or logic 1
value; that is, the input value cannot be a weak 0, a weak 1, or a Z. Then, for every combination of 0 and 1 to the
inputs, trace through the circuit (based on the operations of the two CMOS transistors) to determine the value
obtained at every node in the circuit. When two different values are merged together at the same point in the circuit,
we will use the table in Figure 5.6 to determine the resulting value.
Example 5.1: Analyzing the XOR CMOS circuit
Analyze the CMOS circuit shown in Figure 5.15. For this discussion, the words “top-right,” “top-middle,”
“bottom-middle,” and “bottom-right” are used to refer to the four transistors in the circuit.
Figure 5.16(a) shows the analysis of the circuit with the inputs A = 0 and B = 0. The top-right PMOS transistor
is enabled with a 0 from input A; however, the source for this transistor is a 0 from input B, and this produces a weak
0 at the output of this transistor. In the figure, the arrow denotes that the transistor is enabled, and the label “w 0” at
its output denotes that the output value is a weak 0. For the top-middle PMOS transistor, it is also enabled but with a
0 from input B. The source for this transistor is a 0 from A, and so, the output is again a weak 0. The bottom-middle
NMOS transistor is enabled with a 1 from B'. Since the source is a 0 from A, this transistor outputs a 0. For the
bottom-right NMOS transistor, the 0 from A disables it, and so, a Z value appears at its output. The outputs of these
four transistors are joined together at the point of the circuit output. At this common point, two weak 0’s, a 0, and a
Z are combined together. Referring to Figure 5.6, combining these four values together results in an overall value of
a 0. Hence, the circuit outputs a 0 for the input combination A = 0 and B = 0.
Figure 5.16(b), (c), and (d) show the analysis of the circuit for the remaining three input combinations. The
outputs for all four input combinations match exactly those of the 2-input XOR gate.
♦
0 B 0 1 A B 0 A w0
w0
0 1 Output 0 Z
Z Z Output 1
Z 1 0 (a) (b) 156 Digital Logic and Microprocessor Design with VHDL 0 Chapter 5 - Implementation Technologies B 1 B Z
1 1
w1 A Z Output 1 1 Z
Z A w1 Output 0
0 1 0 (c) (d) Figure 5.16. Analysis of the CMOS XOR gate circuit: (a) shows the analysis for the inputs AB = 00. All the transistor
outputs are annotated with the resulting output value. The letter “w” is used to signify that it is a weak value; (b)
through (d) show the analysis for the remaining input combinations, 01, 10, and 11, respectively.
Example 5.2: Analyzing a CMOS circuit with a short
The CMOS circuit shown below is modified slightly from the XOR circuit from Example 5.1; the top-right
PMOS transistor is replaced with a NMOS transistor. Let us perform an analysis of this circuit using the inputs A = 1
and B = 0.
0 B 1 A 0
1
w1 Output Short
w1 1 The top-right NMOS transistor is enabled with a 1 from input A. The source for this NMOS transistor is a 0
from input B, and so, it outputs a 0. The top-middle PMOS transistor is also enabled, but with a 0 from input B. The
source for this PMOS transistor is a 1 from input A, and so, it outputs a 1. It is not necessary to continue with the
analysis of the remaining two transistors, because at the common output, we already have a 0 (from the top-right
NMOS transistor) combining with a 1 (from the top-middle PMOS transistor) producing a short circuit.
♦ 5.6 Using ROMs to Implement a Function Memory is used for storing binary data. This stored data, however, can be interpreted as being the
implementation of a combinational circuit. A combinational circuit expressed as a Boolean function in canonical
form is implemented in the memory by storing data bits in appropriate memory locations. Any type of memory, such
as ROM (read-only memory), RAM (random access memory), PROM (programmable ROM), EPROM (erasable
PROM), EEPROM (electrically erasable PROM), and so on, can be used to implement combinational circuits. Of
course, non-volatile memory is preferred, since you do want your circuit to stay intact after the power is removed.
In order to understand how combinational circuits are implemented in ROMs, we need to first understand the
internal circuitry of the ROM. ROM circuit diagrams are drawn more concisely by the use of a new logic symbol to
represent a logic gate. Figure 5.17 shows this new logic symbol for an AND gate and an OR gate with multiple inputs.
Instead of having multiple input lines drawn to the gate, the input lines are replaced with just one line going to the
gate. The multiple input lines are drawn perpendicular to this one line. To actually connect an input line to the gate,
an explicit connection point (•) must be drawn where the two lines cross. For example, in Figure 5.17(a) the AND
gate has only two inputs; whereas in (b), the OR gate has three inputs.
= =
(a) (b) 157 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies Figure 5.17 Array logic symbol for: (a) AND gate; (b) OR gate. OR Array A3
A2
A1
A0 OR Array 0
1
2
3
4
5
6
4-to-16 7
Decoder 8
9
10
11
12
13
14
15 A3
A2
A1
A0 0
1
2
3
4
5
6
4-to-16 7
Decoder 8
9
10
11
12
13
14
15 D3 D2 D1 D0 D3 D2 D1 D0 (a) (b) Figure 5.18 Internal circuit for a 16 × 4 ROM: (a) with no connections made; (b) with connections made.
The circuit diagram for a 16 × 4 ROM having 16 locations, each being 4-bits wide, is shown in Figure 5.18(a).
A 4-to-16 decoder is used to decode the four address lines, A3, A2, A1, and A0, to the 16 unique locations. Each output
of the decoder is a location in the memory. Recall that the decoder operation is such that, when a certain address is
presented, the output having the index of the binary address value will have a 1, while the rest of the outputs will
have a 0.
Four OR gates provide the four bits of data output for each memory location. The area for making the
connections between the outputs of the decoder with the inputs of the OR gates is referred to as the OR array. When
no connections are made, the OR gates will always output a 0, regardless of the address input. With connections
made as in Figure 5.18(b), the data output of the OR gates depends on the address selected. For the circuit in Figure
5.18(b), if the address input is 0000, then the decoder output line 0 will have a 1. Since there are no connections
made between the decoder output line 0 and any of the four OR gate inputs, the four OR gates will output a 0.
Therefore, the data stored in location 0 is 0000 in binary. If the address input is 0001, then the decoder output line 1
will have a 1. Since this line is connected to the inputs of the two OR gates for D1 and D0, therefore, D1 and D0 will
both have a 1, while D3 and D2 will both have a 0. Hence, the data stored in location 1 is 0011. In the circuit of
Figure 5.18(b) the value stored in location 2 is 1101.
A 16 × 4 ROM can be used to implement a 4-variable Boolean function as follows. The four variables are the
inputs to the four address lines of the ROM. The 16 decoded locations become the 16 possible minterms for the 4variable function. For each 1-minterm in the function, we make a connection between that corresponding decoder
output line that matches that minterm number with the input of an OR gate. It does not matter which OR gate is used,
as long as one OR gate is used to implement one function. Hence, up to four functions with a total of four variables
can be implemented in a 16 × 4 ROM, such as the one shown in Figure 5.18(a). Larger sized ROMs, of course, can
implement larger and more functions.
From Figure 5.18(b), we can conclude that the function associated with the OR gate output D0, is F = Σ(1,2).
That is, minterms 1 and 2 are the 1-minterms for this function, while the rest of the minterms are the 0-minterms.
Similarly, the function for D1 has only minterm 1 as its 1-minterm. The functions for D2 and D3 both have only
minterm 2 as its 1-minterm.
ROMs are programmed during the manufacturing process and cannot be programmed afterwards. As a result,
using ROMs to implement a function is only cost effective if a large enough quantity is needed. For small quantities, 158 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies EPROMs or EEPROMs are preferred. Both EPROMs and EEPROMs can be programmed individually using an
inexpensive programmer connected to the computer. The memory device is inserted into the programmer. The bits
to be stored in each location of the memory device are generated by the development software. This data file is then
transferred to the programmer, which then actually writes the bits into the memory device. Furthermore, both
EPROMs and EEPROMs can be erased and reprogrammed with different data bits.
Example 5.3: Using a 16 × 4 ROM to implement Boolean functions
Implement the following two Boolean functions using the 16 × 4 ROM circuit shown in Figure 5.18.
F1 (w,x,y,z) = w'x'yz + w'xyz' + w'xyz + wx'y'z' + wx'yz' + wxyz'
F2 (w,x,y,z) = w'x'y'z' + w'x
For F1, the 1-minterms are m3, m6, m7, m8, m10, and m14. For F2, the 1-minterms are m0, m4, m5, m6, and m7.
Notice that in F2, the term w'x expands out to four minterms. The implementation is shown in the circuit connection
below. We arbitrarily pick D0 to implement F1 and D1 to implement F2. OR Array w A3
x A2
y A1
z A0 0
1
2
3
4
5
6
4-to-16 7
Decoder 8
9
10
11
12
13
14
15 D 3 D2 D1 D0
F2 F1 5.7 ♦ Using PLAs to Implement a Function Using ROMs or EPROMs to implement a combinational circuit is very wasteful because usually many locations
in the ROM are not used. Each storage location in a ROM represents a minterm. In practice, only a small number of
these minterms are the 1-minterms for the function being implemented. As a result, the ROM implementing the
function is usually quite empty.
Programmable logic arrays (PLAs) are designed to reduce this waste by not having all of the minterms “builtin” as in ROMs, but rather, allowing the user to specify only the minterms that are needed. PLAs are designed
specifically for implementing combinational circuits.
The internal circuit for a 4 × 8 × 4 PLA is shown in Figure 5.19. The main difference between the PLA circuit
and the ROM circuit is that for the PLA circuit, an AND array is used instead of a decoder. The input signals are
available both in the inverted and non-inverted forms. The AND array allows the user to specify only the product
terms needed by the function; namely, the 1-minterms. The OR array portion of the circuit is similar to that of the
ROM, allowing the user to specify which product terms to sum together. Having four OR gates will allow up to four
functions to be implemented in a single device. 159 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies In addition, the PLA has an output array which provides the capability to either invert or not invert the value at
the output of the OR gate. This is accomplished by connecting one input of the XOR gate to either a 0 or a 1. By
connecting one input of the XOR gate to a 1, the output of the XOR gate is the inverse of the other input.
Alternatively, connecting one input of the XOR gate to a 0, the output of the XOR gate is the same as the other input.
This last feature allows the implementation of the inverse of a function in the AND/OR arrays, and then finally
getting the function by inverting it.
The actual implementation of a combinational circuit into a PLA device is similar to writing data bits into a
ROM or other memory device. A PLA programmer connected to a computer is used. The development software
allows the combinational circuit to be defined and then transferred and programmed into the PLA device.
A3 A2 A1 A0 OR Array AND Array
Output
Array
0 1 F3 F2 F1 F0 Figure 5.19 Internal circuit for a 4 × 8 × 4 PLA.
Example 5.4: Using a 4 × 8 × 4 PLA to implement a full adder circuit
Implement the full adder circuit in a 4 × 8 × 4 PLA. The truth table for the full adder from Section 4.2.1 is
shown here.
xi
0
0
0
0
1
1
1
1 yi
0
0
1
1
0
0
1
1 ci
0
1
0
1
0
1
0
1 160 ci+1
0
0
0
1
0
1
1
1 si
0
1
1
0
1
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies In the PLA circuit shown next, the three inputs, xi, yi, and ci, are connected to the PLA inputs, A2, A1, and A0,
respectively. The first four rows of the AND array implement the four 1-minterms of the function ci+1, while the next
three rows of the AND array implement the first three 1-minterms of the function si. The last minterm, m7, is shared
by both functions, and therefore, it does not need to be duplicated. The two functions, ci+1 and si, are mapped to the
PLA outputs, F1 and F0, respectively. Since the two functions are implemented directly (i.e. not the inverse of the
functions), the XOR gates for both functions are connected to 0. A3 xi
A2 yi
A1 ci
A0 OR Array AND Array
Output
Array
0 F3 F2 F1 F0
ci+1 si 1 ♦ Example 5.5: Using a 4 × 8 × 4 PLA to implement a function
Implement the following function in a 4 × 8 × 4 PLA.
F (w,x,y,z) = Σ(0, 1, 3, 4, 5, 6, 9, 10, 11, 15)
This four variable function has ten 1-minterms. Since the 4 × 8 × 4 PLA can accommodate only eight minterms,
we need to implement the inverse of the function, which will have only six 1-minterms (16 – 10 = 6). The inverse of
the function can then be inverted back to the original function at the output array by connecting one of the XOR
inputs to a 1, as shown here.
F' = Σ(2, 7, 8, 12, 13, 14)
= w'x'yz' + w'xyz + wx'y'z' + wxy'z' + wxy'z + wxyz' 161 Digital Logic and Microprocessor Design with VHDL w x y Chapter 5 - Implementation Technologies z OR Array AND Array
Output
Array
0 1 F
Another way to implement the above function in the PLA is to first minimize it. The following K-map shows
that the function reduces to
F = w'y' + x'z + w'xz' + wyz + wx'y
F w'y' yz wx 00 01
0 00 1 01 1 11
10 x'z
11
1 1
4 10
3 5 7 1
12 2 1
6 w'xz' 1
13 14 wyz 10 15 wx'y 1
8 9
1 11
1 1 With only five product terms, the function can be implemented directly without having to be inverted as shown
in the following circuit. 162 Digital Logic and Microprocessor Design with VHDL w x y Chapter 5 - Implementation Technologies z OR Array AND Array
Output
Array
0
F 5.8 1 ♦ Using PALs to Implement a Function Programmable array logic (PAL®) devices are similar to PLAs, except that the OR array for the PAL is not
programmable but rather, fixed by the internal circuitry. Hence, they are not as flexible in terms of implementing a
combinational circuit.
The internal circuit for a 4 × 4 PAL is shown in Figure 5.20. The OR gate inputs are fixed; whereas, the AND
gate inputs are programmable. Each output section is from the OR of the three product terms. This means that each
function can have, at most, three product terms. To make the device a little bit more flexible, the output F3, is fed
back to the programmable inputs of the AND gates. With this connection, up to five product terms are possible for
one function. 163 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies A3 A2 A1 A0 F3 F2 F1 F0 Figure 5.20. Internal circuit for a 4 × 4 PAL device.
Example 5.6: Using a 4 × 4 PAL to implement functions
Implement the following three functions given in sum-of-minterms format using the PAL circuit of Figure 5.20.
F1 (w,x,y,z) = w'x'yz + wx'yz'
F2 (w,x,y,z) = w'x'yz + wx'yz' + w'xy'z' + wxyz
F3 (w,x,y,z) = w'x'y'z' + w'x'y'z + w'x'yz' + w'x'yz
Function F1 has two product terms, and it can be implemented directly in one PAL section. F2 has four product
terms, and so, it cannot be implemented directly. However, we note that the first two product terms are the same as
F1. Hence, by using F1, it is possible to reduce F2 from four product terms to three as shown here.
F2 (w,x,y,z) = w'x'yz + wx'yz' + w'xy'z' + wxyz
= F1 + w'xy'z' + wxyz
F3 also has four product terms, but these four product terms can be reduced to just one by minimizing the
equation as shown here.
F3 (w,x,y,z) = w'x'y'z' + w'x'y'z + w'x'yz' + w'x'yz
= w'x' (y'z' + y'z + yz' + yz)
= w'x'
The connections for these three functions are shown in the following PAL circuit. Notice that, for functions F1
and F3, there are unused AND gates. Since there are no inputs connected to them, they output a 0, which does not
affect the output of the OR gate. 164 Digital Logic and Microprocessor Design with VHDL w x y Chapter 5 - Implementation Technologies z F1 F2 F3 ♦ 5.9 Complex Programmable Logic Device (CPLD) Using ROMs, PLAs, and PALs to implement a combinational circuit is fairly straightforward and easy to do.
However, to implement a sequential circuit or a more complex combinational circuit may require more sophisticated
and larger programming devices. The complex programmable logic device (CPLD) is capable of implementing a
circuit with upwards of 10,000 logic gates.
The CPLD contains many PAL-like blocks that are connected together using a programmable interconnect to
form a matrix. The PAL-like blocks in the CPLD are called macrocells as shown in Figure 5.21. Each macrocell has
a programmable-AND-fixed-OR array similar to a PAL device for implementing combinational logic operations. The
XOR gate in the macrocell circuit, shown in Figure 5.21, will either invert or not invert the output from the
combinational logic. Furthermore, a flip-flop (discuss in Chapter 6) is included to provide the capability of
implementing sequential logic operations. The flip-flop can be bypassed using the multiplexer for combinational
logic operations.
Groups of 16 macrocells are connected together to form the logic array blocks. Multiple logic array blocks are
linked together using the programmable interconnect, as shown in Figure 5.22. Logic signals are routed between the
logic array blocks on the programmable interconnect. This global bus is a programmable path that can connect any
signal source to any destination on the CPLD.
The input/output (I/O) blocks allow each I/O pin to be configured individually for input, output, or bidirectional operation. All I/O pins have a tri-state buffer that is controlled individually. The I/O pin is configured as
an input port if the tri-state buffer is disabled; otherwise, it is an output port.
Figure 5.23 shows some of the main features of the Altera MAX7000 CPLD. Instead of needing a separate
programmer to program the CPLD, all MAX devices support in-system programmability through the IEEE JTAG
interface. This allows designers to program the CPLD after it is mounted on a printed circuit board. Furthermore, the
device can be reprogrammed in the field. CPLDs are non-volatile, so once they are programmed with a circuit, the
circuit remains implemented in the device even when the power is removed. 165 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies D Q Clock D Q Clock Figure 5.21 Circuit for the logic array block with two macrocells.
I/O
Block I/O
Block I/O
Block I/O
Block Programmable Interconnect Logic
Array
Block Logic
Array
Block Logic
Array
Block I/O
Block Programmable Interconnect I/O
Block Programmable Interconnect Logic
Array
Block I/O
Block Figure 5.22 Internal circuit for a complex programmable logic device (CPLD). 166 I/O
Block Digital Logic and Microprocessor Design with VHDL Feature
Usable logic gates
Macrocells
Logic array blocks
User I/O pins Chapter 5 - Implementation Technologies MAX7000 CPLD
10,000
512
32
212 FLEX10K FPGA
250,000
N/A
1,520
470 Figure 5.23. Features of the Altera MAX7000 CPLD and the FLEX10K250 FPGA. 5.10 Field Programmable Gate Array (FPGA)
Field programmable gate arrays (FPGAs) are complex programmable logic devices that are capable of
implementing up to 250,000 logic gates and up to 40,960 RAM bits, as featured by the Altera FLEX10K250 FPGA
chip (see Figure 5.23). The internal circuitry of the FLEX10K FPGA is shown in Figure 5.24. The device contains
an embedded array and a logic array. The embedded array is used to implement memory functions and complex
logic functions, such as microcontroller and digital signal processing. The logic array is used to implement general
logic, such as counters, arithmetic logic units, and state machines.
I/O
Block I/O
Block I/O
Block I/O
Block Embedded
Array
I/O
Block LAB I/O
Block
Logic Array
Block LAB Column Interconnect LE
LE Column Interconnect Logic Array I/O
Block Row Interconnect EAB LAB LAB I/O
Block Row Interconnect LAB LAB EAB LAB LAB Embedded
Array Block
I/O
Block I/O
Block I/O
Block I/O
Block Figure 5.24. FLEX10K FPGA circuit.
The embedded array consists of a series of embedded array blocks (EAB). When implementing memory
functions, each EAB provides 2,048 bits, which can be used to create RAM, dual-port RAM, or ROM. EABs can be
used independently, or multiple EABs can be combined to implement larger functions.
The logic array consists of multiple logic array blocks (LAB). Each LAB contains eight logic elements (LE) and
a local interconnect. The LE shown in Figure 5.25 is the smallest logical unit in the FLEX10K architecture. Each LE
consists of a 4-input look-up table (LUT) and a programmable flip-flop. The 4-input LUT is a function generator
made from a 16-to-1 multiplexer that can quickly compute any function of four variables. (Refer to Section 4.8 for
how multiplexers are used to implement Boolean functions.) The four input variables are connected to the four 167 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies select lines of the multiplexer. Depending on the values of these four variables, the value from one of the 16
multiplexer inputs is passed to the output. There are 16 1-bit registers connected to the 16 multiplexer inputs to
supply the multiplexer input values. Depending on the function to be implemented, the content of the 1-bit registers
is set to a 0 or a 1. It is set to a 1 for all the 1-minterms of a four variable function, and to a 0 for all the 0-minterms.
The LUT in Figure 5.25 implements the four variable function F(w,x,y,z) = Σ(0, 3, 5, 6, 7, 12, 13, 15). The
programmable flip-flop can be configured for D, T, JK, or SR operations, and is used for sequential circuits. For
combinational circuits, the flip-flop can be bypassed using the 2-to-1 multiplexer.
4-input LUT
1011000011101001 Variable A
Variable B
Variable C
Variable D 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Register Bypass 16-to-1 MUX D Set To Interconnect Q Clock
Clear Register
Bypass Select Programmable
Flip-flop
Figure 5.25. Logic element circuit with a 4-input LUT and a programmable register.
All the EABs, LABs, and I/O elements, are connected together via the FastTrack interconnect, which is a series
of fast row and column buses that run the entire length and width of the device. The interconnect contains
programmable switches so that the output of any block can be connected to the input of any other block.
Each I/O pin in an I/O element is connected to the end of each row and column of the interconnect and can be
used as either an input, output, or bi-directional port. 5.11 Summary Checklist
Voltage levels
Weak 0, weak 1
NMOS
NMOS truth table
PMOS
PMOS truth table
High-impedance Z
Transistor circuits for basic gates
PLD
ROM circuit implementation
PLA circuit implementation
PAL circuit implementation
CPLD
FPGA 5.12 Problems
5.1
5.2
5.3
5.4
5.5 Draw the CMOS circuit for a 2-input NOR gate.
Draw the CMOS circuit for a 2-input OR gate.
Draw the CMOS circuit for a 3-input NAND gate.
Draw the CMOS circuit for a 3-input NOR gate.
Draw the CMOS circuit for an AND gate by using two NMOS transistors for the 0 half of the circuit and two
PMOS transistors for the 1 half of the circuit. 168 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies Answer
Vcc A
Output Vcc
Vcc B 5.6 Draw the CMOS circuit for a 3-input AND gate.
5.7 Derive the truth table for the following CMOS circuits. There are five possible values: 1, 0, Z, weak 1, weak 0,
and short.
a)
A B
Output C
Answer:
A
0
0
0
0
1
1
1
1 B
0
0
1
1
0
0
1
1 C
0
1
0
1
0
1
0
1 Output
Weak 0
1
Z
Z
Short
1
1
Z b) 169 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies Vcc
Vcc A
B F
Vcc
Vcc Answer:
AF
0Z
1B
This is a tri-state buffer.
c)
B
A
Vcc Output Answer:
A
0
0
1
1 B
0
1
0
1 Output
Short
Weak 0
Short
Weak 1 d)
A B Output C
Answer:
A
0
0 B
0
0 C
0
1 Output
Z
0 170 Digital Logic and Microprocessor Design with VHDL 0
0
1
1
1
1 1
1
0
0
1
1 0
1
0
1
0
1 Chapter 5 - Implementation Technologies Z
Z
1
0
1
Z 5.8 Synthesize a CMOS circuit that realizes the following truth table.
A
0
0
1
1 B
0
1
0
1 Output
0
0
1
0 Answer:
Vcc A B Vcc A Output
B 5.9 Synthesize a CMOS circuit that realizes the following truth table.having two inputs and one output. Use as few
transistors as possible.
A
0
0
1
1 B
0
1
0
1 Output
0
0
1
Short Answer: B
A Output or 171 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies Vcc B A Output 5.10 Use one 16 × 4 ROM (4 address lines, 16 entries, 4 data lines) to implement the following functions. Label all
the lines clearly.
f1 = w'xy'z + w'xz
f2 = w
f3 = xy' + xyz
Answer: OR Array w A3
x A2
y A1
z A0 0
1
2
3
4
5
6
4-to-16 7
Decoder 8
9
10
11
12
13
14
15 f3 f2 f1 5.11 Use one 16 × 4 ROM (4 address lines, 16 entries, 4 data lines) to implement the following functions. Label all
the lines clearly.
f1 = w x' y' z + w x' y z' + w' x y' z'
f2 = x y + w' z + w x' y
Answer: 172 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies OR Array
0
1
2
3
4
5
6
4-to-16 7
Decoder 8
9
10
11
12
13
14
15 w A3
x A2
y A1
z A0 f2 f1 5.12 Use one 4 × 8 × 4 PLA to implement the following two functions:
F1(w,x,y,z) = wx'y'z + wx'yz' + wxy'
F2(w,x,y,z) = wx'y + x'y'z
Answer:
w x y z OR Array AND Array
Output
Array
0
F2 F1
5.13 Use one 4 × 8 × 4 PLA to implement the following two functions: 173 1 Digital Logic and Microprocessor Design with VHDL Chapter 5 - Implementation Technologies F1(w,x,y,z) = Σ(0,2,3,4,5,6,11,12,13,14,15)
F2(w,x,y,z) = Σ(1,2,3,5,7,9)
Answer:
w x y z OR Array AND Array
Output
Array
0
F2 F1 174 1 Chapter 6 Latches and Flip-Flops Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops So far, we have been looking at the design of combinational circuits. We will now turn our attention to the
design of sequential circuits. Recall that the outputs of sequential circuits are dependent on not only their current
inputs (as in combinational circuits), but also on all their past inputs. Because of this necessity to remember the
history of inputs, sequential circuits must contain memory elements.
The car security system from Section 2.9 is an example of a combinational circuit. In that example, the siren is
turned on when the master switch is on and someone opens the door. If you close the door afterwards, then the siren
will turn off immediately. For a more realistic car security system, we would like the siren to remain on even if you
close the door after it was first triggered. In order for this modified system to work correctly, the siren must be
dependent on not only the master switch and the door switch but also on whether the siren is currently on or off. In
other words, this modified system is a sequential circuit that is dependent on both the current and on the past inputs
to the system.
In order to remember this history of inputs, sequential circuits must have memory elements. Memory elements,
however, are just like combinational circuits in the sense that they are made up of the same basic logic gates. What
makes them different is in the way these logic gates are connected together. In order for a circuit to “remember” its
current value, we have to connect the output of a logic gate directly or indirectly back to the input of that same gate.
We call this a feedback loop circuit, and it forms the basis for all memory elements. Combinational circuits do not
have any feedback loops.
Latches and flip-flops are the basic memory elements for storing information. Hence, they are the fundamental
building blocks for all sequential circuits. A single latch or flip-flop can store only one bit of information. This bit of
information that is stored in a latch or flip-flop is referred to as the state of the latch or flip-flop. Hence, a single
latch or flip-flop can be in either one of two states: 0 or 1. We say that a latch or a flip-flop changes state when its
content changes from a 0 to a 1 or vice versa. This state value is always available at the output. Consequently, the
content of a latch or a flip-flop is the state value, and is always equal to its output value.
The main difference between a latch and a flip-flop is that for a latch, its state or output is constantly affected by
its input as long as its enable signal is asserted. In other words, when a latch is enabled, its state changes
immediately when its input changes. When a latch is disabled, its state remains constant, thereby, remembering its
previous value. On the other hand, a flip-flop changes state only at the active edge of its enable signal, i.e., at
precisely the moment when either its enable signal rises from a 0 to a 1 (referred to as the rising edge of the signal),
or from a 1 to a 0 (the falling edge). However, after the rising or falling edge of the enable signal, and during the
time when the enable signal is at a constant 1 or 0, the flip-flop’s state remains constant even if the input changes.
In a microprocessor system, we usually want changes to occur at precisely the same moment. Hence, flip-flops
are used more often than latches, since they can all be synchronized to change only at the active edge of the enable
signal. This enable signal for the flip-flops is usually the global controlling clock signal.
Historically, there are basically four main types of flip-flops: SR, D, JK, and T. The major differences between
them are the number of inputs they have and how their contents change. Any given sequential circuit can be built
using any of these types of flip-flops (or combinations of them). However, selecting one type of flip-flop over
another type to use in a particular sequential circuit can affect the overall size of the circuit. Today, sequential
circuits are designed mainly with D flip-flops because of their ease of use. This is simply a tradeoff issue between
ease of circuit design versus circuit size. Thus, we will focus mainly on the D flip-flop. Discussions about the other
types of flip-flops can be found in Section 6.14.
In this chapter, we will look at how latches and flip-flops are designed and how they work. Since flip-flops are
at the heart of all sequential circuits, a good understanding of their design and operation is very important in the
design of microprocessors. 6.1 Bistable Element Let us look at the inverter. If you provide the inverter input with a 1, the inverter will output a 0. If you do not
provide the inverter with an input (that is neither a 0 nor a 1), the inverter will not have a value to output. If you
want to construct a memory circuit using the inverter, you would want the inverter to continue to output the 0 even
after you remove the 1 input. In order for the inverter to continue to output a 0, you need the inverter to self-provide
its own input. In other words, you want the output to feed back the 0 to the input. However, you cannot connect the
output of the inverter directly to its input, because you will have a 0 connected to a 1 and so creating a short circuit. 176 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops The solution is to connect two inverters in series, as shown in Figure 6.1. This circuit is called a bistable
element, and it is the simplest memory circuit. The bistable element has two symmetrical nodes labeled Q and Q',
both of which can be viewed as either an input or an output signal. Since Q and Q' are symmetrical, we can
arbitrarily use Q as the state variable, so that the state of the circuit is the value at Q. Let us assume that Q originally
has the value 0 when power is first applied to the circuit. Since Q is the input to the bottom inverter, therefore, Q' is
a 1. A 1 going to the input of the top inverter will produce a 0 at the output Q, which is what we started off with.
Hence, the value at Q will remain at a 0 indefinitely. Similarly, if we power-up the circuit with Q = 1, we will get Q'
= 0, and again, we get a stable situation with Q remaining at a 1 indefinitely. Thus, the circuit has two stable states:
Q = 0 and Q = 1; hence, the name “bistable.”
Q Q'
Figure 6.1 Bistable element circuit.
We say that the bistable element has memory because it can remember its state (i.e., keep the value at Q
constant) indefinitely. Unfortunately, we cannot change its state (i.e., cannot change the value at Q). We cannot just
input a different value to Q, because it will create a short circuit by connecting a 0 to a 1. For example, let us assume
that Q is currently 0. If we want to change the state, we need to set Q to a 1, but in so doing we will be connecting a
1 to a 0, thus creating a short. Another way of looking at this problem is that we can think of both Q and Q' as being
the primary outputs, which means that the circuit does not have any external inputs. Therefore, there is no way for us
to input a different value. 6.2 SR Latch In order to change the state for the bistable element, we need to add external inputs to the circuit. The simplest
way to add extra inputs is to replace the two inverters with two NAND gates, as shown in Figure 6.2(a). This circuit is
called an SR latch. In addition to the two outputs Q and Q', there are two inputs S' and R' for set and reset,
respectively. Just like the bistable element, the SR latch can be in one of two states: a set state when Q = 1, or a reset
state when Q = 0. Following the convention, the primes in S and R denote that these inputs are active-low (i.e., a 0
asserts them, and a 1 de-asserts them).
To make the SR latch go to the set state, we simply assert the S' input by setting it to 0 (and de-asserting R'). It
doesn’t matter what the other NAND gate input is, because 0 NAND anything gives a 1, hence Q = 1, and the latch is
set. If S' remains at 0 so that Q (which is connected to one input of the bottom NAND gate) remains at 1, and if we
now de-assert R' (i.e., set R' to a 1) then the output of the bottom NAND gate will be 0, and so, Q' = 0. This situation
is shown in Figure 6.2(d) at time t0. From this current situation, if we now de-assert S' so that S' = R' = 1, the latch
will remain in the set state because Q' (the second input to the top NAND gate) is 0, which will keep Q = 1, as shown
at time t1. At time t2, we reset the latch by making R' = 0 (and S' = 1). With R' being a 0, Q' will go to a 1. At the top
NAND gate, 1 NAND 1 is 0, thus forcing Q to go to 0. If we de-assert R' next so that, again, we have S' = R' = 1, this
time the latch will remain in the reset state, as shown at time t3.
Notice the two times (at t1 and t3) when both S' and R' are de-asserted (i.e., S' = R' = 1). At t1, Q is at a 1;
whereas, at t3, Q is at a 0. Why is this so? What is different between these two times? The difference is in the value
of Q immediately before those times. The value of Q right before t1 is 1; whereas, the value of Q right before t3 is 0.
When both inputs are de-asserted, the SR latch remembers its previous state. Previous to t1, Q has the value 1, so at
t1, Q remains at a 1. Similarly, previous to t3, Q has the value 0, so at t3, Q remains at a 0.
S' Q R' Q' S'
0
0
1
1
1 177 R'
0
1
0
1
1 Q
×
×
×
0
1 Qnext
1
1
0
0
1 Qnext'
1
0
1
1
0 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops (a) (b)
S'
R' S' Q Q
Q' Q' R' Undefined
Undefined
t0 t1 t2 t3 t4 t5 t6 (d) (c) Figure 6.2 SR latch: (a) circuit using NAND gates; (b) truth table; (c) logic symbol; (d) sample trace.
If both S' and R' are asserted (i.e., S' = R' = 0), then both Q and Q' are equal to a 1, as shown at time t4, since 0
anything gives a 1. Note that there is nothing wrong with having Q equal to Q'. It is just because we named
these two points Q and Q' that we don’t like them to be equal. However, we could have used another name say, P
instead of Q'. NAND If one of the input signals is de-asserted earlier than the other, the latch will end up in the state forced by the
signal that is de-asserted later, as shown at time t5. At t5, R' is de-asserted first, so the latch goes into the set state
with Q = 1, and Q' = 0.
A problem exists if both S' and R' are de-asserted at exactly the same time, as shown at time t6. Let us assume
for a moment that both gates have exactly the same delay and that the two wire connections between the output of
one gate to the input of the other gate also have exactly the same delay. Currently, both Q and Q' are at a 1. If we set
S' and R' to a 1 at exactly the same time, then both NAND gates will perform a 1 NAND 1 and will both output a 0 at
exactly the same time. The two 0’s will be fed back to the two gate inputs at exactly the same time, because the two
wire connections have the same delay. This time around, the two NAND gates will perform a 1 NAND 0 and will both
produce a 1 again at exactly the same time. This time, two 1’s will be fed back to the inputs, which again will
produce a 0 at the outputs, and so on and on. This oscillating behavior, called the critical race, will continue
indefinitely until one outpaces the other. If the two gates do not have exactly the same delay then, the situation is
similar to de-asserting one input before the other, and so, the latch will go into one state or the other. However, since
we do not know which is the faster gate, therefore, we do not know which state the latch will end up in. Thus, the
latch’s next state is undefined.
Of course, in practice, it is next to impossible to manufacture two gates and make the two connections with
precisely the same delay. In addition, both S' and R' need to be de-asserted at exactly the same time. Nevertheless, if
this circuit is used in controlling some mission-critical device, we don’t want even this slim chance to happen.
In order to avoid this non-deterministic behavior, we must make sure that the two inputs are never de-asserted at
the same time. Note that we do want the situation when both of them are de-asserted, as in times t1 and t3, so that the
circuit can remember its current content. We want to de-assert one input after de-asserting the other, but just not deasserting both of them at exactly the same time. In practice, it is very difficult to guarantee that these two signals are
never de-asserted at the same time, so we relax the condition slightly by not having both of them asserted together.
In other words, if one is asserted, then the other one cannot be asserted. Therefore, if both of them are never asserted
simultaneously, then they cannot be de-asserted at the same time. A minor side benefit for not having both of them
asserted together is that Q and Q' are never equal to each other. Recall that, from the names that we have given these
two nodes, we do want them to be inverses of each other.
From the above analysis, we obtain the truth table in Figure 6.2(b) for the NAND implementation of the SR latch.
In the truth table, Q and Qnext actually represent the same point in the circuit. The difference is that Q is the current
value at that point, while Qnext is the new value to be updated in the next time period. Another way of looking at it is
that Q is the input to a gate, and Qnext is the output from a gate. In other words, the signal Q goes into a gate,
propagates through the two gates, and arrives back at Q as the new signal Qnext. Figure 6.2(c) shows the logic symbol
for the SR latch. 178 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops The SR latch can also be implemented using NOR gates, as shown in Figure 6.3(a). The truth table for this
implementation is shown in Figure 6.3(b). From the truth table, we see that the main difference between this
implementation and the NAND implementation is that for the NOR implementation, the S and R inputs are active-high,
so that setting S to 1 will set the latch, and setting R to 1 will reset the latch. However, just like the NAND
implementation, the latch is set when Q = 1, and reset when Q = 0. The latch remembers its previous state when S =
R = 0. When S = R = 1, both Q and Q' are 0. The logic symbol for the SR latch using NOR implementation is shown
in Figure 6.3(c).
R Q Q' S
(a) S
0
0
0
1
1 R
0
0
1
0
1 Q
0
1
×
×
× Qnext
0
1
0
1
0 Qnext'
1
0
1
0
0 (b) S Q R Q' (c) Figure 6.3 SR latch: (a) circuit using NOR gates; (b) truth table; (c) logic symbol. 6.3 SR Latch with Enable The SR latch is sensitive to its inputs all the time. In other words, Q will always change when either S or R is
asserted. It is sometimes useful to be able to disable the inputs so that asserting them will not cause the latch to
change state but to keep its current state. Of course, this is achieved by de-asserting both S and R. Hence, what we
want is just one enable signal that will de-assert both S and R. The SR latch with enable (also known as a gated SR
latch) shown in Figure 6.4(a) accomplishes this by adding two extra NAND gates to the original NAND gate
implementation of the latch. These two new NAND gates are controlled by the enable input, E, which determines
whether the latch is enabled or disabled. When E = 1, the circuit behaves like the normal NAND implementation of
the SR latch except that the new S and R inputs are active-high rather than active-low. When E = 0, then S' = R' = 1,
and the latch will remain in its previous state regardless of the S and R inputs. The truth table and the logic symbol
for the SR latch with enable is shown in Figure 6.4(b) and (c), respectively.
A typical operation of the latch is shown in the sample trace in Figure 6.4(d). Between t0 and t1, E = 0, so
changing the S and R inputs do not affect the output. Between t1 and t2, E = 1, and the trace is similar to the trace of
Figure 6.2(d) except that the input signals are inverted. 179 Digital Logic and Microprocessor Design with VHDL S Chapter 6 – Latches and Flip-Flops E
0
0
1
1
1
1
1 S'
Q E
Q'
R R' S
×
×
0
0
0
1
1 R
×
×
0
0
1
0
1 (a) Q
0
1
0
1
×
×
× Qnext
0
1
0
1
0
1
1 Qnext'
1
0
1
0
1
0
1 (b)
E
S S
E
R Q R Q' Q Undefined Q' Undefined t0 t1 t2 (d) (c) Figure 6.4 SR latch with enable: (a) circuit using NAND gates; (b) truth table; (c) logic symbol; (d) sample trace. 6.4 D Latch Recall from Section 6.2 that the disadvantage with the SR latch is that we need to ensure that the two inputs, S
and R, are never de-asserted at exactly the same time, and we said that we can guarantee this by not having both of
them asserted. This situation is prevented in the D latch by adding an inverter between the original S' and R' inputs.
This way, S' and R' will always be inverses of each other, and so, they will never be asserted together. The circuit
using NAND gates and the inverter is shown in Figure 6.5(a). There is now only one input D (for data). When D = 0,
then S' = 1 and R' = 0, so this is similar to resetting the SR latch by making Q = 0. Similarly, when D = 1, then S' = 0
and R' = 1, and Q will be set to 1. From this observation, we see that Qnext always gets the same value as the input D,
and is independent of the current value of Q. Hence, we obtain the truth table for the D latch, as shown in Figure
6.5(b).
Comparing the truth table for the D latch shown in Figure 6.5(b) with the truth table for the SR latch shown in
Figure 6.2(b), it is obvious that we have eliminated not just one, but three rows, where S' = R'. The reason for adding
the inverter to the SR latch circuit was to eliminate the row where S' = R' = 0. However, we still need to have the
other two rows where S' = R' = 1 in order for the circuit to remember its current value. By not being able to set both
S' and R' to 1, this D latch circuit has now lost its ability to remember. Qnext cannot remember the current value of Q,
instead it will always follow D. The end result is like having a piece of wire where the output is the same as the
input!
S'
Q D R' Q'
(a) D
0
1 Q
×
× Qnext
0
1 Qnext'
1
0 (b) Figure 6.5 D latch: (a) circuit using NAND gates; (b) truth table; (c) logic symbol. 180 D Q
Q' (c) Digital Logic and Microprocessor Design with VHDL 6.5 Chapter 6 – Latches and Flip-Flops D Latch with Enable In order to make the D latch remember the current value, we need to connect Q (the current state value) back to
the input D, thus creating another feedback loop. Furthermore, we need to be able to select whether to loop Q back
to D or input a new value for D. Otherwise, like the bistable element, we will not be able to change the state of the
circuit. One way to achieve this is to use a 2-input multiplexer to select whether to feedback the current value of Q
or pass an external input back to D. The circuit for the D latch with enable (also known as a gated D latch) is
shown in Figure 6.6(a). The external input becomes the new D input, the output of the multiplexer is connected to
the original D input, and the select line of the multiplexer is the enable signal E.
When the enable signal E is asserted (E = 1), the external D input passes through the multiplexer, and so Qnext
(i.e., the output Q) follows the D input. On the other hand, when E is de-asserted (E = 0), the current value of Q
loops back as the input to the circuit, and so Qnext retains its last value independent of the D input.
When the latch is enabled, the latch is said to be open, and the path from the input D to the output Q is
transparent. In other words, Q follows D. Because of this characteristic, the D latch with enable circuit is often
referred to as a transparent latch. When the latch is disabled, it is closed, and the latch remembers its current state.
The truth table and the logic symbol for the D latch with enable are shown in Figure 6.6(b) and (c). A sample trace
for the operation of the D latch with enable is shown in Figure 6.6(d). Between t0 and t1, the latch is enabled with E
= 1, so the output Q follows the input D. Between t1 and t2, the latch is disabled, so Q remains stable even when D
changes.
An alternative way to construct the D latch with enable circuit is shown in Figure 6.7. Instead of using the 2input multiplexer, as shown in Figure 6.6(a), we start with the SR latch with enable circuit of Figure 6.4(a), and
connect the S and R inputs together with an inverter. The functional operations of these two circuits are identical. S'
Q D 0
y
1s E
0
0
1
1 Q' R' D
×
×
0
1 Q
0
1
×
× Qnext
0
1
0
1 Qnext'
1
0
1
0 E (b) (a)
E
D D Q Q
Q' E Q'
t0 (c) t1 t2 (d) Figure 6.6 D latch with enable: (a) circuit; (b) truth table; (c) logic symbol; (d) sample trace. 181 t3 Digital Logic and Microprocessor Design with VHDL D S Chapter 6 – Latches and Flip-Flops S'
Q E
Q'
R R' Figure 6.7 D latch with enable circuit using four NAND gates. 6.6 Clock Latches are known as level-sensitive because their outputs are affected by their inputs as long as they are
enabled. Their memory state can change during this entire time when the enable signal is asserted. In a computer
circuit, however, we do not want the memory state to change at various times when the enable signal is asserted.
Instead, we like to synchronize all of the state changes to happen at precisely the same moment and at regular
intervals. In order to achieve this, two things are needed: 1) a synchronizing signal, and 2) a memory circuit that is
not level-sensitive. The synchronizing signal, of course, is the clock, and the non-level-sensitive memory circuit is
the flip-flop.
The clock is simply a very regular square wave signal, as shown in Figure 6.8. We call the edge of the clock
signal when it changes from 0 to 1 the rising edge. Conversely, the falling edge of the clock is the edge when the
signal changes from 1 to 0. We will use the symbol
to denote the rising edge and
for the falling edge. In a
computer circuit, either the rising edge or the falling edge of the clock can be used as the synchronizing signal for
writing data into a memory element. This edge signal is referred to as the active edge of the clock. In all of our
examples, we will use the rising clock edge as the active edge. Therefore, at every rising edge, data will be clocked
or stored into the memory element.
A clock cycle is the time from one rising edge to the next rising edge or from one falling edge to the next falling
edge. The speed of the clock, measured in hertz (Hz), is the number of cycles per second. Typically, the clock speed
for a microprocessor in an embedded system runs around 20 MHz, while the microprocessor in a personal computer
runs upwards of 2 GHz and higher. A clock period is the time for one clock cycle (seconds per cycle), so it is just
the inverse of the clock speed.
The speed of the clock is determined by how fast a circuit can produce valid results. For example, a two-level
combinational circuit will have valid results at its output much sooner than, say, an ALU can. Of course, we want
the clock speed to be as fast as possible, but it can only be as fast as the slowest circuit in the entire system. We want
the clock period to be the time it takes for the slowest circuit to get its input from a memory element, operate on the
data, and then write the data back into a memory element. More will be said on this in later sections.
Figure 6.9 shows a VHDL description of a clock-divider circuit that roughly cuts a 25 MHz clock down to 1 Hz.
One Clock Cycle Falling Edge Rising Edge Figure 6.8 Clock signal.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY Clockdiv IS PORT (
Clk25Mhz: IN STD_LOGIC; 182 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops Clk: OUT STD_LOGIC);
END Clockdiv;
ARCHITECTURE Behavior OF Clockdiv IS
CONSTANT max: INTEGER := 25000000;
CONSTANT half: INTEGER := max/2;
SIGNAL count: INTEGER RANGE 0 TO max;
BEGIN
PROCESS
BEGIN
WAIT UNTIL Clk25Mhz'EVENT and Clk25Mhz = '1';
IF count < max THEN
count <= count + 1;
ELSE
count <= 0;
END IF;
IF count < half THEN
Clk <= '0';
ELSE
Clk <= '1';
END IF;
END PROCESS;
END Behavior;
Figure 6.9 VHDL behavioral description of a clock-divider circuit. 6.7 D Flip-Flop Unlike the latch, a flip-flop is not level-sensitive, but rather edge-triggered. In other words, data gets stored
into a flip-flop only at the active edge of the clock. An edge-triggered D flip-flop achieves this by combining in
series a pair of D latches. Figure 6.10(a) shows a positive-edge-triggered D flip-flop, where two D latches are
connected in series. A clock signal Clk is connected to the E input of the two latches: one directly, and one through
an inverter.
The first latch is called the master latch. The master latch is enabled when Clk = 0 because of the inverter, and
so QM follows the primary input D. However, the signal at QM cannot pass over to the primary output Q, because
the second latch (called the slave latch) is disabled when Clk = 0. When Clk = 1, the master latch is disabled, but the
slave latch is enabled so that the output from the master latch, QM, is transferred to the primary output Q. The slave
latch is enabled all the while that Clk = 1, but its content changes only at the rising edge of the clock, because once
Clk is 1, the master latch is disabled, and the input to the slave latch, QM, will be constant. Therefore, when Clk = 1
and the slave latch is enabled, the primary output Q will not change because the input QM is not changing.
The circuit shown in Figure 6.10(a) is called a positive-edge-triggered D flip-flop because the primary output Q
on the slave latch changes only at the rising edge of the clock. If the slave latch is enabled when the clock is low
(i.e., with the inverter output connected to the E of the slave latch), then it is referred to as a negative-edgetriggered flip-flop. The circuit is also referred to as a master-slave D flip-flop because of the two D latches used in
the circuit.
Figure 6.10(b) shows the operation table for the D flip-flop. The
symbol signifies the rising edge of the
clock. When Clk is either at 0 or 1, the flip-flop retains its current value (i.e., Qnext = Q). Qnext changes and follows
the primary input D only at the rising edge of the clock. The logic symbol for the positive-edge-triggered D flip-flop
is shown in Figure 6.10(c). The small triangle at the clock input indicates that the circuit is triggered by the edge of
the signal, and so it is a flip-flop. Without the small triangle, the symbol would be that for a latch. If there is a circle
in front of the clock line, then the flip-flop is triggered by the falling edge of the clock, making it a negative-edgetriggered flip-flop. Figure 6.10(d) shows a sample trace for the D flip-flop. Notice that when Clk = 0, QM follows D,
and the output of the slave latch, Q, remains constant. On the other hand, when Clk = 1, Q follows QM, and the
output of the master latch, QM, remains constant. 183 Digital Logic and Microprocessor Design with VHDL D QM Q E Clk D Master Q
Q' Clk
0
0
1
1 Q E Q' D Chapter 6 – Latches and Flip-Flops Q' Slave D
×
×
×
×
0
1 Q
0
1
0
1
×
× (a) Qnext
0
1
0
1
0
1 Qnext'
1
0
1
0
1
0 (b)
Clk D Q D
QM Clk Q' Q
t0 t1 t2 (c) t3 (d) Figure 6.10 Master-slave positive-edge-triggered D flip-flop: (a) circuit using D latches; (b) operation table; (c)
logic symbol; (d) sample trace.
Figure 6.11 compares the different operations between a latch and a flip-flop. In Figure 6.11(a), we have a D
latch with enable, a positive-edge-triggered D flip-flop, and a negative-edge-triggered D flip-flop, all having the
same D input and controlled by the same clock signal. Figure 6.11(b) shows a sample trace of the circuit’s
operations. Notice that the gated D latch, Qa, follows the D input as long as the clock is high (between times t0 and t1
and times t2 and t3). The positive-edge-triggered flip-flop, Qb, follows the D input only at the rising edge of the clock
at time t2, while the negative-edge-triggered flip-flop, Qc, follows the D input only at the falling edge of the clock at
times t1 and t3.
D D Q Clk E Qa Q' Clk
D Q Qb D
Qa Clk D
Clk Q' Q Qb
Qc Qc
t0 t1 t2 t3 Q' (b) (a) Figure 6.11 Comparison of a gated latch, a positive-edge-triggered flip-flop, and a negative-edge-triggered flip-flop:
(a) circuit; (b) sample trace. 6.7.1 * Alternative Smaller Circuit
Not all master-slave flip-flops are edge-triggered. For instance, using two SR latches to construct a master-slave 184 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops flip-flop results in a flip-flop that is level-sensitive. Conversely, an edged-triggered D flip-flop can be constructed
using SR latches instead of the master-slave D latches.
The circuit shown in Figure 6.12 shows how a positive-edge-triggered D flip-flop can be constructed using
three interconnected SR latches. The advantage of this circuit is that it uses only 6 NAND gates (26 transistors) as
opposed to 11 gates (38 transistors) for the master-slave D flip-flop shown in Figure 6.10(a). The operation of the
circuit is as follows. When Clk = 0, the outputs of gates 2 and 3 will be 1 (since 0 NAND x = 1). With n2 = n3 = 1, this
will keep the output latch (comprising of gates 5 and 6) in its current state. At the same time, n4 = D' since one input
to gate 4 is n3, which is a 1 (1 NAND x = x'). Similarly, n1 = D since n2 = 1, and the other input to gate 1 is n4, which
is D' (again 1 NAND x = x').
When Clk changes to 1, n2 will be equal to D' because 1 NAND n1 = n1', and n1 = D. Similarly, n3 will be equal to
D when Clk changes to 1 because the other two inputs to gate 3 are both D'. Therefore, if Clk = 1 and D = 0, then n2
(which is equal to D') will be 1 and n3 (which is equal to D) will be 0. With n2 = 1 and n3 = 0, this will de-assert S'
and assert R', thus resetting the output latch Q to 0. On the other hand, if Clk = 1 and D = 1, then n2 (which is equal
to D') will be 0 and n3 (which is equal to D) will be 1. This will assert S' and de-assert R'; thus setting the output
latch Q to 1. So at the rising edge of the Clk signal, Q will follow D.
The setting and resetting of the output latch occurs only at the rising edge of the Clk signal, because once Clk is
at a 1 and remains at a 1, changing D will not change n2 or n3. The reason, as noted in the previous paragraph, is that
n2 and n3 are always inverses of each other. Furthermore, the following argument shows that both n2 and n3 will
remain constant even if D changes. Let us first assume that n2 is a 0. If n2 = 0, then n3 (the output of gate 3) will
always be a 1 (since 0 NAND x = 1), regardless of what n4 (the third input to gate 3) may be. Hence, if n4 (the output
of gate 4) cannot affect n3, then D (the input to gate 4) also cannot affect either n2 or n3. On the other hand, if n2 = 1,
then n3 = 0 (n3 = n2'). With a 0 from n3 going to the input of gate 4, the output of gate 4 at n4 will always be a 1 (0
NAND x = 1), regardless of what D is. With the three inputs to gate 3 being all 1’s, n3 will continue to be 0.
Therefore, as long as Clk = 1, changing D will not change n2 or n3. And if n2 and n3 remain stable, then Q will also
remain stable for the entire time that Clk is 1.
Set Latch
1 n1 (D) 2 n2 S'
5 Q 6 Q' (D') Clk
(D) n3 3 R'
n4 (D') 4 D Output Latch Reset Latch
Figure 6.12. Positive-edge-triggered D flip-flop. 6.8 D Flip-Flop with Enable
So far, with the construction of the different memory elements, it seems like every time we add a new feature
we have also lost a feature that we need. The careful reader will have noticed that, in building the D flip-flop, we
have again lost the most important property of a memory element—it can no longer remember its current content!
At every active edge of the clock, the D flip-flop will load in a new value. So how do we get it to remember its
current value and not load in a new value?
The answer, of course, is exactly the same as what we did with the D latch, and that is by adding an enable
input, E, through a 2-input multiplexer, as shown in Figure 6.13(a). When E = 1, the primary input D signal will
pass to the D input of the flip-flop, thus updating the content of the flip-flop at the active edge. When E = 0, the 185 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops current content of the flip-flop at Q is passed back to the D input of the flip-flop, thus keeping its current value.
Notice that changes to the flip-flop value occur only at the active edge of the clock. Here, we are using the rising
edge as the active edge. The operation table and the logic symbol for the D flip-flop with enable is shown in Figure
6.13(b) and (c) respectively. D 0
y
1s D E
Clk Clk Q
Q' Q
Q' Clk
0
0
1
1 E
×
×
×
×
0
0
1
1 D
×
×
×
×
×
×
0
1 Q
0
1
0
1
0
1
×
× Qnext
0
1
0
1
0
1
0
1 Qnext'
1
0
1
0
1
0
1
0 (b) (a) D Q Clk
E Q' (c) Figure 6.13 D flip-flop with enable: (a) circuit; (b) operation table; (c) logic symbol. 6.9 Asynchronous Inputs Flip-flops (as we have seen so far) change states only at the rising or falling edge of a synchronizing clock
signal. Many circuits require the initialization of flip-flops to a known state that is independent of the clock signal.
Sequential circuits that change states whenever a change in input values occurs that is independent of the clock are
referred to as asynchronous sequential circuits. Synchronous sequential circuits, on the other hand, change states
only at the active edge of the clock signal. Asynchronous inputs usually are available for both flip-flops and latches,
and they are used to either set or clear the storage element’s content that is independent of the clock.
Figure 6.14(a) shows a gated D latch with asynchronous active-low Set' and Clear' inputs, and (b) is the logic
symbol for it. Figure 6.14(c) is the circuit for the D edge-triggered flip-flop with asynchronous Set' and Clear'
inputs, and (d) is the logic symbol for it. When Set' is asserted (set to 0) the content of the storage element is set to 1
immediately (i.e., without having to wait for the next rising clock edge), and when Clear' is asserted (set to 0) the
content of the storage element is set to 0 immediately. Set'
D S Set' Q
D Q E
E
Q'
Clear' Q'
Clear' R
(b) (a) 186 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops Set' Q Set'
D Q Clk
Q' Clk Q'
Clear' D
Clear'
(d) (c) Figure 6.14 Storage elements with asynchronous inputs: (a) D latch with active-low set and clear; (b) logic symbol
for (a); (c) D edge-triggered flip-flop with active-low set and clear; (d) logic symbol for (c). 6.10 Description of a Flip-Flop
Combinational circuits can be described with either a truth table or a Boolean equation. For describing the
operation of a flip-flop or any sequential circuit in general, we use a characteristic table, a characteristic equation, a
state diagram, or an excitation table, as discussed in the following subsections. 6.10.1 Characteristic Table
The characteristic table specifies the functional behavior of the flip-flop. It is a simplified version of the flipflop’s operational table by only listing how the state changes at the active clock edge. The table has the flip-flop’s
input signal(s) and current state (Q) listed in the input columns, and the next state (Qnext) listed in the output column.
Qnext' is always assumed to be the inverse of Qnext, so it is not necessary to include this output column. The clock
signal is also not included in the table, because it is a signal that we do not want to modify. Nevertheless, the clock
signal is always assume to exist. Furthermore, since all state changes for a flip-flop (i.e., changes to Qnext) occur at
the active edge of the clock, therefore it is not necessary to list the situations from the operation table for when the
clock is at a constant value.
The characteristic table for the D flip-flop is shown in Figure 6.15(a). It has two input columns (the input signal
D, and the current state Q) and one output column for Qnext. From the operation table for the D flip-flop shown in
Figure 6.10(b), we see that there are only two rows where Qnext is affected during the rising clock edge. Hence, these
are the only two rows inserted into the characteristic table.
The characteristic table is used in the analysis of sequential circuits to answer the question of what is the next
state, Qnext, when given the current state, Q, and input signals (D in the case of the D flip-flop). 6.10.2 Characteristic Equation
The characteristic equation is simply the Boolean equation that is derived directly from the characteristic
table. Like the characteristic table, the characteristic equation specifies the flip-flop’s next state, Qnext, as a function
of its current state, Q, and input signals. The D flip-flop characteristic table has only one 1-minterm, which results in
the simple characteristic equation for the D flip-flop shown in Figure 6.15(b). 187 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops 6.10.3 State Diagram
A state diagram is a graph with nodes and directed edges connecting the nodes, as shown in Figure 6.15(c).
The state diagram graphically portrays the operation of the flip-flop. The nodes are labeled with the states of the
flip-flop, and the directed edges are labeled with the input signals that cause the transition to go from one state of the
flip-flop to the next. Figure 6.15(c) shows the state diagram for the D flip-flop. It has two states, Q = 0 and Q = 1,
which correspond to the two values that the flip-flop can contain. The operation of the D flip-flop is such that when
it is in state 0, it will change to state 1 if the input D is a 1; otherwise, if the input D is a 0, then it will remain in state
0. Hence, there is an edge labeled D = 1 that goes from state Q = 0 to Q = 1, and a second edge labeled D = 0 that
goes from state Q = 0 back to itself. Similarly, when the flip-flop is in state 1, it will change to state 0 if the input D
is a 0; otherwise, it will remain in state 1. These two conditions correspond to the remaining two edges that go out
from state Q = 1 in the state diagram. 6.10.4 Excitation Table
The excitation table is like the mirror image of the characteristic table by exchanging the input signal
column(s) with the output (Qnext) column. The excitation table shows what the flip-flop’s inputs should be in order to
change from the flip-flop’s current state to the next state desired. In other words, the excitation table answers the
question of what the flip-flop’s inputs should be when given the current state that the flip-flop is in and the next state
that we want the flip-flop to go to. This table is used in the synthesis of sequential circuits.
Figure 6.15(d) shows the excitation table for the D flip-flop. As can be seen, this table can be obtained directly
from the state diagram. For example, using the state diagram of the D flip-flop from Figure 6.15(c), if the current
state is Q = 0 and we want the next state to be Qnext = 0, then the D input must be a 0 as shown by the label on the
edge that goes from state 0 back to itself. On the other hand, if the current state is Q = 0 and we want the next state
to be Qnext = 1, then the D input must be a 1.
D
0
1 Q
×
× Qnext
0
1 Qnext = D
(b) (a)
Q
0
0
1
1 D=1
D=0
Q=0 Q=1
D=1
D=0
(c) Qnext
0
1
0
1 D
0
1
0
1 (d) Figure 6.15 Description of a D flip-flop: (a) characteristic table; (b) characteristic equation; (c) state diagram; (d)
excitation table. 6.11 * Timing Issues
So far in our discussion of latches and flip-flops, we have ignored timing issues and the effects of propagation
delays. In practice, timing issues are very important in the correct design of sequential circuits. Consider again the D
latch with enable circuit from Section 6.5 and redrawn in Figure 6.16(a). Signals from the inputs require some delay
to propagate through the gates and finally to reach the outputs.
Assuming that the propagation delay for the inverter is one nanosecond (ns) and 2 ns for the NAND gates, the
timing trace diagram would look like Figure 6.16(b) with the signal delays taken into consideration. The arrows
denote which signal edge causes another signal edge. The number next to an arrow denotes the number of
nanoseconds in delay for the resulting signal to change. 188 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops At time t1, signal D drops to 0. This causes R to rise to 1 after a 1 ns delay through the inverter. The D edge also
causes S' to rise to 1, but after a delay of 2 ns through the NAND gate. After that, R' drops to 0 2 ns after R rises to 1.
This in turn causes Q' to rise to 1 after 2 ns, followed by Q dropping to 0.
At time t2, signal E drops to 0, disabling the circuit. As a result, when D rises to 1 at time t3, both Q and Q' are
not affected.
At time t4, signal E rises to 1 and re-enables the circuit. This causes S' to drop to 0 after 2 ns. R' remains
unchanged at 1 since the two inputs to the NAND gate, E and R, are 1 and 0, respectively. With S' asserted and R' deasserted, the latch is set with Q rising to 1 2 ns after S' drops to a 0. This is followed by Q' dropping to 0 after
another 2 ns.
E
D/S
D S R S'
Q 1 1 2 2 S'
2 R' E 2 2 Q
2 Q'
R 2 2 Q' R' t1 (a) t2 t3 t4 (b) Figure 6.16 D latch with enable: (a) circuit; (b) timing diagram with delays.
Furthermore, for the D latch circuit to latch in the data from input D correctly, there is a critical window of time
right before and right after the falling edge of the enable signal, E, that must be observed. Within this time frame, the
input signal, D, must not change. As shown in Figure 6.17, the time before the falling edge of E is referred to as the
setup time, tsetup, and the time after the falling edge of E is referred to as the hold time, thold. The length of these two
times is dependent on the implementation and manufacturing process and can be obtained from the component data
sheet.
E
D
tsetup thold Figure 6.17 Setup and hold times for the gated D latch. 6.12 Car Security System—Version 2
In Section 2.9, we designed a combinational circuit for a car security system where the siren will come on when
the master switch is on and either the door switch or the vibration switch is also on. However, as soon as both the
door switch and the vibration switch are off, the siren will turn off immediately even though the master switch is still
on. In reality, what we really want is to have the siren remain on, even after both the door and vibration switches are
off. In order to do so, we need to remember the state of the siren. In other words, for the siren to remain on, it should
be dependent not only on whether the door or the vibration switch is on, but also on the fact that the siren is
currently on.
We can use the state of a SR latch to remember the state of the siren (i.e., the output of the latch will drive the
siren). The state of the latch is driven by the conditions of the input switches. The modified circuit, as shown in
Figure 6.18, has an SR latch (in addition to its original combinational circuit) for remembering the current state of 189 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops the siren. The latch is set from the output of the combinational circuit. The latch’s reset is connected to the master
switch so that the siren can be turned off immediately.
A sample timing trace of the operation of this circuit is shown in Figure 6.19. At time 0, the siren is off, even
though the door switch is on, because the master switch is off. At time 300 ns, the siren is turned on by the door
switch since the master switch is also on. At time 500 ns, both the door and the vibration switches are off, but the
siren is still on because it was turned on previously. The siren is turned off by the master switch at time 600 ns.
D
V
M S S'
Q Siren R' Figure 6.18 Modified car security system circuit with memory. Figure 6.19 Sample timing trace of the modified car security system circuit with memory. 6.13 VHDL for Latches and Flip-Flops
6.13.1 Implied Memory Element
VHDL does not have any explicit object for defining a memory element. Instead, the semantics of the language
provide for signals to be interpreted as a memory element. In other words, the memory element is declared
depending on how these signals are assigned.
Consider the VHDL code in Figure 6.20. If Enable is 1, then Q gets the value of D; otherwise, Q gets a 0. In this
code, Q is assigned a value for all possible outcomes of the test in the IF statement. With this construct, a
combinational circuit is produced.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY no_memory_element IS PORT (
D, Enable: IN STD_LOGIC;
Q: OUT STD_LOGIC);
END no_memory_element;
ARCHITECTURE Behavior OF no_memory_element IS
BEGIN
PROCESS(D, Enable)
BEGIN
IF Enable = '1' THEN
Q <= D;
ELSE 190 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops Q <= '0';
END IF;
END PROCESS;
END Behavior;
Figure 6.20 Sample VHDL description of a combinational circuit.
If we remove the ELSE and the statement in the ELSE part, as shown in Figure 6.21, then we have a situation
where no value is assigned to Q if Enable is not 1. The key point here is that the VHDL semantics stipulate that, in
cases where the code does not specify a value of a signal, the signal should retain its current value. In other words,
the signal must remember its current value, and in order to do so, a memory element is implied. 6.13.2 VHDL Code for a D Latch with Enable
Figure 6.21 shows the VHDL code for a D latch with enable. If Enable is 1, then Q gets the value of D.
However, if Enable is not 1, the code does not specify what Q should be; therefore, Q retains its current value by
using a memory element. This code produces a latch and not a flip-flop, because Q follows D as long as Enable is 1
and not only at the active edge of the Enable signal. The process sensitivity list includes both D and Enable, because
either one of these signals can cause a change in the value of the Q output.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY D_latch_with_enable IS PORT (
D, Enable: IN STD_LOGIC;
Q: OUT STD_LOGIC);
END D_latch_with_enable;
ARCHITECTURE Behavior OF D_latch_with_enable IS
BEGIN
PROCESS(D, Enable)
BEGIN
IF Enable = '1' THEN
Q <= D;
END IF;
END PROCESS;
END Behavior;
Figure 6.21 VHDL code for a D latch with enable. 6.13.3 VHDL Code for a D Flip-Flop
Figure 6.22 shows the behavioral VHDL code for a positive-edge-triggered D flip-flop. The only difference
here is that Q follows D only at the rising edge of the clock, and it is specified here by the condition “Clock' EVENT
AND Clock = '1'.” The ' EVENT attribute refers to any changes in the qualifying Clock signal. Therefore, when this
happens and the resulting Clock value is a 1, we have, in effect, a condition for a positive or rising clock edge.
Again, the code does not specify what is assigned to Q when the condition in the IF statement is false, so it implies
the use of a memory element. Note also that the process sensitivity list contains only the clock signal, because it is
the only signal that can cause a change in the Q output.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY D_flipflop IS PORT (
D, Clock: IN STD_LOGIC;
Q: OUT STD_LOGIC); 191 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops END D_flipflop;
ARCHITECTURE Behavior OF D_flipflop IS
BEGIN
PROCESS(Clock)
-- sensitivity list is used
BEGIN
IF Clock’EVENT AND Clock = '1' THEN
Q <= D;
END IF;
END PROCESS;
END Behavior;
Figure 6.22 Behavioral VHDL code for a positive-edge-triggered D flip-flop using an IF statement.
Another way to describe a flip-flop is to use the WAIT statement instead of the IF statement as shown in Figure
6.23. When execution reaches the WAIT statement, it stops until the condition in the statement is true before
proceeding. The WAIT statement, when used in a process block for synthesis, must be the first statement in the
process. Note also that the process sensitivity list is omitted, because the WAIT statement implies that the sensitivity
list contains only the clock signal.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY D_flipflop IS PORT (
D, Clock: IN STD_LOGIC;
Q: OUT STD_LOGIC);
END D_flipflop;
ARCHITECTURE Behavioral OF D_flipflop IS
BEGIN
PROCESS
-- sensitivity list is not used if WAIT is used
BEGIN
WAIT UNTIL Clock’EVENT AND Clock = '0';
-- negative edge triggered
Q <= D;
END PROCESS;
END Behavioral;
Figure 6.23 Behavioral VHDL code for a negative-edge-triggered D flip-flop using a WAIT statement.
Alternatively, we can write a structural VHDL description for the positive-edge-triggered D flip-flop, as shown
in Figure 6.24. This VHDL code is based on the circuit for a positive-edge-triggered D flip-flop, as given in Figure
6.12.
The simulation trace for the positive-edge-triggered D flip-flop is shown in Figure 6.25. In the trace, before the
first rising edge of the clock at time 100 ns, both Q and Q' (QN) are undefined because nothing has been stored in
the flip-flop yet. Immediately after this rising clock edge at 100 ns, Q gets the value of D, and QN gets the inverse.
At 200 ns, D changes to 1, but Q does not follow D immediately but is delayed until the next rising clock edge at
300 ns. At the same time, QN drops to 0. At 400 ns when D drops to 0, Q again follows it at the next rising clock
edge at 500 ns.
-- define the operation of the 2-input NAND gate
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY NAND_2 IS PORT (
I0, I1: IN STD_LOGIC;
O: OUT STD_LOGIC); 192 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops END NAND_2;
ARCHITECTURE Dataflow_NAND2 OF NAND_2 IS
BEGIN
O <= I0 NAND I1;
END Dataflow_NAND2;
-- define the structural operation of the SR latch
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY SRlatch IS PORT (
SN, RN: IN STD_LOGIC;
Q, QN: BUFFER STD_LOGIC);
END SRlatch;
ARCHITECTURE Structural_SRlatch OF SRlatch IS
COMPONENT NAND_2 PORT (
I0, I1 : IN STD_LOGIC;
O : OUT STD_LOGIC);
END COMPONENT;
BEGIN
U1: NAND_2 PORT MAP (SN, QN, Q);
U2: NAND_2 PORT MAP (Q, RN, QN);
END Structural_SRlatch;
-- define the operation of the 3-input NAND gate
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY NAND_3 IS PORT (
I0, I1, I2: IN STD_LOGIC;
O: OUT STD_LOGIC);
END NAND_3;
ARCHITECTURE Dataflow_NAND3 OF NAND_3 IS
BEGIN
O <= NOT (I0 AND I1 AND I2);
END Dataflow_NAND3;
-- define the structural operation of the D flip-flop
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY positive_edge_triggered_D_flipflop IS PORT (
D, Clock: IN STD_LOGIC;
Q, QN: BUFFER STD_LOGIC);
END positive_edge_triggered_D_flipflop;
ARCHITECTURE StructuralDFF OF positive_edge_triggered_D_flipflop IS
SIGNAL N1, N2, N3, N4: STD_LOGIC;
COMPONENT SRlatch PORT (
SN, RN: IN STD_LOGIC;
Q, QN: BUFFER STD_LOGIC);
END COMPONENT; 193 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops COMPONENT NAND_2 PORT (
I0, I1: IN STD_LOGIC;
O: OUT STD_LOGIC);
END COMPONENT;
COMPONENT NAND_3 PORT (
I0, I1, I2: IN STD_LOGIC;
O: OUT STD_LOGIC);
END COMPONENT;
BEGIN
U1: SRlatch PORT MAP (N4, Clock, N1, N2);
U2: SRlatch PORT MAP (N2, N3, Q, QN);
U3: NAND_3 PORT MAP (N2, Clock, N4, N3);
U4: NAND_2 PORT MAP (N3, D, N4);
END StructuralDFF; ----- set latch
output latch
reset latch
reset latch Figure 6.24 Structural VHDL code for a positive-edge-triggered D flip-flop. Figure 6.25 Simulation trace for the positive-edge-triggered D flip-flop. 6.13.4 VHDL Code for a D Flip-Flop with Enable and Asynchronous Set and Clear
Figure 6.26 shows the VHDL code for a positive-edge-triggered D flip-flop with enable and asynchronous
active-high set and clear inputs. The two asynchronous inputs are checked independently of the clock event. When
either the Set or the Clear input is asserted with a 1 (active-high), Q is immediately set to 1 or 0, respectively,
independent of the clock. If Enable is asserted with a 1, then Q follows D at the rising edge of the clock; otherwise,
Q keeps its previous content. Figure 6.27 shows the simulation trace for this flip-flop. Notice in the trace that when
either Set or Clear is asserted (at 100 ns and 200 ns, respectively) Q changes immediately. However, when Enable is
asserted at 400 ns, Q doesn’t follow D until the next rising clock edge at 500 ns. Similarly, when D drops to 0 at 600
ns, Q doesn’t change immediately but drops at the next rising edge at 700 ns. At 800 ns, when D changes to a 1, Q
does not follow the change at the next rising edge at 900 ns, because Enable is now de-asserted.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY d_ff IS PORT (
Clock: IN STD_LOGIC;
Enable: IN STD_LOGIC;
Set: IN STD_LOGIC;
Clear: IN STD_LOGIC;
D: IN STD_LOGIC;
Q: OUT STD_LOGIC);
END d_ff;
ARCHITECTURE Behavioral OF d_ff IS
BEGIN 194 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops PROCESS(Clock,Set,Clear)
BEGIN
IF (Set = '1') THEN
Q <= '1';
ELSIF (Clear = '1') THEN
Q <= '0';
ELSIF (Clock'EVENT AND Clock = '1') THEN
IF Enable = '1' THEN
Q <= D;
END IF;
END IF;
END PROCESS;
END Behavioral;
Figure 6.26 VHDL code for a positive-edge-triggered D flip-flop with active-high enable and asynchronous set and
clear inputs. Figure 6.27 Simulation trace for the positive-edge-triggered D flip-flop with active-high enable and asynchronous
set and clear inputs. 6.14 * Other Flip-Flop Types
There are basically four main types of flip-flops: D, SR, JK, and T. The major differences in these flip-flop
types are in the number of inputs they have and how they change states. Like the D flip-flop, each type can also have
different variations, such as active-high or -low inputs, whether they change state at the rising or falling edge of the
clock signal, and whether they have any asynchronous inputs. Any given sequential circuit can be built using any of
these types of flip-flops or combinations of them. However, selecting one type of flip-flop over another type to use
in a particular circuit can affect the overall size of the circuit. Today, sequential circuits are designed primarily with
D flip-flops only because of their simple operation. Of the four flip-flop’s characteristic equations, the characteristic
equation for the D flip-flop is the simplest. 6.14.1 SR Flip-Flop
Like SR latches, SR flip-flops are useful in control applications where we want to be able to set or reset the data
bit. However, unlike SR latches, SR flip-flops change their content only at the active edge of the clock signal.
Similar to SR latches, SR flip-flops can enter an undefined state when both inputs are asserted simultaneously.
When the two inputs are de-asserted, then the next state is the same as the current state. The characteristic table,
characteristic equation, state diagram, circuit, logic symbol, and excitation table for the SR flip-flop are shown in
Figure 6.28.
The SR flip-flop truth table shown in Figure 6.28(a) is for active-high set and reset signals. Hence, the flip-flop
state, Qnext, is set to 1 when S is asserted with a 1, and Qnext is reset to 0 when R is asserted with a 1. When both S and
R are de-asserted with a 0, the flip-flop remembers its current state. From the truth table, we get the following Kmap for Qnext, which results in the characteristic equation shown in Figure 6.28(b). 195 Digital Logic and Microprocessor Design with VHDL Qnext Chapter 6 – Latches and Flip-Flops SR
00 Q 01 11
× 1 1 1 × 0 10 1 S R'Q Notice that the SR flip-flop circuit shown in Figure 6.28 (d) uses the D flip-flop. The signal for asserting the D
input of the flip-flop is generated by the combinational circuit that is derived from the characteristic equation of the
SR flip-flop, namely D = Qnext = S + R'Q.
S
0
0
0
0
1
1
1
1 R
0
0
1
1
0
0
1
1 Q
0
1
0
1
0
1
0
1 Qnext
0
1
0
0
1
1
×
× Qnext'
1
0
1
1
0
0
×
× Qnext = S + R'Q (b) (a)
SR=10
SR=00 or 01
Q=0 R
S
Clk Q=1 (c) (d)
Q
0
0
1
1 Q Clk
R Q' Clk SR=01 S Q Q' SR=00 or 10 Q D Q' (e) Qnext
0
1
0
1 S
0
1
0
× R
×
0
1
0 (f) Figure 6.28 SR flip-flop: (a) characteristic table; (b) characteristic equation; (c) state diagram; (d) circuit; (e) logic
symbol; (f) excitation table. 6.14.2 JK Flip-Flop
The operation of the JK flip-flop is very similar to the SR flip-flop. The J input is just like the S input in the SR
flip-flop in that, when asserted, it sets the flip-flop. Similarly, the K input is like the R input where it resets the flipflop when asserted. The only difference is when both inputs, J and K, are asserted. For the SR flip-flop, the next
state is undefined; whereas, for the JK flip-flop, the next state is the inverse of the current state. In other words, the
JK flip-flop toggles its state when both inputs are asserted. The characteristic table, characteristic equation, state
diagram, circuit, logic symbol, and excitation table for the JK flip-flop are shown in Figure 6.29. 196 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops 6.14.3 T Flip-Flop
The T flip-flop has one input, T (which stands for toggle), in addition to the clock. When T is asserted (T = 1),
the flip-flop state toggles back and forth at each active edge of the clock, and when T is de-asserted, the flip-flop
keeps its current state. The characteristic table, characteristic equation, state diagram, circuit, logic symbol, and
excitation table for the T flip-flop are shown in Figure 6.30.
J
0
0
0
0
1
1
1
1 K
0
0
1
1
0
0
1
1 Q
0
1
0
1
0
1
0
1 Qnext
0
1
0
0
1
1
1
0 Qnext'
1
0
1
1
0
0
0
1 Qnext = K'Q + JQ' (b) (a)
JK=10 or 11
JK=00 or 01
Q=0 J
K Q=1
JK=00 or 10 Q D Clk Clk Q Q' Q' JK=01 or 11
(c) J (d)
Q
0
0
1
1 Q Clk
K Q' (e) Qnext
0
1
0
1 J
0
1
×
× K
×
×
1
0 (f) Figure 6.29 JK flip-flop: (a) characteristic table; (b) characteristic equation; (c) state diagram; (d) circuit; (e) logic
symbol; (f) excitation table. 197 Digital Logic and Microprocessor Design with VHDL T
0
0
1
1 Q
0
1
0
1 Qnext
0
1
1
0 Chapter 6 – Latches and Flip-Flops Qnext'
1
0
0
1 Qnext = TQ' + T'Q = T ⊕ Q (a) (b) T=1
T=0
Q=0 Q Q Q' D T
Clk Q=1 Q' Clk T=0
T=1
(c) T (d)
Q
0
0
1
1 Q Clk
Q' (e) Qnext
0
1
0
1 T
0
1
1
0 (f) Figure 6.30 T flip-flop: (a) characteristic table; (b) characteristic equation; (c) state diagram; (d) circuit; (e) logic
symbol; (f) excitation table. 6.15 Summary Checklist
Feedback loop
Bistable element
Latch
Flip-flop
Clock
Level-sensitive, active edge, rising / falling edge, clock cycle
SR latch
SR latch with enable
D latch
D latch with enable
D flip-flop
Characteristic table, characteristic equation, state diagram circuit, excitation table
Asynchronus inputs
VHDL implied memory element
SR flip-flop
Characteristic table, characteristic equation, state diagram circuit, excitation table
JK flip-flop
Characteristic table, characteristic equation, state diagram circuit, excitation table
T flip-flop
Characteristic table, characteristic equation, state diagram circuit, excitation table 198 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops 6.16 Problems
1. Draw the SR latch with enable similar to that shown in Figure 6.4 but using NOR gates to implement the SR
latch. Derive the truth table for this circuit.
Answer
R R Q
E
Q'
S E
0
0
1
1
1
1
1
2. S S
×
×
0
0
0
1
1 R
×
×
0
0
1
0
1 Q
0
1
0
1
×
×
× Qnext'
1
0
1
0
1
0
0 Qnext
0
1
0
1
0
1
0 Draw the D latch using NOR gates
Answer
Q Q' D
3. Draw the D latch with enable similar to the circuit in Figure 6.6 (a) but use NAND gates instead of the
multiplexer.
Answer
D S S'
Q E Q'
R R' 4. Draw the master-slave negative edge-triggered D flip-flop circuit. 5. Derive the truth table for the negative edge-triggered D flip-flop. 6. Draw the circuit for the SR flip-flop with enable. 199 Digital Logic and Microprocessor Design with VHDL Chapter 6 – Latches and Flip-Flops 7. Derive the truth table for the SR flip-flop with enable. 8. Write the behavioral VHDL code for the SR flip-flop with enable using the IF clock’EVENT statement. 9. Do questions 6, 7, and 8 for the JK flip-flop. 10. Do questions 6, 7, and 8 for the T flip-flop.
11. Complete the following timing diagram for the following circuit. Assume that the signal delay through the NOR
gates is 3 nanoseconds, and the delay through the NOT gate is 1 nanosecond.
D' Q Q' D
D D' Q Q' Time (ns) 0 1 2 3 200 4 5 6 7 8 Chapter 7 Sequential Circuits Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits In the previous chapter, we looked at the design and operation of flip-flops – the most fundamental memory
element used in microprocessor circuits. We saw that a single flip-flop is capable of remembering only one bit of
information or one bit of history. In order for a sequential circuit to remember more inputs and a longer history, the
circuit must contain more flip-flops. This collection of (D) flip-flops used to remember the complete history of past
inputs is referred to as the state memory. The entire content of the state memory at a particular time forms a binary
encoding that represents the complete history of inputs up to that time. We refer to this binary encoding in the state
memory at one particular instance of time as the state of the system at that time.
The output signals from a sequential circuit are generated by the output logic circuit. Recall that the outputs of
sequential circuits are dependent on their past and current inputs. Since all the inputs are “remembered” in the form
of states in the state memory, therefore, we can say that the outputs are dependent on the content of the state
memory. Therefore, the output logic is simply a combinational circuit that is dependent on the content of the state
memory, and may or may not be dependent on the current inputs. The output signals that the output logic generates
constitute the actions or operations that are performed by the sequential circuit. Hence, a sequential circuit can
perform different operations in different states simply by generating different output signals.
If we want a sequential circuit to perform, say, four different operations, then we will need four states – one
operation per state. Of course if several operations can be performed in parallel, then we can assign them to one
state. But for now, just to keep things simple, we will simply assign one operation per state. Furthermore, there
might be an operation where we may want to repeat it for, say, a hundred times. Instead of assigning this same
operation to one hundred different states, we will want to use just one state, and have some form of looping
capabilities to repeat that state a hundred times.
Thus, a sequential circuit operates by transitioning from one state to the next, generating different output
signals. The part inside a sequential circuit that is responsible for determining what next state to go to is called the
next-state logic circuit. Based on the current state that the system is in (i.e., the past inputs), and the current inputs,
the next-state logic will determine what the next state should be. This statement, in fact, is equivalent to saying that
the outputs are dependent on the past and current inputs, since a state is used to remember the past inputs, and it also
determines the outputs to be generated. The next-state logic, however, is just a combinational circuit that takes the
contents of the state memory flip-flops and the current inputs as its inputs. The outputs from the next-state logic are
used to change the contents of the state memory flip-flops. The circuit changes state when the contents of the state
memory change, and this happens at the active edge of every clock cycle since values are written into a flip-flop at
the active clock edge.
The speed at which a sequential circuit sequences through the states is determined by the speed of the clock
signal. The state memory flip-flops are always enabled, so at every active edge of the clock, a new value is stored
into the flip-flops. The limiting factor for the clock speed is in the time that it takes to perform all the operations that
are assigned to a particular state. All data operations assigned to a state must finish their operations within one clock
period so that the results can be written into registers at the next active clock edge.
A sequential circuit is also known as a finite-state machine (FSM) because the size of the state memory is
finite, and therefore, the total number of different possible states is also finite. A sequential circuit is like a machine
that operates by stepping through a sequence of states. Although there is only a finite number of different states, the
FSM can, however, go to any of these states as many times as necessary. Hence, the sequence of states that the FSM
can go through can be infinitely long.
The control unit inside the microprocessor is a finite-state machine, therefore, in order to be able to construct a
microprocessor, we need to understand the construction and operation of FSMs. In this chapter, we will first look at
how to precisely describe the operation of a finite-state machine using state diagrams. Next, we will look at the
analysis and synthesis of finite-state machines. 7.1 Finite-State-Machine (FSM) Models In the introduction, we mentioned that the output logic circuit is dependent on the content of the state memory,
and may or may not be dependent on the current inputs. The fact that the output logic may or may not be dependent
on the current inputs gives rise to two different FSM models.
Figure 7.1 (a) shows the general schematic for the Moore FSM where its outputs are dependent only on its
current state, i.e. on the content of the state memory. Figure 7.1 (b) shows the general schematic for the Mealy FSM 202 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits where its outputs are dependent on both the current state of the machine and also the current inputs. The only
difference between the two figures is that for the Moore FSM, the output logic circuit only has the current state as its
input, whereas, for the Mealy FSM, the output logic circuit has both the current state and the input signals as its
inputs. In both models, we see that the inputs to the next-state logic are the primary input signals and the current
state of the machine. The next-state logic circuit generates values to change the contents of the state memory. Since
the state memory is made up of one or more D flip-flops, and the content of the D flip-flop changes to whatever
value is at its D input at the next active clock edge, therefore, to change a state, the next-state logic circuit simply
has to generate values for all the D inputs for all the flip-flops. These D input values are referred to as the excitation
values, since they “excite” or cause the D flip-flops to change states. input signals Next-state
Logic
Circuit excitation
Clock State
Memory
register current state Output
Logic
Circuit output signals (a) input signals Next-state
Logic
Circuit excitation
Clock State
Memory
register current state Output
Logic
Circuit output signals (b)
Figure 7.1. Finite-state machine models: (a) Moore FSM; (b) Mealy FSM.
Figure 7.2 (a) and (b) show a sample circuit of a Moore and Mealy FSM respectively. The two circuits are
identical except for their outputs. For the Moore FSM, the output circuit is a 2-input AND gate that gets its input
values from the outputs of the two D flip-flops. Remember that the state of the FSM is represented by the content of
the state memory, which are the contents of the flip-flops. The content (or state) of a flip-flop is represented by the
value at the Q (or Q' ) output. Hence, this circuit is only dependent on the current state of the machine.
For the Mealy FSM, the output circuit is a 3-input AND gate. In addition to getting its two inputs from the flipflops, the third input to this AND gate is connected to the primary input C. With this one extra connection, this output
circuit is dependent on both the current state and the input, thus making it a Mealy FSM.
For both circuits, the state memory consists of two D flip-flops. Having two flip-flops, four different
combinations of values can be represented. Hence, this finite-state machine can be in any one of four different states.
The state that this FSM will go to next depends on the value at the D inputs of the flip-flops.
Every flip-flop in the state memory requires a combinational circuit to generate a next-state value for its
input(s). Since we have two D flip-flops, each having one (D) input, therefore, the next-state logic circuit consists of
two combinational circuits; one for input D0 and one for D1. The inputs to these two combinational circuits are the
Q’s, which represent the current state of the flip-flops, and the primary input C. Notice that it is not necessary for the
input C to be an input to all the combinational circuits. In the sample circuit, only the bottom combinational circuit
is dependent on the input C. 203 Digital Logic and Microprocessor Design with VHDL C Chapter 7 - Sequential Circuits Input D1 Q1 Y Clk
Clear D0 Output
Q'1 Q0 Clk
Clear Clock
Reset
Next-state logic Q'0 State memory Output logic (a)
C Input D1 Q1 Y Clk
Clear D0 Output
Q'1 Q0 Clk
Clear Clock
Reset
Next-state logic Q'0 State memory Output logic (b)
Figure 7.2. Sample finite-state machine circuits: (a) Moore; (b) Mealy. 7.2 State Diagrams State diagrams are used to precisely describe the operation of finite-state machines. A state diagram is a
deterministic graph with nodes and directed edges connecting the nodes. There is one node for every state of the
FSM, and these nodes are labeled with its state name or encoding. For every state transition of the FSM there is a
directed edge connecting two nodes. The directed edge originates from the node for current state that the FSM is
transitioning from, and goes to the node for the next state that the FSM is transitioning to. Edges may or may not
have labels on them. Edges for unconditional transitions from one state to another will not have a label. In this case,
only one edge can originate from that node. Conditional transitions from a state will have two outgoing edges for 204 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits each input signal condition. The two edges from this state will have the corresponding input signal conditions
labeled on them – one edge with the label for when the condition is true, and the other edge with the label for when
the condition is false. If there is more than one input signal, then all possible input conditions must be labeled on the
outgoing edges from the state. The graph is deterministic because from any node, it should show which is the next
node to go to for any input combination. If an edge is not labeled, or if not all possible input conditions are labeled
on the outgoing edges from the same state, then these missing conditions are don’t care conditions.
Figure 7.3 (a) shows a sample state diagram having four states, one input signal C, and one output signal Y. The
four states are labeled with the four encoded binary values 00, 01, 10, and 11. In this book, we will always use state
0 as the starting or reset state unless stated otherwise. There are three unconditional transitions, i.e., edges with no
labels, from state 00 to 01, 10 to 00, and 11 to 00. There is one conditional transition from state 01 to 10 or 11. For
this conditional transition from state 01, if the condition (C = 0) is true then the transition from 01 to 10 is made.
Otherwise, if the condition (C = 0) is false, that is (C = 0)' is true or (C = 1) is true, then the transition from 01 to 11
is made.
The output signal Y in Figure 7.3 (a) is labeled inside each node denoting that the output is dependent only on
the current state. For example, when the FSM is in state 01, the output Y is set to a 1, whereas, in state 11, Y is set to
a 0. Hence, this state diagram is for a Moore FSM.
The operation of the FSM based on the state diagram in Figure 7.3 (a) goes as follows. After reset, the FSM
starts from state 00. When it is in state 00, it outputs a 0 for Y. At the next clock cycle, the FSM unconditionally
transitions to state 01 and outputs a 1 for Y. Next, the FSM will either go to state 10 or 11 in the next clock cycle
depending on the condition (C = 0). If the condition (C = 0) is true, then the FSM will go to state 10 and output a 0
for Y, otherwise, it will go to state 11 and also output a 0 for Y. From either state 10 or 11, the FSM will
unconditionally transition back to state 00 at the next clock cycle. The FSM will always go to a new state at the
beginning of the next active clock edge.
Figure 7.3 (b) shows a slightly different state diagram from the one in Figure 7.3 (a). Instead of labeling the
output signal Y inside a node, it is labeled on the edges. What this means is that the output is dependent on both the
current state, i.e., the state in which the edge originates from, and the input signal C. For example, when the FSM is
in state 01, if the FSM takes the left edge for the condition (C = 0) to state 10, then it will output a 0 for Y. However,
if the FSM takes the right edge for the condition (C = 0)' to state 11, then it will output a 1 for Y. Hence, this second
state diagram is for a Mealy FSM.
Figure 7.3 (c) shows a state diagram having five states, two input signals, and no output signals. In practice, all
FSMs should have output signals, otherwise, they don’t do anything useful. The five states in this state diagram are
given the logical state names s0, s1, s2, s3, and s4. The two input signals are A and B. Again, we will use the state
name with subscript 0, namely s0, as the starting state. From state s0, there is one unconditional edge going to state
s1. This unlabeled edge is equivalent to having the label AB=××, meaning that this edge is taken for any combination
of the two input signals. From state s1, there are four outgoing edges labeled with the four different combinations of
the two input signals. State s2 has only two outgoing edges. However, the two labels on them cover the four possible
input conditions since B is don’t care in both cases. State s3 has only three outgoing edges, but again the labels on
them cover all four input conditions.
00
Y=0 00
Y=0
Y=0 01
Y=1
(C = 0)
10
Y=0 Y=0
01
(C = 0)
Y=0 (C = 0)'
11
Y=0 10 205 (C = 0)'
Y=1
11 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits (a) (b) s0
AB=00
s4 s1 AB=01
AB=10 AB=11 AB=10 AB=00
s3 AB=0× s2
AB=1× AB=×1
(c) Figure 7.3. Sample state diagrams: (a) a Moore FSM with four states, one input signal C, and one output signal Y;
(b) a Mealy FSM with four states, one input signal C, and one output signal Y; (c) a FSM with five states and two
input signals A and B.
As you can see, a state diagram is very similar to a computer program flow chart where the nodes are for the
statements or data operations, and the edges are for the control of the program sequence. Because of this similarity,
we should be able to convert any program to a state diagram. Example 7.1 shows how to convert a simple C-style
pseudo-code to a state diagram.
Example 7.1
Derive the state diagram based on the following pseudo-code.
x=5
while (x ≠ 0){
output x
x=x–1
}
The pseudo-code has three data operation statements and one conditional test. Each data operation statement is
assigned to a node (state) as shown in Figure 7.4 (a). Each node is given a name for the state, and is annotated with
the statement to be executed in that state. At this point, instead of labeling the nodes with the actual binary encoding
for the state, it is better to just give it a name. The actual encoding of the state can be done later on in the synthesis
process.
Next, we assign directional edges to the diagram based on the sequence of execution. Starting from state s0
where the statement x = 5 is executed, the program then tests for the condition (x ≠ 0). If the condition is true, then
the output statement is executed, otherwise, the loop (and the program) is terminated. Referring to Figure 7.4 (b),
there are two outgoing edges from state s0. The edge from s0 to s1 has the label (x ≠ 0), i.e., if the condition (x ≠ 0) is
true, then this edge is taken, and so it will go to state s1 to execute the output statement. On the other hand, if the
condition is false, the loop needs to be terminated. Since there is no statement after the loop, therefore, we have to
add an extra no-operation state s3 to the state diagram for it to go to. The edge from s0 to s3 is labeled (x ≠ 0)',
meaning that the edge is taken when the condition (x ≠ 0) is false.
After executing the output statement, the decrement statement is executed. This sequence is reflected in the
unconditional edge going from state s1 to s2. After executing the decrement statement in s2, the condition (x ≠ 0) in 206 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits the while loop is again tested. If the condition is true, it will take the edge with the label (x ≠ 0) back to state s1 to
repeat the loop. If the condition is false, it will take the edge with the label (x ≠ 0)' to state s3. From state s3, it
unconditionally loops back to itself, thus, going nowhere and doing nothing.
♦ s0
x=5 s0
x=5 s1
output x (x ≠ 0) (x ≠ 0)' (x ≠ 0) s3
s2
x=x-1 (x ≠ 0)' (a) s1
output x s2
x=x-1 (b) Figure 7.4. State diagram for Example 7.1: (a) data operations assigned to nodes; (b) complete state diagram with
the transitional edges. 7.3 Analysis of Sequential Circuits Very often we are given a sequential circuit and need to know its operation. The analysis of sequential circuits
is the process in which we are given a sequential circuit (such as the ones in Figure 7.2), and we want to obtain a
precise description of the operation of the circuit by deriving the state diagram for it. The steps for the analysis of
sequential circuits are as follows:
1.
2.
3.
4.
5.
6. Derive the excitation equations from the next-state logic circuit.
Derive the next-state equations by substituting the excitation equations into the flip-flop’s characteristic
equations.
Derive the next-state table from the next-state equations.
Derive the output equations from the output logic circuit.
Derive the output table from the output equations.
Draw the state diagram from the next-state table and the output table. 7.3.1 Excitation Equation
The excitation equations are the equations for the next-state logic circuit in the FSM. In other words, they are
just the input equations to the state memory flip-flops in the FSM. Since the next-state logic is a combinational
circuit, therefore, deriving the excitation equations is just an analysis of a combinational circuit as discussed in
Section 3.1.2. The next-state logic circuit that is derived by these equations “excites” the flip-flops by causing them
to change states, hence the name “excitation equation”. These equations provide the signals to the inputs of the flipflops, and are expressed as a function of the current state and the inputs to the FSM. The current state is determined
by the current contents of the flip-flops, that is, the flip-flops’ output signal Q (and Q' ). There is one equation for
each flip-flop’s input.
The following are two sample excitation equations for the two D flip-flops used in the circuit from Figure
7.2 (a). Equation (1) is from the next-state logic circuit for the D1 input of flip-flop 1, and equation (2) is from the
next-state circuit for the D0 input of flip-flop 0.
D1 = Q1'Q0
D0 = Q1'Q0' + CQ1' (1)
(2) 207 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits 7.3.2 Next-state Equation
The next-state equations specify what the flip-flops’ next state is going to be depending on two things: 1) the
inputs to the flip-flops, and 2) the functional behavior of the flip-flops. The inputs to the flip-flops are provided by
the excitation equations as discussed in Section 7.3.1 above. The functional behavior of a flip-flop, as you recall
from Section 6.10.2, is described formally by its characteristic equation. The characteristic equation tells us what
Qnext ought to be, that is, what the next state ought to be, depending on the current state and current inputs. Thus, to
derive the next-state equations, we substitute the excitation equations into the corresponding flip-flop’s
characteristic equations.
For example, the characteristic equation for the D flip-flop (from Section 6.10.2) is
Qnext = D
Therefore, substituting the two excitation equations (1) and (2) from above into the characteristic equation for the D
flip-flop will give us the following two next-state equations
Q1next = D1 = Q1'Q0
Q0next = D0 = Q1'Q0' + CQ1' (3)
(4) 7.3.3 Next-state Table
The next-state table is simply the truth table as derived from the next-state equations. It lists for every
combination of the current state (the Q) values and input values, what the next state (the Qnext) values should be.
These next state values are obtained by substituting the current state and input values into the appropriate next-state
equations.
Figure 7.5 shows the next-state table as obtained from the two next-state equations (3) and (4) from
Section 7.3.2 above. Having two flip-flops, Q1Q0, there are four encodings, 00, 01, 10, and 11, for the current state.
There is one input signal C, with the two possible values, 0 and 1. The entries in the table are the next state values
Q1next Q0next. For each entry, the leftmost bit is for the Q1 flip-flop, and the rightmost bit is for the Q0 flip-flop. These
next state values are obtained from substituting the current state values Q1Q0, and input value C into the next-state
equations (3) and (4).
For example, to get the Q1next value for the top left entry (the left bit in the blue entry), we substitute the current
state values Q1 = 0 and Q0 = 0, and the input value C = 0 into equation (3) giving
Q1next = Q1'Q0
= 0' • 0
=1•0
=0
Substituting the same values into equation (4) will give us the Q0next value for that same top left entry.
Q1next = Q1'Q0' + CQ1'
= 0' • 0' + 0 • 0'
=1+0
=1
The rest of the entries in the next-state table are obtained in the same manner by substituting the corresponding
values for Q1, Q0, and C into the two next-state equations.
The top left entry tells us that if the current state is 00, and the input signal C is a 0, then the next state that the
FSM will go to is 01. From the current state 00, if the input signal C is a 1, the next state is also 01. This means that
the transition from state 00 to 01 does not depend on the input condition C, so this is an unconditional transition.
From state 01, there are two conditional transitions: the FSM will transition to state 10 if the condition C = 0 is true,
otherwise if C = 1, it will transition to state 11. From either state 10 or 11, the FSM will go to state 00
unconditionally. 208 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Current State
Q1Q0
00
01
10
11 Next State
Q1next Q0next
C=0 C=1
01
01
10
11
00
00
00
00 Figure 7.5. A next-state table with four states and one input signal C. 7.3.4 Output Equation
The output equations are the equations derived from the combinational output logic circuit in the FSM.
Depending on the type of FSM (Moore or Mealy), the output equations can be dependent on just the current state or
on both the current state and the inputs.
For the Moore circuit of Figure 7.2 (a), the output equation is
Y = Q1'Q0 (5) For the Mealy circuit of Figure 7.2 (b), the output equation is
Y = CQ1'Q0 (6) A typical FSM will have many output signals, and so there will be one equation for every output signal. 7.3.5 Output Table
Like the next-state table, the output table is the truth table that is derived from the output equations. The output
tables for the Moore and Mealy FSMs are slightly different from each other. For the Moore FSM, the output table
lists for every combination of the current state what the output values should be. Whereas for the Mealy FSM, the
output table lists for every combination of the current state and input values what the output values should be. These
output values are obtained by substituting the current state and input values into the appropriate output equations.
Figure 7.6 (a) and (b) show the output tables for the Moore and Mealy FSMs as derived from the output
equations (5) and (6) respectively from Section 7.3.4 above. For the Moore FSM, the output signal Y is dependent
only on the current state value Q1Q0, whereas, for the Mealy FSM, the output signal Y is dependent on both the
current state and input C.
Current State
Q1Q0 Output
Y Current State
Q1Q0 00
01
10
11 0
1
0
0 00
01
10
11 (a) Output
Y
C=0 C=1
0
0
0
1
0
0
0
0
(b) Figure 7.6. Output table: (a) for Moore FSM; (b) for Mealy FSM. 7.3.6 State Diagram
The last step in the analysis is to derive the state diagram. The state diagram is obtained directly from the nextstate table and the output table. 209 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits The next-state table from Figure 7.5 shows that there are four states in the state diagram. For each next state
entry in the table, there is a corresponding edge going from that current state to that next state. The corresponding
input condition is the label for that edge.
The state diagram shown in Figure 7.3 (a) is derived from the next-state table from Figure 7.5, and the Moore
output table from Figure 7.6 (a). The state diagram shown in Figure 7.3 (b) is derived from the same next-state table
from Figure 7.5, but using the Mealy output table from Figure 7.6 (b). 7.3.7 Example: Analysis of a Moore FSM
We will now illustrate the complete process of analyzing a Moore FSM with an example.
Example 7.2
Figure 7.7 shows a simple sequential circuit. Comparing this circuit with the general FSM schematic in Figure
7.1, we conclude that this is a Moore type FSM since the output logic consists of a 2-input AND gate that is
dependent only on the current state Q1Q0. We will follow the above six steps to do a detail analysis of this circuit.
C Input D1 Q1 Y Clk
Clear D0 Output Q'1 Q0 Clk
Clear Clock
Reset
Next-state logic Q'0 State memory Output logic Figure 7.7. A simple Moore finite-state machine.
Step 1 is to derive the excitation equations, which are the equations for the next-state logic circuit. These
equations are dependent on the current state of the flip-flops Q1 and Q0, and the input C. One equation is needed for
every data input of all the flip-flops in the state memory. Our sample circuit has two flip-flops having the two inputs
D1, and D0, so we get the two excitation equations as shown in Figure 7.8 (a). These two equations are obtained
from analyzing the two combinational circuits that provide the inputs D1 and D0 to the two flip-flops. For this
particular example, both of these combinational circuits are simple two level sum-of-products circuits.
Step 2 is to derive the next-state equations. These equations tell us what the next-state is going to be given the
inputs to the flip-flops, and the functional behavior of the flip-flops. One equation is needed for every flip-flop. The
functional behavior of the flip-flop is described by its characteristic equation, which for the D flip-flop, is Qnext = D.
The inputs to the flip-flops are just the excitation equations derived from step 1. Hence, we simply substitute the
excitation equation into the characteristic equation for each flip-flop to obtain the next-state equation for that flipflop. With two flip-flops in the example, we get two next-state equations, one for Q1next and one for Q0next. Figure
7.8 (b) shows these two next-state equations.
Step 3 is to derive the next-state table. The next-state values in the table are obtained by substituting every
combination of current state and input values into the next-state equations obtained in step 2. In our example, there 210 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits are two flip-flops, Q1 and Q0, and input C. Hence the table will have eight next-state entries. There are two bits for
every entry – the first bit for Q1next, and the second for Q0next.
For example, to find the Q1next value for the current state Q1Q0 = 00 and C = 1 (the blue entry), we substitute the
values Q1 = 0, Q0 = 0 and C = 1 into the equation Q1next = C'Q1 + Q1Q0' + CQ1'Q0 = (1' • 0) + (0 • 0' ) + (1 • 0' • 0)
to get the value 0. Similarly, we get Q0next by substituting the same values for Q1, Q0, and C into the equation Q0next =
C'Q0 + CQ0' = (1' • 0) + (1 • 0' ) to get the value 1. The resulting next-state table for our example is shown in Figure
7.8 (c).
Step 4 is to derive the output equations from the output logic circuit. One output equation is needed for every
output signal. For our example, there is only one output signal Y that is dependent only on the current state of the
machine. The output equation for Y as derived from the circuit diagram is shown in Figure 7.8 (d).
Step 5 is to derive the output table. Just like the next-state table, the output table is obtained by substituting all
possible combinations of the current state values into the output equation(s) for the Moore FSM. The output table for
our Moore FSM example is shown in Figure 7.8 (e).
Step 6 is to draw the state diagram, which is derived directly from the next-state and output tables. Every state
in the next-state table will have a corresponding node labeled with the state encoding in the state diagram. For every
next state entry in the next-state table, there will be a corresponding directed edge. This edge originates from the
node labeled with the current state and ends at the node labeled with the next state entry. The edge is labeled with
the corresponding input conditions.
For example, in the next-state table, when the current state Q1Q0 is 00, the next state Q1next Q0next is 01 for the
input C = 1. Hence, in the state diagram, there is a directed edge from node 00 to node 01 with the label C = 1. For a
Moore FSM, the outputs are dependent only on the current state, thus the output values from the output table are
included inside each node in the state diagram. The complete state diagram for our example is shown in Figure
7.8 (f).
A sample timing diagram for the execution of the circuit is shown in Figure 7.8 (g). The two D flip-flops used
in the circuit are positive edge-triggered flip-flops so they change their states at each rising clock edge. Initially, we
assume that these two flip-flops are both in state 0. The first rising clock edge is at time t0. Normally, the flip-flops
will change state at this time, however, since C is a 0, the flip-flops’ values remain constant. At time t1, C changes to
a 1, so that at the next rising clock edge at time t2, the flip-flop values Q1Q0 changes to 01. At the next two rising
clock edges, t3 and t4, the value for Q1Q0 changes to 10, then 11 respectively. At time t4 when Q1Q0 = 11, the output
Y also changes to a 1 since Y = Q1 • Q0. At time t5, input C drops back down to a 0 but the output Y remains at a 1.
Q1Q0 remains the same at 11 through the next rising clock edge since C is 0. At time t6, C changes back to a 1 and so
at the next rising clock edge at time t7, Q1Q0 increments again to 00 and the cycle repeats.
When C = 1, the FSM cycles through the four states in order repeatedly. When C = 0, the FSM stops at the
current state until C is asserted again. If we interpret the four state encodings as a decimal number, then we can
conclude that the circuit of Figure 7.7 is for a modulo-4 up counter that cycles through the four values 0, 1, 2, and 3.
The input C enables or disables the counting.
♦ 211 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits D1 = C'Q1 + Q1Q0' + CQ1'Q0
D0 = C'Q0 + CQ0'
(a) Y = Q1Q0 Q1next = D1 = C'Q1 + Q1Q0' + CQ1'Q0
Q0next = D0 = C'Q0 + CQ0' (d) (b)
Current State
Q 1Q 0
00
01
10
11 Next State
Q1next Q0next
C=0
C=1
00
01
01
10
10
11
11
00 Current State
Q1Q0
00
01
10
11 (c) C =0 Q 1 Q 0 = 00
Y =0 C =1 C =1 Output
Y
0
0
0
1 (e) Q 1 Q 0 = 01
Y =0 C =0 Clk
C C =1 Q1
Q0 C =0 Q 1 Q 0 = 11
Y =1 C =1 Q 1 Q 0 = 10
Y =0 C =0 Y
t0 t1 t2 (f) t3 t4 t5 t6 t7 (g) Figure 7.8. Analysis of a Moore FSM: (a) excitation equations; (b) next-state equations; (c) next-state table; (d)
output equation; (e) output table; (f) state diagram; (g) timing diagram. 7.3.8 Example: Analysis of a Mealy FSM
Example 7.3 illustrates the process for performing an analysis on a Mealy FSM.
Example 7.3
Figure 7.9 shows a simple Mealy FSM. This circuit is exactly like the one in Figure 7.7 except that the output
circuit, which in this example is just one 3-input AND gate, is dependent on not only the current state Q1Q0, but also
on the input C.
The analysis for this circuit goes exactly like the one for the Moore FSM in Example 7.2 up to creating the nextstate table in step 3. The only difference is in deriving the output equation and output table for steps 4 and 5. For a
Mealy FSM, the output equation is dependent on both the current state and the input value. Since the circuit has only
one output signal, we obtain the output equation that is dependent on C as shown in Figure 7.10 (a). Figure 7.10 (b)
shows the resulting output table obtained by substituting all possible values for Q1, Q0, and C into the output
equation.
For the state diagram, we cannot put the output value inside a node since the output value is dependent on the
current state and the input value. Thus, the output value is placed on the edge that corresponds to the current state 212 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits value and input value as shown in Figure 7.10 (c). Output signal Y is 0 for all edges except for the one originating
from state 11 having the input condition C = 1. On this one edge, Y is a 1.
C Input D1 Q1 Y Clk
Clear D0 Output
Q'1 Q0 Clk
Clear Clock
Reset
Next-state logic Q'0 State memory Figure 7.9. A simple Mealy finite-state machine. Y = CQ1Q0
(a)
Current State
Q1Q0
00
01
10
11 Output
Y
C=0
C=1
0
0
0
0
0
0
0
1
(b) 213 Output logic Digital Logic and Microprocessor Design with VHDL C =0
Y =0 Q 1 Q 0 = 00 C =1
Y =0 C =1
Y =1
C =0
Y =0 Q 1 Q 0 = 11 Q 1 Q 0 = 01 Chapter 7 - Sequential Circuits C =0
Y =0 C C =1
Y =0
C =1
Y =0 Clk Q1
Q0 Q 1 Q 0 = 10 C =0
Y =0 Y
t0 (c) t1 t2 t3 t4 t5 t6 t7 (d) Figure 7.10. Analysis of a Mealy FSM: (a) output equation; (b) output table; (c) state diagram; (d) timing diagram.
A sample timing diagram is shown in Figure 7.10 (d). This diagram is exactly the same as the one for the Moore
FSM shown in Figure 7.8 (g) up to time t5. At time t5, input C drops to a 0, and so output Y also drops to a 0 since
Y = C • Q1 • Q0. At time t6, C rises back up to a 1, and so Y also rises to a 1 immediately. Since the output circuit is a
combinational circuit, Y does not change at the active edge of the clock, but changes immediately when the inputs
change. At time t7 when Q1Q0 changes to 00, Y again changes back to a 0.
Except for the difference in how this circuit generates the output signal Y, this Mealy FSM behaves exactly the
same as the Moore FSM from Example 7.2 in the way that it changes from one state to the next. This, of course, is
due to the fact that both next-state tables are identical. Thus, this Mealy FSM circuit is also a modulo-4 up counter.♦ 7.4 Synthesis of Sequential Circuits The synthesis of sequential circuits is just the reverse of the analysis of sequential circuits. In synthesis, we
start with what is usually an ambiguous functional description of the circuit that we want. From this description, we
need to come up with the precise operation of the circuit using a state diagram. The state diagram allows us to
construct the next-state and output tables. From these two tables, we get the next-state and output equations, and
finally the complete FSM circuit.
During the synthesis process, there are many possible circuit optimizations in terms of the circuit size, speed,
and power consumption that can be performed. Circuit optimization is discussed in Section 7.8. In this section, we
will focus only on synthesizing a functionally correct sequential circuit.
The steps for the synthesis of sequential circuits are as follows:
1.
2.
3.
4.
5.
6.
7. Produce a state diagram from the functional description of the circuit.
Derive the next-state table from the state diagram.
Convert the next-state table to the implementation table.
Derive the excitation equations for each flip-flop input from the implementation table.
Derive the output table from the state diagram.
Derive the output equations from the output table.
Draw the FSM circuit diagram based on the excitation and output equations. 7.4.1 State Diagram
The first step in the sequential circuit synthesis process is to derive the state diagram for it. The circuit to be
built is usually described using an ambiguous natural language. Not only does the language itself create
uncertainties, in many cases the description of the circuit is also incomplete. This incomplete description arises
when not all possible situations of an event or behavior are specified. In order to translate an ambiguous description
into a precise state diagram, the designer must have a full understanding of the functional behavior of the circuit in
question. In addition, the designer may need some ingenuity and creativity to fill in the missing gaps. Meaningful
assumptions need to be made and stated clearly, and ambiguous situations need to be clarified. This is the one step 214 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits in the design process where there is no clear-cut answer for it. In this step, we rely on the knowledge and expertise
of the designer to come up with a correct and meaningful state diagram.
Instead of using a natural language to describe the circuit, a more precise method can be used. Other ways to
describe a circuit more precisely include the use of a hardware description language such as VHDL, a state action
table, or an ASM chart. The use of ASM charts and state action tables are described in Chapter 10.
In this section, we will construct a FSM circuit based on the C style pseudo-code shown in Figure 7.11. Do not
try to interpret the logical execution of the code because it does not perform anything meaningful. Furthermore, this
section is not about optimizing the code by modifying it to make it shorter, although optimizing the code this way
may produce a smaller FSM circuit. In this section, the focus is on learning how to convert any given pseudo-code,
as is, to a FSM circuit that realizes it. Section 7.8 discusses how to optimize sequential circuits.
repeat
Y=0
if (B
Y=
else
Y=
}
Y=1
} {
-- s0
= 0){
0 -- s1 1 -- s2
-- s3 Figure 7.11. C style pseudo-code for synthesis.
The pseudo-code shown in Figure 7.11 contains four signal assignment statements – two Y = 0, and two Y = 1.
We assign one state to each of the four signal assignment statements. The first Y = 0 is assigned to state s0, the
second Y = 0 is assigned to state s1, and so on, as shown in the pseudo-code.
After the first Y = 0 statement, the if statement conditionally determines whether to execute the second Y = 0
statement or the Y = 1 statement. Hence, from state s0, there is one edge going to state s1, and one edge going to state
s2. The labels on these two edges are the conditions for the if statement. The edge going to state s1 has the label
(B = 0), and the edge going to state s2 has the label (B = 1). From either state s1 or state s2, state s3 is executed,
hence, there are two unconditional edges from these two states to s3. Finally, because of the unconditional repeat
loop, there is an unconditional edge from s3 going back to state s0. The resulting state diagram is shown in Figure
7.12 (a). 7.4.2 Next-state Table
Given a state diagram, it is easy to derive both the next-state and output tables from it. Since the next-state and
output tables, and the state diagram portrait the same information but depicted in a different format, therefore, it
requires only a straightforward translation from one to the other.
Figure 7.12 (b) shows the next-state table for the state diagram shown in (a). The row labels are the current state
and the column labels are the input conditions. The table entries are the next states. Translating directly from the
state diagram, from current state s0, if B is a 0, then the next state is s1. Correspondingly, in the next-state table, the
entry for the intersection of the current state s0 and input B = 0 is s1. 215 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits s0
Y=0
B=0 B=1 s1
Y=0 s2
Y=1 s3
Y=1
(a)
Next State
Q1next Q0next
B=0
B=1
s1 01
s2 10
s3 11
s3 11
s3 11
s3 11
s0 00
s0 00 Current State
Q1Q0
s0 00
S1 01
s2 10
s3 11 Implementation
D1 D0
B=0
B=1
01
10
11
11
11
11
00
00 Current State
Q 1Q 0
00
01
10
11 (b) (c) D1 D0 Q1Q0
B
00 01 0 1 1 0 1 1 1 1 1 11 10 B D1 = (Q1 ⊕ Q0) + BQ1' Q1Q0
00
1 01 11 10 1 1 1 1 D1 = (Q1 ⊕ Q0) + B'Q1'
(d)
Current State
Q1Q0 Output
Y s0 00
S1 01
s2 10
s3 11 0
0
1
1
(e) 216 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits B D1 Q1 Y Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset Q'0 (f)
Figure 7.12. (a) A simple state diagram; (b) next-state table; (c) implementation table using D flip-flops; (d)
excitation equations; (e) output table; (f) FSM circuit.
In the next-state table, the actual encoding for the states is also given. To encode the four states, two flip-flops,
Q1 and Q0, are required. In the example, the encoding given to the four states, s0, s1, s2, and s3, is just the four
different combinations of the two flip-flop values, 00, 01, 10, and 11 respectively. Using different encoding schemes
can give different results in terms of circuit size, speed, and power consumption. This optimization technique is
further discussed in Section 7.8.2. 7.4.3 Implementation Table
The implementation table is derived from the next-state table. Whereas, the next-state table is independent of
the flip-flop type used, the implementation table is dependent on the choice of flip-flop used. A FSM can be
implemented using any one of the four different types of flip-flops (as discussed in Section 6.11) or combinations of
them. Using different flip-flops or combinations of flip-flops can produce different size circuits but with the same
functionality. The current trend in microprocessor design is to use only D flip-flops because of their ease of use. We
will, likewise, use only D flip-flops in our synthesis of sequential circuits. Section 7.8.3 discusses how sequential
circuits are synthesized with other types of flip-flops.
The implementation table shows what the flip-flop inputs ought to be in order to realize the next-state table. In
other words, it shows the necessary inputs for the flip-flops that will produce the next states as given in the nextstate table. The next-state table answers the question of what is the next state of the flip-flop given the current state
of the flip-flop and the input values. The implementation table, on the other hand, answers the question of what
should the input(s) to the flip-flop be in order to realize the corresponding next state given in the next-state table.
The flip-flop inputs that we are concerned with are the synchronous inputs. For the D flip-flop, this is just the D
input. For the other flip-flop types, they are the S and R inputs for the SR flip-flop; the J and K inputs for the JK flipflop; and the T input for the T flip-flop. We do not consider the asynchronous inputs such as the Set and Clear
inputs, nor do we consider the clock input signal.
Hence, to derive the implementation table using D flip-flops, we need to determine the value that must be
assigned to the D input such that it will cause the corresponding Qnext value given in the next-state table. However,
since the characteristic equation for the D flip-flop (i.e. the equation that describes the operation of the D flip-flop as
given in Section 6.10.2) is
Qnext = D 217 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits therefore, the values for Qnext and D are the same.
Thus, the entries in the implementation table using D flip-flops are identical to the entries in the next-state table.
The only difference between the two tables is in the meaning of the entries. In the next-state table as shown in
Figure 7.12 (b), the label for the entries is Qnext for the next state to go to, whereas, in the implementation table as
shown in Figure 7.12 (c), the label for the entries is D for the input to the D flip-flop. Since there are two flip-flops,
Q1 and Q0, each having one input D, hence the implementation table has the two corresponding inputs D1 and D0.
The leftmost bit is for flip-flop 1 and the rightmost bit is for flip-flop 0. Note that if one of the other types of flipflops is used, the two tables will not be the same as discuss in Section 7.8.3. 7.4.4 Excitation Equation and Next-state Circuit
Recall that the excitation equations are the equations for the flip-flop’s synchronous inputs. There is one
excitation equation for every input of every flip-flop. Remember that we do not include the asynchronous inputs and
the clock input. The excitation equations are dependent on the current state encodings, i.e., the contents of the flipflops, and the primary FSM input signals.
The excitation equations are what caused the flip-flops in the state memory to change state. The circuit that is
derived from these equations is the next-state circuit in the FSM. The next-state circuit is a combinational circuit,
and so deriving this circuit is the same as synthesizing any other combinational circuit as discussed in Section 3.2.
The implementation table derived from the previous step is just the truth table for the excitation equations. For
our example, we need two equations for the two flip-flop inputs, D1 and D0. In the example, extracting the leftmost
bit in every entry in the implementation table will give us the truth table for D1, and therefore, the excitation
equation for D1. Similarly, extracting the rightmost bit in every entry in the implementation table will give us the
truth table and excitation equation for D0. The truth table, in the form of a K-map, and the excitation equations for
D1 and D0 are given in Figure 7.12 (d). 7.4.5 Output Table and Equation
The output table and output equations are used to derive the output circuit in the FSM. The output table can be
obtained directly from the state diagram. In the state diagram of Figure 7.12 (a), the output signal Y is dependent
only on the state. In states s0 and s1, Y is assigned the value 0. In states s2 and s3, Y is assigned a 1. The resulting
output table is shown in Figure 7.12 (e).
The output equation as derived from the output truth table is simply
Y = Q1 7.4.6 FSM Circuit
Using Figure 7.2 (a) as a template, our FSM circuit requires two D flip-flops for its state memory. The number
of flip-flops to use was determined when the states were encoded. The type of flip-flops to use was determined when
deriving the implementation table. The next-state circuit is drawn from the excitation equations, while the output
circuit is drawn from the output equation. Connecting these three parts, state memory, next-state circuit, and output
circuit, together produces the final FSM circuit shown in Figure 7.12 (f). 7.4.7 Examples: Synthesis of Moore FSMs
We will now illustrate the synthesis of Moore FSMs with two examples. Example 7.4 illustrates the synthesis of
a simple Moore FSM. Example 7.5 illustrates the synthesis of a Moore FSM that is more typical of what the control
unit of a microprocessor is like.
Example 7.4
For our first synthesis example, we will design a modulo-6 up counter using D flip-flops having a count enable
input C, and an output signal Y that is asserted when the count is equal to five. The count is to be represented
directly by the contents of the flip-flops. 218 Digital Logic and Microprocessor Design with VHDL C= 0 Chapter 7 - Sequential Circuits C= 1 Q2Q1Q0 = 000
Y=0 Q2Q1Q0 = 001
Y=0 C= 1 Q2Q1Q0 = 101
Y=1 C= 1 Q2Q1Q0 = 010
Y=0 C= 0 C= 0 C= 1 C= 1
C= 0 C= 0 Q2Q1Q0 = 100
Y=0 Q2Q1Q0 = 011
Y=0 C= 1 C= 0 (a)
Next State
Q2next Q1next Q0next
C=0
C=1
000
001
001
010
010
011
011
100
100
101
101
000 Current State
Q 2Q 1Q 0
000
001
010
011
100
101 Current State
Q2Q1Q0
000
001
010
011
100
101 (b)
D2
Q1Q0 01 11 00 00 1 1 01 1 11 10 CQ2'Q1'Q0 D0 CQ2 Q1Q0 00 01 11 10 1 11 1 10
CQ2'Q1Q0 Q1Q0 00 01 1 1 C'Q2'Q1 Q2'Q1Q0' D2 = Q2Q1'Q0' + C'Q2Q1' + CQ2'Q1Q0
D1 = C'Q2'Q1 + Q2'Q1Q0' + CQ2'Q1'Q0
D0 = C'Q1'Q0 + C'Q2'Q0 + CQ1'Q0' + CQ2'Q0'
(d) 219 CQ2'Q0' 01 1 11 10 1 11 01
1 CQ1'Q0' CQ2
00 00 10
C'Q2Q1' (c) D1 Q2Q1'Q0' CQ2 Implementation
D2 D1 D0
C=0
C=1
000
001
001
010
010
011
011
100
100
101
101
000 1 1 1 10
C'Q2'Q0 1 C'Q1'Q0 Digital Logic and Microprocessor Design with VHDL Current State
Q2Q1Q0 Output
Y 000
001
010
011
100
101 0
0
0
0
0
1 Chapter 7 - Sequential Circuits Y = Q2Q1'Q0
(e) C D2 Q2 Y Clk
Clear D1 Q'2 Q1 Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset Q'0 (f)
Figure 7.13. Synthesis of a Moore FSM for Example 7.4: (a) state diagram; (b) next-state table; (c) implementation
table; (d) K-maps and excitation equations; (e) output table and equation; (f) FSM circuit.
Step 1 is to construct the state diagram. From the above functional description, we need to construct a state
diagram that will show the precise operation of the circuit. A modulo-6 counter counts from zero to five, and then
back to zero. Since the count is represented by the flip-flop values and we have six different counts (from zero to
five), we will need three flip-flops (Q2, Q1, Q0) that will produce the sequence 000, 001, 010, 011, 100, 101, 000, …
when C is asserted, otherwise, when C is de-asserted, the counting stops. In other words, from state 000, which is
count = 0, there will be an edge that goes to state 001 with the label C = 1. From state 001, there is an edge that goes
to state 010 with the label C = 1, and so on. For the counting to stop at each count, there will be edges at each state
that loop back to the same state with the label C = 0. Furthermore, we want to assert Y in state 101, so in this state,
we set Y to a 1. For the rest of the states, Y is set to a 0. Hence, we obtain the state diagram in Figure 7.13 (a) for a
modulo-6 up counter.
Step 2 is to derive the next-state table, which is a direct translation from the state diagram. We have three flipflops Q2, Q1, and Q 0, and one primary input C. The current states for the flip-flops are listed down the rows, while
the input is listed across the columns. The entries are the next states. For each entry in the next-state table, we need
to determine what the next state is for each of the three flip-flops, so there are three bit values, Q2next, Q1next, and
Q0next for each entry. For example, if the current state is Q2Q1Q0 = 010 and the input is C = 1, then the next state
Q2nextQ1nextQ0next is 011. The next-state table is shown in Figure 7.13 (b). 220 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Step 3 is to convert the next-state table to its implementation table. Since for the D flip-flop, the implementation
table is the same as the next-state table, we can simply use the next-state table and just re-label the entry heading as
shown in Figure 7.13 (c).
Step 4 is to derive the excitation equations for all the flip-flop inputs in terms of the current state and the
primary input. These equations are obtained directly from the implementation table. In the example, there are three
flip-flops with the three inputs D2, D1, and D0, which correspond to the three bits in the entries in the implementation
table. To derive the equation for D2, we consider just the leftmost bit in each entry for the truth table for D2. Looking
at all the leftmost bits, there are four 1-minterms giving the canonical equation
D2 = C'Q2Q1'Q0' + C'Q2Q1'Q0 + CQ2'Q1Q0 + CQ2Q1'Q0'
Similarly, the equation for D1 is derived from considering just the middle bit for all the entries, and the equation
for D0 from the rightmost bit. Since these equations will be used to construct the next-state circuit, they should be
simplified. The three K-maps and simplified excitation equations for D2, D1, and D0 are shown in Figure 7.13 (d).
Steps 5 and 6 are to derive the output table and equation. There is one equation for every output signal. Since
the value of Y is labeled inside each node, it is therefore dependent only on the current state. From the state diagram,
Y is asserted only in state 101, so Y has a 1 only in that current state entry, while the rest of them are 0’s. The output
table and equation are shown in Figure 7.13 (e).
Finally, we can draw the circuit for the FSM. We know that the circuit is a Moore FSM that uses three D flipflops for its state memory having one primary input C and one output Y. The next-state function circuit is derived
from the three excitation equations for D2, D1, and D0. The output function circuit is derived from the output
equation for Y. The full circuit is shown in Figure 7.13 (f).
♦
Example 7.5
In this example, we will synthesize a Moore FSM that is more typical of what the control unit of a
microprocessor is like. We start with the state diagram as shown in Figure 7.14 (a). Each state is labeled with a state
name, s0, s1, s2, and s3, and has two output signals x and y. There are also two conditional status signals Start and
(n=9) labeled on four of the edges, while the rest of the edges do not have any conditions. From state s0, the
conditional edge labeled Start is taken when Start = 1, otherwise, the edge labeled Start' is taken. Similarly, from
state s2, the edge with the label (n = 9) is taken when the condition is true, that is, when the value of variable n is
equal to nine. If n is not equal to nine, then the edge with the label (n = 9)' is taken.
Two flip-flops Q0 and Q1 are needed in order to encode the four states. For simplicity, we will use the binary
value of the index of the state name to be the encoding for that state. For example, the encoding for state s0 is Q1Q0
= 00 and the encoding for state s1 is Q1Q0 = 01, and so on.
From the above analysis, we are able to derive the next-state table as shown in Figure 7.14 (b). The four current
states for Q1Q0 are listed down the four rows. The four columns are for the four combinations of the two conditional
signals Start and (n=9). For example, the column with the value Start, (n=9) = 10 means Start = 1 and (n=9) = 0.
The condition (n=9) = 0 means that the condition (n=9) is false which means (n=9)' is true. The entries in the table
are the next states, Q1next Q0next, for the two flip-flops.
For example, looking at the state diagram, from state s2 we go back to state s1 when the condition (n=9)' is true
independent of the Start condition. Hence, in the next-state table, for the current state row s2 (10), the two next-state
entries for when the condition (n=9)' is true is s1 (01). The condition “(n=9)' is true” means (n=9) = 0. This
corresponds to the two columns with the labels 00 and 10, that is, Start can be either 0 or 1, while (n=9) is 0.
Using D flip-flops to implement the FSM, we get the implementation table shown in Figure 7.14 (c). The
implementation table and the next-state table are identical when D flip-flops are used. The only difference between
them is the meaning given to the entries. For the next-state table, the entries are the next state of the flip-flops,
whereas for the implementation table, the entries are the inputs to the flip-flops. They are the input values necessary
to get to that next state. Again, since the next state is equal to the input value (Qnext = D) for a D flip-flop, therefore,
the entries in these two tables are the same. 221 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Start' Current
State
Q1Q0 s0
x=0
y=1 00
s0 00
s2 10
s1 01
s0 00 s0 00
s1 01
s2 10
s3 11 Start Next State
Q1next Q0next
Start, (n=9)
01
10
s0 00 s1 01
s2 10 s2 10
s3 11 s1 01
s0 00 s0 00 s1
x=1
y=1 (b) s2
x=1
y=1 Current
State
Q1Q0 (n = 9)' s3
x=1
y=0 Implementation
D1 D0
Start, (n=9)
01
10
s0 00 s1 01
s2 10 s2 10
s3 11 s1 01
s0 00 s0 00 00
s0 00
s2 10
s1 01
s0 00 s0 00
s1 01
s2 10
s3 11 (n = 9) (c) (a) D1 Start,(n=9)
Q1Q0
00 01 D0
11 10 1 1 1 1 1 Q1Q0 00
01 Start,(n=9)
00 01 Q1'Q0 10
1 1 1 01 11
10 11
1 00
1 11
s1 01
s2 10
s3 11
s0 00 11
10 Q1Q0'(n=9) 1 Q1Q0' D1 = Q1'Q0 + Q1Q0'(n=9) 1 Q0'Start D0 = Q1Q0' + StartQ0'
(d) 222 11
s1 01
s2 10
s3 11
s0 00 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits x Current State
Q1Q0 Output
xy 00
01
10
11 01
11
11
10 y Q0 Q1 0 1 1 1 x = Q1 + Q0 (e) 1 1 1 1 1 0
0 1 0 Start (n=9) 0 Q0 Q1 1 0 y = ( Q 1Q 0) '
(f) Input
signals x
D1 Q1 y Clk
Clear D0 Output
signals Q'1 Q0 Clk
Clear Clock
Reset
Next-state logic Q'0 State memory Output logic (g)
Figure 7.14. Synthesis of a Moore FSM for Example 7.5: (a) state diagram; (b) next-state table; (c) implementation
table; (d) excitation equations and K-maps for D1 and D0; (e) output table; (f) output equations and K-maps; (g)
FSM circuit.
The excitation equations are derived from the implementation table. There is one excitation equation for every
data input of every flip-flop used. Since we have two D flip-flops, therefore, we have two excitation equations; one
for D1 and the second for D0. The equations are dependent on the four variables Q1, Q0, Start, and (n=9). We look at
the implementation table as one having two truth tables merged together, one truth table for D1 and one for D0. Since
the two bits in the entries are ordered D1D0, therefore, for the D1 truth table, we look at only the leftmost D1 bit in
each entry, and for the D0 truth table, we look at only the rightmost D0 bit. Extracting the two truth tables from the
implementation table in this manner, we obtain the two K-maps and corresponding excitation equations for D1 and
D0 as shown in Figure 7.14 (d). The excitation equations allow us to derive the next-state combinational circuit.
The output table is obtained from the output signals given in the state diagram. The output table is just the truth
table for the two output signals x and y. The output signal equations derived from the output table are dependent on
the current state Q1Q0. The output table, K-maps and output equations are shown in Figure 7.14 (e) and (f).
From the excitation and output equations, we can easily produce the next-state and output circuits, and the
resulting FSM circuit shown in Figure 7.14 (g).
♦ 223 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits 7.4.8 Example: Synthesis of a Mealy FSM
The next example illustrates the synthesis of a Mealy FSM. You will find that this process is almost identical to
the synthesis of a Moore FSM with the one exception of deriving the output equations. The outputs for a Mealy
FSM are dependent on both the current state and the input signals, whereas, for the Moore FSM, they are only
dependent on the current state.
Example 7.6
In this example, we will synthesize a Mealy FSM based on the state diagram shown in Figure 7.15 (a) using D
flip-flops. The four states are already encoded with the values of the two flip-flops. There are two conditional input
signals (x=0) and (x=y). Since these are conditions, the equal sign means the test for equality. There is one output
signal A, which can be set to either a 0 or a 1 value. The equal sign here means assignment. Notice that what makes
this a Mealy FSM state diagram is the fact that the outputs are associated with the edges and not the nodes. Current State
Q1Q0
00
01
10
11 00
(x=0)'
A=0 (x=0)
A=1
01 (b) 10
A=0 Current State
Q1Q0 A=1
(x=y)
A=0 11 00
01
10
11 (x=y )'
A=1 D1 00 1
1 1 D0
11 10 Q1Q0 1 01 (x=0), (x=y)
00 01 01 1 1 1 1 D1 = Q1'Q0 + Q1Q0' + Q1' (x=0)'
= (Q1 ⊕ Q0) + Q1' (x=0)' 1 10 1 1 11 1 1 1 11 10 1 00 11
10 Implementation
D1D0
(x=0), (x=y)
00 01 10 11
10 10 01 01
11 11 11 11
11 11 11 11
01 00 01 00
(c) (a) (x=0), (x=y)
Q1Q0
00 01 Next State
Q1nextQ0next
(x=0), (x=y)
00 01 10 11
10 10 01 01
11 11 11 11
11 11 11 11
01 00 01 00 1 1 1 1
1 1 1 D0 = Q1'Q0 + Q1Q0' + Q1' (x=0) + Q0(x=0)' (x=y)'
= (Q1 ⊕ Q0) + Q1' (x=0) + Q0(x=0)' (x=y)'
(d) 224 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits A
Q1Q0 Current State
Q1Q0
00
01
10
11 Output
A
(x=0), (x=y)
00 01 10 11
0
0
1
1
0
0
0
0
1
1
1
1
1
0
1
0 (x=0), (x=y)
00 01 10 1 00 11 1 1 1 01
11 1 10 1 1 1 A = Q1Q0 + Q1(x=y)' + Q1'Q0' (x=0)
(f) (e)
(x=0) (x=y) D1 Q1 Clk
Clear D0 Q'1 A Q0 Clk
Clock
Reset Clear Q'0 (g)
Figure 7.15. Synthesis of a Mealy FSM for Example 7.6: (a) state diagram; (b) next-state table; (c) implementation
table; (d) excitation equations and K-maps for D1 and D0; (e) output table; (f) output equation and K-map; (g) FSM
circuit.
Deriving the next-state and implementation tables for a Mealy FSM is exactly the same as for a Moore FSM.
The next-state and implementation tables for this example are shown in Figure 7.15 (b) and (c). The excitation
equations and K-maps for D1 and D0 are shown in (d).
The output table as shown in Figure 7.15 (e) is slightly different from the output tables for Moore FSMs. In
addition to the output signal A being dependent on the current state Q1Q0, it is also dependent on the two input
signals (x=0) and (x=y). Hence the table has four columns for the four possible combinations of the two input
signals. The entries in the table are the values for A.
Looking at the state diagram in Figure 7.15 (a), we see that from state 00, output signal A is assigned the value 1
when the condition (x=0) is true, otherwise it is assigned a 0. Since the condition (x=y) is not labeled on these two
edges going out from state 00, therefore, the output is independent to this condition from state 00. Hence, in row 00,
the two entries under the two columns with the label 00 and 01, are both 0; whereas, the two entries under the two
columns 10 and 11 are both 1. 225 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Using the output table as the truth table, we are able to derive the K-map and output equation for A as shown in
Figure 7.15 (f). Notice that the equation is also dependent on the two input signals.
Again, using the excitation and output equations, we are able to draw the final FSM circuit shown in Figure
7.15 (g). 7.5 Unused State Encodings and the Encoding of States In a real world situation, the number of states used in the state diagram is most likely not a power of two. For
example, the state diagram shown in Figure 7.13 (a) for the modulo-6 counter uses six states. To encode six states,
we need at least three flip-flops since two flip-flops can encode only four different combinations. However, three
flip-flops give eight different combinations. So two combinations are not used. The question is what do we do with
these unused encodings? In the next-state table, what next state values do we assigned to these unused states? Do we
just ignore them?
If the FSM can never be in any of the unused states, then it does not matter what their next states are. In this
case, we can put “don’t care” values for their next states. The resulting next-state circuit may be smaller because of
the “don’t care” values.
But what if, by chance, the FSM enters one of these unused states? The operation of the FSM will be
unpredictable because we do not know what the next state is. Well, this is not exactly true because even though we
started with the “don’t cares,” we have mapped them to a fixed excitation equation. So these unused states do have
definite next states. It is just that these next states are not what we wanted. Hence, the resulting FSM operation will
be incorrect if it ever enters one of the unused states. If this FSM is used in a mission critical control unit, we do not
want even this slight chance to occur.
One solution is to use the initialization or starting state as the next state for these unused state encodings. This
way, the FSM will restart from the beginning if it ever enters one of these unused states.
So far, we have been using the sequential binary value to encode the states in order, for example, state s0 is
encoded as 00, state s1 as 01, state s2 as 10, and so on. However, there is no reason why we cannot use a different
encoding for the states. In fact, we do want to use a different encoding if it will result in a smaller circuit.
Example 7.7 shows a FSM with an unused state encoding, and the encoding of one state differently.
Example 7.7
In this example, we will synthesize a FSM for the one-shot circuit first discussed in Section 3.5.1. Recall that
the one-shot circuit outputs a single short pulse when given an input of arbitrary time length. In this FSM circuit, the
length of the single short pulse will be one clock cycle. The state diagram for this circuit is shown in Figure 7.16 (a).
State s0, encoded as 00, is the reset state, and the FSM waits for a key press in this state. When a switch is
pressed, the FSM goes to state s1, encoded as 01, to output a single short pulse. From s1, the FSM unconditionally
goes to state s2, encoded as 11, to turn off the one-shot pulse. Hence, the pulse only lasts for one clock cycle,
irregardless of how long the key is pressed. To break the loop, and wait for another key press, the FSM has to wait
for the release of the key in state s2. When the key is released, the FSM goes back to state s0 to wait for another key
press.
This state diagram uses two bits to encode the three states, hence state encoding 10 is not used. The state
diagram shows that if the FSM enters state 10, it will unconditionally go to the reset state 00 in the next clock cycle.
Furthermore, we have encoded state s2 as 11 instead of 10 for the index two.
The corresponding next-state table is shown in Figure 7.16 (b). Using D flip-flops to implement this FSM, the
implementation table, again, is like the next-state table. Therefore, we can use the next-state table directly to derive
the two excitation equations for D1 and D0 as shown in (c). The output table and output equation is shown in (d), and
finally the complete FSM circuit in (e). 226 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Keypressed'
s0
00 unused
10 s1
01 Keypressed' output Oneshot s2
11 Keypressed Keypressed (a) Current State
Q1Q0 Next State
Q1nextQ0next
Keypressed
0
00
11
00
00 00
01
11
10 1
01
11
11
00 (b)
D1 D0 Keypressed
0
1
Q1Q0
00
01
11 Q1'Q0
1 Keypressed
0
1 Q1'Keypressed 00 Q1Q0 Q1'Q0 01 1 Q0Keypressed 11 1 10 1
1 1 Q0Keypressed
1 10 D1 = Q1'Q0 + Q0Keypressed D0 = Q1'Keypressed + Q1'Q0 + Q0Keypressed
(c) Current State
Q 1Q 0 Output
Oneshot 00
01
11
10 0
1
0
0
(d) 227 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Keypressed D1 Q1 Clk
Clear D0 Q'1 Oneshot Q0 Clk
Clear Clock
Reset Q'0 (e)
Figure 7.16. FSM for one-shot circuit: (a) state diagram; (b) next-state table; (c) excitation equations and K-maps
for D1 and D0; (d) output table and output equation; (e) FSM circuit. 7.6 Designing a Car Security System—Version 3 We will revisit the car security system example from Chapters 2 and 6. Recall that in the first version (Chapter
2) the circuit is a combinational circuit. The problem with a combinational circuit is that once the alarm is triggered,
by lets say opening the door, the alarm can be turned off immediately by closing the door again. However, what we
want is that once the alarm is triggered, it should remain on even after closing the door, and the only way to turn it
off is to turn off the master switch.
This requirement suggests that we need a sequential circuit instead where the output is dependent on not only
the current input switch settings but also on the current state of the alarm. Thus, we are able to come up with the
state diagram as shown in Figure 7.17 (a). In addition to the three input switches M, D and V for Master, Door, and
Vibration, we need two states, 1 and 0, to depict whether the siren is on or off respectively. If the siren is currently
on, i.e. in the 1 state, then it will remain in that state as long as the master switch is still on, so it doesn’t matter
whether the door is now close or open. This is represented by the edge that goes from state 1 and loops back to state
1 with the label MDV=1××. From the on state, the only way to turn off the siren is to turn off the master switch. This
is represented by the edge going from state 1 to state 0 with the label MDV=0××. If the siren is currently off, it is
turned on when the master switch is on, and either the door switch or the vibration switch is on. This is represented
by the edge going from state 0 to state 1 with the labels MDV=101,110, or 111. Finally, from the off state, the siren
will remain off when either the master switch remains off, or if the master switch is on but none of the other two
switches are on. This is represented by the edge from state 0 looping back to state 0.
The state diagram is translated to the corresponding next-state table and implementation table using one D flipflop as shown in Figure 7.17 (b). Again the next-state table and implementation table are the same except that the
entries for the next-state table are for the next states, and the entries for the implementation table are for the inputs to
the flip-flop. Doing a 4-variable K-map on the implementation table gives us the excitation equation shown in
Figure 7.17 (c). The final circuit for this car security system is shown in Figure 7.17 (d). The circuit uses one D flipflop. The next-state circuit is derived from the excitation equation, which produces the signal for the D input of the
flip-flop. The output of the flip-flop directly drives the siren. 228 Digital Logic and Microprocessor Design with VHDL MDV=0xx, 100 Chapter 7 - Sequential Circuits MDV=1xx 101, 110, 111 0 Siren = 0 1 Siren = 1 MDV=0xx
(a)
Next State Current
State
Q 000
0
0 0
1 001
0
0 (D flip-flop Implementation)
Qnext (D)
M,D,V
010 011 100 101 110 111
0
0
0
1
1
1
0
0
1
1
1
1
(b) D0
DV
00 MQ0 01 11 10 1 1 1 1 1 1 1 00
01
11
10 D0 = Q0M + MV + MD = Q0M + M(V + D)
(c) D0 D
V
M
Clock
Reset Q0 Siren Clk
Clear (d)
Figure 7.17. Car security system – version 3: (a) state diagram; (b) next-state / implementation table; (c) K-map and
excitation equation; (d) circuit. 7.7 VHDL for Sequential Circuits Writing VHDL code for sequential circuits is usually done at the behavioral level. The advantage of writing
behavioral VHDL code is that we do not need to manually synthesize the circuit. The synthesizer will automatically
produce the netlist for the circuit from the behavioral code.
In order to write the behavioral VHDL code for a sequential circuit, we need to use the information from the
state diagram for the circuit. The main portion of the code contains two processes: a next-state-logic process, and an
output-logic process. The edges (both conditional and unconditional) from the state diagram are used to derive the 229 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits next-state-logic process, which will generate the next-state logic circuit. The output signal information in the state
diagram is used to derive the process for the output logic.
We will now illustrate the behavioral VHDL coding of sequential circuits with several examples.
Example 7.8
In this example, we will write the behavioral VHDL code for the Moore FSM of Example 7.2. The state
diagram for the example from Figure 7.8 is repeated here in Figure 7.18. Since the synthesizer will automatically
take care of the state encoding, therefore, the states only need to be labeled with their logical names. The behavioral
VHDL code for this Moore FSM based on this state diagram and output table is shown in Figure 7.19. C =0 s0
Y=0 C =1 C =1 C =0 s3
Y=1 s1
Y=0 C =0 C =1 C =1 s2
Y=0 C =0 Figure 7.18. State diagram for Example 7.8.
The entity section declares the primary I/O signals for the circuit. There is the global input clock and reset
signals. The clock signal determines the speed in which the sequential circuit will transition from one state to the
next. The reset signal initializes all the state memory flip-flops to zero. In addition to the standard global clock and
reset signals, the entity section also declares all the input and output signals. For this example, there is an input
signal C, and an output signal Y; both of which are of type STD_LOGIC.
The architecture section starts out with using the TYPE statement to define the four states, s0, s1, s2, and s3, used
in the state diagram. The SIGNAL statement declares the signal state to store the current state of the FSM. There are
two processes in the architecture section that execute concurrently: the next-state-logic process, and the output-logic
process. As the name suggests, the next-state process defines the next-state logic circuit that is inside the control
unit, and the output logic process defines the output logic circuit inside the control unit. The main statement within
these two processes is the CASE statement that determines what the current state is.
In the next-state-logic process, the current state of the FSM is initialized to s0 on reset. The CASE statement is
executed only at the rising clock edge because of the test (clock'EVENT AND clock = '1') in the IF statement. Hence,
the state signal is assigned a new state value at every rising clock edge. The new state value is, of course, dependent
on the current state and input signals, if any. For example, if the current state is s0, the case for s0 is selected. From
the state diagram, we see that when in state s0, the next state is dependent on the input signal C. Hence, in the code,
an IF statement is used. If C is 1 then the new state s1 is assigned to the signal state, otherwise, s0 is assigned to state.
For the latter case, even though we are not changing the state value s0, we still make that assignment to prevent the
VHDL synthesizer from using a memory element for the state signal. Recall from Section 6.13.1 that VHDL
synthesizes a signal using a memory element if the signal is not assigned a value for all possible cases. The rest of
the cases in the CASE statement are written similarly based on the remaining edges in the state diagram.
In the output-logic process, all the output signals must be assigned a value in every case. Again, the reason is
that we do not want these output signals to come from memory elements. In the FSM model, the output circuit is a
combinational circuit, and so it should not contain any memory elements. For each state in the CASE statement in the
output process, the values assigned to each of the output signal are taken directly from the output table. For this
example, there is only one output signal Y.
A sample simulation trace of this sequential circuit is shown in Figure 7.20. In the simulation trace, between
times 100ns and 800ns when R is de-asserted and C is asserted, the state changes at each rising clock edge (at times
300ns, 500ns, and 700ns.) At time 700ns when the current state is s3, we see that the output signal Y is also asserted. 230 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits At time 800ns, input C is de-asserted, as a result, the FSM did not change state at the next rising clock edge at time
900ns.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY MooreFSM IS PORT (
clock: IN STD_LOGIC;
reset: IN STD_LOGIC;
C: IN STD_LOGIC;
Y: OUT STD_LOGIC);
END MooreFSM;
ARCHITECTURE Behavioral OF MooreFSM IS
TYPE state_type IS (s0, s1, s2, s3);
SIGNAL state: state_type;
BEGIN
next_state_logic: PROCESS (clock, reset)
BEGIN
IF (reset = '1') THEN
state <= s0;
ELSIF (clock'EVENT AND clock = '1') THEN
CASE state IS
WHEN s0 =>
IF C = '1' THEN
state <= s1;
ELSE
state <= s0;
END IF;
WHEN s1 =>
IF C = '1' THEN
state <= s2;
ELSE
state <= s1;
END IF;
WHEN s2=>
IF C = '1' THEN
state <= s3;
ELSE
state <= s2;
END IF;
WHEN s3=>
IF C = '1' THEN
state <= s0;
ELSE
state <= s3;
END IF;
END CASE;
END IF;
END PROCESS;
output_logic: PROCESS (state)
BEGIN
CASE state IS
WHEN s0 =>
Y <= '0'; 231 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits WHEN s1 =>
Y <= '0';
WHEN s2 =>
Y <= '0';
WHEN s3 =>
Y <= '1';
END CASE;
END PROCESS;
END Behavioral;
Figure 7.19. Behavioral VHDL code of a Moore FSM for Example 7.7. Figure 7.20. Simulation trace of a Moore FSM for Example 7.8.
Example 7.9
This example shows how a Mealy FSM is written using behavioral VHDL code. We will use the Mealy FSM
from Example 7.3. The state diagram for this FSM is shown in Figure 7.10. This FSM is very similar to the one
from the previous example except that the generation of the output signal Y is also dependent on the input signal C.
The VHDL code is shown in Figure 7.21. In this code, we see that the next-state-logic process is identical to the
previous FSM code. In the output-logic process, the only difference is in state s3 where an IF statement is used to
determine the value of the input signal C. The output signal Y is assigned a value depending on the result of this test.
The simulation trace for this Mealy FSM is shown in Figure 7.22. Notice that the only difference between this
trace and the one from the previous example is in the Y signal between times 800ns and 1us. During this time
period, the input signal C is de-asserted. In the previous trace, this has no effect on Y, however, for the Mealy FSM
trace, Y is also de-asserted.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY MealyFSM IS PORT (
clock: IN STD_LOGIC;
reset: IN STD_LOGIC;
C: IN STD_LOGIC;
Y: OUT STD_LOGIC);
END MealyFSM;
ARCHITECTURE Behavioral OF MealyFSM IS
TYPE state_type IS (s0, s1, s2, s3);
SIGNAL state: state_type;
BEGIN
next_state_logic: PROCESS (clock, reset)
BEGIN
IF (reset = '1') THEN
state <= s0; 232 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits ELSIF (clock'EVENT AND clock = '1') THEN
CASE state is
WHEN s0 =>
IF C = '1' THEN
state <= s1;
ELSE
state <= s0;
END IF;
WHEN s1 =>
IF C = '1' THEN
state <= s2;
ELSE
state <= s1;
END IF;
WHEN s2=>
IF C = '1' THEN
state <= s3;
ELSE
state <= s2;
END IF;
WHEN s3=>
IF C = '1' THEN
state <= s0;
ELSE
state <= s3;
END IF;
END CASE;
END IF;
END PROCESS;
output_logic: PROCESS (state, C)
BEGIN
CASE state IS
WHEN s0 =>
Y <= '0';
WHEN s1 =>
Y <= '0';
WHEN s2 =>
Y <= '0';
WHEN s3 =>
IF (C = '1') THEN
Y <= '1';
ELSE
Y <= '0';
END IF;
END CASE;
END PROCESS;
END Behavioral;
Figure 7.21. Behavioral VHDL code for the Mealy FSM of Example 7.9. 233 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Figure 7.22. Simulation trace for the Mealy FSM of Example 7.9.
Example 7.10
This is another example of a Moore FSM written using behavioral VHDL code. This FSM is from Example 7.5,
and the state diagram for this example is shown in Figure 7.14. The behavioral VHDL code for this FSM is shown in
Figure 7.23, and the simulation trace in Figure 7.24.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY MooreFSM IS PORT(
clock: IN STD_LOGIC;
reset: IN STD_LOGIC;
start, neq9: IN STD_LOGIC;
x,y: OUT STD_LOGIC);
END MooreFSM;
ARCHITECTURE Behavioral OF MooreFSM IS
TYPE state_type IS (s0, s1, s2, s3);
SIGNAL state: state_type;
BEGIN
next_state_logic: PROCESS (clock, reset)
BEGIN
IF (reset = '1') THEN
state <= s0;
ELSIF (clock'EVENT AND clock = '1') THEN
CASE state IS
WHEN s0 =>
IF start = '1' THEN
state <= s1;
ELSE
state <= s0;
END IF;
WHEN s1 =>
state <= s2;
WHEN s2 =>
IF neq9 = '1' THEN
state <= s3;
ELSE
state <= s1;
END IF;
WHEN s3 =>
state <= s0;
END CASE;
END IF;
END PROCESS; 234 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits output_logic: PROCESS (state)
BEGIN
CASE state IS
WHEN s0 =>
x <= '0';
y <= '1';
WHEN s1 =>
x <= '1';
y <= '1';
WHEN s2 =>
x <= '1';
y <= '1';
WHEN s3 =>
x <= '1';
y <= '0';
END CASE;
END PROCESS;
END Behavioral;
Figure 7.23. Behavioral VHDL code for the Moore FSM of Example 7.10. Figure 7.24. Simulation trace for the Moore FSM of Example 7.10. 7.8 * Optimization for Sequential Circuits In designing any digital circuit, in addition to getting a functionally correct circuit, we like to optimize it for
size, speed, and power consumption. In this section, we will briefly discuss some of the issues involved. A full
treatment of optimization for sequential circuits is beyond the scope of this book.
Since sequential circuits also contain combinational circuit parts (the next-state logic and the output logic),
these parts should also be optimized following the optimization procedures for combinational circuits as discussed in
Section 4.4. Some basic choices for sequential circuit optimization include state reduction, state encoding, and
choice of flip-flop types. 7.8.1 State Reduction
Sequential circuits with fewer states most likely will result in a smaller circuit since the number of states
directly translates to the number of flip-flops needed. Fewer flip-flops imply a smaller state memory for the FSM.
Furthermore, fewer flip-flops also mean fewer flip-flop inputs, so the number of excitation equations needed is also
reduced. This of course means that the next-state circuit will be smaller.
There are two levels in which we can reduce the number of states. At the pseudo-code description level, we can
try to optimize the code by shortening the code if possible. We can also assign two or more data operations to the
same state. 235 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits After obtaining a state diagram, we may still be able to reduce the number of states by removing equivalent
states. If two states are equivalent, we can remove one of them, and instead use the other equivalent state. The
resulting FSM will still be functionally equivalent. Two states are said to be equivalent if the following two
conditions are true:
1.
2. Both states produce the same output for every input.
Both states have the same next state for every input. 7.8.2 State Encoding
When initially drawing the state diagram for a sequential circuit, it is preferred to keep the state names
symbolic. However, these state names must be eventually encoded with a unique bit string. State encoding is the
process of determining how many flip-flops are required to represent the states in the next-state table or state
diagram, and to assign a unique bit string combination to each named state. In all the examples presented so far, we
have been using the straight binary encoding scheme where n flip-flops are needed to encode 2n states. For example,
for four states, state s0 gets the encoding 00, state s1 gets the encoding 01, s2 gets 10, and s3 gets the encoding 11.
However, this scheme does not always lead to the smallest FSM circuit. Other encoding schemes are minimum bit
change, prioritized adjacency, and one-hot encoding.
For the minimum bit change scheme, binary encodings are assigned to the states in such a way that the total
number of bit changes for all state transitions is minimized. In other words, if every edge in the state diagram is
assigned a weight that is equal to the number of bit change between the source encoding and the destination
encoding of that edge, this scheme would select the one that minimizes the sum of all these edge weights.
For example, given a four-state state diagram shown in Figure 7.25 (a), the minimum bit change scheme would
use the encoding shown in (b) and not the encoding shown in (c). In both (b) and (c), the number of bit change
between the encodings of two states joined by an edge is labeled on that edge. For example, in (b), the number of bit
change between state s1 = 01 and s2 = 11 is 1. The encoding used in (b) has a smaller sum of all the edge weights
than the encoding used in (c).
Notice that even though the encoding of Figure 7.25 (b) produces the smallest total edge weight, there are
several other ways to encode these four states that will also produce the same total edge weight. For example,
assigning 00 to s1 instead of to s0, 01 to s2 instead of s1, 11 to s3, and 10 to s0. s0 s1 00 1 1 s3 s2
(a) 10 01 1 1
(b) 11 00 1 2 11 01 2 1 10 (c) Figure 7.25. Minimum bit change encoding: (a) a four-state state diagram; (b) encoding with a total weight of 4; (c)
encoding with a total weight of 6.
For the prioritized adjacency scheme, adjacent states to any state s are given certain priorities. Encodings are
assigned to these adjacent states such that those with a higher priority will have an encoding that has fewer bit
change from the encoding of state s than those adjacent states with a lower priority.
In the one-hot encoding scheme, each state is assigned one flip-flop. A state is encoded with its flip-flop having
a 1 value while all the other flip-flops have a 0 value. For example, the one-hot encoding for four states would be
0001, 0010, 0100, and 1000. 236 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits 7.8.3 Choice of Flip-Flops
A FSM can be implemented using any of the four types of flip-flops, SR, D, JK, and T (see Section 6.13) or any
combinations of them. Using different flip-flops can produce a smaller circuit but with the same functionality. The
decision as to what types of flip-flops to use is reflected in the implementation table. Whereas, the next-state table is
independent of the flip-flop types used, the implementation table is dependent on these choices of flip-flops.
The implementation table answers the question of what the flip-flop inputs should be in order to realize the
next-state table. In order to do this, we need to use the excitation table for the selected flip-flop(s). Recall that the
excitation table is used to answer the question of what the inputs should be when given the current state that the flipflop is in and the next state that we want the flip-flop to go to. So to get the entries for the implementation table, we
substitute the next-state values from the next-state table with the corresponding entry in the excitation table.
For example, if we have the following next-state table
Next State
Q1next Q0next
C=0
C=1
00
00
10
10
01
11
00
00 Current
State
Q1Q0
00
01
10
11 and we want to use the SR flip-flop to implement the circuit, we would convert the next-state table to the
implementation table as follows. First, the next state column headings from the next-state table (Q1nextQ0next) are
changed to the corresponding flip-flop input names (S1R1S0R0). Since the SR flip-flop has two inputs, therefore, each
next-state bit Qnext is replaced with two input bits SR. This is done for all the flip-flops used as shown below
Current State
Q 1Q 0 Implementation
S1R1S0R0
C=0
C=1 00
01
10
11 10__ To derive the entries in the implementation table, we will need the excitation table for the SR flip-flop (from
Section 6.13.1) shown below
Q
0
0
1
1 Qnext
0
1
0
1 S
0
1
0
× R
×
0
1
0 For example, if the current state for flip-flop one is Q1 = 0 and the next state Q1next = 1, we would do a table
lookup in the excitation table for QQnext = 01. The corresponding two input bits are SR = 10. Hence, we would
replace the 1 bit for Q1next in the next-state table with the two input bits S1R1 = 10 in the same entry location in the
implementation table. Proceeding in this same manner for all the next-state bits in the next-state table entries, we
obtain the complete implementation table below 237 Digital Logic and Microprocessor Design with VHDL Current State
Q 1Q 0
00
01
10
11 Chapter 7 - Sequential Circuits Implementation
S1R1S0R0
C=0
C=1
0×0×
0×0×
1001
1001
0110
×010
0101
0101 Once we have the implementation table, deriving the excitation equations and drawing the next-state circuit are
identical for all flip-flop types.
The output table and output equations are not affected by the change in flip-flop types, and so they remain
exactly the same too.
Example 7.11
In this example, we will design a modulo-6 up counter using T flip-flops. This is similar to Example 7.4 but
using T flip-flops instead of D flip-flops. The next-state table for the modulo-6 up counter as obtained from
Example 7.4 is shown in Figure 7.26 (a). The excitation table for the T flip-flop as derived in Section 6.14.3 is
shown in Figure 7.26 (b).
The implementation table is obtained from the next-state table by substituting each next-state bit with the
corresponding input bit of the T flip-flop. This is accomplished by doing a table look-up from the T flip-flop
excitation table.
For example, in the next-state table for the current state Q2Q1Q0 = 010 and the input C = 1, we want the next
state Q2next Q1next Q0next to be 011. The corresponding entry in the implementation table shown in Figure 7.26 (c)
using T flip-flops would be T2T1T0 = 001 because for flip-flop2 we want its content to go from Q2 = 0 to Q2next = 0.
The excitation table tells us that to realize this change, the T2 input needs to be a 0. Similarly, for flip-flop1 we want
its content to go from Q1 = 1 to Q1next = 1, and again the T1 input needs to be a 0 to realize this change. Finally, for
flip-flop0 we want its content to go from Q0 = 0 to Q0next = 1, this time, we need T0 to be a 1. Continuing in this
manner for all the entries in the next-state table, we obtain the implementation table shown in Figure 7.26 (c).
From the implementation table, we obtain the excitation equations just like before. For this example, we have
the three input bits T2, T1 and T0, which results in the three equations. These equations are dependent on the four
variables Q2, Q1, Q0, and C. The three K-maps and excitation equations for T2, T1, and T0 are shown in Figure 7.26
(d). The output equation is the same as before (see Figure 7.13 (e)). Finally, the complete modulo-6 up counter
circuit is shown in Figure 7.26 (e).
Comparing this circuit with the circuit from Example 7.3 shown in Figure 7.13 (f) where D flip-flops are used,
it is obvious that using T flip-flops for this problem result in a much smaller circuit than using D flip-flops.
♦ 238 Digital Logic and Microprocessor Design with VHDL Next State
Q2next Q1next Q0next
C=0
C=1
000
001
001
010
010
011
011
100
100
101
101
000 Current
State
Q2Q1Q0
000
001
010
011
100
101 Qnext
0
0
1
1 Chapter 7 - Sequential Circuits Qnext'
0
1
0
1 Implementation
T2 T1 T0
C=0
C=1
000
001
000
011
000
001
000
111
000
001
000
101 Current State
Q 2Q 1Q 0 T
0
1
1
0 000
001
010
011
100
101 (b) (a)
T2 (c)
T1 CQ2
00 01 11 10 00 0 0 0 0 01 0 0 1 0 11 0 × × 10 0 × × Q1 Q0 CQ2Q0
CQ1Q0 T0 CQ2
00 01 11 10 00 0 0 0 0 01 0 0 0 1 1 11 0 × × 0 10 0 × × T2 = CQ2Q0 + CQ2Q1 Q1 Q0 CQ2'Q0 CQ2
00 01 11 10 00 0 0 1 1 01 0 0 1 1 1 11 0 × × 1 0 10 0 × × 1 Q1Q0 C T1 = CQ2'Q0 T0 = C (d)
C T2 Q2 y Clk
Clear T1 Q'2 Q1 Clk
Clear T0 Q'1 Q0 Clk
Clock
Reset Clear Q'0 (e)
Figure 7.26. Synthesis of a FSM for Example 7.11: (a) next-state table; (b) excitation table for the T flip-flop; (c)
implementation table using T flip-flops; (d) K-maps and excitation equations; (e) FSM circuit. 7.9 Summary Checklist 239 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits State diagram
State encoding
Output signal
Conditional edge
Next-state table
Implementation table
Excitation equation
Output table
Output equation
Next-state logic
State memory
Output logic
FSM circuit
Unused state encoding
Be able to derive the state diagram from an arbitrary pseudo-code circuit description
Be able to derive the next-state table from a state diagram
Be able to derive the implementation table from a next-state table
Be able to derive the excitation equations from an implementation table
Be able to derive the output table from a state diagram
Be able to derive the output equations from an output table
Be able to derive the FSM circuit from the excitation and output equations 7.10 Problems
7.1. Analyze the following FSMs and derive the state diagram for it:
a) C is an input, and a and b are outputs.
C a
D1 Q1 Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset
Answer
Excitation / next-state equations:
Q1next = D1 = CQ1'Q0' + C'Q1'Q0
Q0next = D0 = Q1'Q0' + CQ0'
Output equations: 240 Q'0 b Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits a = Q1'Q0
b = Q1Q0
Next-state and output tables: Current State
Q1Q0
00
01
10
11 Next State
Q1next Q0next
C=0 C=1
01
11
10
00
00
01
00
00 Output
a
0
1
0
0 b
0
0
0
1 State diagram: C=1 00
ab=00 C=0
C=1 C=×
11
ab=01 01
ab=10 C=0
C=1 C=0
10
ab=00 b) C is an input, and a and b are outputs.
C a
D1 Q1 Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset
Answer 241 Q'0 b Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Excitation / next-state equations:
Q1next = D1 = CQ1'Q0' + C'Q1'Q0
Q0next = D0 = Q1'Q0' + CQ0'
Output equations:
a = CQ1'Q0
b = CQ0
Next-state and output tables:
Current State
Q1Q0
00
01
10
11 Next State / Output
Q1next Q0next / ab
C=0
C=1
01 / 00
11 / 00
10 / 00
00 / 11
00 / 00
01 / 00
00 / 00
00 / 01 State diagram:
00
ab=00
C=0
C=1
ab=00
ab=00
C=0
C=1
ab=01
ab=11
C=1
11
01
C=0
ab=00
C=1
ab=00 C=0
ab=00
10 c) A and B are inputs, and X and Y are outputs. 242 Digital Logic and Microprocessor Design with VHDL A Chapter 7 - Sequential Circuits B X
D1 Q1 Y Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset
Answer
Excitation / next-state equations:
Q1next = D1 = AQ1 + BQ1'Q0
Q0next = D0 = A' Q0' = A'Q0' + AQ0
Next-state table: Current State
Q1 Q0
00
01
10
11 00
01
00
01
00 Next State
Q1next Q0next
AB =
01 10
01 00
10 01
01 10
00 11 11
00
11
10
11 Output equations:
X = Q1 + Q0'
Y = (Q1' Q0)' = Q1 + Q0'
Output table:
Current State
Q1 Q0
00
01
10
11 Output
X Y 1
0
1
1 1
0
1
1 State diagram: 243 Q'0 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits AB = 1x AB = 10
AB = 0x 00
X=Y=1 AB = 0x 01
X=Y=0 AB = 00 AB = 11 AB = 01
AB = 0x 11
X=Y=1 10
X=Y=1 AB = 1x AB = 1x d) (Z≠0) is an input, and ClrX, LoadY, inZ, LoadX, stat1, LoadZ, and subtract are outputs.
(Z≠0) D1 Q1 ClrX
LoadY
inZ
LoadX
stat1
LoadZ
subtract Clk
Clear D0 Q'1 Q0 Clk Clock
Reset Clear Q'0 Answer:
Excitation / next-state equations:
D1 = Q1next = Q0 + (Z≠0)'
D0 = Q0next = Q1 + Q0'
Output equations:
ClrX = LoadY = inZ = Q1'Q0'
LoadX = stat1 = Q1'Q0
LoadZ = Q0'
subtract = Q1Q0'
Next-state table / Output table:
Current
State
Q1Q0
00
01
10 Next State
Q1next Q0next
(Z≠0) = 0
(Z≠0) = 1
11
01
10
10
11
01 Outputs
ClrX
1
0
0 244 LoadX
0
1
0 LoadY
1
0
0 LoadZ
1
0
1 inZ
1
0
0 stat1
0
1
0 subtract
0
0
1 Digital Logic and Microprocessor Design with VHDL 11 11 11 Chapter 7 - Sequential Circuits 0 0 0 0 State diagram: 0
(Z≠0)
ClrX=0
LoadX=1
LoadY=0
LoadZ=0
inZ=0
stat1=1
subtract=0 ClrX=1
LoadX=0
LoadY=1
LoadZ=1
inZ=1
stat1=0
subtract=0
(Z≠0)' 1 3 (Z≠0)
(Z≠0)'
2
ClrX=0
LoadX=0
LoadY=0
LoadZ=1
inZ=0
stat1=0
subtract=1 e) Start is an input, and LoadN, and LoadM are outputs. 245 ClrX=0
LoadX=0
LoadY=0
LoadZ=0
inZ=0
stat1=0
subtract=0 0 0 0 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Start LoadN
D2 Q2 LoadM Clk
Clear D1 Q'2 Q1 Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset
Answer: Excitation equations:
D2 = Q2'Q0 + Q1'
D1 = Q2'Q0' + Q2'Q1'Q0 + S'Q2Q1'Q0'
D0 = Q1Q0' + S'Q1'Q0' + SQ2Q0
Next-state equations:
Q2next = D2 = Q2'Q0 + Q1'
Q1next = D1 = Q2'Q0' + Q2'Q1'Q0 + Start'Q2Q1'Q0'
Q0next = D0 = Q1Q0' + Start'Q1'Q0' + StartQ2Q0
Next-state table: Current State
Q2Q1Q0
000
001
010
011
100
101
110
111 Next State
Q2nextQ1nextQ0next
Start =
0
1
111
110
110
110
011
011
100
100
111
100
100
101
001
001
000
001 Output equations:
LoadN = Q2Q1Q0 246 Q'0 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits LoadM = Q2'Q1Q0'
State diagram
LoadN=0
LoadM=0
000 LoadN=0
LoadM=0
001 Start'
Start' Start
Start 111
LoadN=1
LoadM=0 010
LoadN=0
LoadM=1 110
LoadN=0
LoadM=0 7.2. 011
LoadN=0
LoadM=0
Start'
LoadN=0
LoadM=0
101 Start'
100
LoadN=0
LoadM=0
Start Start Analyze the following FSMs and derive the state diagram for it:
a) C is an input, and a and b are outputs.
C a
T1 Q1 Clk
Clear T0 Q'1 Q0 Clk
Clear Clock
Reset
Answer
Excitation equations:
T1 = CQ1'Q0' + C'Q1'Q0 247 Q'0 b Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits T0 = Q1'Q0' + CQ0'
Characteristic equation for the T flip-flop:
Qnext = T ⊕ Q
Next-state equations:
Q1next = T1 ⊕ Q1
= (CQ1'Q0' + C'Q1'Q0) ⊕ Q1
= (CQ1'Q0' + C'Q1'Q0)'Q1 + (CQ1'Q0' + C'Q1'Q0)Q1'
= (CQ1'Q0' )' (C'Q1'Q0)' Q1 + CQ1'Q0' + C'Q1'Q0
= (C'+Q1+Q0) (C+Q1+Q0' ) Q1 + CQ1'Q0' + C'Q1'Q0
= C'Q1 + C'Q1Q0' + CQ1 + Q1 + Q1Q0' + CQ1Q0 + Q1Q0 + CQ1'Q0' + C'Q1'Q0
= Q1 + CQ1'Q0' + C'Q1'Q0
Q0next = T0 ⊕ Q0
= (Q1'Q0' + CQ0' ) ⊕ Q0
= (Q1'Q0' + CQ0' )' Q0 + (Q1'Q0' + CQ0' )Q0'
= (Q1'Q0' )' (CQ0' )' Q0 + Q1'Q0' + CQ0'
= (Q1+Q0) (C'+Q0) Q0 + Q1'Q0' + CQ0'
= C'Q1Q0 + Q1Q0 + C'Q0 + Q0 + Q1'Q0' + CQ0'
= Q0 + Q1'Q0' + CQ0'
Output equations:
a = Q1'Q0
b = Q1Q0
Next-state and output tables:
Next State
Q1next Q0next
C=0 C=1
01
11
11
01
10
11
11
11 Current State
Q1Q0
00
01
10
11 Output
a
0
1
0
0 b
0
0
0
1 State diagram: C=1
C=× 11
ab=01 00
ab=00 C=0 C=1 248 01
ab=10 C=0
10
ab=00 b) C is an input, and a and b are outputs. C=0 C=1 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits C a
J1 Q1 Clk
K1 Clr J0 Q'1 Q0 Clk
K0 Clock
Reset
Answer:
Excitation equations:
J1 = CQ1'Q0' + C'Q1'Q0
K1 = C'Q1
J0 = Q1'Q0'
K0 = CQ0
Characteristic equation for the JK flip-flop:
Qnext = K'Q + JQ'
Next-state equations:
Q1next = K1'Q1 + J1Q1'
= (C'Q1)'Q1 + (CQ1'Q0' + C'Q1'Q0)Q1'
= (C+Q1' )Q1 + CQ1'Q0' + C'Q1'Q0
= CQ1 + CQ1'Q0' + C'Q1'Q0
Q0next = K0'Q0 + J0Q0'
= (CQ0)'Q0 + (Q1'Q0' )Q0'
= (C' +Q0' )Q0 + Q1'Q0'
= C'Q0 + Q1'Q0'
Output equations:
a = Q1'Q0
b = Q1Q0
Next-state and output tables: 249 Clr Q'0 b Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Next State
Q1next Q0next
C=0 C=1
01
11
11
00
00
10
01
10 Current State
Q1Q0
00
01
10
11 Output
a
0
1
0
0 b
0
0
0
1 State diagram:
00
ab=00 C=1 C=0
C=1 C=0
11
ab=01 01
C=0 ab=10
C=0 C=1 C=1
10
ab=00 c) C is an input, and a and b are outputs.
C a
S1 Q1 Clk
R1 Clr S0 Q'1 Q0 Clk
R0 Clock
Reset
Answer:
Excitation equations:
S1 = CQ1'Q0' + C'Q1'Q0
R1 = C'Q1
S0 = Q1'Q0'
R0 = CQ0
Next-state equations:
Q1next = S1 + R1'Q1 250 Clr Q'0 b Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits = CQ1'Q0' + C'Q1'Q0 + (C'Q1)'Q1
= CQ1'Q0' + C'Q1'Q0 + (C + Q1' )Q1
= CQ1'Q0' + C'Q1'Q0 + CQ1 + Q1'Q1
= CQ1'Q0' + C'Q1'Q0 + CQ1
Q0next = S0 + R0'Q0
= Q1'Q0' + (CQ0)'Q0
= Q1'Q0' + (C' + Q0' )Q0
= Q1'Q0' + C'Q0 + Q0'Q0
= Q1'Q0' + C'Q0 Next-state table:
Next State
Q1next Q0next
C=0 C=1
01
11
11
00
00
10
01
10 Current State
Q1Q0
00
01
10
11
State diagram: C=0
00 C=1
C=0 01 C=1
C=0
11 C=0
10 C=1
C=1 d) C is an input, and a and b are outputs. 251 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits C a
S1 Q1 b Clk
R1 Clr J0 Q'1 Q0 Clk
K0 Clr Q'0 Clock
Reset
7.3. Synthesize a FSM circuit using D flip-flops for the following state diagrams:
a)
A=0 00 01
A=1
A=1 A=0 A=0 A=× 11 10
A=1 Answer:
Next-state table:
Current State
Q 1Q 0
00
01
10
11 Next State
Q1next Q0next
A=0 A=1
11
10
01
11
01
01
00
10 Implementation table:
Current State
Q 1Q 0
00
01
10
11 Next State
J1K1S0R0
A=0 A=1
1×10 1×0×
0××0 1××0
×110 ×110
×101 ×001 252 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits Excitation equations:
J1 = Q0' + A
K1 = Q0' + A'
S0 = (Q1 ⊕ Q0) + Q0'A'
R 0 = Q 1Q 0
FSM circuit: A J0 Q0 Clk
Q'
K0
Clear 0 S0 Q0 Clk
Q'
R0
Clear 0 Clock
Reset
b) 7.4. Use JK flip-flops to synthesize a FSM circuit for the state diagrams in Problem 7.3. 7.5. Use SR flip-flops to synthesize a FSM circuit for the state diagrams in Problem 7.3. 7.6. Use T flip-flops to synthesize a FSM circuit for the state diagrams in Problem 7.3. 7.7. Use a JK flip-flop for flip-flop 1, and a T flip-flop for flip-flop 2 to synthesize a FSM circuit for the state
diagrams in Problem 7.3. 7.8. Design a modulo-4 up/down counter using D flip-flops. The count is represented by the content of the flipflops. The circuit has a Count signal and an Up signal. The counter counts when Count is asserted, and stops
when Count is de-asserted. The Up signal determines the direction of the count. When Up is asserted, the
count increments by one at each clock cycle. When Up is de-asserted, the count decrements by one at each
clock cycle. 7.9. Design a modulo-5 up counter using D flip-flops similar to Problem 7.8, but without the Up signal. 7.10. Design a modulo-5 up/down counter using D flip-flops similar to Problem 7.8.
7.11. Design a modulo-4 up counter using T flip-flops.
Answer:
Next-state table: 253 Digital Logic and Microprocessor Design with VHDL Current State
Q1 Q0
00
01
10
11 Chapter 7 - Sequential Circuits Next State
Q1next Q0next
C=0 C=1
00
01
01
10
10
11
11
00 Implementation table:
Current State
Q1 Q0
00
01
10
11 Next State
T1 T0
C=0 C=1
00
01
00
11
00
01
00
11 Excitation equations:
T1 = CQ0
T0 = C
Circuit:
C T1 Q1 Clk
Q'1 T0
Clk Q0 Clk
Q'0 7.12. Design a FSM that counts the following decimal sequence
3, 7, 2, 6, 3, 7, 2, 6, …
The count is to be represented directly by the contents of the D flip-flops. The counting starts when the control
input C is asserted and stops whenever C is de-asserted. Assume that the next-state from all unused states is the
state for the first count in the sequence, i.e. the state for 3.
Answer:
Since we’re using the flip-flop content to represent the count and the largest number is 7, therefore, we need
three (3) bits even though there are only four numbers in the sequence.
State diagram: 254 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits C=0 C=0
011 C=1 111
C=1 xxx
all unused
state C=1
110 010
C=1
C=0 C=0
Next-state table and implementation table:
Current State
Q2Q1Q0
000
001
010
011
100
101
110
111 Next State / Implementation
Q2next Q1next Q0next / D2D1D0
C=0
C=1
011
011
011
011
010
110
011
111
011
011
011
011
110
011
111
010 Excitation equations:
D2 = Q2'Q1C + Q2Q1C'
D1 = 1
D0 = Q1' + Q2'Q0 + Q0C' + Q2Q0'C
Circuit: 255 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits C D2 Q2 Clk
Q'2 '1' D1 Q1 Clk
Q'1 D0 Q0 Clk
Q'0
Clk 7.13. Design a counter that counts in the following sequence:
1, 4, 6, 7, 1, 4, 6, 7, ….
The count is to be represented directly by the contents of three D flip-flops. The counter is enabled by the input
C. The count stops when C = 0. The next-state from all unused states are undefined.
7.14. Repeat Example 7.6 but encode state s2 as 10 instead of 11, and the unused state is 11. See if the resulting
FSM circuit is larger or smaller.
7.15. Repeat Problem 7.13, but use a JK flip-flop, a D flip-flop, and a SR flip-flop in this order starting from the
most significant bit for the three flip-flops.
Answer:
Next-state table:
Current State
Q2Q1Q0
001
100
110
111 Next State
Q2next Q1next Q0next
C=0
C=1
001
100
100
110
110
111
111
001 Excitation tables for the three flip-flops:
Q
0
0 Qnext
0
1 J
0
1 K
×
× D
0
1 S
0
1 R
×
0 256 Digital Logic and Microprocessor Design with VHDL 1
1 ×
× 0
1 1
0 0
1 0
× Chapter 7 - Sequential Circuits 1
0 Implementation table:
Implementation
J2 K2 D1 S0 R0
C=0
C=1
0×00
1×001
×000×
×010×
×010×
×0110
×01×0
×10×0 Current State
Q2Q1Q0
001
100
110
111 K-maps and excitation equations:
J2
Q1Q0 CQ2 × 01 0 × × 0 × 11 × 1 0 × × 10 × 1 1 × × 0 0 × 01 × × × × × 11 × 0 1 × 10 × 0 0 × × × 01 0 × × 1 × × × × × × Q1Q0 CQ0 J2 = C K2 = CQ0
R0 CQ2
00 01 11 10 00 × 0 0 × 01 × × × 0 11 × × × × 10 × 0 1 × S0 = CQ1 10 1 00 × 10 CQ1 11 0 10 10 Q1Q0 01 × 11 11 11 S0 00 01 01 00 01 11 10 00 CQ1' × × × × 01 0 × × 1 11 × 0 0 × 10 × × 0 × R0 = CQ1' Circuit: 257 Q1Q0 CQ0'
C'Q1 D1 = CQ0' + C'Q1 CQ2 Q1Q0 CQ2 00 00 00
00 C D1 K2 CQ2 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits C J2 Q2 Clk
K2 Q'2 D1 Q1 Clk
Q'1
S0 Q0 Clk
R0 Q'0 Clk 7.16. Manually design and implement on the UP2 board the following FSM circuit. Make the LEDs in the 7segment display move in a clockwise direction around in a circle, i.e. turn on and off the LED segments in
this order: segment a, b, c, d, e, f, a, b, etc.
7.17. Manually design and implement on the UP2 board the following FSM circuit. Similar to Problem 7.16, but
make one 7-segment LED display in a clockwise direction, and the other in an anti-clockwise direction.
7.18. Manually design and implement on the UP2 board the following FSM circuit. Similar to Problem 7.16, but
make it so that each time when a push button switch is pressed, the display changes directions.
7.19. Manually design and implement on the UP2 board the following FSM circuit. Input from the eight DIP
switches. Output on the 7-segment the decimal number that represents the number of DIP switches that are in
the on position.
7.20. Manually design and implement on the UP2 board a FSM circuit for controlling three switches, T1, T2, and
T3, and three lights L1, L2, and L3. Each light is turned on by the corresponding switch, for example, T1 turns
on L1. Initially, all switches are off. The first switch that is pressed will turn on its corresponding light. When
the first light is turned on, it will remain on, while the other two lights remain off, and they are unaffected by
subsequent switch presses until reset.
Answer: Q1 Q0
00
01
10
11 000
00
01
10
11 001
01
01
10
11 Q1next Q0next
T3 T2 T1
010 011 100 101
10
00
11
00
01
01
01
01
10
10
10
10
11
11
11
11 258 110
00
01
10
11 111
00
01
10
11 Digital Logic and Microprocessor Design with VHDL Chapter 7 - Sequential Circuits T 3 T2 T1 L1
D1 Q1 L2 Q'1 L3 Clk
Clear D0 Q0 Clk
Clear Q'0 Reset
Clock 7.21. Design a FSM circuit for controlling a simple home security system. The operation of the system is as
follows.
Inputs: - Front gate switch (FS)
- Motion detector switch (MS)
- Asynchronous Reset switch (R)
- Clear switch (C) Outputs: - Front gate melody (FM)
- Motion detector melody (MM) • When the reset switch (R) is asserted, the FSM goes to the initialization state (S_init) immediately. The
encoding for the initialization state is zeros for all the flip-flops. • From state S_init, the FSM unconditionally goes to the wait state (S_wait). • From state S_wait, the FSM waits for one of the four switches to be activated. All the switches are active
high so when a switch is pressed or activated, it sends out a 1. The following actions are taken when a
switch is pressed: • When FS is pressed, the FSM goes to state S_front. In state S_front, the front gate melody is turned on by
setting FM = 1. The FSM remains in state S_front until the clear switch is pressed. Once the clear switch is
pressed, the FSM goes back to S_wait. • When MS is activated, the FSM goes to state S_motion. In state S_motion, MM is turned on with a 1. MM
will remain on for two more clock periods and then it will go back to S_wait. • From any state, as soon as the reset switch is pressed, the FSM immediately goes back to state S_init. • Pressing the clear switch only affects the FSM when it is in state S_front. The clear switch has no effect on
the FSM when it is in any other states. • Any unused state encoding will have S_init as their next state. 259 Chapter 8 Standard Sequential Components Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components In a computer system, we usually want to store more than one bit of information. More over, we may want to
group several bits together and consider them as one unit, such as an integer is made up of eight bits. In Chapter 6,
we presented the circuits for latches and flip-flops for storing one bit of information. In this chapter, we will look at
registers for storing multiple bits of information as a unit. Registers also are made more versatile by adding extra
functionalities, such as counting and shifting, to it. We will also look at the design of counters and shift registers.
Very often, computer circuits may need to store several values at the same time. Instead of using several
separate registers, we may want to combine these registers together. Register files and memories are like an array of
registers for storing multiple values. In this chapter, we will also look at the construction of register files and
memory circuits.
Similar to the standard combinational components, these sequential components are used in almost every digital
circuit. Hence, rather than having to redesign them each time that they are needed, they usually are available in
standard libraries. 8.1 Registers When we want to store a byte of data, we need to combine eight flip-flops together and have them work
together as a unit. A register is just a circuit with two or more D flip-flops connected together in such a way that
they all work exactly the same way and are synchronized by the same clock and enable signals. The only difference
is that each flip-flop in the group is used to store a different bit of the data.
Figure 8.1(a) shows a 4-bit register with parallel load and asynchronous clear. Four D flip-flops with activehigh enable and asynchronous clear are used. Notice in the circuit that the control inputs, Clk, E, and Clear, for all of
the flip-flops are connected, respectively, in common; so that when a particular input is asserted, all of the flip-flops
will behave in exactly the same way. The 4-bit input data is connected to D0 through D3, while Q0 through Q3 serve
as the 4-bit output data for the register. When the active-high load signal Load is asserted (i.e., Load = 1), the data
presented on the D lines is stored into the register (the four flip-flops) at the next rising edge of the clock signal.
When Load is de-asserted, the content of the register remains unchanged. The register can be asynchronously
cleared (i.e., setting all of the Qi’s to 0 immediately, without having to wait for the next active clock edge) by
asserting the Clear line. The content of the register is always available on the Q output lines, so no control line is
required for reading the data from the register. Figure 8.1(b) and (c) show the operation table and the logic symbol,
respectively, for this 4-bit register.
Figure 8.2 shows the VHDL code for the 4-bit register with active-high Load and Clear signals. Notice that the
coding is very similar to that for the single D flip-flop. The main difference is that the data inputs and outputs are 4bits wide. A sample simulation trace for the register is shown in Figure 8.3. At time 100 ns, even though Load is
asserted, the register is not written with the D input value of 5, because Clear is asserted. Between times 200 ns and
400 ns, Load is de-asserted, so even though Clear is de-asserted, the register still is not loaded with the input value
of 5. At time 400 ns, Load is asserted, but the input data is not loaded into the register immediately (as can be seen
by Q being a 0). The loading occurs at the next rising edge of the clock at 500 ns when Q changes to 5. At time 600
ns, Clear is asserted, and so, Q is reset to 0 immediately, without having to wait until the next rising clock edge at
700 ns. 261 Digital Logic and Microprocessor Design with VHDL D3 Chapter 8 - Standard Sequential Components D2 D1 D0 Clear
Clear
D3
Q3 Clear
D2
Q2 Clk Clear
D1
Q1 Clk E Clear
D0
Q0 Clk E Clk E E Clock
Load
Q3 Q2 Q1 Q0 (a) D3
Clear
1
0
0 Load
×
0
1 Operation
Reset register to zero asynchronously
No change
Load in a value at rising clock edge D2 D1 D0 Clear 4-bit Register
with Parallel Load and
Load Asynchronous Clear Clock
Q3
Q2
Q1
Q0 (b) (c) Figure 8.1 A 4-bit register with parallel load and asynchronous clear: (a) circuit; (b) operation table; (c) logic
symbol.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY reg IS GENERIC (size: INTEGER := 3);-- size of the register
PORT (
Clock, Clear, Load: IN STD_LOGIC;
D: IN STD_LOGIC_VECTOR(size DOWNTO 0);
Q: OUT STD_LOGIC_VECTOR(size DOWNTO 0));
END reg;
ARCHITECTURE Behavior OF reg IS
BEGIN
PROCESS(Clock, Clear)
BEGIN
IF Clear = '1' THEN
Q <= (OTHERS => '0');
ELSIF (Clock'EVENT AND Clock = '1') THEN
IF Load = '1' THEN
Q <= D;
END IF;
END IF;
END PROCESS;
END Behavior;
Figure 8.2 VHDL code for a 4-bit register with active-high Load and Clear signals. 262 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Figure 8.3 Sample simulation trace for the 4-bit register. 8.2 Shift Registers Similar to the combinational shifter and rotator circuits, there are the equivalent sequential shifter and rotator
circuits. The circuits for the shift and rotate operations are constructed exactly the same. The only difference in the
sequential version is that the operations are performed on the value that is stored in a register rather than directly on
the input value. The main usage for a shift register is for converting from a serial data input stream to a parallel data
output or vice versa. For a serial-to-parallel data conversion, the bits are shifted into the register at each clock cycle,
and when all of the bits (usually eight bits) are shifted in, the 8-bit register can be read to produce the eight bit
parallel output. For a parallel-to-serial conversion, the 8-bit register is first loaded with the input data. The bits are
then individually shifted out, one bit per clock cycle, on the serial output line. 8.2.1 Serial-to-Parallel Shift Register
Figure 8.4(a) shows a 4-bit serial-to-parallel converter. The input data bits come in on the Serial_in line at a rate
of one bit per clock cycle. When Shift is asserted, the data bits are loaded in one bit at a time. In the first clock cycle,
the first bit from the serial input stream, Serial_in, gets loaded into Q3, while the original bit in Q3 is loaded into Q2,
Q2 is loaded into Q1, and so on. In the second clock cycle, the bit that is in Q3 (i.e., the first bit from the Serial_in
line) gets loaded into Q2, while Q3 is loaded with the second bit from the Serial_in line. This continues for four
clock cycles until four bits are shifted into the four flip-flops, with the first bit in Q0, second bit in Q1, and so on.
These four bits are then available for parallel reading through the output Q. Figure 8.4(b) and (c) show the operation
table and the logic symbol, respectively, for this shift register.
The structural VHDL code for a 4-bit serial-to-parallel shift register is shown in Figure 8.5. The code is written
at the structural level. The operation of a D flip-flop with enable is first defined. The ARCHITECTURE section for the
ShiftReg entity uses four PORT MAP statements to instantiate four D flip-flops. These four flip-flops then are
connected together using the internal signals, N0, N1, N2, and N3, such that the output of one flip-flop is connected to
the input of the next flip-flop. These four internal signals also connect to the four output signals, Q0 to Q3, for the
register output. Note that we cannot use the output signals, Q0 to Q3, to directly connect the four flip-flops together,
since output signals cannot be read.
A sample simulation trace of the serial-to-parallel shift register is shown in Figure 8.6. At the first rising clock
edge at time 100 ns, the Serial_in bit is a 0, so there is no change in the 4 bits of Q, since they are initialized to 0’s.
At the next rising clock edge at time 300 ns, the Serial_in bit is a 1, and it is shifted into the leftmost bit of Q.
Hence, Q has the value of 1000. At time 500 ns, another 1 bit is shifted in, giving Q the value of 1100. At time 700
ns, a 0 is shifted in, giving Q the value of 0110. Notice that as bits are shifted in, the rightmost bits are lost. At time
900 ns, Shift is de-asserted, so the 1 bit in the Serial_in line is not shifted in. Finally, at time 1.1 µs, another 1 bit is
shifted in. 263 Digital Logic and Microprocessor Design with VHDL Serial_in D3 Q3 Chapter 8 - Standard Sequential Components D2 Clk Q2 D1 Clk E Q1 D0 Clk E Q0 Clk E E Clock
Shift
Q3 Q2 Q1 Q0 (a) Serial_in
Shift
0
1 Operation Shift No change
One bit from Serial_in is shifted in Q3 (b) 4-bit
Serial-to-Parallel
Shift Register
Q2 Q1 Clock
Q0 (c) Figure 8.4 A 4-bit serial-to-parallel shift register: (a) circuit; (b) operation table; (c) logic symbol.
-- D flip-flop with enable
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY D_flipflop IS
PORT(D, Clock, E : IN STD_LOGIC;
Q : OUT STD_LOGIC);
END D_flipflop;
ARCHITECTURE Behavior OF D_flipflop IS
BEGIN
PROCESS(Clock)
BEGIN
IF (Clock'EVENT AND Clock = '1') THEN
IF (E = '1') THEN
Q <= D;
END IF;
END IF;
END PROCESS;
END Behavior;
-- 4-bit shift register
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY ShiftReg IS
PORT(Serial_in, Clock, Shift : IN STD_LOGIC;
Q : OUT STD_LOGIC_VECTOR(3 downto 0));
END ShiftReg;
ARCHITECTURE Structural OF ShiftReg IS
SIGNAL N0, N1, N2, N3 : STD_LOGIC; 264 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components COMPONENT D_flipflop PORT (D, Clock, E : IN STD_LOGIC;
Q : OUT STD_LOGIC);
END COMPONENT;
BEGIN
U1: D_flipflop
U2: D_flipflop
U3: D_flipflop
U4: D_flipflop
Q(3) <= N3;
Q(2) <= N2;
Q(1) <= N1;
Q(0) <= N0;
END Structural; PORT
PORT
PORT
PORT MAP
MAP
MAP
MAP (Serial_in,
(N3, Clock,
(N2, Clock,
(N1, Clock, Clock,
Shift,
Shift,
Shift, Shift, N3);
N2);
N1);
N0); Figure 8.5 Structural VHDL code for a 4-bit serial-to-parallel shift register. Figure 8.6 Sample simulation trace for the 4-bit serial-in-parallel-out shift register of Figure 8.5. 8.2.2 Serial-to-Parallel and Parallel-to-Serial Shift Register
For both the serial-to-parallel and parallel-to-serial operations, we perform the same left-to-right shifting of bits
through the register. The only difference between the two operations is whether we want to perform a parallel read
after the shifting or a parallel write before the shifting. For the serial-to-parallel operation, we want to perform a
parallel read after the bits have been shifted in. On the other hand, for the parallel-to-serial operation, we want to
perform a parallel write first and then shift the bits out as a serial stream.
We can implement both operations into the serial-to-parallel circuit from the previous section simply by adding
a parallel load function to the circuit, as shown in Figure 8.7(a). The four multiplexers work together for selecting
whether we want the flip-flops to retain the current value, load in a new value, or shift the bits to the right by one bit
position. The operation of this circuit is dependent on the two select lines, SHSel1 and SHSel0, which control which
input of the multiplexers is selected. The operation table and logic symbol are shown in Figure 8.7(b) and (c),
respectively. The behavioral VHDL code and a sample simulation trace for this shift register are shown in Figure 8.8
and Figure 8.9, respectively. 265 Digital Logic and Microprocessor Design with VHDL D3 Chapter 8 - Standard Sequential Components D2 D1 D0 Serial_in
SHSel1
SHSel0 Serial_out
3210
s1
s0 y 3210
s1
s0 y D3 Q3 3210
s1
s0 y D2 Clk Q2 3210
s1
s0 y D1 Clk Q1 D0 Clk Q0 Clk Clock
Q3 Q2 Q1 Q0 (a) D3
Serial_in
SHSel1
0
0
1
1 SHSel0
0
1
0
1 Operation
No operation (i.e., retain current value)
Parallel load in new value
Shift right
Rotate right D2
D1
D0
Serial_out
4-bit
Serial-to-Parallel
SHSel1 and Parallel-to-Serial
Clock
Shift Register
SHSel0
Q3
Q2
Q1
Q0 (c) (b) Figure 8.7 A 4-bit serial-to-parallel and parallel-to-serial shift register: (a) circuit; (b) operational table; (c) logic
symbol.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY shiftreg IS PORT (
Clock: IN STD_LOGIC;
SHSel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
Serial_in: IN STD_LOGIC;
D: IN STD_LOGIC_VECTOR(3 DOWNTO 0);
Serial_out: OUT STD_LOGIC;
Q: OUT STD_LOGIC_VECTOR(3 DOWNTO 0));
END shiftreg;
ARCHITECTURE Behavioral OF shiftreg IS
SIGNAL content: STD_LOGIC_VECTOR(3 DOWNTO 0);
BEGIN
PROCESS(Clock)
BEGIN
IF (Clock'EVENT AND Clock='1') THEN
CASE SHSel IS
WHEN "01" => -- load
content <= D;
WHEN "10" => -- shift right, pad with bit from Serial_in
content <= Serial_in & content(3 DOWNTO 1);
WHEN OTHERS =>
NULL;
END CASE; 266 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components END IF;
END PROCESS;
Q <= content;
Serial_out <= content(0);
END Behavioral;
Figure 8.8 Behavioral VHDL code for a 4-bit serial-to-parallel and parallel-to-serial shift register. Figure 8.9 Sample trace for the 4-bit serial-to-parallel and parallel-to-serial shift register. 8.3 Counters Counters, as the name suggests, are for counting a sequence of values. However, there are many different types
of counters depending on the total number of count values, the sequence of values that it outputs, whether it counts
up or down, and so on. The simplest is a modulo-n counter that counts the decimal sequence 0, 1, 2, … up to n-1 and
back to 0. Some typical counters are described next.
Modulo-n counter: Counts from decimal 0 to n – 1 and back to 0. For example, a modulo-5 counter sequence
in decimal is 0, 1, 2, 3, and 4.
Binary coded decimal (BCD) counter: Just like a modulo-n counter, except that n is fixed at 10. Thus, the
sequence is always from 0 to 9.
n-bit binary counter: Similar to modulo-n counter, but the range is from 0 to 2n – 1 and back to 0, where n is
the number of bits used in the counter. For example, a 3-bit binary counter sequence in decimal is 0, 1, 2, 3,
4, 5, 6, and 7.
Gray-code counter: The sequence is coded so that any two consecutive values must differ in only one bit. For
example, one possible 3-bit gray-code counter sequence is 000, 001, 011, 010, 110, 111, 101, and 100.
Ring counter: The sequence starts with a string of 0 bits followed by one 1 bit, as in 0001. This counter simply
rotates the bits to the left on each count. For example, a 4-bit ring counter sequence is 0001, 0010, 0100,
1000, and back to 0001.
We will now look at the design of several counters. 267 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components 8.3.1 Binary Up Counter
An n-bit binary counter can be constructed using a modified n-bit register where the data inputs for the register
come from an incrementer (adder) for an up counter, and a decrementer (subtractor) for a down counter. To get to
the next up-count sequence from the value that is stored in a register, we simply have to add a 1 to it. We can use the
full adder discussed in Section 4.2.1 as the input to the register, but we can do better. The full adder adds two
operands plus the carry. But what we want is just to add a 1, so the second operand to the full adder is always a 1.
Since the 1 can also be added in via the carry-in signal of the adder, we really do not need the second operand input.
This modified adder that only adds one operand with the carry-in is called a half adder (HA). Its truth table is
shown in Figure 8.10(a). We have a as the only input operand, cin and cout are the carry-in and carry-out signals,
respectively, and s is the sum of the addition. In the truth table, we are simply adding a plus cin to give the sum s and
possibly a carry-out, cout. From the truth table, we obtain the two equations for cout and s shown in Figure 8.10(b).
The HA circuit is shown in Figure 8.10(c) and its logic symbol in (d).
a
0
0
1
1 cin
0
1
0
1 cout
0
0
0
1 s
0
1
1
0 cout = a cin
s = a ⊕ cin (a) (b) cin
a cout cout cin
s HA a s (d) (c)
Figure 8.10 Half adder: (a) truth table; (b) equations; (c) circuit; (d) logic symbol. Several half adders can be daisy-chained together, just like with the full adders to form an n-bit adder. The
single operand input a comes from the register. The initial carry-in signal, c0, is used as the count enable signal,
since a 1 on c0 will result in incrementing a 1 to the register value, and a 0 will not. The resulting 4-bit binary up
counter circuit is shown in Figure 8.11(a), along with its operation table and logic symbol in (b) and (c). As long as
Count is asserted, the counter will increment by 1 on each clock pulse until Count is de-asserted. When the count
reaches 2n – 1 (which is equivalent to the binary number with all 1’s) the next count will revert back to 0, because
adding a 1 to a binary number with all 1’s will result in an overflow on the Overflow bit, and all of the original bits
will reset to 0. The Clear signal allows an asynchronous reset of the counter to 0. Count
Overflow cout cin C3 HA a s D3 cout D2 Clk C2 HA a s Q3 cin cin C1 cout s HA a s D1 Q2 Clk Clear cout D0 Q1 Clk Clear cin
HA a Q0 Clk Clear Clear Clock
Clear
Q2 Q3
(a) 268 Q1 Q0 Digital Logic and Microprocessor Design with VHDL Clear
1
0
0 Count
×
0
1 Chapter 8 - Standard Sequential Components Clear
Count Operation
Reset counter to zero
No change
Count up 4-bit Binary
Up Counter
Q3 Q2 Q1 Overflow
Clock
Q0 (c) (b) Figure 8.11 A 4-bit binary up counter with asynchronous clear: (a) circuit; (b) operation table; (c) logic symbol.
The behavioral VHDL code for the 4-bit binary up counter is shown in Figure 8.12. The statement
is needed in order to perform additions on STD_LOGIC_VECTORs. The internal
signal value is used to store the current count. When Clear is asserted, value is assigned the value “0000” using the
expression OTHERS => '0'. Otherwise, if Count is asserted, then value will be incremented by 1 on the next rising
clock edge. Furthermore, the count in value is assigned to the counter output, Q, using the concurrent statement
Q <= value, because it is outside the PROCESS block. A sample simulation trace is shown in Figure 8.13.
USE IEEE.STD_LOGIC_UNSIGNED.ALL LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL; -- need this to add STD_LOGIC_VECTORs ENTITY counter IS PORT (
Clock: IN STD_LOGIC;
Clear: IN STD_LOGIC;
Count: IN STD_LOGIC;
Q : OUT STD_LOGIC_VECTOR(3 DOWNTO 0));
END counter;
ARCHITECTURE Behavioral OF counter IS
SIGNAL value: STD_LOGIC_VECTOR(3 DOWNTO 0);
BEGIN
PROCESS (Clock, Clear)
BEGIN
IF Clear = '1' THEN
value <= (OTHERS => '0');
-- 4-bit vector of 0, same as "0000"
ELSIF (Clock'EVENT AND Clock='1') THEN
IF Count = '1' THEN
value <= value + 1;
END IF;
END IF;
END PROCESS;
Q <= value;
END Behavioral;
Figure 8.12 Behavioral VHDL code for a 4-bit binary up counter. 269 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Figure 8.13 Simulation trace for the 4-bit binary up counter. 8.3.2 Binary Up-down Counter
We can design an n-bit binary up-down counter just like the up counter, except that we need both an adder and a
subtractor for the data input to the register. The half adder-subtractor (HAS) truth table is shown in Figure 8.14(a).
The Down signal is to select whether we want to count up or down. Asserting Down (setting to 1) will count down.
The top half of the table is exactly the same as the HA truth table. For the bottom half, we are performing a
subtraction of a – cin, where s is the difference of the subtraction, and cout is a 1 if we need to borrow. For example,
for 0 – 1, we need to borrow, so cout is a 1. When we borrow, we get a 2; and 2 – 1 = 1, so s is also a 1. The two
resulting equations for cout and s are shown in Figure 8.14(b). The circuit and logic symbol for the half addersubtractor are shown in Figure 8.14(c) and (d).
Down
0
0
0
0
1
1
1
1 a
0
0
1
1
0
0
1
1 cin
0
1
0
1
0
1
0
1 cout
0
0
0
1
0
1
0
0 s
0
1
1
0
0
1
1
0 cout = Down' a cin + Down a' cin = (Down ⊕ a) cin
s = Down' (a ⊕ cin) + Down (a ⊕ cin) = a ⊕ cin (a) (b) Down
cin
a cout Down
cout cin
s HAS a s
(c) (d) Figure 8.14 Half adder-subtractor (HAS): (a) truth table; (b) equations; (c) circuit; (d) logic symbol.
We can simply replace the HAs with the HASs in the up counter circuit to get the up-down counter circuit, as
shown in Figure 8.15(a). Its operation table and logic symbol are shown in Figure 8.15(b) and (c). Again, the
Overflow signal is asserted each time the counter rolls over from 1111 back to 0000.
The VHDL code for the up-down counter, shown in Figure 8.16, is similar to the up counter code but with the
additional logic for the Down signal. If Down is asserted, then value is decremented by 1, otherwise it is
incremented by 1. To make the code a little bit different, the counter output signal, Q, is declared as an integer that
ranges from 0 to 15. This range, of course, is the range for a 4-bit binary value. Furthermore, the storage for the
current count, value, is declared as a variable of type integer rather than as a signal. Notice also, that the signal
assignment statement, Q <= value, is put inside the PROCESS block. Instead of being a concurrent statement (when it 270 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components was placed outside the PROCESS block in Figure 8.12), it is now a sequential statement. A sample simulation trace is
shown in Figure 8.17. Count
Down
Overflow C4 C3 Down
cout
cin
HAS a
s D3 Q3 Down
cout
cin
HAS a
s D2 Clk C2 Q2 D1 Clk Clear Down
cout
cin
HAS a
s C1 Q1 Down
cout
cin
HAS a
s D0 Clk Clear Q0 Clk Clear Clear Clock
Clear
Q3 Q2 Q1 Q0 (a) Clear
1
0
0
0 Count
×
0
1
1 Down
×
×
0
1 Clear
Count
Down Operation
Reset counter to zero
No change
Count up
Count down 4-bit Binary
Up-down Counter
Q3 Q2 Q1 Overflow
Clock Q0 (c) (b) Figure 8.15 A 4-bit binary up-down counter with asynchronous clear: (a) circuit; (b) operation table; (c) logic
symbol.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY udcounter IS PORT (
Clock: IN STD_LOGIC;
Clear: IN STD_LOGIC;
Count: IN STD_LOGIC;
Down: IN STD_LOGIC;
Q: OUT INTEGER RANGE 0 TO 15);
END udcounter;
ARCHITECTURE Behavioral OF udcounter IS
BEGIN
PROCESS (Clock, Clear)
VARIABLE value: INTEGER RANGE 0 TO 15;
BEGIN
IF (Clear = '1') THEN
value := 0;
ELSIF (Clock'EVENT AND Clock='1') THEN
IF (Count = '1') THEN
IF (Down = '0') THEN
value := value + 1;
ELSE 271 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components value := value - 1;
END IF;
END IF;
END IF;
Q <= value;
END PROCESS;
END Behavioral;
Figure 8.16 VHDL code for a 4-bit binary up-down counter. Figure 8.17. Simulation trace for the 4-bit binary up-down counter. 8.3.3 Binary Up-down Counter with Parallel Load
To make the binary counter more versatile, we need to be able to start the count sequence with any number
other than zero. This is accomplished easily by modifying our counter circuit to allow it to load in an initial value.
With the value loaded into the register, we can now count starting from this new value. The modified counter circuit
is shown in Figure 8.18(a). The only difference between this circuit and the up-down counter circuit shown in Figure
8.15(a) is that a 2-input multiplexer is added between the s output of the HAS and the Di input of the flip-flop. By
doing this, the input of the flip-flop can be selected from either an external input value (if Load is asserted) or the
next count value from the HAS output (if Load is de-asserted). If the HAS output is selected, then the circuit works
exactly like before. If the external input is selected, then whatever value is presented on the input data lines will be
loaded into the register. The operational table and logic symbol for this circuit are shown in Figure 8.18(b) and (c).
We have kept the Clear line, so that the counter can still be initialized to 0 at anytime. Notice that there is a
timing difference between asserting the Clear line to reset the counter to 0, as opposed to loading in a 0 by asserting
the Load line and setting the data input to a 0. In the first case, the counter is reset to 0 immediately after the Clear is
asserted, while the latter case will reset the counter to 0 at the next rising edge of the clock.
This counter can start with whatever value is loaded into the register, but it will always count up to 2n – 1,
where n is the number of bits for the register. This is when the register contains all 1’s. When the counter reaches the
end of the count sequence, it will always cycle back to 0, and not to the initial value that was loaded in. However,
we can add a simple comparator to this counter circuit so that the count sequence can start or end with any number
in between and cycle back to the new starting value, as shown in the next section. 272 Digital Logic and Microprocessor Design with VHDL D3 Chapter 8 - Standard Sequential Components D2 D1 D0 Count
Down
Overflow C4 Down
cout
cin C3 Down
cout
cin s HAS a
1
s C2 s HAS a 0 1
s y C1 Down
cout
cin Down
cout
cin s HAS a 0 1
s y s HAS a 0 1
s y 0
y Load
D3 Q3 D2 Clk Q2 D1 Clk Clear Q1 D0 Clk Clear Q0 Clk Clear Clear Clock
Clear
Q3 Q2 Q1 Q0 (a) Clear
1
0
0
0
0 Load
×
0
0
0
1 Count
×
0
1
1
× Down
×
×
0
1
× Operation
Reset counter to zero
No change
Count up
Count down
Load value (b) Clear
Count
Down
Load D3 D2 D1 D0 4-bit Binary Overflow
Up-down Counter
with Parallel Load Clock
Q3 Q2 Q1 Q0 (c) Figure 8.18. A 4-bit binary up-down counter with parallel load and asynchronous clear: (a) circuit; (b) operation
table; (c) logic symbol. 8.3.4 BCD Up Counter
A limitation with the binary up-down counter with parallel load is that it always counts up to 2n – 1 for an n bit
register and then cycles back to zero. If we want the count sequence to end at a number less than 2n – 1, we need to
use an equality comparator to test for this new ending number. The comparator compares the current count value
that is in the register with this new ending number. When the counter reaches this new ending number, the
comparator asserts its output.
The counter can start from a number that is initially loaded in. However, if we want the count sequence to cycle
back to this new starting number each time, we need to assert the Load signal at the end of each count sequence and
reload this new starting number. The output of the comparator is connected to the Load line, so that when the
counter reaches the ending number, it will assert the Load line and loads in the starting number. Hence, the counter
can end at a new ending number and cycles back to a new starting number.
The binary coded decimal (BCD) up counter counts from 0 to 9 and then cycles back to 0. The circuit for it is
shown in Figure 8.19. The heart of the circuit is just the 4-bit binary up-down counter with parallel load. A 4-input
AND gate is used to compare the count value with the number 9. When the count value is 9, the AND gate comparator
outputs a 1 to assert the Load line. Once the Load line is asserted, the next counter value will be the value loaded in
from the counter input D. Since D is connected to all 0’s, the counter will cycle back to 0 at the next rising clock
edge. The Down line is connected to a 0, since we only want to count up. 273 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components 0
Clear
Count
0 Clear
Count
Down
Load 0 0 0 D3 D2 D1 D0 4-bit Binary Overflow
Up-down Counter
with Parallel Load Clock
Q3 Q2 Q1 Q0 Q3 Q2 Q1 Q0 Figure 8.19 BCD up counter.
In order for the timing of each count to be the same, we must use the Load operation to load in the value 0,
rather than using the Clear operation. If we connect the output of the AND gate to the Clear input instead of the Load
input, we will still get the correct count sequence. However, when the count reaches 9, it will change to a 0 almost
immediately, because when the output of the AND gate asserts the asynchronous Clear signal, the counter is reset to
0 right away and not at the next rising clock edge.
Example 8.1: Constructing an up counter circuit
This example uses the 4-bit binary up-down counter with parallel load to construct an up counter circuit that
counts from 3 to 8 (in decimal), and back to 3.
The circuit for this counter, shown in Figure 8.20, is almost identical to the BCD up counter circuit. The only
difference is that we need to test for the number 8 instead of 9 as the last number in the sequence, and the first
number to load in is a 3 instead of a 0. Hence, the inputs to the AND gate for comparing with the binary counter
output is 1000, and the number for loading in is 0011.
♦ 0
Clear
Count
0 Clear
Count
Down
Load 0 1 1 D3 D2 D1 D0 4-bit Binary Overflow
Up-down Counter
with Parallel Load Clock
Q3 Q2 Q1 Q0 Q3 Q2 Q1 Q0 Figure 8.20 Counter for Example 8.1. 8.3.5 BCD Up-down Counter
We can get a BCD up-down counter by modifying the BCD up counter circuit slightly. The counter counts from
0 to 9 for the up sequence and 9 down to 0 for the down sequence. For the up sequence, when the count reaches 9,
the Load line is asserted to load in a 0 (0000 in binary). For the down sequence, when the count reaches 0, the Load
line is asserted to load in a 9 (1001 in binary). 274 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components The BCD up-down counter circuit is shown in Figure 8.21. Two 5-input AND gates acting as comparators are
used. The one labeled “Up” will output a 1 when Down is de-asserted (i.e., counting up), and the count is 9. The one
label “Dn” will output a 1 when Down is asserted, and the count is 0. The Load signal is asserted by either one of
these two AND gates. Four 2-to-1 multiplexers are used to select which of the two starting values, 0000 or 1001, is to
be loaded in when the Load line is asserted. The select lines for these four multiplexers are connected in common to
the Down signal, so that when the counter is counting up, 0000 is loaded in when the counter wraps around, and
1001 is loaded in when the counter wraps around while counting down. It should be obvious that the two values,
0000 and 1001, can also be loaded in without the use of the four multiplexers. 10 Clear
Count
Down Clear
Count
Down
Load 00 00 10 1
s 1
s 1
s 1
s 0
y D3 0
y D2 0
y D1 0
y D0 4-bit Binary Overflow
Up-down Counter
with Parallel Load Clock
Q3 Q2 Q1 Q0 Q3 Q2 Q1 Q0 Up
Dn Figure 8.21 BCD up-down counter.
Example 8.2: Constructing an up-down counter circuit
This example uses the 4-bit binary up-down counter with parallel load to construct an up-down counter circuit
that outputs the sequence, 2, 5, 9, 13, and 14, repeatedly.
The 4-bit binary counter can only count numbers consecutively. In order to output numbers that are not
consecutive, we need to design an output circuit that maps from one number to another number. The required
sequence has five numbers, so we will first design a counter to count from 0 to 4. The output circuit will then map
the numbers, 0, 1, 2, 3, and 4 to the required output numbers, 2, 5, 9, 13, and 14, respectively.
The inputs to the output circuit are the four output bits of the counter, Q3, Q2, Q1, and Q0. The outputs from this
circuit are the modified four bits, O3, O2, O1, and O0, for representing the five output numbers. The truth table and
the resulting output equations for the output circuit are shown in Figure 8.22(a) and (b), respectively. The easiest
way to see how the output equations are obtained is to use a K-map and put in all of the don’t-cares. The complete
counter circuit is shown in Figure 8.22(c).
♦ 275 Digital Logic and Microprocessor Design with VHDL Decimal
Q3 Q2 Q1 Q0
Input
0
0
0
0
0
1
0
0
0
1
2
0
0
1
0
3
0
0
1
1
4
0
1
0
0
Rest of the Combinations Decimal
Output
2
5
9
13
14 Chapter 8 - Standard Sequential Components O3 O2 O1 O0 0
0
1
1
1
× 0
1
0
1
1
× 1
0
0
0
1
× 0
1
1
1
0
× O0 = Q1 + Q0
O1 = Q1'Q0'
O2 = Q2 + Q0
O3 = Q2 + Q1 (b) (a) 0
Clear
Count
0 Clear
Count
Down
Load 0 0 0 D3 D2 D1 D0 4-bit Binary Overflow
Up-down Counter
with Parallel Load Clock
Q3 Q2 Q1 Q0 O3
O2
O1
O0
(c)
Figure 8.22 Counter for Example 8.2. 8.4 Register Files When we want to store several numbers concurrently in a digital circuit, we can use several individual registers
in the circuit. However, there are times when we want to treat these registers as a unit, similar to addressing the
individual locations of an array or memory. So, instead of having several individual registers, we want to have an
array of registers. This array of registers is known as a register file. In a register file, all of the respective control
signals for the individual registers are connected in common. Furthermore, all of the respective data input and output
lines for all of the registers are also connected in common. For example, the Load lines for all of the registers are
connected together, and all of the d3 data lines for all of the registers are connected together. So the register file has
only one set of input lines and one set of output lines for all of the registers. In addition, address lines are used to
specify which register in the register file is to be accessed.
In a microprocessor circuit requiring an ALU, the register file usually is used for the source operands of the
ALU. Since the ALU usually takes two input operands, we like the register file to be able to output two values from
possibly two different locations of the register file at the same time. So, a typical register file will have one write
port and two read ports. All three ports will have their own enable and address lines. When the read enable line is
de-asserted, the read port will output a 0. On the other hand, when the read enable line is asserted, the content of the
register specified by the read address lines is passed to the output port. The write enable line is used to load a value
into the register specified by the write address lines. 276 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components The logic symbol for a 4 × 8 register file (four registers, each being 8-bits wide) is shown in Figure 8.23. The 8bit write port is labeled In, and the two 8-bit read ports are labeled Port A and Port B. WE is the active-high write
enable line. To write a value into the register file, this line must be asserted. The WA1 and WA0 are the two address
lines for selecting the write location. Since there are four locations in this register file, two address lines are needed.
The RAE line is the read enable line for Port A. The two read address select lines for Port A are RAA1 and RAA0. For
Port B, we have the Port B enable line, RBE, and the two address lines, RBA1 and RBA0.
8 In
WE
Clock
WA1
4×8
WA0
Register File
RAE
RBE
RAA1
RBA1
RAA0 Port
Port RBA0
A
B
8 8 Figure 8.23 Logic symbol for a 4 × 8 register file.
The register circuit from Figure 8.1 does not have any control for the reading of the data to the output port. In
order to control the output of data, we can use a 2-input AND gate to enable or disable each of the data output lines,
Qi. We want to control all the data output lines together, therefore, one input from all of the 2-input AND gates are
connected in common. When this common input is set to a 0, all the AND gates will output a 0. When this common
input is set to a 1, the output for all of the AND gates will be the value from the other input. An alternative to using
AND gates to control the read ports is to use tri-state buffers. Instead of outputting a 0 when disabled, the tri-state
buffers will have a high impedance.
Our register file has two read ports, that is, two output controls for each register. So, instead of having just one
2-input AND gate per output line, Qi, we need to connect two AND gates to each output line: one for Port A, and one
for Port B. An 8-bit wide register file cell circuit will have eight AND gates for Port A, and another eight AND gates
for Port B, as shown in Figure 8.24. AE and BE are the read enable signals for Port A and Port B, respectively. For
each read port, the read enable signal is connected in common to one input of all of the eight AND gates. The second
input from each of the eight AND gates connects to the eight output lines, Q0 to Q7. Write Port
8
Load D0-7
8-bit Q0-7
8
Register 1 AE
8
×8 8 Read
Port_A 8 Read
Port_B BE
18
×8 Figure 8.24 An 8-bit wide register file cell with one write port and two read ports.
For a 4 × 8 register file, we need to use four 8-bit register file cells. In order to select which register file cell we
want to access, three decoders are used to decode the addresses, WA1, WA0, RAA1, RAA0, RBA1, and RBA0. One
decoder is used for the write addresses, WA1 and WA0; one for the Port A read addresses, RAA1 and RAA0; and one
for the Port B read addresses, RBA1 and RBA0. The decoders’ outputs are used to assert the individual register file
cell’s write line, Load, and read enable lines, AE, and BE. The complete circuit for the 4 × 8 register file is shown in
Figure 8.25. The respective read ports from each register file cell are connected to the external read port through a 4input × 8-bit OR gate. 277 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components For example, to read from Register 3 through Port B, the RBE line has to be asserted, and the Port B address
lines, RBA0 and RBA1, have to be set to 112 (for Register 3). The data from Register 3 will be available immediately
on Port B. To write a value to Register 2, the write address lines, WA0 and WA1, are set to 102, and then the write
enable line, WE, is asserted. The data at input D is then written into Register 2 at the next active (rising) clock edge.
Since all three decoders can be enabled at the same time, therefore, the two read operations and the write operation
can all be asserted together.
Input D
8 Write
Port 8
WA0 A0 WA1 A1 2-to-4 Y1
Decoder Y
2 WE E Y0 Load D0-7
8-bit Q0-7
8
Register 1 AE
8
×8 1 Write
Port 8
Load D0-7
8-bit Q0-7
8
Register Load D0-7
8-bit Q0-7
8
Register ×8 Load D0-7
8-bit Q0-7
8
Register
Clock Read
Port B 1 1 Y3 E RAE Read
Port A Y0 A0 RBA0 8 Y1 2-to-4 A1
Y2 Decoder RBA1 Read
Port B Y3 RBE 8 8 Read
Port A 8 Read
Port B BE
8
×8 RAA1 Read
Port B AE
8
×8 RAA0 8 BE
18 1 A0 Y1 2-to-4 A1
Y2 Decoder 8 AE
8
×8 Y0 Read
Port A BE
18 ×8 Write
Port 8 8 AE
18 ×8 Write
Port 8 Read
Port A BE
8
×8 Y3 8 ×8 E ×8
8 Port A 8 Port B Figure 8.25 A 4 × 8 register file circuit with one write port and two read ports.
In terms of the timing issues, the data on the read ports are available immediately after the read enable line is
asserted, whereas, the write occurs at the next active (rising) edge of the clock. Because of this, the same register can
be accessed for both reading and writing at the same time; that is, the read and write enable lines can be asserted at
the same time using the same read and write address. When this happens, then the value that is currently in the
register is read through the read port, and a new value will be written into the register at the next rising clock edge.
This timing is shown in Figure 8.26. The important point to remember is that, when the read and write operations are
performed at the same time on the same register, the read operation always reads the current value stored in the
register and never the new value that is to be written in by the write operation. The new value written in is available
only after the next rising clock edge. 278 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components current value in register new value in register one clock cycle
clock signal
write register
with new value WE and RAE
asserted read current
value in register
Figure 8.26 Read and write timings for a register file cell.
The VHDL code for the 4 × 8 register file is shown in Figure 8.27. The main code is composed of three
processes: the write process and the two read port processes. These three processes are similar to three concurrent
statements in that they are executed in parallel. The write process is sensitive to the clock, and because of the IF
clock statement in the process, a write occurs only at the rising edge of the clock signal. The two read port processes
are not sensitive to the clock but only to the read enable and read address signals. So the read data is available
immediately when these lines are asserted. The function CONV_INTEGER(WA) converts the STD_LOGIC_VECTOR WA
to an integer so that the address can be used as an index into the RF array.
A sample simulation trace is shown in Figure 8.28. In the simulation trace, both the write address, WA, and Port
A read address, RAA, are set to Register 3. At 0 ns, the input data, D, is 5. With write enable, WE, asserted, the data 5
is stored into RF(3) at the next rising edge of the clock, which happens at 100 ns. When RAE is asserted at 200 ns,
the data 5 from RF(3) is available on Port A immediately. At 400 ns, both WE and RAE are asserted at the same
time. The current data 5 from RF(3) appears immediately on Port A. However, the new data 7 is written into RF(3)
at 500 ns, the next rising clock edge. The new data 7 is available on Port A only after time 500 ns.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL; -- needed for CONV_INTEGER() ENTITY regfile IS port(
clock: IN std_logic;
--clock
WE: IN std_logic;
--write enable
WA: IN std_logic_vector(1 DOWNTO 0);
--write address
D: IN std_logic_vector(7 DOWNTO 0);
--input
RAE, RBE: IN std_logic;
--read enable ports A & B
RAA, RBA: IN std_logic_vector(1 DOWNTO 0);
--read address port A & B
PortA, PortB: OUT std_logic_vector(7 DOWNTO 0));--output port A & B
END regfile;
ARCHITECTURE Behavioral OF regfile IS
SUBTYPE reg IS std_logic_vector(7 DOWNTO 0);
TYPE regArray IS array(0 to 3) OF reg;
SIGNAL RF: regArray;
--register file contents
BEGIN
WritePort: PROCESS (clock)
BEGIN
IF (clock'EVENT AND clock='1') THEN
IF (WE = '1') THEN
RF(CONV_INTEGER(WA)) <= D;
-- fn to convert from vector to integer
END IF;
END IF;
END PROCESS; 279 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components ReadPortA: PROCESS (RAA, RAE)
BEGIN
-- Read Port A
IF (RAE = '1') THEN
PortA <= RF(CONV_INTEGER(RAA)); -- fn to convert from vector to integer
ELSE
PortA <= (others => '0');
END IF;
END PROCESS;
ReadPortB: PROCESS (RBE, RBA)
BEGIN
-- Read Port B
IF (RBE = '1') THEN
PortB <= RF(CONV_INTEGER(RBA)); -- fn to convert from vector to integer
ELSE
PortB <= (others => '0');
END IF;
END PROCESS;
END Behavioral;
Figure 8.27 VHDL code for a 4 × 8 register file with one write port and two read ports. Figure 8.28 Sample simulation trace for the 4 × 8 register file. 8.5 Static Random Access Memory Another main component in a computer system is memory. This can refer to either random access memory
(RAM) or read-only memory (ROM). We can make memory the same way we make the register file but with more
storage locations. However, there are several reasons why we don’t want to. One reason is that we usually want a lot
of memory and we want it very cheap, so we need to make each memory cell as small as possible. Another reason is
that we want to use a common data bus for both reading data from, and writing data to the memory. This implies
that the memory circuit should have just one data port (and not two or three like the register file) for both reading
and writing of data.
The logic symbol, showing all of the connections for a typical RAM chip is shown in Figure 8.29(a). There is a
set of data lines, Di, and a set of address lines, Ai. The data lines serve for both input and output of the data to the
location that is specified by the address lines. The number of data lines is dependent on how many bits are used for
storing data in each memory location. The number of address lines is dependent on how many locations are in the
memory chip. For example, a 512-byte memory chip will have eight data lines (8 bits = 1 byte) and nine address
lines (29 = 512). 280 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components In addition to the data and address lines, there are usually two control lines: chip enable (CE), and write enable
(WR). In order for a microprocessor to access memory, either with the read operation or the write operation, the
active-high CE line must first be asserted. Asserting the CE line enables the entire memory chip. The active-high
WR line selects which of the two memory operations is to be performed. Setting WR to a 0 selects the read operation,
and data from the memory is retrieved. Setting WR to a 1 selects the write operation, and data from the
microprocessor is written into the memory. Instead of having just the WR line for selecting the two operations, read
and write, some memory chips have both a read enable and a write enable line. In this case, only one line can be
asserted at any one time. The memory location in which the read and write operations are to take place, of course, is
selected by the value of the address lines. The operation of the memory chip is shown in Figure 8.29(b).
An-1
2n × m
RAM A1
A0 CE
WR Dm-1 CE
0
1
1 D1 D0 WR
×
0
1 Operation
None
Read from memory location selected by address lines
Write to memory location selected by address lines (b) (a)
Figure 8.29 A 2 × m RAM chip: (a) logic symbol; (b) operation table.
n Notice in Figure 8.29(a) that the RAM chip does not require a clock signal. Both the read and write memory
operations are not synchronized to the global system clock. Instead the data operations are synchronized to the two
control lines, CE and WR. Figure 8.30(a) shows the timing diagram for a memory write operation. The write
operation begins with a valid address on the address lines, followed immediately by the CE line being asserted.
Shortly after, valid data must be present on the data lines, and then the WR line is asserted. As soon as the WR line is
asserted, the data that is on the data lines is then written into the memory location that is addressed by the address
lines.
A memory read operation also begins with setting a valid address on the address lines, followed by CE going
high. The WR line is then pulled low, and shortly after, valid data from the addressed memory location is available
on the data lines. The timing diagram for the read operation is shown in Figure 8.30(b).
Each bit in a static RAM chip is stored in a memory cell similar to the circuit shown in Figure 8.31(a). The main
component in the cell is a D latch with enable. A tri-state buffer is connected to the output of the D latch so that it
can be selectively read from. The Cell enable signal is used to enable the memory cell for both reading and writing.
For reading, the Cell enable signal is used to enable the tri-state buffer. For writing, the Cell enable together with the
Write enable signals are used to enable the D latch so that the data on the Input line is latched into the cell. The logic
symbol for the memory cell is shown in Figure 8.31 (b). Address Valid Address Data Address Valid Address Data Valid Data CE CE WR Valid Data WR
(a) (b) Figure 8.30 Memory timing diagram: (a) read operation; (b) write operation. 281 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Cell
enable Cell
enable
Input D Q Output Input E MC Output Write
enable Write
enable (b) (a)
Figure 8.31 Memory cell: (a) circuit; (b) logic symbol. To create a 4 × 4 static RAM chip, we need sixteen memory cells forming a 4 × 4 grid, as shown in Figure 8.32.
Each row forms a single storage location, and the number of memory cells in a row determines the bit width of each
location. So all of the memory cells in a row are enabled with the same address. Again, a decoder is used to decode
the address lines, A0 and A1. In this example, a 2-to-4 decoder is used to decode the four address locations. The CE
signal is for enabling the chip, specifically to enable the read and write functions through the two AND gates. The
internal WE signal, asserted when both the CE and WR signals are asserted, is used to assert the Write enables for all
of the memory cells. The data comes in from the external data bus, Di, through the input buffer and to the Input line
of each memory cell. The purpose of using an input buffer for each data line is so that the external signal coming in
only needs to drive just one device (the buffer) rather than having to drive several devices (i.e., all of the memory
cells in the same column). Which row of memory cells actually gets written to will depend on the given address. The
read operation requires CE to be asserted and WR to be de-asserted. This will assert the internal RE signal, which in
turn will enable the four output tri-state buffers at the bottom of the circuit diagram. Again, the location that is read
from is selected by the address lines.
The VHDL code for a 16 × 4 RAM chip is shown in Figure 8.33. The bi-directional data port, D, is declared as
that it can be read from and written to. The actual memory content is stored in the variable mem, which is
an array of size 16 of type STD_LOGIC_VECTOR. BUFFER so 282 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components 0
MC MC MC MC MC MC MC MC MC MC MC MC MC MC MC MC 1
A1
A0 2-to-4
Decoder
2 3
WR WE CE
RE
Input
Buffers Output
Buffers D3 D2 D1 D0 Figure 8.32 A 4 × 4 RAM chip circuit.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_ARITH.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL; -- needed for CONV_INTEGER() ENTITY memory IS PORT (
CE, WR: IN STD_LOGIC;
--chip enable, write enable
A: IN STD_LOGIC_VECTOR(3 DOWNTO 0);
--address
D: BUFFER STD_LOGIC_VECTOR(3 DOWNTO 0));
--data
END memory;
ARCHITECTURE Behavioral OF memory IS
BEGIN
PROCESS (CE, WR)
SUBTYPE cell IS STD_LOGIC_VECTOR(3 DOWNTO 0);
TYPE memArray IS array(0 TO 15) OF cell;
VARIABLE mem: memArray;
--memory contents
VARIABLE ctrl: STD_LOGIC_VECTOR(1 DOWNTO 0);
BEGIN
ctrl := CE & Wr; -- group signals for CASE decoding
CASE ctrl IS
WHEN "10" => -- read
D <= mem(CONV_INTEGER(A));-- fn TO convert from bit vector TO integer
WHEN "11" => -- write
mem(CONV_INTEGER(A)) := D;-- fn TO convert from bit vector TO integer
WHEN OTHERS =>
-- invalid or not enable
D <= (OTHERS => 'Z');
END CASE; 283 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components END PROCESS;
END Behavioral;
Figure 8.33 VHDL code for a 16 × 4 RAM chip. 8.6 * Larger Memories In general, there is always a need for larger memories. Because of product availability constraints, we need to
construct these larger memories from multiple, smaller memory chips. Larger memory requirements can be for
either more memory locations, wider bit widths for each location, or both. 8.6.1 More Memory Locations
For example, we may want to have a 1 K × 8-bit memory built using multiple 256 × 8-bit memory chips. Using
such small numbers is archaic, but you get the idea. In this case, we would need four of these 256 × 8-bit memory
chips, since 1 K = 4 × 256. A 256 × 8-bit memory chip has eight address lines, since 28 = 256. To decode four chips,
we need an additional two address lines to enable which of the four chips we want to address. Thus, we need a total
of ten address lines with the first eight, A0 to A7, connected, respectively, in common directly to the eight address
lines on the four chips, and the last two lines, A8 and A9, connected to the address inputs of a 2-to-4 decoder. The
four outputs from the decoder are used to assert the chip enable, CE, line of the four memory chips, RAM0 to RAM3.
The data lines and the write enable lines are all connected, respectively, in common. The circuit is shown in Figure
8.34(a).
The 256-byte memory chip, RAM0, is enabled when the address bits, A8 and A9, are 00. Hence, the address
range for RAM0 is from 0 to 255 (0000000000 to 0011111111 in binary). Similarly, RAM1 is enabled when the
address bits, A8 and A9, are 01. Hence, the address range for RAM1 is from 256 to 511 (0100000000 to 0111111111
in binary). The address range for RAM2 is from 512 to 767 (1000000000 to 1011111111 in binary), and the address
range for RAM3 is from 768 to 1023 (1100000000 to 1111111111 in binary).
A particular memory location is accessed as follows. If we want to write to memory location 717, which is
binary 1011001101, the Y2 line of the decoder would be asserted, since bits 8 and 9 are “10.” This Y2 line in turn
asserts the CE line of the RAM2 chip, while the remaining RAM chips are disabled. Finally, within the RAM2 chip
that is enabled, location 205, which is binary 11001101 from bits 0 to 7 of the original address, is selected. Location
205 in the third RAM chip is location 717 for the entire memory, since 256 + 256 + 205 = 717. 8.6.2 Wider Bit Width
We may also want to have wider bit width for each memory location made from smaller ones. For example, we
may want to have a memory that is 512 locations × 16-bits wide made from 256 × 8-bit memory chips. Again, we
would need four 256-byte memory chips, but connected as shown in Figure 8.34(b). For 512 locations, only nine
address lines are needed, with the first eight, A0 to A7, connected, respectively, in common directly to the eight
address lines on the four chips, and the last line, A8, connected to the address input of a 1-to-2 decoder. For a 16-bit
wide data bus, we need to connect two 8-bit wide chips in parallel so that each two similar 8-bit wide location in the
two chips can be combined together to form a 16-bit wide location. Since these two chips need to work together,
their chip enable, CE, lines must be connected in common and asserted by the same output from the decoder.
Memory chips RAM0 and RAM2 are for storing the data bits D0 to D7, while memory chips RAM1 and RAM3
are for storing the data bits D8 to D15. The address range for RAM0 and RAM1 is from 0 to 255 (000000000 to
011111111 in binary), and the address range for RAM2 and RAM3 is from 256 to 511 (100000000 to 111111111 in
binary). 284 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Address: 0-255 Y0
A8
A9 A0
A1 Address: 256-511 Y1 2-to-4
Decoder Y2
E WR
A7 CE
RAM0
256 × 8
8
A0 D7
D0 WR
A7 CE
RAM1
256 × 8
8
D0
A0 D7
Address: 512-767 Address: 0-255 8 Y0
A8 A0 8
1-to-2
Decoder Y1 Y3 E E 8
Address: 768-1023 D7-0
D15-8 Address: 256-511
WR
A7 CE
RAM3
256 × 8
8
D0
A0 D7 16 WR
A7 CE
RAM2
256 × 8
8
D0
A0 D7
8 8 WR
A7 CE
RAM3
256 × 8
8
D0
A0 D7 WR
A7 CE
RAM0
256 × 8
8
A0 D7
D0
8 8 WR
A7 CE
RAM2
256 × 8
8
A0 D7
D0 E WR
A7 CE
RAM1
256 × 8
8
A0 D7
D0 D7-0
D15-8 A7-0 WR D15-0 8 A7-0 WR D7-0
(b) (a) Figure 8.34 Larger memory made from smaller memory chips: (a) a 1 K × 8-bit memory made from four 256 × 8bit memory chips; (b) a 512 × 16-bit memory made from four 256 × 8-bit memory chips.
Example 8.3: Building larger memory using smaller RAM chips
Build a 2 M-byte memory using 512 K-byte RAM chips.
A 512 K-byte RAM chip has 9 address lines, A0 to A8, because 29 = 512 K. Since 4 × 512 K = 2 M, therefore,
we need to use four 512 K-byte RAM chips. In order to select from these four RAM chips, we need two more
address lines, A9 and A10. Hence, the system must have at least 11 address lines. The first 9 address lines, A0 to A8,
are connected directly to the four RAM chips. The last two address lines, A9 and A10, are connected to a 2-to-4
decoder. The four outputs of the decoder are connected to the chip enables (CE) for the four RAM chips. The eight
data lines and the write enable lines are all connected, respectively, in common. The circuit is shown in Figure 8.35.♦ 285 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Address: 0-511
WR
A8 CE
RAM0
256 × 8
8
D0
A0 D7 Y0
A9
A10 A0
A1 Address: 512-1023
WR
A8 CE
RAM1
256 × 8
8
D0
A0 D7 Y1 2-to-4
Decoder Address: 1024-1535 Y2
E 8 8 WR
A8 CE
RAM2
256 × 8
8
D0
A0 D7 E
Y3 Address: 1536-2047 8 WR
A8 CE
RAM3
256 × 8
8
D0
A0 D7
8 A8-0 WR D7-0 Figure 8.35 A 2 M-byte memory circuit for Example 8.3.
Example 8.4: Determining an address range
What is the address range for the Y5 line in the following circuit? A10
A11
A12
A13 Y0 A0
A1
A2
A3 Y1
Y2
4-to-8
Decoder Y3
Y4
Y5 E E Y6
Y7 286 Digital Logic and Microprocessor Design with VHDL Chapter 8 - Standard Sequential Components Y5 is asserted when the address lines, A13, A12, A11, and A10, are 0101. The lowest address is when the ten loworder address bits, A9 to A0, are all 0’s, and the highest address is when these ten bits are all 1’s. Hence, the address
♦
range for Y5 is from 01010000000000 to 01011111111111 in binary or 5120 to 6143 in decimal. 8.7 Summary Checklist
Registers
Serial-to-Parallel Shift Registers
Parallel-to-Serial Shift Registers
Binary counters
Binary up-down counters
BCD counters
BCD up-down counters
Counters for random sequences
Register files
Random access memories (RAM)
Building more memory locations using smaller RAM chips
Building wider bit-width memories using smaller RAM chips 8.8 Problems 8.1. The 4-bit binary up counter VHDL code shown in Figure 8.12 does not have the Overflow output signal.
Modify the code to include the Overflow signal. 8.2. For the BCD up counter circuit shown in Figure 8.19, what happens if the output of the AND gate comparator
is connected to the Clear signal instead of to the Load signal? Will it produce the same waveform? Explain
your observations. 8.3. In the BCD up-down counter circuit shown in Figure 8.21, four 2-input multiplexers are used to select the
correct value to be loaded in. Modify the circuit so that the multiplexers are not needed. 8.4. Write the behavioral VHDL code for the BCD up-down counter. 8.5. Use the 4-bit binary up-down counter with parallel load to construct an up-down counter circuit that counts
from 0 to 7 decimal and back to 0. 8.6. Use the 4-bit binary up-down counter with parallel load to construct an up-down counter circuit that counts
from 5 to 13 decimal and back to 5. 8.7. Use the 4-bit binary up-down counter with parallel load to construct an up-down counter circuit that outputs
the sequence: 7, 12, 19, 36, 42, 58, and 57, repeatedly. 8.8. Use the 4-bit binary up-down counter with parallel load to construct an up-down counter circuit that outputs
the sequence: 4, 8, 5, 3, 16, and 7, repeatedly. 8.9. Write the behavioral VHDL code for the BCD up-down counter. 8.10. Write the structural VHDL code for the BCD up-down counter based on the circuit diagram shown in Figure
8.21. Use the 4-bit binary up-down counter VHDL code as a component.
8.11. What are the valid address ranges for the Y5 and Y7 lines in the following circuits? 287 Digital Logic and Microprocessor Design with VHDL A5
A6
A7
A8 Y0 A0
A1
A2
A3 Y1
Y2
4-to-8
Decoder A5
A8
A9
A12 Chapter 8 - Standard Sequential Components Y0 A0
A1
A2
A3 Y3 Y1
Y2
4-to-8
Decoder Y4
Y5 E E Y6 A5
A14
A9
A12 Y0 A0
A1
A2
A3 Y3 Y1
Y2
4-to-8
Decoder Y4
Y5 E E Y6 Y7 Y4
Y5 E E Y6 Y7 (a) Y3 (b) Y7
(c) 8.12. Design a 32 M-byte memory using 4 M-byte RAM chips. Label all of the signals clearly.
8.13. Design an 8 M-byte memory using 2 M × 4-bit RAM chips. Label all of the signals clearly.
8.14. Manually design and implement on the UP2 board an FSM circuit for writing the value 13 into location 2 of a
4 × 8 register file, then read location 2 through Port A, and display the number as binary on the eight LEDs.
8.15. Manually design and implement on the UP2 board the following FSM circuit for controlling a 4 × 8 register
file. For input, use two DIP switches to specify the register file location and another eight DIP switches to
specify the data input. Use a push-button for the write enable signal. For output, use the eight LEDs. The
eight output LEDs continuously display the content of the current selected register file location. When the
push-button is pressed, the data input is loaded into the selected location. 288 Chapter 9 Datapaths Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths In Chapter 4, we learned how to design functional units for performing single, simple data operations, such as
the adder for adding two numbers or the comparator for comparing two values. The next logical question to ask is
how do we design a circuit for performing more complex data operations or operations that involve multiple steps?
For example, how do we design a circuit for adding four numbers or a circuit for adding a million numbers? For
adding four numbers, we can connect three adders together, as shown in Figure 9.1(a). However, for adding a
million numbers, we really don’t want to connect a million minus one adders together like that. Instead, we want a
circuit with just one adder and to use it a million times. A datapath circuit allows us to do just that, that is, for
performing operations involving multiple steps. Figure 9.1(b) shows a simple datapath using one adder to add as
many numbers as we want. In order for this to be possible, a register is needed to store the temporary result after
each addition. The temporary result from the register is fed back to the input of the adder so that the next number
can be added to the current sum.
In this chapter, we will look at the design of datapaths. Recall that the datapath is the second main part in a
microprocessor. The datapath is responsible for the manipulation of data. It includes (1) functional units such as
adders, shifters, multipliers, ALUs, and comparators, (2) registers and other memory elements for the temporary
storage of data, and (3) buses, multiplexers, and tri-state buffers for the transfer of data between the different
components in the datapath, and the external world. From the microprocessor road map figure at the beginning of
this chapter, we see that external data enters the datapath through the data input lines. Results from the datapath
operations are provided through the data output lines. These signals serve as the primary input/output data ports for
the microprocessor.
In order for the datapath to function correctly, appropriate control signals must be asserted at the right time.
Control signals are needed for all of the select and control lines for all of the components used in the datapath. This
includes all of the select lines for multiplexers, ALUs, and other functional units having multiple operations; all of
the read/write enable signals for registers and register files; address lines for register files; and enable signals for tristate buffers. Thus, the operation of the datapath is determined by which control signals are asserted or de-asserted
and at what time. In a microprocessor, these control signals are generated by the control unit. Number 1 Number 2 Number 3 Number 4 Numbers from 1
to 1 Million + +
+
Register + (a) (b) Figure 9.1 Circuits to add several numbers: (a) combinational circuit to add four numbers; (b) datapath to add one
million numbers.
Some of the control signals generated by the control unit are dependent on the data that is being manipulated
within the datapath. (For example, the result of a conditional test with a number that is stored in a register.) Hence,
in order for the control unit to generate these control signals correctly, the datapath needs to supply status signals to
the control unit. These status signals are usually from the output of comparators. The comparator tests for a given
logical condition between two data values in the datapath. These values are obtained either from memory elements
or directly from the output of functional units, or are hardwired as constants. The status signals provide information
for the control unit to determine what state to go to next. For example, in a conditional loop situation, the status
signal provides the result of the condition being tested, and tells the control unit whether to repeat or exit the loop.
Since the datapath performs all of the functional operations of a microprocessor (and the microprocessor is for
solving problems) therefore, the datapath must be able to perform all of the operations that are required to solve the 290 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths given problem. For example, if the problem requires the addition of two numbers, the datapath, therefore, must
contain an adder. If the problem requires the storage of three temporary variables, the datapath must have three
registers. However, even with these requirements, there are still many options as to what actually is implemented in
the datapath. For example, an adder can be implemented as a single adder circuit, or as part of the ALU. These
functional units can be used many times. Registers can be separate register units or combined in a register file.
Furthermore, two temporary variables can share the same register if they are not needed at the same time.
Datapath design is also referred to as register-transfer level (RTL) design. In a register-transfer level design,
we look at how data is transferred from one register to another, or back to the same register. If the same data is
written back to a register without any modifications, then nothing has been accomplished. Therefore, before writing
the data to a register, the data usually passes through one or more functional units, and gets modified.
The sequence of RTL operations—read data from a register, modify data by functional units, and write result to
a register—is referred to as a register-transfer operation. Every register-transfer operation must complete within
one clock cycle (which is equivalent to one state of the FSM, since the FSM changes state at every clock cycle).
Furthermore, in a single register-transfer operation, a functional unit cannot be used more than once. However, the
same functional unit can be used more than once if it is used by different register-transfer operations. In other words,
a functional unit can be used only once in the same clock cycle, but can be used again in a different clock cycle.
We will now look at how datapaths are designed, and how they are used to solve problems. First, we will look
at the design of dedicated datapaths for solving single specific problems, and then we will look at general datapaths
where they can be used for solving different problems. 9.1 Designing Dedicated Datapaths The goal for designing a dedicated datapath is to build a circuit for solving a single specific problem. In this
chapter, we will specify the problem in the form of an algorithm. We will use C-style pseudocodes to write the
algorithms. The logical interpretation of the algorithm is irrelevant in what we are trying to do, so when given a
certain segment of code, we will just take the code as is and will not optimize it in any manner.
In a register-transfer level design, we focus on how data move from register to register via some functional units
where they are modified. In the design process, we need to decide on the following issues:
•
•
•
• What kind of registers to use, and how many are needed?
What kind of functional units to use, and how many are needed?
Can a certain functional unit be shared between two or more operations?
How are the registers and functional units connected together so that all of the data movements specified by
the algorithm can be realized? Since the datapath is responsible for performing all of the data operations, it must be able to perform all of the
data manipulation statements and conditional tests specified by the algorithm. For example, the assignment
statement:
A=A+3
takes the value that is stored in the variable A, adds the constant 3 to it, and stores the result back into A. Note that
whatever the initial value of A is here is irrelevant since that is a logical issue. In order for the datapath to perform
the data operation specified by this statement, the datapath must have a register for storing the value A. Furthermore,
there must be an adder for performing the addition. The constant 3 can be hardwired into the circuit as a binary
value.
The next question to ask is how do we connect the register, the adder, and the constant 3 together so that the
execution of the assignment statement can be realized. Recall from Section 8.1 that a value stored in a register is
available at the Q output of the register. Since we want to add A + 3, we connect the Q output of the register to the
first operand input of the adder, and connect the constant 3 to the second operand input. We want to store the result
of the addition back into A (i.e., back into the same register), therefore, we connect the output of the adder to the D
input of the register, as shown in Figure 9.2(a). 291 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths The storing of the adder result into the register is accomplished by asserting the Load control signal of the
register (i.e., asserting Aload). This ALoad signal is an example of what we have been referring to as the datapath
control signal. This control signal controls the operation of this datapath. The control unit, which we will talk about
in the next chapter, will control this signal by either asserting or de-asserting it.
The actual storing of the value into the register, however, does not occur immediately when ALoad is asserted.
Since the register is synchronous to the clock signal, therefore, the actual storing of the value occurs at the next
active clock edge. Because of this, the new value of A is not available at the Q output of the register during the
current clock cycle, but is available at the beginning of the next clock cycle.
As another example, the datapath shown in Figure 9.2(b) can perform the execution of the statement:
A=B+C
where B and C are two different variables stored in two separate registers, thus providing the two operand inputs to
the adder. The output of the adder is connected to the D input of the A register for storing the result of the adder. D7-0
Load 8-bit Register
B
Clock Q7-0 D7-0
Load 8-bit Register
A
Clock Q7-0 ALoad
Clock '3'
8 8 D7-0
Load 8-bit Register
C
Clock Q7-0 8 8 8 + + 8 ALoad
Clock D7-0
Load 8-bit Register
A
Clock Q7-0 (b) (a) Figure 9.2 Sample datapaths: (a) for performing A = A + 3; (b) for performing A = B + C.
The execution of the statement is realized simply by asserting the ALoad signal, and the actual storing of the
value for A occurs at the next active edge of the clock. During the current clock cycle, the adder will perform the
addition of B and C, and the result from the adder must be ready and available at its output before the end of the
current clock cycle so that, at the next active clock edge, the correct value will be written into A. Since we are not
writing any values to register B or C, we do not need to control the two Load signals for them.
If we want a single datapath that can perform both of the statements:
A=B+C
and
A=A+3
we will need to combine the two datapaths in Figure 9.2 together.
Since A is the same variable in the two statements, only one register for A is needed. However, register A now
has two data sources: one from the first adder for B + C, and the second from the second adder for A + 3. The
problem is that two or more data sources cannot be connected directly together to one destination, as shown in
Figure 9.3(a) because their signals will collide, resulting in incorrect values. The solution is to use a multiplexer to 292 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths select which of the two sources to pass to register A. The correct datapath using the multiplexer is shown in Figure
9.3(b).
Both statements assign a value to A, so ALoad must be asserted for the execution of both statements. The actual
value that is written into A, however, depends on the selection of the multiplexer. If Amux is asserted, then the result
from the bottom adder (i.e., the result from A + 3) is stored into A; otherwise, the result from the top adder is stored
into A. Since the two adders are combinational circuits and the value from a register is always available at its output,
therefore, the results from the two additions are always available at the two inputs of the multiplexer. But depending
on the Amux control signal, only one value will be passed through to the A register.
Notice that the datapath does not show which statement is going to be executed first. The sequence in which
these two statements are executed depends on whether the signal Amux is asserted first or de-asserted first. If this
datapath is part of a microprocessor, then the control unit would determine when to assert or de-assert this Amux
control signal, since it is the control unit that performs the sequencing of datapath operations.
Furthermore, notice that these two statements cannot be executed within the same clock cycle. Since both
statements write to the same register, and a register can only latch in one value at an active clock edge, only one
result from one adder can be written into the register in one clock cycle. The other statement will have to be
performed in another clock cycle, but not necessarily the next cycle. D7-0
Load 8-bit Register
B
Clock Q7-0 D7-0
Load 8-bit Register
C
Clock Q7-0 8 D7-0
Load 8-bit Register
B
Clock Q7-0 8 D7-0
Load 8-bit Register
C
Clock Q7-0 8 8 + +
8 8 D7-0
Load 8-bit Register
A
Clock Q7-0 ALoad
Clock 1 Amux Error 0 D7-0
Load 8-bit Register
A
Clock Q7-0 ALoad
Clock
'3' 8 8 '3'
8 8 8 + 8 + (a) (b) Figure 9.3 Datapath for performing A = A + 3 and A = B + C: (a) without multiplexer—wrong; (b) with
multiplexer—correct.
Example 9.1: Designing a dedicated datapath
Design a datapath that can execute the two statements:
A=B+C
and 293 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths A=A+3
using only one adder.
The only difference between this datapath and the one in Figure 9.3(b) is that it should use only one adder. So
starting with this one adder, in order to execute the first statement, the first operand input to the adder is from
register B, and the second operand input to the adder is from register C. However, to execute the second statement,
the two input operands to the adder are register A and the constant 3. Since both input operands have two different
sources, again we must use a multiplexer for each of them. The output of the two multiplexers will connect to the
two adder input operands, as shown in Figure 9.4. For both statements, the result of the addition is stored in register
A, therefore, the output of the adder connects to the input of the A register.
Notice that the two select lines for the two multiplexers can be connected together. This is possible because the
two operands B and C for the first statement are connected to input 0 of the two multiplexers, respectively, and the
two operands A and 3 for the second statement are connected to input 1 of the two multiplexers, respectively. Thus,
de-asserting the Mux select signal will pass the two correct operands for the first statement, and likewise, asserting
the Mux select signal will pass the two correct operands for the second statement. We want to reduce the number of
control signals for the datapath as much as possible, because (as we will see in the next chapter) minimizing the
number of control signals will minimize the size of the output circuit in the control unit.
♦ D7-0
Load 8-bit Register
A
Clock Q7-0 ALoad D7-0
Load 8-bit Register
B
Clock Q7-0 8 D7-0
Load 8-bit Register
C
Clock Q7-0 8 8 Clock '3'
8 1 0 1 0 Mux
8 8 + Figure 9.4 Datapath for performing A = A + 3 and A = B + C using only one adder. 9.1.1 Selecting Registers
In most situations, one register is needed for each variable used by the algorithm. However, if two variables are
not used at the same time, then they can share the same register. If two or more variables share the same register,
then the data transfer connections leading to the register and out from the register usually are made more complex,
since the register now has more than one source and destination. Having multiple destinations is not too big of a
problem, since we can connect all of the destinations to the same source.1 However, having multiple sources will
require a multiplexer to select one of the several sources to transfer to the destination. Figure 9.5 shows a circuit
with a register having two sources—one from an external input and one from the output of an adder. A multiplexer
is needed in order to select which one of these two sources is to be the input to the register. 1 This is true only theoretically. In practice, there are fan-in (multiple sources with one destination) and fan-out (one
source with multiple destinations) limits that must be observed. 294 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Input 1 + 0 Reg
Figure 9.5 Circuit of a register with two sources.
After deciding how many registers are needed, we still need to determine whether to use a single register file
containing enough register locations, separate individual registers, or a combination of both for storing the variables
in. Furthermore, registers with built-in special functions, such as shift registers and counters, can also be used. For
example, if the algorithm has a FOR loop statement, a single counter register can be used to not only store the count
variable but also to increment the count. This way, not only do we reduce the component count, but the amount of
datapath connections between components is also reduced. Decisions for selecting the type of registers to use will
affect how the data transfer connections between the registers and functional units are connected. 9.1.2 Selecting Functional Units
It is fairly straightforward to decide what kind of functional units are required. For example, if the algorithm
requires the addition of two numbers, then the datapath must include an adder. However, we still need to decide
whether to use a dedicated adder, an adder–subtractor combination, or an ALU (which has the addition operation
implemented). Of course, these questions can be answered by knowing what other data operations are needed by the
algorithm. If the algorithm has only an addition and a subtraction, then you may want to use the adder–subtractor
combination unit. On the other hand, if the algorithm requires several addition operations, do we use just one adder
or several adders?
Using one adder may decrease the datapath size in terms of number of functional units, but it may also increase
the datapath size because more complex data transfer paths are needed. For example, if the algorithm contains the
following two addition operations:
a=b+c
d=e+f
Using two separate adders will result in the datapath shown in Figure 9.6(a); whereas, using one adder will require
the use of two extra 2-to-1 multiplexers to select which register will supply the input to the adder operands, as
shown in Figure 9.6(b). Furthermore, this second datapath requires two extra control signals for the two
multiplexers. In terms of execution speed, the datapath on the left can execute both addition statements
simultaneously within the same clock cycle, since they are independent of each other. However, the datapath on the
right will have to execute these two additions sequentially in two different clock cycles, since there is only one adder
available. The final decision as to which datapath to use is up to the designer.
b b c e 1 f + c 0 + d a (a) d (b) 295 f 1 + a e 0 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.6 Datapaths for realizing two addition operations: (a) using two separate adders; (b) using one adder. 9.1.3 Data Transfer Methods
There are several methods in which the registers and functional units can be connected together so that the
correct data transfers between the different units can be made.
Multiple Sources
If the input to a unit has more than one source, then a multiplexer can be used to select which one of the
multiple sources to use. The sources can be from registers, constant values, or outputs from other functional units.
Figure 9.7 shows two such examples. In Figure 9.7(a), the left operand of the adder has four sources: two from
registers, one from the constant 1, and one from the output of an ALU. In Figure 9.7(b), register a has two sources:
one from the constant 1 and one from the output of an adder.
a b 3
s1
s0 '1' ALU
+ '1' 210 1 + 0 a (a) (b) Figure 9.7 Examples of multiple sources using multiplexers: (a) an adder operand having four sources; (b) a register
having two sources.
Multiple Destinations
A source having multiple destinations does not require any extra circuitry. The one source can be connected
directly to the different destinations, and all of the destinations where the data is not needed would simply ignore the
data source. For example, in Figure 9.6(b), the output of the adder has two destinations: register a, and register d. If
the output of the adder is for register a, then the Load line for register a is asserted, while the Load line for register d
is not; and if the output of the adder is for register d, then the Load line for register d is asserted, while the Load line
for register a is not. In either case, only the correct register will take the data while the other units simply will ignore
the data.
This also works if one of the destinations is a combinational functional unit. In this case, the functional unit will
take the source data and manipulates it. However, the output of the functional unit will not be used (that is not stored
in any register) so functionally, it doesn’t matter that the functional unit worked on the source, because the result is
not stored. However, it does require power for the functional unit to manipulate the data, so if we want to reduce the
power consumption, we would want the functional unit to not manipulate the data at all. This, however, is a power
optimization issue that is beyond the scope of this book.
Tri-state Bus
Another scheme where multiple sources and destinations can be connected to the same data bus is through the
use of tri-state buffers. The point to note here is that, when multiple sources are connected to the same bus, only one
source can output at any one time. If two or more sources output to the same bus at the same time, then there will be
data conflicts. This occurs when one source outputs a 0 while another source outputs a 1. By using tri-state buffers
to connect between the various sources and the common data bus, we want to make sure that only one tri-state buffer
is enabled at any one time, while the rest of them are all disabled. Tri-state buffers that are disabled output highimpedance Z values, so no data conflicts can occur. 296 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.8 shows a tri-state bus with five units (three registers, an ALU, and an adder) connected to it. An
advantage of using a tri-state bus is that the bus is bi-directional, so that data can travel in both directions on the bus.
Connections for data going from a component to the bus need to be tri-stated, while connections for data going from
the bus to a component need not be. Notice also that data input and output of a register both can be connected to the
same tri-state bus; whereas, the input and output of a functional unit (such as the adder or ALU) cannot be connected
to the same tri-state bus. ALU a b Common Data Bus c + Figure 9.8 Multiple sources using tri-state buffers to share a common data bus. 9.1.4 Generating Status Signals
Although it is the control unit that is responsible for the sequencing of statement execution, the datapath,
however, must supply the results of the conditional tests for the control unit so that the control unit can determine
what statement to execute next. Status signals are the results of the conditional tests that the datapath supplies to the
control unit. Every conditional test that the algorithm has requires a corresponding status signal. These status signals
usually are generated by comparators.
For example, if the algorithm has the following IF statement
IF (A = 0) THEN …
the datapath must, therefore, have an equality comparator that compares the value from the A register with the
constant 0, as shown in Figure 9.9(a). The output of the comparator is the status signal for the condition (A = 0).
This status signal is a 1 when the condition (A = 0) is true; otherwise, it is a 0. Recall from Section 4.10 that the
circuit for the equality comparator with the constant 0 is simply a NOR gate, therefore, we can replace the black box
for the comparator with just an 8-input NOR gate, as shown in Figure 9.9(b). D7-0
Load 8-bit Register
A
Clock Q7-0 D7-0
Load 8-bit Register
A
Clock Q7-0 8 (A = 0 ) = 8 '0' (A = 0 ) (a) (b) Figure 9.9 Comparators for generating the status signal (A = 0).
There are times when an actual comparator is not needed for generating a status signal. For example, if we want
a status signal for testing whether a number is an odd number, as in the following IF statement
IF (A is an odd number) THEN … 297 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths we can simply use the A0 bit of the 8-bit number from register A as the status signal for this condition, since all odd
numbers have a 1 in the zero bit position. The generation of this status signal is shown in Figure 9.10. D7-0
Load 8-bit Register
A
Clock Q7-0
8 A0 (A is an odd number) Figure 9.10 Comparator for generating the status signal (A is an odd number). 9.2 Using Dedicated Datapaths Any given datapath will have a number of control signals. By asserting or de-asserting these control signals at
different times, the datapath can perform different register-transfer operations. Since the execution of an operation
requires the correct assertion and de-assertion of all of the control signals together, we would like to think of all of
them as a unit rather than as individual signals. All of the control signals for a datapath, when grouped together, are
referred to as a control word. Hence, a control word will have one bit for each control signal in the datapath. One
register-transfer operation of a datapath, therefore, is determined by the values set in one control word, and so, we
can specify the datapath operation simply by specifying the bit string for the control word. Each control word
operation will take one clock cycle to perform. By combining multiple control words together in a certain sequence,
the datapath will perform the specified operations in the order given.
Example 9.2: Deriving control words for a datapath
The datapath in Figure 9.4, having the two control signals ALoad and Mux, was designed to execute the two
statements: A = A + 3 and A = B + C. The control word for this datapath, therefore, has two bits—one for each
control signal. The ordering of these two bits at this point is arbitrary; however, once decided, we need to be
consistent with the order. The two control words for performing the two statements are shown in Figure 9.11.
Control word 1 specifies the control word bit string for executing the statement, A = A + 3. This is accomplished
by asserting both the ALoad and the Mux signals. Control word 2 is for executing the statement, A = B + C, by
asserting ALoad and de-asserting Mux.
♦
Control
Word
1
2 Instruction ALoad Mux A=A+3
A=B+C 1
1 1
0 Figure 9.11 Control words for the datapath in Figure 9.4 for performing the two statements: A = A + 3 and
A = B + C.
In order for the datapath to operate automatically, the control unit will have to generate these control signals
correctly at the appropriate time. In the following chapters, we will learn how to construct the control unit and then
combine it with the datapath to form a microprocessor. 9.3 Examples of Dedicated Datapaths We will now illustrate the design of dedicated datapaths with several examples. The datapaths produced in the
examples are by no mean the only correct datapaths for solving each of the problems. Just like writing a computer
program, there are many ways of doing it. 298 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths 9.3.1 Simple IF-THEN-ELSE
Example 9.3: Simple IF-THEN-ELSE
In this example, we want to construct a 4-bit-wide dedicated datapath for solving the simple IF-THEN-ELSE
algorithm shown in Figure 9.12. To create a datapath for the algorithm, we need to look at all of the data
manipulation statements in the algorithm, since the datapath is responsible for manipulating data. These data
manipulation instructions are the register-transfer operations. In most cases, one data manipulation instruction is
equivalent to one register-transfer operation. However, some data manipulation instructions may require two or
more register-transfer operations to realize.
The algorithm uses two variables, A and B; therefore, the datapath should have two 4-bit registers—one for each
variable. Line 1 of the algorithm inputs a value into A. In order to realize this operation, we need to connect the data
input signals to the input of register A, as shown in Figure 9.13. By asserting the ALoad signal, the data input value
will be loaded into register A at the next active clock edge.
Line 2 of the algorithm tests the value of A with the constant 5. The datapath in Figure 9.13 uses a 4-input AND
gate for the equality comparator with the four input bits connected as 0101 to the four output bits of register A.
Since 5 in decimal is 0101 in binary, bits 0 and 2 are not inverted for the two 1’s in the bit string, while bits 1 and 3
are inverted for the two 0’s. With this connection, the AND gate will output a 1 when the input is a 5. The output of
this comparator is the 1-bit status signal for the condition (A = 5) that the datapath sends to the control unit.
1
2
3
4
5
6
7 INPUT A
IF (A = 5) THEN
B=8
ELSE
B = 13
END IF
OUTPUT B Figure 9.12 Algorithm for solving the simple IF-THEN-ELSE problem of Example 9.3.
Input '8' '13' Muxsel
4 ALoad 1 0
4 D3-0
Load 4-bit Register
A
Clock Q3-0 BLoad D3-0
Load 4-bit Register
B
Clock Q3-0 Clock
(A = 5 ) A3 4 4 Out A0
Output
Figure 9.13 Dedicated datapath for solving the simple IF-THEN-ELSE problem of Example 9.3.
Given the status signal for the comparison (A = 5), the control unit will decide whether to execute line 3 or line
5 of the algorithm. This decision is done by the control unit and not by the datapath. The datapath is responsible only
for all of the register-transfer operations. Lines 3 and 5 require loading either an 8 or a 13 into register B. In order to
be able to select one data from several sources, a multiplexer is needed—in this case, a 2-to-1 multiplexer is used.
One input of the multiplexer is connected to the constant 8 and the other to the constant 13. The output of the
multiplexer is connected to the input of register B, so that one of the two constants can be loaded into the register.
Again, which constant is to be loaded into the register is dependent on the condition in line 2. Knowing the result of 299 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths the test from the status signal, the control unit will generate the correct signal for the multiplexer select line, Muxsel.
The actual loading of the value into register B is accomplished by asserting the BLoad signal.
Finally, the algorithm outputs the value from register B in line 7. This is accomplished by connecting a tri-state
buffer to the output of the B register. To output the value, the control unit asserts the enable line, Out, on the tri-state
buffer, and the value from the B register will be passed to the data output lines.
Notice that the complete datapath shown in Figure 9.13 consists of two separate circuits. This is because the
algorithm does not require the values of A and B to be used together. A question we might ask is whether we can
connect the output of the comparator to the multiplexer select signal so that the status signal (A = 5) directly controls
Muxsel. Logically, this is alright, since if the condition (A = 5) is true, then the status signal is a 1. Assigning a 1 to
Muxsel will select the 1 input of the multiplexer, thus passing the constant 8 to register B. Otherwise, if the condition
(A = 5) is false, then Muxsel will get a 0 from the comparator, and the constant 13 will pass through the multiplexer.
The advantage of doing this is that the datapath will generate one less status signal and requires one less control
signal from the control unit. However, in some situations, we need to be careful with the timing when we use status
signals to directly control the control signals.
Figure 9.14 shows the control words for performing the statements in Figure 9.12 using the datapath in Figure
9.13. Control word 1 executes the instruction INPUT A. To do this, the ALoad signal is asserted, and the data value at
the input port will be loaded into the register at the next active clock edge. For this instruction, we do not need to
load a value into the B register; hence, BLoad is de-asserted for this control word. Furthermore, because of this, it
does not matter what the multiplexer outputs, so Muxsel can be a don’t-care value. For control words 2 and 3, we
want to load one of the two constants into B; therefore, BLoad is asserted for both of these control words, and the
value for Muxsel determines which constant is loaded into B. When Muxsel is asserted, the constant 8 is passed to
the input of the B register, and when it is de-asserted, the constant 13 is passed to the register. Control word 4 asserts
the Out signal to enable the tri-state buffer, thus outputting the value from the B register.
♦
Control
Word
1
2
3
4 Instruction ALoad Muxsel BLoad Out INPUT A
B=8
B = 13
OUTPUT B 1
0
0
0 ×
1
0
× 0
1
1
0 0
0
0
1 Figure 9.14 Control words for solving the simple IF-THEN-ELSE problem of Example 9.3 9.3.2 Counting 1 to 10
Example 9.4: Counting 1 to 10
Construct a 4-bit-wide dedicated datapath to generate and output the numbers from 1 to 10. The algorithm for
this counting problem is shown in Figure 9.15.
From the algorithm, we see that again we need a 4-bit register for storing the value for i. For line 3, an adder can
be used for incrementing i. Both lines 1 and 3 write a value into i, thus providing two sources for the register. Our
first inclination might be to use a 2-input multiplexer. However, notice that loading a 0 into a register is equivalent
to clearing the register with the asynchronous Clear line, as long as the timing is correct. The resulting datapath is
shown in Figure 9.16(a). For line 1, we assert the Clear signal to initialize i to 0, and for line 3, we assert the iLoad
signal to load in the result from the adder, which adds a 1 to the current value of i. Asserting Out will output i. The
status signal for the conditional test (i ≠ 10) is realized by the 4-input NAND gate, where the four input bits of the
NAND gate are connected to the four output lines from the register as 1010 binary for the constant decimal 10.
Alternatively, instead of using a separate register and adder, we can use a single 4-bit up counter to implement
the entire algorithm, as shown in Figure 9.16(b). Again, initializing i to 0 is accomplished by asserting the Clear
signal. To increment i for line 3, we simply assert the Count signal. Generating the status signal and outputting the
count are the same as before. 300 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths The control words for the two different datapaths in Figure 9.16 are shown in Figure 9.17, respectively. For
both cases, asserting the Clear signal will initialize i to 0. To increment i for the datapath in Figure 9.16(a), we need
to assert iLoad. This will load in the value from the output of the adder, which is the result for i + 1, since one
operand of the adder is i, and the other operand is the constant 1. For the datapath in Figure 9.16(b), we simply have
to assert Count to increment i. The internal counter will increment the content in the register. Control word 3 asserts
the Out signal, which asserts the enable signal on the tri-state buffer, thus passing the content of the register to the
output port.
Note that control words 2 and 3 (corresponding to lines 3 and 4 in the algorithm, respectively) must be executed
ten times in order to output the ten numbers. The looping in the algorithm is implemented in the control unit, and we
will see in the next chapter how it is done.
♦
1
2
3
4
5 i=0
WHILE (i ≠ 10){
i=i+1
OUTPUT i
} Figure 9.15 Algorithm for solving the counting problem of Example 9.4.
'1'
+ iLoad
Clear
Clock D3-0
Load 4-bit Register
Clear
i
Clock Q3-0 i3 (i ≠ 10) 4-bit Up
Count Counter
Clear
i
Clock Q3-0
4
i Count
Clear
Clock 3 (i ≠ 10) i0 4 Out i0
Out Output
Output (b) (a) Figure 9.16 Dedicated datapath for solving the counting problem of Example 9.4: (a) using a separate adder and
register; (b) using a single up counter.
Control
Word
1
2
3 Instruction iLoad Clear i=0
i=i+1
OUTPUT i 0
1
0 1
0
0 Control
Word
1
2
3 Out
0
0
1 (a) Instruction Count Clear Out i=0
i=i+1
OUTPUT i 0
1
0 1
0
0 0
0
1 (b) Figure 9.17 Control words for solving the counting problem of Example 9.4: (a) using the datapath in Figure
9.16(a); (b) using the datapath in Figure 9.16(b). 301 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths 9.3.3 Summation of n Down to 1
Example 9.5: Summation of n down to 1
Construct an 8-bit dedicated datapath to generate and add the numbers from n down to 1, where n is an 8-bit
user-input number. The datapath should output the sum of the numbers when done and notify external devices that
the calculation is completed by asserting a Done signal. The algorithm is shown in Figure 9.18.
1
2
3
4
5
6
7 sum = 0
INPUT n
WHILE (n ≠ 0){
sum = sum + n
n=n-1
}
OUTPUT sum Figure 9.18 Algorithm for solving the summation problem of Example 9.5.
We first note that we need to have two 8-bit registers with load function for storing the two variables, n and
sum. The register for sum should also include a Clear function for initializing it to 0. The register for n should also
be a down counter for decrementing n. A separate adder is used for the addition operation. The resulting dedicated
datapath is shown in Figure 9.19.
For initializing sum, we simply can assert the Clear line. Asserting nLoad will input a value for n. Asserting
sumLoad will load into register sum the value from the output of the adder, which is the summation of sum plus n.
Decrementing n by 1 is accomplished by asserting nCount. Finally, asserting Out will enable the tri-state buffer, thus
outputting the value from the sum register. The Out signal is also used as a Done signal to notify external devices
that the calculation is completed. The comparator for the WHILE loop condition (n ≠ 0) is an 8-input OR gate. The
output of this OR gate is the status signal for the control unit.
♦ The control words for this example are shown in Figure 9.20.
Input nLoad
nCount D7-0
Load 8-bit Down
Count Counter
n
Clock Q7-0 sumLoad
Clear
Clock
(n ≠ 0 ) D7-0
Load 8-bit Register
Clear sum
Clock Q7-0
8 8 + Out
Output Done Figure 9.19 Dedicated datapath for solving the summation problem of Example 9.5. 302 8 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5 Chapter 9 - Datapaths Instruction nLoad nCount sumLoad Clear Out sum = 0
INPUT n
sum = sum + n
n=n–1
OUTPUT sum 0
1
0
0
0 0
0
0
1
0 0
0
1
0
0 1
0
0
0
0 0
0
0
0
1 Figure 9.20 Control words for solving the summation problem of Example 9.5. 9.3.4 Factorial of n
Example 9.6: Factorial of n
Design an 8-bit dedicated datapath for evaluating the factorial of n. The factorial of n is defined to be the
product of 1 × 2 × 3 × … × n. Figure 9.21 shows the algorithm for solving the factorial of n where n is an 8-bit user
input number. The datapath should notify external devices that the calculation is completed by asserting a Done
signal when the data at the output port is valid.
1
2
3
4
5
6 INPUT n
product = 1
WHILE (n > 1) {
product = product * n
n=n–1
}
OUTPUT product Figure 9.21 Algorithm for solving the factorial problem of Example 9.6.
After analyzing the algorithm, we conclude that the following registers and functional units are needed for the
datapath:
•
•
•
•
• An 8-bit down counter with parallel load for storing the variable n and for decrementing n. The parallel
load feature will allow for the input of n.
An 8-bit register with load for storing the variable product.
A multiply functional unit.
A greater-than-one comparator for returning the status signal to the control unit.
A tri-state buffer for output. The complete dedicated datapath is shown in Figure 9.22. The input port is connected to the input of the n
register. By asserting nLoad, the value at the input port is loaded into register n, thus realizing the instruction in line
1. Since this n register is also a down counter, by asserting nCount, we will have executed line 5.
Register product has two sources, one from line 2, and the second from line 4. A 2-to-1 multiplexer is used to
select from these two sources using the productMux select line. For line 2, the constant 1 is connected to the 1-input
of the multiplexer, so that, by asserting both productMux and productLoad, the constant 1 will be loaded into the
register. For line 4, one operand of the multiply function is product, and the other operand is n. Hence, the output of
the product register is connected to one input of the multiply unit. Since n is stored in the counter, therefore, the
output of the counter is connected to the second input of the multiply unit. The result of the multiply is assigned
back into the variable product. Thus, in the datapath, we needed a connection from the output of the multiply
functional unit back to the input of the register where product is stored via the 2-to-1 multiplexer.
The tri-state buffer, connected to the output of the product register, is needed for the final output of the result.
The Out signal for enabling the tri-state buffer is also used as the Done signal. Finally, the comparator generates the
status signal for the condition (n > 1) to the control unit. One input to the comparator comes from the counter where
n is stored, and the other input is the constant 1. 303 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Input '1'
1 productMux
productLoad
nLoad
nCount D7-0
Load 8-bit Down
Counter
Count
n
Clock Q7-0 0 D7-0
Load 8-bit Register
product
Clock Q7-0 Clock 8 '1' 8 8 > * (n > 1)
Out
Done Output Figure 9.22 Dedicated datapath for solving the factorial problem of Example 9.6.
The control words for evaluating the factorial using the dedicated datapath shown in Figure 9.22 has five
control signals: productMux, productLoad, nLoad, nCount, and Out, as shown in Figure 9.23(a).
Control word 1 loads n into the counter register by asserting the nLoad signal. We are not loading into the
product register, so the productLoad signal is de-asserted, and therefore, it doesn’t matter what the productMux
signal is. Both nCount and Out are de-asserted, since we don’t want to decrement n, and we are not done evaluating.
Control word 2 initializes product to 1. Since the constant 1 is connected to the 1-input of the multiplexer, we
need to set productMux to 1. The productLoad signal has to be asserted to actually load the constant 1 into the
register.
Control word 3 multiplies product with n. Since the value stored in a register is always available at its output,
the combinational multiplier circuit constantly is performing the multiplication of these two numbers. The output
from the multiplier is stored back into the product register simply by setting productMux to 0 and asserting the
productLoad signal.
Control word 4 decrements n by asserting the nCount signal. Since we are using a down counter, asserting
nCount will decrement the value stored in the register. Again, we are not loading a value into the product register, so
the value of productMux does not matter.
Control word 5 outputs the value stored in the product register. This is accomplished simply by asserting the
Out signal. The value stored in the product register, which is available at its output, is passed through the tri-state
buffer to the output port. Since the Out signal is also connected to the Done signal, asserting Out will also assert
Done.
Notice that with this dedicated datapath, several instructions can be performed in parallel, as shown by the
control words in Figure 9.23(b). Specifically, inputting n and initializing product to 1 can be performed
simultaneously by control word 1. In the datapath circuit, the path that goes from the data input port to the counter
register for storing n is separated completely from the path that goes from the constant 1 to the register for storing
product. Because of this, control word 1 can assert both the productLoad and the nLoad signals together. Asserting
productLoad and productMux will load in the constant 1, while nLoad will load in n.
Similarly, the multiplication and the decrement instructions can be performed together by control word 2. Here,
productLoad is asserted and productMux is de-asserted to store the result from the multiplier, while nCount is
asserted to decrement n. We will see in the next chapter that, by reducing the number of control words, the circuit
for the control unit will be smaller.
♦ 304 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5 Chapter 9 - Datapaths Instruction productMux productLoad nLoad nCount Out INPUT n
product = 1
product = product * n
n=n–1
OUTPUT product ×
1
0
×
× 0
1
1
0
0 1
0
0
0
0 0
0
0
1
0 0
0
0
0
1 (a)
Control
Word
1
2
3 Instruction productMux productLoad nLoad nCount Out INPUT n, product = 1
product = product * n, n = n – 1
OUTPUT product 1
0
× 1
1
0 1
0
0 0
1
0 0
0
1 (b)
Figure 9.23 Control words for solving the factorial problem of Example 9.6: (a) using five control words; (b) using
only three control words. 9.3.5 Count 0’s and 1’s
Example 9.7: Count 0’s and 1’s
In this example, we want to construct an 8-bit dedicated datapath for solving the following problem:
Input an 8-bit number. Output a 1 if the number has the same number of 0’s and 1’s, otherwise, output a 0.
(E.g., the number 10111011 will output a 0; whereas, the number 10100011 will output a 1.)
The algorithm for solving the problem is shown in Figure 9.24. The WHILE loop is executed eight times using
the counteight variable for the 8 bits in the input number n. For each bit in n, if it is a 1, the variable countbit is
incremented, otherwise, it is decremented. At the end of the WHILE loop, if countbit is equal to 0, then there are the
same number of 0’s and 1’s in n.
INPUT n
countbit = 0
// for counting the number of zero and one bits
counteight = 0
// for looping eight times
WHILE (counteight ≠ 8) {
IF (n0 = 1) THEN
// test whether bit 0 of n is a 1
countbit = countbit + 1
ELSE
countbit = countbit – 1
END IF
n = n >> 1
// shift n right one bit
counteight = counteight + 1;
}
IF (countbit = 0) THEN
OUTPUT 1
ELSE
OUTPUT 0
END IF
ASSERT Done
Figure 9.24 Algorithm for solving the count 0’s and 1’s problem of Example 9.7. 305 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths After analyzing the algorithm, we conclude that the following registers and functional units are needed for the
datapath:
•
•
•
•
• An 8-bit shifter with parallel load register for storing and shifting n.
A 4-bit up counter for counteight.
A 4-bit up-down counter for countbit.
A “not equal to 8” comparator for looping eight times.
An “equal to 0” comparator for testing with countbit. The dedicated datapath for implementing the algorithm is shown in Figure 9.25. Notice that there are no
connections between the shifter and the two counters; they are completely separate circuits. To extract bit 0 of n and
test whether it is equal to a 1 or not, we only have to connect to the least significant bit, n0, of the shifter, and no
active component is necessary. To test for (counteight ≠ 8), we use a 4-input NAND gate with the three least
significant input bits inverted. When counteight is equal to eight, the NAND gate outputs a 0, which serves as the
ending loop condition. Using the Down signal, the countbit register can be either incremented or decremented. Since
register countbit is keeping track of the number of 0’s and 1’s, if it is a 0 at the end, it means that n has the same
number of 0’s and 1’s. The NOR gate will output a 1 if countbit is a 0. Whether the output of the NOR gate actually is
outputted will depend on whether the tri-state buffer is enabled or not. When the control unit asserts the Out signal
to enable the tri-state buffer, this 1 signal also serves as the Done signal to inform the external user that the
computation is completed. In other words, when the Done signal is asserted, the output value is the correct result for
the computation.
The control words for this example are shown in Figure 9.26. Notice that there are several control words with
don’t-care values. For example, in control word 2 for initializing countbit to 0, it does not matter what the setting for
the Down signal, is since we are not incrementing or decrementing the count.
♦
Input
8
D7-0
Load 8-bit RightShift shift Register
n
Clock nLoad
nShift n0
(n 0 = 1 )
4-bit Up
Count Counter
Clear counteight
Clock Q3-0
Q4 eightCount 3 (counteight ≠ 8) Q0
Down
bitCount
Clear
Clock Down 4-bit Up-down
Count Counter
Clear countbit
Clock Q3-0
4 Out
Done Output Figure 9.25 Dedicated datapath for solving the count 0’s and 1’s problem of Example 9.7. 306 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5
6
7 Chapter 9 - Datapaths Instruction nLoad nShift eightCount Down bitCount Clear Out INPUT n
countbit = 0, counteight = 0
countbit = countbit + 1
countbit = countbit – 1
n = n >> 1
counteight = counteight + 1
OUTPUT 1, OUTPUT 0, ASSERT Done 1
0
0
0
0
0
0 0
0
0
0
1
0
0 ×
0
0
0
0
1
0 ×
×
0
1
×
×
× ×
0
1
1
0
0
0 ×
1
0
0
0
0
0 0
0
0
0
0
0
1 Figure 9.26 Control words for solving the count 0’s and 1’s problem of Example 9.7. 9.4 General Datapaths As the name suggests, a general datapath is more general than dedicated datapath in the sense that it can be used
to solve various problems instead of just one specific problem, as long as it has all of the required functional units
and enough registers for storing all of the temporary data. The idea of using a general datapath is that we can use a
“ready-made” datapath circuit to solve a given problem without having to modify it. The tradeoff is a time-versusspace issue. On one hand, we do not need the extra time to design a dedicated datapath. On the other hand, the
general datapath may contain more features than what the problem requires, so it not only increases the size of the
circuit but also consumes more power than necessary.
Figure 9.27 shows an example of a simple, general datapath. It contains one functional unit, the ALU, and one
8-bit register for storing data. The input to the A operand of the ALU can be either from an external data input or the
constant 1, as selected by the multiplexer select line IE. The B operand of the ALU is always from the content of the
register. The operation of the ALU is determined by the three control lines: ALU2, ALU1, and ALU0, as defined in the
ALU operation table. The design of the ALU was discussed in Section 4.5. The register has a load capability for
loading the output of the ALU into the register when the Load signal line is asserted. The register can also be reset to
0 asynchronously by asserting the Clear signal line. The content of the register can be passed to the external data
output port by asserting the output enable line, OE, of the tri-state buffer. We assume here that the buses for
transferring the data between the components are 8-bits wide. All the control lines, of course, are 1-bit wide. Data
Input '1'
8 6 IE 1 8
0
8 ALU2
0
0
0
0
1
1
1
1 A
B
ALU2
ALU1 ALU
ALU0 5
4
3 8
D7-0
Load 8-bit Register
Clear
Clock Q7-0 2
1
Clock 0 8 OE
8 Data
Output
Figure 9.27 A simple, general datapath circuit. 307 ALU1
0
0
1
1
0
0
1
1 ALU0
0
1
0
1
0
1
0
1 Operation
Pass through A
A AND B
A OR B
NOT A
A+B
A–B
A+1
A–1 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths There are seven control signals (numbers 0 to 6) for controlling the operations of this simple, general datapath.
When grouped together, they form the control word for this datapath. Just like the dedicated datapath, various
operations can be performed by this datapath either by asserting or de-asserting these control signals at different
times.
For example, to load a value from the external data input to the register, we would set the control word as
follows:
Control word
Value IE
6
1 ALU2
5
0 ALU1
4
0 ALU0
3
0 Load
2
1 Clear
1
0 OE
0
0 By setting IE = 1, we select the external input to pass through the multiplexer. From the ALU operation table,
we see that setting the ALU control lines: ALU2, ALU1, and ALU0, to 000 selects the pass-through operation, which
passes the data from the A operand input of the ALU to the output of the ALU with no modifications. Finally, setting
Load = 1 loads the value from the output of the ALU into the register. Thus, we have stored the input value into the
register. We do not want to clear the register nor output the value from the register, so both Clear and OE are set to
0’s.
Note that the actual writing of the register occurs at the next active edge of the clock. Thus, the new value is not
available from the register until the beginning of the next clock cycle. If we read from the register during the current
clock cycle, we would get the original value that was in the register and not the new value that was just entered in. In
other words, if we had set OE to 1 in the above control word, we would be reading the register in the current clock
cycle and, thus, be outputting the original value found in the register rather than the new value from the data input.
We will look at this important issue again in Section 9.6. 9.5 Using General Datapaths Using a general datapath is exactly the same as using a dedicated datapath. We simply have to assert (or deassert) the appropriate control signals at the right time so that the data transfer between registers is correct to realize
a data manipulation statement. The following example shows how we can use the general datapath shown in Figure
9.27 to solve a problem.
Example 9.8: Using a general datapath to output the numbers from 1 to 10
To see how a general datapath is used to perform a computation, let us write the control words to generate and
output the numbers from 1 to 10 using the general datapath shown in Figure 9.27. This example, of course, is
identical to Example 9.4, except that we are using a different datapath. Hence, only the control words will be
different. The algorithm is repeated here in Figure 9.28(a), and the control words for using this general datapath to
solve the problem are shown in Figure 9.28(b).
1
2
3
4
5 i=0
WHILE (i ≠ 10){
i=i+1
OUTPUT i
}
(a) Control
Word
1
2
3 Instruction
i=0
i=i+1
OUTPUT i IE
6
×
0
× ALU2 ALU1 ALU0
5–3
×××
100 (add)
×××
(b) 308 Load
2
0
1
0 Clear
1
1
0
0 OE
0
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.28 Generate and output the numbers from 1 to 10: (a) algorithm; (b) control words for the datapath in
Figure 9.27 using three control words.
Control word 1 initializes i to 0 by asserting the Clear signal for the register. The ALU is not needed in this
operation, so it doesn’t matter what the inputs to the ALU are or the operation that is selected for the ALU to
perform. Hence, the four control lines: IE (for selecting the input), ALU2, ALU1, and ALU0 (for selecting the ALU
operation) are all set to don’t-cares. Load is de-asserted because we do not need to store the output of the ALU to the
register. At this time, we also do not want to output the value from the register, so the output control line OE is also
de-asserted.
Control word 2 increments i, so we need to add a 1 to the value that is stored in the register. Although, the ALU
has an increment operation, we cannot use it because the ALU was designed such that the operation increments the
A operand rather than the B operand, and this datapath is connected such that the output of the register goes to the B
operand only. Now, we can modify the ALU to have an increment B operation, or we can modify the datapath so
that the output of the register can be routed to the A operand of the ALU. However, both of these solutions require
modifications to the datapath, and this defeats the purpose of using a general datapath. If we are going to modify this
datapath, we may as well modify it even further to get a dedicated data path. So instead, what we can do is to use the
ALU addition operation (100) to increment the value stored in the register by one. We can get a 1 to the A operand
by setting IE to 0, since the 0-input line of the multiplexer is tied to the constant 1. The B operand will have the
register value. Finally, we need to load the result of the addition back into the register, so the Load line is asserted.
Control word 3 outputs the value from the register. Again, we don’t care about the inputs to the ALU and the
operation of the ALU, so there is no new value to load into the register. We definitely do not want to clear the
register. We simply want to output the value from the register, so we just assert OE by setting it to a 1.
Just like for the dedicated datapath, the general datapath must also provide the status signal for the condition
(i ≠ 10) to the control unit. Using this status signal, the control unit will determine whether to repeat control words 2
and 3 in the loop or to terminate. We must, therefore, add a comparator to this general datapath to test whether the
condition (i ≠ 10) is true or not. The output of this comparator is the status signal that the datapath sends back to the
control unit. Since variable i is stored in the register, we should connect the output of the register to the input of this
comparator, as shown in Figure 9.29.
Data
Input '1'
8 6 IE 1 8
0
8 A
B
ALU2
ALU1 ALU
ALU0 5
4
3 8
D7-0
Load 8-bit Register
Clear
Clock Q7-0
Q7
8 2
1
Clock
(i ≠ 10) Q0 0 8 8 OE
8 Data
Output
Figure 9.29 The comparator circuit for generating the status signal (i ≠ 10) added to the general datapath. 309 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths The simulation trace for this counting example is shown in Figure 9.30. Notice that two cycles are needed for
each count, as shown in the output signal in the trace—the first cycle for control word 2 where the output shows a Z
value, and the second cycle for control word 3 where the actual count appears at the output. These two cycles are
repeated ten times for the ten numbers. For example, at 500 ns (at the beginning of the first of the two clock cycles),
Load = 1 and OE = 0. The current content of the register is 1. Since OE = 0, so the output is Z. At 700 ns (the
beginning of the second of the two clock cycles), the register is updated with the value 2. Load is de-asserted and
OE is asserted, and the number 2 is output.
This simulation trace was obtained by manually asserting and de-asserting the seven datapath control signals at
each clock cycle. This is only because we wanted to test out the datapath, and we have not yet constructed the
control unit for generating these control signals automatically. What we really need to do eventually is to construct
the control unit based on the control words from Figure 9.28(b). The control unit will generate the appropriate
control signals for the datapath for each clock cycle.
♦ Figure 9.30 Simulation trace for Example 9.8 using the three control words shown in Figure 9.28(b). 9.6 A More Complex General Datapath When a particular general datapath does not contain all of the functional units and/or registers needed to
perform all of the required operations specified in the algorithm that we are trying to implement, then we need to
select a more complex datapath. When working with general datapaths, the goal is to find the simplest and smallest
datapath that matches the requirements of the problem as closely as possible. Example 9.9 shows the need for
selecting a more complex datapath.
Example 9.9: Determining the need for a complex datapath
Let us use the simple datapath of Figure 9.27 to generate and add the numbers from n down to 1, where n is an
8-bit user input number, and output the sum of these numbers. The algorithm for doing this is shown in Figure 9.31.
The algorithm requires the use of two variables, n for the input that counts down to 0, and sum for adding up the
total. This means that we need two registers in the datapath, unless we want the user to enter the numbers from n
down to 1 manually and just use the one register to store the sum. Thus, we conclude that the datapath of Figure 9.27
cannot be used to implement this algorithm.
♦
In order to implement the algorithm of Figure 9.31 we need a slightly more complex datapath that includes at
least two registers. One possible datapath is shown in Figure 9.32(a). The main difference between this datapath and
the previous one is that a register file (RF) with four locations is used instead of having just one register. The
register file, as discussed in Section 8.4, has one write port and two read ports. To access a particular port, the enable
line for that port must be asserted and the address for the location set up. The designated lines are WE for write
enable, RAE for read Port A enable, and RBE for read Port B enable, WA1,0 for the write address, RAA1,0 for the read
Port A address, and RBA1,0 for the read Port B address. The read Ports A and B can be read simultaneously, and they
are connected to the two input operands A and B of the ALU, respectively. The ALU operations are specified in
Figure 9.32(b). The result of the ALU is passed through a shifter whose operations are specified in Figure 9.32(c).
Although the shifter is not needed by the algorithm of Figure 9.31, it is available in this datapath. The output of the 310 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths shifter is routed back to the register file via the multiplexer, or it can be output externally by enabling the output tristate buffer. The datapath width is again 8-bits wide.
1
2
3
4
5
6
7 sum = 0
INPUT n
WHILE (n ≠ 0){
sum = sum + n
n=n-1
}
OUTPUT sum Figure 9.31 Algorithm to generate and sum the numbers from n down to 1.
ALU2
0
0
0
0
1
1
1
1 Data Input
8
15 IE 8
1 0 8
WE
WA 4 x 8
RAE RF RBE
RAA
RBA 14
13-12
11
10-9
Clock A Operation
Pass through A
A AND B
A OR B
NOT A
A+B
A–B
A+1
A–1 8 (b)
A
B
ALU2
ALU1 ALU
ALU0 5
4
3 8 8 SH1
Shifter
SH0 2
1 0 ALU0
0
1
0
1
0
1
0
1 B 8 8
7-6 ALU1
0
0
1
1
0
0
1
1 S H1
0
0
1
1 OE
8 S H0
0
1
0
1 Operation
Pass through
Shift left and fill with 0
Shift right and fill with 0
Rotate right Data Output
(c) (a) Figure 9.32 Complex general datapath with a register file: (a) circuit; (b) ALU operations; (c) Shifter operations.
Example 9.10: Using complex datapath to sum numbers
The summation algorithm of Figure 9.31 can be implemented using the datapath shown in Figure 9.32. The
control words for manipulating the datapath for this algorithm are shown in Figure 9.33.
Control word 1 initializes sum to 0 by performing a subtraction where the two operands are the same. The
location of the register file (RF) used for the two operands is arbitrary because it doesn’t matter what the value is, as
long as both operands get the same value. We use RF location 00 to store the value of the variable sum. Thus, we
assert all three RF enable lines, and sets the RF write address and both read addresses to location 0. The shifter is not
needed, so the pass-through operation is selected. 311 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5 Instruction
sum = 0
INPUT n
sum = sum + n
n=n–1
OUTPUT sum IE
15
0
1
0
0
× WE
14
1
1
1
1
0 WA1,0
13–12
00
01
00
01
×× Chapter 9 - Datapaths RAE
11
1
0
1
1
1 RAA1,0
10–9
00
××
00
01
00 RBE
8
1
0
1
0
0 RBA1,0
7–6
00
××
01
××
×× ALU2,1,0
5–3
101 (subtract)
×××
100 (add)
111 (decrement)
000 (pass) SH1,0
2–1
00
××
00
00
00 OE
0
0
0
0
0
1 Figure 9.33 Control words to generate and sum the numbers from n down to 1 using the datapath in Figure 9.32.
All of the operations specified by a control word are performed within one clock cycle. The timing for the
operations of this control word is as follows. At the active edge of the clock, the FSM enters the state for this control
word. The appropriate control signals for this control word to the datapath are asserted. Data from RF location 0 is
read for both ports and passed to the ALU. Recall that the register file is constructed such that the data from reading
a port is available immediately, and does not have to wait until the next active clock edge. Since both the ALU and
the shifter are combinational circuits, they will operate within the same clock cycle. The result is written back to RF
location 0 at the next active clock edge. Thus, the updated or new value in RF location 0 is not available until the
beginning of the next clock cycle.
Control word 2 inputs the value n and stores it in RF location 1. To read from the input, we set IE = 1. To write
n into RF location 1, we set WE = 1 and WA1,0 = 01. Both the ALU and the shifter are not used in this control word,
so their select lines are set to don’t-cares.
Control word 3 reads sum through Port A by setting RAE = 1 and RAA1,0 = 00, and n through Port B by setting
RBE = 1 and RBA1,0 = 01. These two numbers are added together by setting the ALU select lines, ALU2,1,0, to 100.
The result of the addition passes through the shifter and the multiplexer, and then it is written back to RF location 0.
Control word 4 decrements n by 1 by using the decrement operation of the ALU (111). From RF location 01, n
is read through Port A and passes to the A operand of the ALU. The ALU decrements the A operand by 1, and the
result is written back to RF location 01.
After decrementing n, the WHILE loop needs to test the value of n. In order for the control unit to loop correctly,
the datapath must generate the correct status signal for the condition (n ≠ 0). The question is where do we connect
the comparator inputs to in the datapath so that we can get the correct value for n? Since n is stored in location 01 of
the register file, our first inclination might be to connect to Port A of the register file, which is also the input to the
ALU. An alternative might be to connect to the output of the ALU. The difference between these two connection
points is that, for the first connection, we would be using the value of n before the decrement, while for the second
connection, we would be using the value of n after the decrement. Furthermore, if we look at these same two
connection points at a different time, say, during the clock cycle for control word 3, then the values are not even for
n, but rather for sum, since control word 3 reads sum through Port A, and the ALU calculates sum + n. Therefore, in
order to know which is the correct connection point, we need to look at the state diagram for this algorithm to
determine in which state is the status signal needed and what data is being passed through the datapath during that
clock cycle. We will come back to this problem in the next chapter.
Control word 5 outputs the result that is stored in sum by reading from RF location 0 via Port A and passing it
through the ALU and shifter. OE is asserted to enable the tri-state buffer for the output.
The simulation trace of the control words is shown in Figure 9.34. Again, the datapath control signals are set
manually until n (stored in RF(1)) reaches 0, at which point OE is asserted, and the summation value 55 appears on
the output.
♦ 312 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.34 Simulation trace for the summation problem control words of Figure 9.33.
Example 9.11: Using a general datapath to implement the multiplication of two numbers
Write the control words for performing the multiplication of two unsigned numbers using the general datapath
shown in Figure 9.32. The two numbers are entered in through the data input port. The multiplication algorithm is
shown in Figure 9.35.
1
2
3
4
5
6
7
8 prod = 0
INPUT A
INPUT B
WHILE (B ≠ 0){
prod = prod + A
B=B-1
}
OUTPUT prod Figure 9.35 Algorithm to multiply two unsigned numbers.
The control words for manipulating the datapath to perform the multiplication of two unsigned numbers are
shown in Figure 9.36.
Control word 1 assigns a 0 (obtained from subtracting a number from itself) to RF(0). RF(0) is used for storing
the variable prod.
Control words 2 and 3 input the two unsigned numbers A and B into RF(1) and RF(2), respectively. IE is set to
1 so that the value comes from Data Input, and because of this, it doesn’t matter what the ALU and the shifter will
do. WE is asserted for the write. WA1,0 are set to the addresses 01 for variable A and 10 for variable B.
Control word 4 performs the addition of prod with A. Variable prod, stored in RF(0), is read through Port A;
and variable A, from RF(1), is read through Port B. The ALU is selected for addition, and the result is passed
through the shifter. With WE being asserted and WA1,0 set to 00, the result is written back into RF(0).
Control word 5 decrements B. Variable B, from RF(2), is read through Port A. The ALU is selected for the
decrement operation, and the result is again passed through the shifter. The result is written back into RF(2).
Control word 6 outputs the result stored in prod by reading RF(0) through Port A, and passing this value
through the ALU and the shifter. The value is passed to the data output port by asserting OE.
♦ 313 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5
6 Instruction
prod = 0
INPUT A
INPUT B
prod = prod + A
B=B–1
OUTPUT prod IE
15
0
1
1
0
0
× WE
14
1
1
1
1
1
0 WA1,0
13–12
00
01
10
00
10
×× Chapter 9 - Datapaths RAE
11
1
0
0
1
1
1 RAA1,0
10–9
00
××
××
00
10
00 RBE
8
1
0
0
1
0
0 RBA1,0
7–6
00
××
××
01
××
×× ALU2,1,0
5–3
101 (subtract)
×××
×××
100 (add)
111 (decrement)
000 (pass) SH1,0
2–1
00
××
××
00
00
00 OE
0
0
0
0
0
0
1 Figure 9.36 Control words to multiply two unsigned numbers using the datapath in Figure 9.32. 9.7 Timing Issues In Figure 9.28(b) of Example 9.8 to generate and output the numbers from 1 to 10, two control words are used
for the addition and output operations. Control word 2 does the addition and writing of the result into the register,
while control word 3 outputs the count stored in i. During the current clock cycle for control word 2, the operation
starts with the constant ‘1’ passing through the multiplexer, followed by the ALU performing the addition i + 1, as
shown in Figure 9.37. The current value of i is available from the output of the register at the beginning of the
current clock cycle. The result from the addition is available shortly before the beginning of the next clock cycle
(next rising clock edge). Recall from Section 6.7 that a value is latched into a flip-flop at the active (rising) edge of
the clock. So, although the Load signal is asserted at the beginning of the current clock cycle, a value is not written
into the register until the next rising clock edge, which is at the beginning of the next clock cycle. Since the addition
result is available before the next rising clock edge, it is this addition result that is written into the register at the
beginning of the next clock cycle.
So for control word 2, we are doing both a read and a write of the same register, i, in one clock cycle for
executing the instruction i = i + 1. The read is for the i that is on the right-hand side of the equal sign, and the write
is for the i that is on the left-hand side of the equal sign. From the above analysis, we see that performing both a read
and a write to the same register in the same clock cycle does not create any data conflict because the reading occurs
at the beginning of the current clock cycle and, therefore, is getting the current value that is in the register. The
writing occurs at the beginning of the next clock cycle, which happens after the reading.
As a result, the value that is available at the output of the register in the current clock cycle is still the value
before the write back, which is the value before the addition of i + 1. If we assert the OE signal in the current clock
cycle to output the register value (that is, do the addition and output operations in the same clock cycle as shown in
control word 2 of Figure 9.38), the output value would be the value before the addition and not the result from after
the addition. According to the algorithm in Figure 9.28(a), the output should be for the value from after the addition.
Because of this, Figure 9.28(b) uses control word 3 (starting at the next clock cycle) to output the value from after
the addition. ALU performs i + 1
Constant '1' passes through multiplexer
Read i from register Result of i + 1 ready
current clock cycle next clock cycle Clock Signal
Start of control word 2
IE de-asserted
Load asserted Start of control word 3
Write register with ALU result 314 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.37 Read and write timings for a control word.
The simulation trace for the two control words of Figure 9.38 (where the addition and output operations are
both done in the same control word) is shown in Figure 9.39. There are two main differences between this
simulation trace and the one in Figure 9.30. The first is that each count now requires only one clock cycle rather than
two, because only control word 2 is repeated ten times. As a result, the time to count to ten is about half (2.3 µs
versus 4.1 µs). The second thing is that the first output value is a 0 and not a 1 (as it should be). The first time that
control word 2 executes is the clock cycle between 100 ns and 300 ns. The incremented value (1) is not written into
the register until at 300 ns, so when OE is asserted before 300 ns, the output value is a 0.
We certainly like the fact that it only requires half the time to execute the algorithm, but outputting an extra zero
at the beginning is not what we wanted. There are several possible solutions, one of which is shown in Figure 9.40.
In Figure 9.40, control signal OE is not asserted in control word 2, which is executed only once at the beginning
before the loop. (Notice that control words 1 and 2 together are equivalent to assigning the constant 1 to i.)
Subsequent executions of control word 3 will have OE asserted together with the addition, and this one we will
repeat ten times. The corrected simulation trace for this set of control words is shown in Figure 9.41. Again, only 2.3
µs are needed for the completion of the algorithm, but this time, the count at the output starts at 1 rather than 0.
Control
Word
1
2 Instruction
i=0
i = i + 1 & OUTPUT i IE
6
×
0 ALU2 ALU1 ALU0
5–3
×××
100 (add) Load
2
0
1 Clear
1
1
0 OE
0
0
1 Figure 9.38 Counting algorithm using two control words for the datapath in Figure 9.27(a). Figure 9.39 Simulation trace for using the two control words from Figure 9.38. Control
Word
1
2
3 Instruction
i=0
i=i+1
i = i + 1 & OUTPUT i IE
6
×
0
0 ALU2 ALU1 ALU0
5–3
×××
100 (add)
100 (add) Load
2
0
1
1 Clear
1
1
0
0 OE
0
0
0
1 Figure 9.40. Optimized control words for the counting algorithm using the datapath in Figure 9.27(a). 315 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths Figure 9.41 Corrected simulation trace for using the three control words from Figure 9.40. 9.8 VHDL for Datapaths In modeling the datapath using VHDL, we need to work at the structural level. First, all of the components used
in the datapath must be described individually. It doesn’t matter whether these components are described at the
behavioral, dataflow, or structural level. These components are then connected together in an enclosing entity using
the structural level method. 9.8.1 Dedicated Datapath
Figure 9.42 shows the VHDL code for the dedicated datapath of Figure 9.25.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_ARITH.ALL;
ENTITY Datapath IS PORT (
Clock: IN STD_LOGIC;
-- primary datapath input
Input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- control signals
nLoad: IN STD_LOGIC;
nShift: IN STD_LOGIC;
eightCount: IN STD_LOGIC;
Clear: IN STD_LOGIC;
bitCount: IN STD_LOGIC;
Down: IN STD_LOGIC;
OutDone: IN STD_LOGIC;
-- status signals
eq8: OUT STD_LOGIC;
N0eq1: OUT STD_LOGIC;
-- primary datapath output
Count: OUT STD_LOGIC_VECTOR(3 DOWNTO 0);
Done: OUT STD_LOGIC;
Output: OUT STD_LOGIC);
END Datapath; 316 -- for debug Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths ARCHITECTURE Structural OF Datapath IS
COMPONENT shiftreg PORT (
Clock: IN STD_LOGIC;
SHSel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
D: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
Q: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END COMPONENT;
COMPONENT counter PORT (
Clock: IN STD_LOGIC;
Clear: IN STD_LOGIC;
Count: IN STD_LOGIC;
Down: IN STD_LOGIC;
Q: OUT INTEGER RANGE 0 TO 15);
END COMPONENT;
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
BEGIN SHSel: STD_LOGIC_VECTOR(1 DOWNTO 0);
ShiftOut: STD_LOGIC_VECTOR(7 DOWNTO 0);
CountbitOut: INTEGER RANGE 0 TO 15;
CounteightOut: INTEGER RANGE 0 TO 15;
Equal: STD_LOGIC;
Up: STD_LOGIC; SHSel <= nShift & nLoad;
U0: shiftreg PORT MAP(Clock,SHSel,Input,ShiftOut);
N0eq1 <= ShiftOut(0);
-- counteight
Up <= '0';
U1: counter PORT MAP(Clock,Clear,eightCount,Up,CounteightOut);
eq8 <= '1' WHEN CounteightOut = 8 ELSE '0';
Count <= CONV_STD_LOGIC_VECTOR(CounteightOut, 4); -- for debug
-- countbit
U2: counter PORT MAP(Clock,Clear,bitCount,Down,CountbitOut);
Equal <= '1' WHEN CountbitOut = 0 ELSE '0';
Output <= Equal WHEN OutDone = '1' ELSE 'Z';
Done <= OutDone;
END Structural;
Figure 9.42 VHDL code for the datapath in Figure 9.25. 9.8.2 General Datapath
Figure 9.43 and Figure 9.44 show the complete VHDL code for modeling the complex, general datapath circuit
from Figure 9.32. Figure 9.43 lists the definitions of all of the components used in the datapath. The VHDL codes
and the detail constructions of the components used in the datapath circuit have been discussed in previous chapters.
Figure 9.44 shows the enclosing entity that connects these components together at the structural level to form the
datapath.
-- 2-to-1 MUX
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY mux2 IS PORT ( 317 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths S: IN STD_LOGIC;
D1, D0: IN STD_LOGIC_VECTOR(7 downto 0);
Y: OUT STD_LOGIC_VECTOR(7 downto 0));
END mux2; -- select line
-- data bus input
-- data bus output ARCHITECTURE Behavioral OF mux2 IS
BEGIN
PROCESS(S, D1, D0)
BEGIN
IF(S = '0' )THEN
Y <= D0;
ELSE
Y <= D1;
END IF;
END PROCESS;
END Behavioral;
-------------------------------------------------------------------------- Register File
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL;
--USE IEEE.STD_LOGIC_ARITH.ALL;
ENTITY regfile IS PORT (
clk: IN STD_LOGIC;
--clock
WE: IN STD_LOGIC;
--write enable
WA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--write address
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
--input
RAE: IN STD_LOGIC;
--read enable ports
RAA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--read address port
RBE: IN STD_LOGIC;
--read enable ports
RBA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--read address port
Aout, Bout: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
--output port
END regfile; A
A
A
A
A &
&
&
&
& B
B
B
B
B ARCHITECTURE Behavioral OF regfile IS
SUBTYPE reg IS STD_LOGIC_VECTOR(7 DOWNTO 0);
TYPE regArray IS array(0 TO 3) OF reg;
SIGNAL RF: regArray;
--register file contents
BEGIN
WritePort: PROCESS (clk)
BEGIN
IF (clk'EVENT AND clk = '1') THEN
IF (WE = '1') THEN
RF(CONV_INTEGER(WA)) <= input;
END IF;
END IF;
END PROCESS;
ReadPortA:
BEGIN
IF (RAE
Aout
ELSE
Aout
END IF; PROCESS (RAE, RAA)
= '1') then
<= RF(CONV_INTEGER(RAA)); -- convert bit VECTOR to integer
<= (others => '0'); 318 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths END PROCESS;
ReadPortB: PROCESS (RBE, RBA)
BEGIN
IF (RBE = '1') then
Bout <= RF(CONV_INTEGER(RBA)); -- convert bit VECTOR to integer
ELSE
Bout <= (others => '0');
END IF;
END PROCESS;
END Behavioral;
-------------------------------------------------------------------------- ALU
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
-- need the following to perform arithmetics on STD_LOGIC_VECTORs
USE IEEE.STD_LOGIC_UNSIGNED.ALL;
ENTITY alu IS PORT (
ALUSel: IN STD_LOGIC_VECTOR(2 DOWNTO 0);
A, B: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
F: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END alu;
ARCHITECTURE Behavior OF alu
BEGIN
PROCESS(ALUSel, A, B)
BEGIN
CASE ALUSel IS
WHEN "000" =>
-F <= A;
WHEN "001" =>
-F <= A AND B;
WHEN "010" =>
-F <= A OR B;
WHEN "011" =>
-F <= NOT A;
WHEN "100" =>
-F <= A + B;
WHEN "101" =>
-F <= A - B;
WHEN "110" =>
-F <= A + 1;
WHEN others =>
-F <= A - 1;
END CASE;
END PROCESS;
END Behavior; -- select for operations
-- input operands
-- output IS pass A through
AND
OR
NOT
add
subtract
increment
decrement -------------------------------------------------------------------------- Shifter
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY shifter IS PORT (
SHSel: IN STD_LOGIC_VECTOR(1 DOWNTO 0); 319 -- select for operations Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- input operands
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0)); -- output
END shifter;
ARCHITECTURE Behavior OF shifter IS
BEGIN
PROCESS(SHSel, input)
BEGIN
CASE SHSel IS
WHEN "00" =>
-- pass through
output <= input;
WHEN "01" =>
-- shift right
output <= input(6 DOWNTO 0) & '0';
WHEN "10" =>
-- shift left
output <= '0' & input(7 DOWNTO 1);
WHEN OTHERS =>
-- rotate right
output <= input(0) & input(7 DOWNTO 1);
END CASE;
END PROCESS;
END Behavior;
-------------------------------------------------------------------------- Tri-state buffer
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY TriStateBuffer IS PORT (
E: IN STD_LOGIC;
D: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
Y: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END TriStateBuffer;
ARCHITECTURE Behavioral OF TriStateBuffer IS
BEGIN
PROCESS (E, D) -- get error message if no d
BEGIN
IF (E = '1') THEN
Y <= D;
ELSE
Y <= (OTHERS => 'Z');
-- to get 8 Z values
END IF;
END PROCESS;
END Behavioral;
Figure 9.43 Components for the datapath of Figure 9.32.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY datapath IS PORT (
clock: IN STD_LOGIC;
input: IN STD_LOGIC_VECTOR( 7 DOWNTO 0 );
IE, WE: IN STD_LOGIC;
WA: IN STD_LOGIC_VECTOR (1 DOWNTO 0);
RAE: IN STD_LOGIC;
RAA: IN STD_LOGIC_VECTOR (1 DOWNTO 0); 320 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths RBE: IN STD_LOGIC;
RBA: IN STD_LOGIC_VECTOR (1 DOWNTO 0);
aluSel: IN STD_LOGIC_VECTOR(2 DOWNTO 0);
shSel: IN STD_LOGIC_VECTOR (1 DOWNTO 0);
OE: IN STD_LOGIC;
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END datapath;
ARCHITECTURE Structural OF datapath IS
COMPONENT mux2 PORT (
S: IN STD_LOGIC;
D1, D0: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
Y: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END COMPONENT; -- select lines
-- data bus input
-- data bus output COMPONENT regfile PORT (
clk: IN STD_LOGIC;
--clock
WE: IN STD_LOGIC;
--write enable
WA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--write address
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
--input
RAE: IN STD_LOGIC;
--read enable ports
RAA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--read address port
RBE: IN STD_LOGIC;
--read enable ports
RBA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
--read address port
Aout, Bout: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
--output port
END COMPONENT;
COMPONENT alu PORT (
ALUSel: IN STD_LOGIC_VECTOR(2 DOWNTO 0);
A, B: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
F: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END COMPONENT; A
A
A
A
A &
&
&
&
& B
B
B
B
B -- select for operations
-- input operands
-- output COMPONENT shifter PORT (
SHSel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
-- select for operations
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- input operands
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0)); -- output
END COMPONENT;
COMPONENT tristatebuffer PORT (
E: IN STD_LOGIC;
D: IN STD_LOGIC_VECTOR(7 downto 0);
Y: OUT STD_LOGIC_VECTOR(7 downto 0));
END COMPONENT;
SIGNAL muxout, rfAout, rfBout: STD_LOGIC_VECTOR( 7 DOWNTO 0 );
SIGNAL aluout, shiftout, tristateout: STD_LOGIC_VECTOR( 7 DOWNTO 0 );
BEGIN
-- doing structural modeling here
U0: mux2 PORT MAP( IE, input, shiftout, muxout );
U1: regfile PORT MAP(clock,WE,WA,muxout,RAE,RAA,RBE,RBA,rfAout,rfBout );
U2: alu PORT MAP( ALUsel, rfAout, rfBout, aluout );
U3: shifter PORT MAP(SHSel,aluout,shiftout);
U4: tristatebuffer PORT MAP(OE, shiftout, tristateout);
output <= tristateout; 321 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths END Structural;
Figure 9.44 Datapath of Figure 9.32 constructed at the structural level. 9.9 Summary Checklist
Datapath
Dedicated datapath
Control signals
Status signals
Control word
General datapath
Register-transfer level design
Timing issues (when a register is updated)
Be able to design a dedicated datapath from an algorithm
Be able to generate the status signals
Be able to use a given datapath by deriving the control words for it to implement an algorithm 9.10 Problems
9.1. Derive the truth table for the circuit in Figure P9.1. The truth table should only have columns for the control
signal inputs and outputs. The data inputs, D0 to D3, are written in the table entries. D3 D2 A0
E 1-to-2
Decoder Y0 1
s D1 D0 0
y 1
s Y1 Subtract +/- OutE Output
Figure P9.1.
Answer:
A0
0
0
0
0
0
0 E
0
0
0
0
1
1 Subtract
0
0
1
1
0
0 OutE
0
1
0
1
0
1 Output
Z
D2 + D 0
Z
D2 - D0
Z
D3 + D 0 322 0
y Digital Logic and Microprocessor Design with VHDL 0
0
1
1
1
1
1
1
1
1 1
1
0
0
0
0
1
1
1
1 1
1
0
0
1
1
0
0
1
1 0
1
0
1
0
1
0
1
0
1 Chapter 9 - Datapaths Z
D3 - D0
Z
D2 + D 0
Z
D2 - D0
Z
D2 + D 1
Z
D2 - D1 9.2. Use schematic entry to implement the general datapath shown in Figure 9.27. 9.3. Test the datapath from Problem 9.2 by performing a simulation on the algorithm shown in Figure 9.28. 9.4. Implement the datapath from Problem 9.2 on the UP2 board. Perform sample tests to show that it operates
correctly. 9.5. Write the VHDL code for the general datapath shown in Figure 9.27. 9.6. Use simulation to show that the synthesized circuits from Problem 9.2 and Problem 9.5 perform functionally
identical to each other. 9.7. Use schematic entry to implement the general datapath shown in Figure 9.32. 9.8. Test the datapath from Problem 9.7 by performing a simulation on the algorithm shown in Figure 9.31. 9.9. Implement the datapath from Problem 9.7 on the UP2 board. Perform sample tests to show that it operates
correctly. 9.10. Test the datapath from Problem 9.7 by performing a simulation on the algorithm shown in Figure 9.35. 9.10.1 Use schematic entry to implement the datapath for Counting 1 to 10
9.11. Example 9.4. 9.10.2 Simulate and test the datapath for Counting 1 to 10
9.12. Example 9.4. 9.10.3 Write the VHDL code for the datapath for Counting 1 to 10
9.13. Example 9.4. 9.10.4 Use schematic entry to implement the datapath for Summation of n Down to 1
9.14. Example 9.5. 9.10.5 Simulate and test the datapath for Summation of n Down to 1
9.15. Example 9.5. 323 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths 9.10.6 Write the VHDL code for the datapath for Summation of n Down to 1
9.16. Example 9.5. 9.10.7 Use schematic entry to implement the datapath for Factorial of n
9.17. Example 9.6. 9.10.8 Simulate and test the datapath for Factorial of n
9.18. Example 9.6. 9.10.9 Write the VHDL code for the datapath for Factorial of n
9.19. Example 9.6. 9.10.10 Use schematic entry to implement the datapath for Count 0’s and 1’s 9.20. Example 9.7. 9.10.11 Simulate and test the datapath for Count 0’s and 1’s 9.21. Example 9.7. 9.10.12 Write the VHDL code for the datapath for Count 0’s and 1’s 9.22. Example 9.7.
9.23. Derive and simulate the control words using the datapath shown in Figure 9.32 for inputting two 8-bit
unsigned numbers, and then output the largest number.
9.24. Derive and simulate the control words using the datapath shown in Figure 9.32 for inputting three 8-bit
unsigned numbers, and then output the largest number, followed by the second largest.
9.25. Derive and simulate the control words using the datapath shown in Figure 9.32 for inputting ten 8-bit
unsigned numbers, and then output the sum of the numbers.
9.26. Derive and simulate the control words using the datapath shown in Figure 9.32 for inputting multiple 8-bit
unsigned numbers until a zero is entered, and then output the number of numbers entered.
9.27. Derive and simulate the control words using the datapath shown in Figure 9.32 for inputting an 8-bit unsigned
number, and then output the number of 1 bits in the number. For example, the number 11010110 has five 1
bits.
9.28. Derive and simulate the control words using the datapath shown in Figure 9.32 for solving the Greatest
Common Divisor (GCD) algorithm shown in Figure P9.28.
WHILE (X ≠ Y){
IF (X < Y) THEN
Y = Y - X;
ELSE
X = X - Y; 324 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths END IF
}
Figure P9.28.
9.29. Design an 8-bit dedicated datapath for the algorithm in Figure P9.29, and write the control words for it. Use
only one adder-subtractor unit for all the addition and subtraction operations. Label clearly all of the control
and status signals.
w=0
x=0
y=0
INPUT z
WHILE (z ≠ 0) {
w=w–2
IF (z is an odd number) THEN
x=x+2
ELSE
y=y+1
END IF
z=z–1
}
Figure P9.29.
Answer:
input
zMux zLoad
yLoad
xLoad
wLoad
Clear
Clock
(z ≠ 0)
(z is odd) 1 8 0 D7-0
Load 8-bit register
Clear
z
Clock Q7-0 D7-0
Load 8-bit register
Clear
y
Clock Q7-0 D7-0
Load 8-bit register
Clear
x
Clock Q7-0 8 D7-0
Load 8-bit register
Clear
w
Clock Q7-0 8
8 8 z0 '1' '2'
3210 LOp1-0 1 s1-0 ROp
8 8 +/- Subtract 325 0 Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5
6 Chapter 9 - Datapaths Instruction zMux zLoad yLoad xLoad wLoad Clear LOp1-0 ROp Subtract w = 0, x = 0, y = 0
INPUT z
w=w–2
x=x+2
y=y+1
z=z–1 ×
1
×
×
×
0 0
1
0
0
0
1 0
0
0
0
1
0 0
0
0
1
0
0 0
0
1
0
0
0 1
0
0
0
0
0 ××
××
00
01
10
11 ×
××
0
0
1
1 ×
×
1
0
0
1 9.30. Design a dedicated datapath for inputting two 8-bit unsigned numbers, and then output the largest number.
The datapath should have only one input port and one output port. Label clearly all of the control and status
signals.
Answer:
Input
8 LoadX
LoadY
Clear
Clock
(X >Y ) D7-0
Load 8-bit register
Clear
X
Q7-0 D7-0
Load 8-bit register
Clear
Y
Q7-0 >
1 SelX 0 OutE
Output
9.31. Design and simulate a dedicated datapath for inputting three 8-bit unsigned numbers, and then output the
largest number, followed by the second largest.
9.32. Design and simulate a dedicated datapath for inputting ten 8-bit unsigned numbers, and then output the sum
of the numbers.
9.33. Design and simulate a dedicated datapath for inputting multiple 8-bit unsigned numbers until a zero is
entered, and then output the number of numbers entered.
9.34. Design and simulate a dedicated datapath for inputting an 8-bit unsigned number, and then output the number
of 1 bits in the number. For example, the number 11010110 has five 1 bits.
9.35. Design and simulate a dedicated datapath for solving the algorithm in Figure P9.35. Use only one adder (i.e.,
no adder-subtractor and no ALU) for all the arithmetic operations. Include the circuits for generating all of the
status signals. The datapath is 4-bits wide.
s1 = 0;
s2 = 0; 326 Digital Logic and Microprocessor Design with VHDL Chapter 9 - Datapaths FOR(i=0; i≠10; i++){
INPUT j;
IF (j is even) THEN
s1++;
ELSE
s2++;
END IF
}
OUTPUT s1;
OUTPUT s2;
Figure P9.35.
9.36. Design and simulate a dedicated datapath for solving the Greatest Common Denominator (GCD) algorithm
shown in Figure P9.36. The two numbers X and Y are 8-bit unsigned numbers. Include the circuits for
generating all the status signals.
WHILE (X ≠ Y){
IF (X < Y) THEN
Y = Y - X;
ELSE
X = X - Y;
END IF
}
Figure P9.36.
9.37. Design and simulate a dedicated datapath for implementing a stack of size 10. When the Push signal is
asserted, the input value is pushed onto the stack. When the Pop signal is asserted, the value at the top of the
stack is output. 327 Chapter 10 Control Units Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units In the last chapter, we saw how a datapath is designed and how it is used to execute a particular algorithm by
specifying the control words to manipulate the datapath at each clock cycle. In that chapter, we tested the datapath
by setting the control word signals manually during simulation. However, to actually have the datapath
automatically operate according to the control words, a control unit is needed to generate the control word signals
automatically at each clock cycle.
The control unit inside the microprocessor is a finite state machine. By stepping through a sequence of states,
the control unit controls the operations of the datapath. For each state that the control unit is in, the output logic that
is inside the control unit will generate all of the appropriate control signals for the datapath to perform one data
operation. These data operations are referred to as register-transfer operations. Each register-transfer operation
consists of reading a value from a register, modifying the value by one or more functional units, and finally, writing
the modified value back into the same or a different register.
Since the control unit is a finite state machine, the circuit for it is derived exactly as discussed in Section 7.3
having the next-state circuit, the state memory register, and the output logic circuit. However, the sequential circuits
synthesized in Section 7.3 are stand-alone FSMs and were not used as part of larger circuits. What we want to do
now is to synthesize these FSMs as control units for microprocessors. The main difference is in the way we define
and use the input and output signals of the FSM. The FSM models shown in Figure 7.1 simply show that the FSM
has input and output signals. For a control unit, we want to be more precise as to where the input signals come from,
where the output signals go to, and what these signals are used for.
There are two types of input signals and two types of output signals. The control inputs are the primary
external input signals to the control unit. These are the external signals for controlling the operation of the
microprocessor. For example, a Start signal will tell the microprocessor to start executing, or a Reset signal will
reset the state memory to the initialization state. The status signals are input signals from the datapath. Usually,
comparators in the datapath generate these status signals for determining execution branches in an algorithm (for
example, to test whether or not to repeat a loop). These two types of input signals together provide information for
the next-state logic circuit in the FSM to determine what next state to go to in the next clock cycle.
The control outputs are the primary output signals from the microprocessor to the external world. For example,
when a microprocessor is finished executing an algorithm, it outputs a Done signal to let the user know that it is
done, and that the data being output by the datapath is valid. The control signals are probably the most important of
all the input/output signals, since these are the signals that directly control the operations of the datapath. In every
clock cycle, the control unit will generate a different set of control signals for the datapath to perform one registertransfer operation.
One of the first things that we do when constructing an FSM circuit is to derive the state diagram. When
deriving the state diagram for the control unit, we have to be very careful with the timings of the register-transfer
operations. The issue here is that, when we write a value into a register, that value is not available until the
beginning of the next clock cycle. Hence, if we read from the register in the current clock cycle, we would be
reading the old value in the register rather than the new value that is being written into the register.
When we were designing general FSMs in Chapter 7, this timing issue was never a problem, because we were
not using the FSMs to control register-transfer operations in a datapath. They are stand-alone FSMs, and so their
input and output signals are independent of each other.
However, the FSMs that we want to design in this chapter are for controlling register-transfer operations in a
datapath. The output signals from these FSMs are control signals for the datapath, and some of them are used to load
registers with new values. These new register values may be used by comparators for the testing of conditions. The
results of these conditional tests are the status signals used by the control unit to determine what next state to go to.
Finally, from the different states, different control signals are generated. Hence, for a control unit, the status (input)
signals and the control (output) signals are dependent on each other. Thus, when status signals are generated, we
need to make sure that they are from tests of the intended register value.
Timing problem issues are resolved during the state diagram derivation step. Once we have the correct timing
state diagram, then deriving the actual control unit circuit is exactly the same as the process described in Chapter 7
for deriving an FSM. To help deal with timing issues, we sometimes prefer to use an algorithmic state machine
(ASM) chart or a state action table to describe the behavior of a control unit rather than using a state diagram. The
use of ASM charts and state action tables is discussed in Section 10.3. 329 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units 10.1 Constructing the Control Unit
In Chapter 9, we learned how to construct and use a datapath to implement an algorithm. All data manipulation
instructions in the algorithm are converted to control words, and each control word is executed in one clock cycle to
perform one register-transfer operation. A control unit is used to generate the appropriate control signals in the
control words so that the datapath can perform all of the required register-transfer operations automatically. These
control signals are the output signals from the output logic circuit that is inside the FSM.
In addition to generating the control signals, the control unit is also needed to control the sequencing of the
instructions in the algorithm. The datapath is responsible only for the manipulation of the data; it only performs the
register-transfer operations. It is the control unit that determines when each register-transfer operation is to be
executed and in what order. The sequencing done by the control unit is established during the derivation of the state
diagram.
The state diagram shows what register-transfer operation is executed in what state and the sequencing of the
execution of these operations. A state is created for each control word, and each state is executed in one clock cycle.
The edges in the state diagram are determined by the sequence in which the instructions in the algorithm are
executed. The sequential execution of instructions is represented by unconditional transitions between states (i.e.,
edges with no labels). Execution branches in the algorithm are represented by conditional transitions from a state
with two outgoing edges: one with the label for when the condition is true and the other with the label for when the
condition is false. If there is more than one condition, then all possible combinations of these conditions must be
labeled on the outgoing edges from every state. These conditions are the status signals generated by the datapath,
and passed to the next-state logic in the FSM.
Once the state diagram is derived, the actual construction of the control unit circuit follows the same procedure
as for the FSM discussed in Section 7.3. We will now illustrate the construction of simple control units with several
examples. 10.1.1 Counting 1 to 10
Example 10.1: Deriving the control unit for the counting problem
In this example, we will construct the state diagram and the control unit for controlling the dedicated datapath
from Example 9.4 to generate and output the numbers from 1 to 10. The algorithm, dedicated datapath, and control
words from Example 9.4 are repeated here in Figure 10.1 for convenience.
1
2
3
4
5 i=0
WHILE (i ≠ 10){
i=i+1
OUTPUT i
}
(a) 330 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units '1'
+ iLoad
Clear
Clock D3-0
Load 4-bit Register
Clear
i
Clock Q3-0 i3 (i ≠ 10) 4 i0
Out
Output
(b)
Control
Word
1
2
3 Instruction iLoad Clear Out i=0
i=i+1
OUTPUT i 0
1
0 1
0
0 0
0
1 (c)
Figure 10.1 The counting problem for Example 10.1: (a) algorithm; (b) dedicated datapath; (c) control words.
From the algorithm shown in Figure 10.1(a), we see that there are three data manipulation instructions: lines 1,
3, and 4. Line 2 is not a data manipulation statement, but rather, it is a control statement. From Example 9.4, we
have derived the three corresponding control words for these three data manipulation instructions for controlling the
dedicated datapath shown in Figure 10.1(b). These three control words are shown here in Figure 10.1(c).
We start by assigning these three control words to three separate states in the state diagram, as shown in Figure
10.2(a). The states are given the symbolic names s0, s1, s2, and s3 and are annotated with the control word and
instruction that each is assigned to execute. In state s0, we want the control unit to generate the control signals for
control word 1 to execute the instruction i = 0. In state s1, we want the control unit to generate the control signals for
control word 2 to execute the instruction i = i + 1. State s2 executes the instruction OUTPUT i. State s3 is added simply
for exiting the WHILE loop and halting the execution of the algorithm.
The sequence in which the states are connected follows the sequence of the instructions in the algorithm. The
FSM starts from the reset state s0, which initializes i to 0. After executing line 1 in the algorithm, the execution of
line 3 depends on the condition in the WHILE loop. Since line 1 is executed in state s0 and line 3 is executed in state
s1, therefore, transitioning from state s0 to s1 depends on the test condition (i ≠ 10). This condition is represented by
the two outgoing edges from state s0: one edge going to state s1 with the label (i ≠ 10) for when the condition is true,
and one edge going to state s3 with the label (i ≠ 10)' for when the condition is false. The execution of line 4 follows
immediately after line 3; hence, there is an unconditional edge from state s1 to s2. From state s2, there are the same
two conditional edges as from state s0 for testing whether to repeat the WHILE loop or not. If the condition (i ≠ 10) is
true, then the FSM will go back to state s1; otherwise, the FSM will exit the WHILE loop and go to state s3. The FSM
halts in state s3 by having an unconditional edge going back to itself. No register-transfer operation is assigned to
state s3.
Having derived the state diagram, the actual construction of the control unit circuit is exactly the same as for
constructing general FSM circuits. The next-state table is shown in Figure 10.2(b). Since there is a total of four
states, two flip-flops are needed to encode them. For simplicity, the straight binary encoding scheme is used for 331 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units encoding the states. Hence, state s0 is encoded as Q1Q0 = 00, state s1 is encoded as Q1Q0 = 01, and so on. In the nextstate table, these four states are assigned to four rows, each labeled with the state name and their encoding.
In addition to the four current states listed down the rows of the table, the next state of the FSM is also
dependent on the status signal for the test condition (i ≠ 10). Thus, we have the last two columns in the table: one
column with the label (i ≠ 10)' for when the condition is false and one column with the label (i ≠ 10) for when the
condition is true.
The two flip-flops and one status signal give us a total of three variables (or 23 different combinations) to
consider in the next-state table. Each next-state entry in the table is obtained from the state diagram by looking at the
corresponding current state and the edges leading out from that state to see what the next state is. For example,
looking at the state diagram shown in Figure 10.2(a), the edge with the label (i ≠ 10) leading out from state s0 goes
to state s1. Correspondingly, in the next-state table, the next-state entry at the intersection of row s0 (00) and the
column labeled (i ≠ 10) has the value s1 (01).
From the next-state table, we get the implementation table, as shown in Figure 10.2(c). Using D flip-flops to
implement the FSM, the implementation table is the same as the next-state table because the characteristic equation
for the D flip-flop is Qnext = D. The only difference between the two tables is that the bits in the entries mean
something different. In the next-state table, the bits in the entries (labeled Q1nextQ0next) are the next states for the FSM
to go to. In the implementation table, the bits (labeled D1D0) are the inputs necessary to realize those next states.
From the implementation table, we derive the excitation equations. The excitation equations are used to derive
the next-state circuit for generating the inputs to the state memory flip-flops. Since we have used two D flip-flops,
two excitation equations (one for D1, and one for D0) are needed, as shown in Figure 10.2(d). The two K-maps for
these two excitation equations are obtained from extracting the corresponding bits from the implementation table.
For example, the leftmost bit in each entry in the implementation table is for the D1 K-map and equation, and the
rightmost bit is for the D0 K-map and equation. These two excitation equations are dependent on the three variables,
Q1, Q0, and (i ≠ 10), which represent the current state and status signal, respectively. Having derived the excitation
equations, it is trivial to draw the next-state circuit based on these equations.
The output logic circuit for the FSM is derived from the control word signals and the states in which the control
words are assigned to. Recall that the control signals control the operation of the datapath, and now we are
constructing the control unit to control the datapath. So what the control unit needs to do is to generate and output
the appropriate control signals in each state to execute the instruction that is assigned to that state. In other words,
the control signals for controlling the operation of the datapath are simply the output signals from the output logic
circuit in the FSM.
To derive the output table, we take the control word table and replace all of the control word numbers with the
actual encoding of the state in which that control word is assigned to. For example, looking at the state diagram
shown in Figure 10.2(a), control word 1 is assigned to state s0. So in the output table, we put in the value 00 instead
of the control word number 1. The value 00 is the encoding that we have given to state s0, and it represents the
current state value for the two flip-flops, Q1, and Q0. Since there is no control word or instruction assigned to the
halting state, s3 (11), all of the control signals for this state can be de-asserted. The output table and the resulting
output equations are shown in Figure 10.2(e) and (f), respectively.
Once we have derived the excitation and output equations, we easily can draw the control unit circuit shown in
Figure 10.2(g). The state memory simply consists of the two D flip-flops with asynchronous clear signals. All of the
asynchronous clear signals are connected to the global Reset signal. Both the next-state logic circuit and the output
logic circuit are combinational circuits and are constructed from the excitation equations and output equations,
respectively.
♦ 332 Digital Logic and Microprocessor Design with VHDL Control Word 1
i=0 Chapter 10 - Control Units Current State
Q1Q0 s0 (i ≠ 10)
Control Word 2
i=i+1 (i ≠ 10)' s0 00
s1 01
s2 10
s3 11 s1 Next State
Q1next Q0next
(i ≠ 10)' (i ≠ 10)
s3 11
s1 01
s2 10
s2 10
s3 11
s1 01
s3 11
s3 11 (i ≠ 10) (b) Control Word 3
OUTPUT i s2
Current State
Q1Q0 (i ≠ 10)'
Halt 00
01
10
11 s3 Implementation
D1 D0
(i ≠ 10)' (i ≠ 10)
11
01
10
10
11
01
11
11
(c) (a) D0 D1 (i ≠ 10)
0
Q1Q0 Q1Q0 1 (i ≠ 10)
0 (i ≠ 10)' 00
01 1 1 11 1 01 1 10 00 Q0 1 1 1 1 1
Q0' 11 1 10 D1 = (i ≠ 10)' + Q0 1
1 1 D0 = Q1 + Q0'
(d) Q1Q0
00
01
10
11 iLoad
0
1
0
0 Clear
1
0
0
0 Out
0
0
1
0 iLoad = Q1'Q0
Clear = Q1'Q0'
Out = Q1Q0'
(f) (e) 333 Q1 Digital Logic and Microprocessor Design with VHDL Status Signal from
the Datapath Next-state
Logic Chapter 10 - Control Units State
Memory Output Logic and
Control Signals to the Datapath (i ≠ 10) D1 Q1 iLoad Clk
Clear Q'1 Clear
Out D0 Q0 Clk
Clock
Reset Clear Q'0 (g)
Figure 10.2 Construction of the control unit for Example 10.1: (a) state diagram; (b) next-state table; (c)
implementation table using D flip-flops; (d) K-maps and excitation equations; (e) output table; (f) output equations
for the three control signals; (g) circuit. 10.1.2 Simple IF-THEN-ELSE
We will now construct the state diagram and the control unit for controlling the dedicated datapath from
Example 9.3 for the simple IF-THEN-ELSE problem. The algorithm, dedicated datapath, and control words from
Example 9.3 are repeated here in Figure 10.3 for convenience. Example 10.2 shows the naive way of creating a state
diagram by simply assigning one state for each control word, just like in Example 10.1. However, we will see from
the example that the state diagram created this way for this problem is incorrect, because the status signal generated
is wrong. Example 10.3 shows the derivation of the corrected state diagram and the construction of the control unit
for the IF-THEN-ELSE problem. 334 Digital Logic and Microprocessor Design with VHDL 1
2
3
4
5
6
7 Chapter 10 - Control Units INPUT A
IF (A = 5) THEN
B=8
ELSE
B = 13
END IF
OUTPUT B
(a) Input '8' '13'
1 Muxsel 0 4 ALoad 4 D3-0
Load 4-bit Register
A
Clock Q3-0 BLoad D3-0
Load 4-bit Register
B
Clock Q3-0 Clock
(A = 5) A3 4 4 Out A0
Output
(b)
Control
Word
1
2
3
4 Instruction ALoad Muxsel BLoad Out INPUT A
B=8
B = 13
OUTPUT B 1
0
0
0 ×
1
0
× 0
1
1
0 0
0
0
1 (c)
Figure 10.3 The IF-THEN-ELSE problem for Example 10.2 and Example 10.3: (a) algorithm; (b) dedicated datapath;
(c) control words.
Example 10.2: Deriving an incorrect state diagram
In this example, we will derive the state diagram for the control unit for controlling the dedicated datapath from
Example 9.3 for solving the IF-THEN-ELSE problem. From the algorithm shown in Figure 10.3(a), we see that there
are four data manipulation instructions: lines 1, 3, 5, and 7. Line 2 is not a data manipulation instruction, but rather,
it is a control statement. From Example 9.3, we have derived the four corresponding control words for these four
data manipulation instructions for controlling the dedicated datapath shown in Figure 10.3(b). These four control
words are repeated here in Figure 10.3(c).
Again, we start by assigning these four control words to four separate states in the state diagram, as shown in
Figure 10.4(a). These four states are given the symbolic names s_input, s_equal, s_notequal, and s_output, and are
annotated with the control word and instruction that is assigned to them. For example, in state s_input, we want the
control unit to generate the control word 1 signals for executing the instruction INPUT A, and in state s_equal, we
want the control unit to generate the control word 2 signals for executing the instruction B = 8. 335 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units After the INPUT A instruction, the execution of the following two instructions, B = 8 and B = 13, depends on the
condition of the IF statement. Since these three instructions are assigned to the three states: s_input, s_equal, and
s_notequal, respectively, therefore, transitioning from state s_input to the other two states depends on the condition
(A = 5) in the IF statement. This conditional execution is represented by the two outgoing edges from state s_input:
one edge going to state s_equal with the label (A = 5) for when the condition is true, and one edge going to state
s_notequal with the label (A = 5)' for when the condition is false. The instruction, OUTPUT B, is executed
unconditionally after executing either of the two instructions, B = 8 or B = 13; therefore, from either state s_equal or
s_notequal, there is an unconditional edge going to state s_output. The algorithm halts after executing OUTPUT B, so
we make the FSM halt in state s_output by unconditionally going back to itself.
According to the algorithm, after inputting a value for A in state s_input, we need to test for the condition
(A = 5). If the condition is true, we go to state s_equal to execute the instruction B = 8; otherwise, we go to state
s_notequal to execute the instruction B = 13. From either state s_equal or s_notequal, the next and final state is
s_output. Let us assume that state s_input is executed in clock cycle 1. In clock cycle 2, either state s_equal or
s_notequal is executed. State s_output is then executed in clock cycle 3.
There are two important points to understand and remember here:
1. At a rising clock edge, a register is loaded with a new value if its load signal is asserted 2. At every rising clock edge, the FSM enters a new state—the next state As with all of our other examples, we use the rising clock edge as the active edge. The reason for point 1 is
because we are using positive edge-triggered D flip-flops with enable in our registers. So if the flip-flop is enabled,
then the input data will be stored into the flip-flop at the next rising clock edge. Point 2 is because we are also using
positive edge-triggered D flip-flops in the state memory register inside the FSM. However, these flip-flops are
always enabled without the need of an enable signal (see Section 6.7). Therefore, at every rising clock edge, a new
value from the next-state logic circuit will be stored into the state memory register, and so, the FSM enters a new
state at every rising clock edge.
If we construct the control unit based on the state diagram shown in Figure 10.4(a), then the following scenario
can occur. At the first rising clock edge, the FSM enters state s_input. Shortly after entering state s_input, the FSM
asserts the control signal ALoad to load in a value for variable A. Since a register is loaded at a rising clock edge,
and the first rising clock edge has passed, therefore, the value for A will be loaded into the register at the next rising
clock edge (i.e., at the beginning of clock cycle 2). However, the FSM needs to go to the next state (either s_equal or
s_notequal) also at the beginning of clock cycle 2. In order for the FSM to know which of the two states to go to, the
FSM must know the result of the test condition (A = 5) while it is still in state s_input. The dilemma here is that the
status signal generated by the comparator for the test (A = 5) is needed in state s_input, and the test uses the input
value of A. However, this value of A is not available until the beginning of the next clock cycle. Therefore, what the
comparator is reading from the register in clock cycle 1 is the old (or current) value of A and not the new input
value.
Figure 10.4(b) shows the timing diagram for this state diagram with the incorrect result. The user inputs a 5.
The diagram shows that the value 5 is loaded into register A at time 200 ns (the beginning of clock cycle 2). Since
the input is a 5, the test for (A = 5) should be true, and the output should be an 8. However, in the timing diagram,
we see that the state changes to s_notequal at time 200 ns. This is because the conditional test is reading A with the
old value of 0 rather than the new value of 5. Hence, the output of 13 is incorrect.
♦ 336 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units s_
input
(A = 5)' Control Word 1
INPUT A
(A = 5) Control Word 3 s_not
equal
B = 13 s_
Control Word 2
equal
B=8 s_
Control Word 4
output OUTPUT B (a) Clock Cycle 1 Clock Cycle 2 Clock Cycle 3 Clock Cycle 4 (b)
Figure 10.4 Example 10.2: (a) incorrect state diagram; (b) incorrect timing diagram.
You may be wondering why the state diagram for Example 10.1 is correct, whereas the state diagram for
Example 10.2 is incorrect. The first two lines of these two algorithms are very similar: The first line assigns a value
to a variable (one with the constant zero, and the other with an input), and the second line is a conditional test with
the value in the variable. So what causes the conditional test for Example 10.1 to be correct, but the conditional test
for Example 10.2 to be incorrect? The reason is because the timing of these two variable assignments is different. In
Example 10.1, the variable i is assigned a 0 by asserting the asynchronous Clear signal for the register. Because of
the asynchronous clear, i will get a 0 immediately and will not have to wait until the next clock cycle; hence, the
conditional test is correct. However, in Example 10.2, we use the ALoad signal to load in a value for the register.
Therefore, A will not get the new value until the beginning of the next clock cycle, but as the conditional test needs
the value in the current clock cycle, hence, the test is incorrect.
Example 10.3: Deriving a correct state diagram and the control unit
In Example 10.2, even though the comparator is testing for the condition (A = 5), it is not getting the correct
value for A in the clock cycle that the test result is needed. In order for the comparator to get the correct value for A,
it needs to wait until the value is loaded into the register at the next clock cycle. One simple way to resolve this
timing error is to add an extra state after inputting the value for A so that the value can be written into the register
before it is read back out for the test. There is no control word assigned to this extra state. This modified state
diagram is shown in Figure 10.5(a) and the corresponding timing diagram with the corrected result in (b). The same
input value of 5 is loaded into the register at time 200 ns at the beginning of clock cycle 2. However, the reading of
the register for the conditional test does not occur until shortly after time 200 ns. Hence, the test result is equal, and
the FSM goes to state s_equal and outputs the correct value of 8. 337 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units s_
input Control Word 1
INPUT A s_
extra
(A = 5)' (A = 5) Control Word 3 s_not
equal
B = 13 s_
Control Word 2
equal
B=8 s_
Control Word 4
output
OUTPUT B (a) Clock Cycle 1 Clock Cycle 2 Clock Cycle 3 Clock Cycle 4 (b)
Figure 10.5 Example 10.3: (a) correct state diagram; (b) correct timing diagram.
Now that we have derived the correct state diagram, the actual construction of the control unit circuit is exactly
the same as before. The next-state table is shown in Figure 10.6(b). Since there is a total of five states, three flipflops are needed to encode them. For example, state s_input is encoded as Q2Q1Q0 = 000, state s_extra is encoded as
Q2Q1Q0 = 001, and so on. The three remaining encodings (101, 110, and 111) are not used. In normal circumstances,
the control unit should never get to one of the unused states. However, due to noises or glitches in the circuit, the
FSM may end up in one of these unused states. Because of this, it is a good idea to set the next state for all of the
unused states to the reset state. On the other hand, if you don’t care what the next state is for these unused states,
then don’t-care values can be used, and this may result in simpler excitation equations. In our next-state table, we
have added three more rows for these three unused states. Their next state for all input conditions is the reset state,
000.
The corresponding implementation table (using D flip-flops) and the resulting excitation equations are shown in
Figure 10.6(c) and (d), respectively. Note that the excitation equations listed in Figure 10.6(d) are not the only
correct equations; there are other possible correct excitation equations. For instance, if don’t-care values are used for
the next states in the three unused states, then the excitation equations may be reduced further.
The output logic circuit for the FSM is based on the control word signals and the states in which the control
words are assigned to. There is one output equation for each control word signal, and these equations are dependent
only on the states of the FSM (i.e., the Q values of the flip-flops). To derive the truth tables for these output
equations, simply take the control word table and replace all of the control word numbers with the actual encoding 338 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units of the state (i.e., the Q values) in which that control word is assigned to. Having the output table, the output
equations can be derived easily, as shown in Figure 10.6(e) and (f).
Once we have derived the excitation and output equations, we can draw the control unit circuit, as shown in
Figure 10.6(g). The state memory simply consists of the three D flip-flops. Both the next-state logic circuit and the
output logic circuit are combinational circuits, and are constructed from the excitation equations and output
equations, respectively.
♦
s_
input Control Word 1
INPUT A s_
extra
(A = 5)' Current State
Q 2Q 1Q 0
(A = 5) Control Word 3 s_not
equal
B = 13 s_input 000
s_extra 001
s_notequal 010
s_equal 011
s_output 100
Unused 101
Unused 110
Unused 111 s_
Control Word 2
equal
B=8 s_
Control Word 4
output
OUTPUT B Next State
Q2next Q1next Q0next
(A = 5)'
(A = 5)
s_extra 001
s_extra 001
s_notequal 010
s_equal 011
s_output 100
s_output 100
s_output 100
s_output 100
s_output 100
s_output 100
000
000
000
000
000
000 (b) (a)
Current State
Q2Q1Q0
000
001
010
011
100
101
110
111 Next State
D2D1D0
(A = 5)' (A = 5)
001
001
010
011
100
100
100
100
100
100
000
000
000
000
000
000 D2 = Q2'Q1 + Q2Q1'Q0'
D1 = Q2'Q1'Q0
D0 = Q2'Q1'Q0' + Q2'Q1' (A = 5) (d) (c)
Q2Q1Q0
000
001
010
011
100
101
110
111 Instruction
INPUT A
No operation
B=8
B = 13
OUTPUT B
No operation
No operation
No operation ALoad
1
0
0
0
0
0
0
0 Muxsel
×
×
1
0
×
×
×
× BLoad
0
0
1
1
0
0
0
0 Out
0
0
0
0
1
0
0
0 ALoad
Muxsel
BLoad
Out = Q2'Q1'Q0'
= Q2'Q1Q0'
= Q2'Q1
= Q2Q1'Q0' (f) (e) 339 Digital Logic and Microprocessor Design with VHDL Next-state
Logic Chapter 10 - Control Units State
Memory D2 Output Logic and
Control Signals to the Datapath Q2 ALoad Clk
Clear D1 Q'2 Muxsel
Q1 BLoad Clk
Clear Q'1 Out
D0 Q0 Clk Clock
Reset Clear Q'0 ( A = 5)
Status Signal from
the Datapath (g)
Figure 10.6 Construction of the control unit for Example 10.3: (a) state diagram; (b) next-state table; (c)
implementation table using D flip-flops; (d) excitation equations; (e) output table; (f) output equations for the four
control signals; (g) circuit. 10.2 Generating Status Signals
In Chapter 9, we discussed how status signals are generated and provided several examples showing how it is
done. In situations where a register value is to be tested, we have connected a comparator to the output of that
register. However, there are situations when connecting a comparator to the output of a register will produce the
wrong status signal because it is comparing with an incorrect value. In Examples 10.2 and 10.3, we saw that, if we
read from a register that has just been written to in the same clock cycle, we will be reading the old value and not the
new value. The solution we used in Example 10.3 was simply to insert an extra state to wait for the value to be
written into the register and thus be made available at the output of the register.
What makes the problem even more difficult (but interesting) is that, during different clock cycles, different
values may pass through the same point in the datapath. For example, the output of a 2-to-1 multiplexer can have
two different values during two different times, because in one clock cycle, the select line for the multiplexer might
be asserted, while in another clock cycle, the select line might be de-asserted—thus passing two different sources to
the output. Another datapath component that can output different values at different times is the register file. By
providing different read addresses to the register file, the register file will output values from different locations.
Thus, we see that, in order to provide the correct value for the comparator to test, we have to look at not only the 340 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units location in the datapath where the value is stored but also the clock cycle in which the value is needed by the
comparator. This timing information is obtained from the state diagram. The state diagram tells us in what state a
status signal is needed, and when the FSM is in that state, what data manipulation instruction is being performed,
and, therefore, what data is available at different points in the datapath. We will now illustrate this with two
examples.
Example 10.4: Generating a correct status signal by connecting the comparator to a different point in the datapath
In Example 10.2, the state diagram shown in Figure 10.4(a) derived for the datapath shown in Figure 10.3(b) is
wrong, because in state s_input (when the status signal for the comparison is needed), the comparator is not getting
the correct input value. The disadvantage to the solution used in Example 10.3 is that an extra clock cycle is needed
for the execution of the algorithm because of the extra state.
Another way to resolve this timing problem is to connect the comparator to a different point in the datapath
instead of adding the extra state to the state diagram. The state diagram in Figure 10.4(a) shows that we need to use
the input value right away in state s_input for the comparison (A = 5). The datapath shown in Figure 10.3(b) has the
comparator connected to the output of register A. Because of this, we need an extra clock cycle to wait until the
input value has been written into the register before it is available to be tested.
Instead of connecting the comparator to the output of the register, we can connect the comparator directly to the
primary input (i.e., to the input of the register), as shown in Figure 10.7. This way, we do not have to wait for the
input value to be written into the register first, but instead, the input value will be available immediately in state
s_input. As a result, the comparator will have the correct value for performing the test in the right clock cycle, and
we will not have to add an extra state to the state diagram.
♦
'8' '13' Input
A3 (A = 5 ) A0
ALoad Muxsel
4 1 0
4 D3-0
Load 4-bit Register
A
Clock Q3-0 BLoad D3-0
Load 4-bit Register
B
Clock Q3-0 Clock
4 4 Out
Output Figure 10.7 Generating the correct status signal for Example 10.4.
Example 10.5: Summation of n down to 1
We will now derive the state diagram and generate the correct status signal for the general datapath from
Example 9.10 for summing the numbers from n down to 1, where n is an input number. The algorithm, general
datapath, and control words from Example 9.10 are repeated here in Figure 10.8. Figure 10.8(c) shows the five
control words for controlling the general datapath shown in Figure 10.8(b) for performing the algorithm shown in
Figure 10.8(a).
Figure 10.9(a) shows a first attempt to derive the state diagram for the algorithm in Figure 10.8(a). The five data
manipulation instructions are assigned to five states. On reset, the FSM starts from state s0, where it initializes sum
to 0 and waits for the Start signal. The Start signal is for telling the control unit when the data input is ready and to
begin the execution of the algorithm. (The Start signal is similar in function to the Enter key found on personal
computer keyboards.) The remaining edges in the state diagram follow the execution sequence in the algorithm.
State s1 executes control word 2 to input the number for n. State s2 executes control word 3 to sum n. State s3
executes control word 4 to decrement n. Finally, state s4 executes control word 5 to output the value for sum. 341 Digital Logic and Microprocessor Design with VHDL 1
2
3
4
5
6
7 Chapter 10 - Control Units sum = 0
INPUT n
WHILE (n ≠ 0){
sum = sum + n
n=n-1
}
OUTPUT sum
(a)
Data Input
Point A
15 IE 1 0 Point B
WE
WA 4 x 8
RAE RF RBE
RAA
RBA 14
13-12
11
10-9
Clock
Point C
8
7-6 A B A
B
ALU2
ALU
ALU1
ALU0 5
4
3
Point D SH1
Shifter
SH0 2
1
Point E
0 OE Data Output
(b)
Control
Word
1
2
3
4
5 Instruction
sum = 0
INPUT n
sum = sum + n
n=n–1
OUTPUT sum IE
15
0
1
0
0
× WE
14
1
1
1
1
0 WA1,0
13–12
00
01
00
01
×× RAE
11
1
0
1
1
1 RAA1,0
10–9
00
××
00
01
00 RBE
8
1
0
1
0
0 RBA1,0
7–6
00
××
01
××
×× ALU2,1,0
5–3
101 (subtract)
×××
100 (add)
111 (decrement)
000 (pass) SH1,0
2–1
00
××
00
00
00 OE
0
0
0
0
0
1 (c)
Figure 10.8 The summation problem for Example 10.5: (a) algorithm; (b) general datapath; (c) control words.
In state s1, we input a value for n and immediately test for the condition (n ≠ 0). Similarly, in state s3, we
perform the instruction n = n – 1 and immediately test for the condition (n ≠ 0). However, we remember from
Example 10.3 that we cannot test for a condition that involves a value that is being written into a register in the same 342 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units clock cycle. If we try to solve this timing problem like we did in Example 10.3, we would get the state diagram
shown in Figure 10.9(b), where an extra wait state, s_wait, is added. On the other hand, Example 10.4 shows that
sometimes it is not necessary to add an extra wait state if we can find a point in the datapath to connect the
comparator so that the correct status signal can be generated.
Start'
s0 Start Start'
Control Word 1
sum = 0 s0 Control Word 1
sum = 0 s1 Control Word 2
INPUT n Start
(n ≠ 0 ) ' Control Word 2
INPUT n s1 (n ≠ 0 ) ' (n ≠ 0 ) s_
wait (n ≠ 0 ) s2 (n ≠ 0 ) Control Word 3
sum = sum + n s2 Control Word 3
sum = sum + n s3 Control Word 4
n=n−1 s3 Control Word 4
n=n−1 s4 Control Word 5
OUTPUT sum (n ≠ 0 )'
Control Word 5
OUTPUT sum s4 (a) (b) Figure 10.9 Two possible state diagrams for solving the summation problem for Example 10.5: (a) without an extra
wait state; (b) with an extra wait state.
In terms of optimization, the state diagram in Figure 10.9(a) would be a better choice if the correct status signal
can be generated because it requires one less state. So the problem is to find a location in the datapath to connect the
comparator so that it will output the correct status signal for the test (n ≠ 0) during the clock cycle when it is needed.
The state diagram in Figure 10.9(a) shows that the status signal (n ≠ 0) is needed in states s1 and s3. Therefore, we
need to find a point in the datapath such that the correct value for n is available during the times when states s1 and
s3 are being executed.
In state s3, the instruction n = n – 1 is being executed. In order for the datapath to execute n = n – 1, the value
for n must be present at the A operand input of the ALU (point C in Figure 10.8(b)), and the result for the decrement
n – 1 will be available at the output of the ALU (point D in Figure 10.8(b)) during the same clock cycle. According
to the state diagram in Figure 10.9(a), we want to test for n after the decrement. If we connect the comparator to the
A operand input of the ALU at point C, we would be comparing the value of n before the decrement during this
clock cycle. Furthermore, the value for n is not even available in the next clock cycle at point C. The reason is that
either state s2 or s4 will be executed in the next clock cycle, and for both of these states, the value for sum is needed.
So in the next clock cycle, the value for sum, and not n, is available at point C. Since the result of the decrement is
available at point D in the current clock cycle, the comparator should be connected to the output of the ALU at point
D. 343 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units The problem now is that we also need n in state s1. In state s1, the instruction INPUT n is executed, and the value
for n is being stored in the register file. Looking at Figure 10.8(b), we see that, in the clock cycle when the datapath
executes the instruction INPUT n, n is available only at the primary data input of the datapath at point A and at the
output of the multiplexer at point B. Since the writing of n to the register file occurs at the next rising clock edge, n
is not going to be available at the output of the register file in that same clock cycle, let alone at the output of the
ALU. So in order for the status signal (n ≠ 0) to be correct in state s1, we need to connect the comparator to either
the primary data input at point A or to the output of the multiplexer at point B.
For state s3, we need to connect the comparator to point D, but for state s1, we need to connect the comparator to
either point A or point B. However, in order to generate the correct status signal for both states, the comparator must
be connected to a point that is correct for both cases. Notice that the execution of the instruction n = n – 1 in state s3
requires the output of the ALU at point D be routed back to the output of the multiplexer at point B within the same
clock cycle, so that the decremented value of n can be written into the register file at the next rising clock edge. This
is possible because both the shifter and the multiplexer are combinational circuits. So, this new decremented value
of n that we want for the comparison in state s3 is not only available at point D, but also available at points E and B.
We can therefore conclude that connecting the comparator to the output of the multiplexer at point B will generate
the correct status signal for the comparison (n ≠ 0) in both states, s1 and s3, as shown in Figure 10.10.
One last point to note is that, since the comparator is a combinational circuit, it constantly is outputting a value.
When the FSM is in a state other than s1 or s3, the comparator would be comparing another value with 0. For
instance, in state s2, the ALU is evaluating sum + n, so the comparator would be comparing the result of sum + n
with 0. However, this will not affect the correct operation of the control unit, because in state s2, the next-state logic
does not use this status signal to determine what the next state is. Therefore, even though the comparator is
generating an incorrect status signal in state s2, the control unit will not use it. When the status signal from the output
of this comparator is needed in state s1 or s3, it will be the result of a correct comparison.
Having derived the correct state diagram shown in Figure 10.9(a) and the correct status signal from the datapath
shown in Figure 10.10, we can now finish the construction of the control unit circuit, as shown in Figure 10.11.
From the state diagram, we can derive the next-state table, as shown in Figure 10.11(b). Three flip-flops, Q2, Q1, and
Q0, are needed to encode the five states. In addition to the current states listed down the rows of the table, the next
state of the FSM also is dependent on the control input signal Start and the status signal (n ≠ 0), from which we get
the four columns. The labels for the four columns are the four combinations of the two input signals Start and
(n ≠ 0). For example, the column labeled 10 means that the input signal Start is true (i.e., Start = 1) and the status
signal (n ≠ 0) is false (i.e., (n ≠ 0) = 0). The derivation of the excitation equations (Figure 10.11(d)), output
equations (Figure 10.11(f)), and finally, the complete control unit circuit (Figure 10.11(g)) follows the same
procedure as for synthesizing a general FSM. The Done control output signal is to notify the external world that the
execution of the algorithm has been completed and that the data at the Data Output is valid.
♦ 344 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units Data Input
Status Signal to the
Control Unit
15 (n ≠ 0) IE 1 0 8
WE
WA 4 x 8
RAE RF RBE
RAA
RBA 14
13-12
11
10-9
Clock A B 8
7-6
5
4
3 A
B
ALU2
ALU
ALU1
ALU0 2
1 SH1
Shifter
SH0 0 OE Data Output
Figure 10.10 Generating the status signal (n ≠ 0) in the datapath for Example 10.5. 345 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units Current State
Q2Q1Q0 Start'
Control Word 1
sum = 0 s0 Start
(n ≠ 0 ) ' Control Word 2
INPUT n s1 (n ≠ 0 )
Control Word 3
sum = sum + n s2 (n ≠ 0 ) 00
s0 000
s4 100
s3 011
s4 100
s4 100
s0 000
s0 000
s0 000 s0 000
s1 001
s2 010
s3 011
s4 100
Unused 101
Unused 110
Unused 111 Implementation
D2 D1 D0
Start, (n ≠ 0)
00
01
10
11
000 000 001 001
100 010 100 010
011 011 011 011
100 010 100 010
100 100 100 100
000 000 000 000
000 000 000 000
000 000 000 000 Current State
Q 2Q 1Q 0
000
001
010
011
100
101
110
111 (n ≠ 0 )'
Control Word 5
OUTPUT sum s4 (c) (a)
D2 Q2 = 0
Start, (n ≠ 0)
Q1Q0
00
01
11 D1 Q2 = 1
10 00 01 11 1 1 Q2 = 0
Start, (n ≠ 0)
01
Q1Q0
11
00 10 1 00 1 Q2 = 1
10 00 01 00 01 1 1 01 1 1 11 1 1 11 1 1 1 1 10 10
Q2'Q0(n ≠ 0)' 11
s1 001
s2 010
s3 011
s2 010
s4 100
s0 000
s0 000
s0 000 (b) Control Word 4
n=n−1 s3 Next State
Q2next Q1next Q0next
Start, (n ≠ 0)
01
10
s0 000 s1 001
s2 010 s4 100
s3 011 s3 011
s2 010 s4 100
s4 100 s4 100
s0 000 s0 000
s0 000 s0 000
s0 000 s0 000 Q2Q1'Q0' 1 Q2'Q1Q0' D2 = Q2Q1'Q0' + Q2'Q0(n ≠ 0)' 1 Q2'Q0(n ≠ 0) D1 = Q2'Q1Q0' + Q2'Q0(n ≠ 0) 346 11 10 Digital Logic and Microprocessor Design with VHDL D0 Q2 = 0
Start, (n ≠ 0)
Q1Q0
00
01
11 10 1 1 1 Chapter 10 - Control Units 1 00 Q2 = 1
00 01 11 10 01
11
10 1 1 Q2'Q1Q0' Q2'Q0'Start D0 = Q2'Q1Q0' + Q2'Q0'Start
(d)
State
Q 2Q 1Q 0
000
001
010
011
100
101
110
111 IE WE WA1 WA0 RAE RAA1 RAA0 RBE RBA1 RBA0 ALU2 ALU1 ALU0 SH1 SH0 OE 15
0
1
0
0
×
0
0
0 14
1
1
1
1
0
0
0
0 13
0
0
0
0
×
0
0
0 12
0
1
0
1
×
0
0
0 11
1
0
1
1
1
0
0
0 10
0
×
0
0
0
0
0
0 9
0
×
0
1
0
0
0
0 8
1
0
1
0
0
0
0
0 7
0
×
0
×
×
0
0
0 6
0
×
1
×
×
0
0
0 5
1
×
1
1
0
0
0
0 4
0
×
0
1
0
0
0
0 3
1
×
0
1
0
0
0
0 2
0
×
0
0
0
0
0
0 1
0
×
0
0
0
0
0
0 0
0
0
0
0
1
0
0
0 (e)
IE
WE
WA1
WA0
RAE
RAA1
RAA0
RBE = Q2'Q1'Q0
= Q2'
=0
= Q2'Q0
= Q2'Q1 + Q1'Q0'
=0
= Q2'Q1Q0
= Q2'Q0' RBA1
RBA0
ALU2
ALU1
ALU0
S H1
S H0
OE =0
= Q2'Q1Q0'
= Q2'Q0' + Q2'Q1
= Q2'Q1Q0
= Q2'Q1'Q0' + Q2'Q1Q0
=0
=0
= Q2Q1'Q0' (f) 347 Digital Logic and Microprocessor Design with VHDL Next-state
Logic Start Chapter 10 - Control Units State
Memory Output Logic and
Control Signals to the Datapath Control
Input RBE IE
D2 Q2 0 Clk RBA1 ALU2 WA0 ALU1
ALU0 Q1 Clk
Clear D0 Q'1 Q0 0 Clk Clock
Reset WA1 Q'2 0
D1 RBA0 RAE Clear WE Clear 0 RAA1 Q'0 RAA0 SH1
SH0
OE (n ≠ 0 )
Status Signal from
the Datapath Done
Control
Output (g)
Figure 10.11 Construction of the control unit for Example 10.5: (a) state diagram; (b) next-state table; (c)
implementation table; (d) K-maps and excitation equations; (e) output table; (f) output equations; (g) circuit. 10.3 Stand-Alone Controllers
So far in the discussion, the control unit is used as a controller for the datapath. However, the control unit can be
independent of the datapath and used as a stand-alone controller for controlling an external device. There are many
electronic devices that use a stand-alone control unit for controlling them (for example, the controllers for a
microwave oven, a soda dispensing machine, a traffic light signal, and a pacemaker, to name just a few). All of these
devices do not require any calculations or data manipulations, so they do not require the use of a datapath. The
controllers in them are used just to turn on or off a switch or a light, or whatever external devices they are made to
control.
For a traffic light controller, there are sets of three lights (green, yellow, and red) to turn on or off and sensors in
the road to see if there are cars in one of the four directions. For a soda dispensing machine, there will be switches to
sense the amount of money put in and the selection made. From these inputs, the controller will determine what
switch to turn on to allow the selected soda to fall out.
We will now illustrate the construction of stand-alone controllers with several examples. 10.3.1 Rotating Lights
Example 10.6: Stand-alone controller for rotating lights 348 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units For a simple, interesting example, let us implement a controller that will make the six peripheral segments in a
7-segement LED light move around in a clockwise or counterclockwise direction, depending on the input of the
switch W.
Since there are six light patterns, one for each LED being on and the rest of them off, we can use six states and
assign one light pattern to one state. Each state will go to the next state in a sequential clockwise direction if the
switch W is pressed and to the previous state in a sequential counterclockwise direction if W is not pressed. The
resulting state diagram is similar to the one for a modulo-6 up-down counter and is shown in Figure 10.12(a). The
six output signals (a, b, c, d, e, and f) correspond to the six peripheral segments in a 7-segment LED. A 1 to the
segment will turn on that LED, while a 0 will turn it off.
The next-state table derived from the state diagram and the corresponding implementation table, using D flipflops, are shown in Figure 10.12(b). The resulting excitation equations (as derived from the implementation table)
are shown in Figure 10.12(c). The output table and the resulting output equations are shown in Figure 10.12(d) and
(e), respectively. The final controller circuit is shown in Figure 10.12(f). To see the lights move, the six output
signals are connected to the six segments of the 7-segment LED, and a push-button switch is connected to the W
input, as shown in Figure 10.12(g).
♦
W
fedcba = 000001 s0 s1
W' fedcba = 000010
W W'
W
W'
fedcba = 100000 s5 s2
W' W W' fedcba = 000100 W W' fedcba = 010000 s4 s3 fedcba = 001000 W
(a)
Current State
Q 2Q 1 Q 0
000
001
010
011
100
101 Next State (Implementation)
Q2next Q1next Q0next (D2 D1 D0)
W'
W
101
001
000
010
001
011
010
100
011
101
100
000
(b) 349 Digital Logic and Microprocessor Design with VHDL D2
Q1Q0 Chapter 10 - Control Units WQ2
00
00 01 11 1 10 1 01 1 11 1 10 D2 = W'Q2'Q1'Q0' + WQ2Q1'Q0' + W'Q2Q1'Q0 + WQ2'Q1Q0
D1
Q1Q0 WQ2
00 01 00 11 10 1 01
11 1
1 10 1 D1 = W'Q2Q1'Q0' + WQ2'Q1'Q0 + W'Q2'Q1Q0 + WQ2'Q1Q0'
D0
Q1Q0 WQ2
00
00 1 01 11 10 1 1 1 01 Q2'Q0' 11
10 Q1'Q0' 1 1 D0 = Q1'Q0' + Q2'Q0'
(c)
Current State
Q2Q1Q0
000
001
010
011
100
101 Output
fedcba
000001
000010
000100
001000
010000
100000
(d) 350 Digital Logic and Microprocessor Design with VHDL f
e
d
c
b
a Chapter 10 - Control Units = Q2Q1'Q0
= Q2Q1'Q0'
= Q2'Q1Q0
= Q2'Q1Q0'
= Q2'Q1'Q0
= Q2'Q1'Q0'
(e) W f
D2 Q2 a e Clk D1 d Q'2 D0 b e c b Q1 Clk
Clear f c Clear d a
Q'1 Q0 Clk
Clear Clock
Reset Q'0 (f)
Common
Vcc
for All
Segments Vcc W
W b
a a f f b e e c Controller
Clock
Reset Clock
Reset d
c d (g)
Figure 10.12 Controller for Example 10.6: (a) state diagram; (b) next-state (implementation) table; (c) K-maps and
excitation equations; (d) output table; (e) output equations; (f) controller circuit; (g) interface circuit. 351 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units 10.3.2 PS/2 Keyboard Controller
Example 10.7: Controller for a PS/2 keyboard
In this next example, we will design and implement a more realistic and practical controller, and that is for the
PS/2 keyboard. The UP2 development board has a 6-pin mini-DIN connector for connecting either a PS/2 keyboard
or a PS/2 mouse to it. We will design and implement the controller for the keyboard on the FLEX chip. The
operation of the keyboard and the controller is as follows.
The communication between the keyboard and the controller (which will be on the FLEX chip) uses two
signals, KeyboardClock and KeyboardData. When there is no activity, that is, when there is no key press on the
keyboard, both KeyboardClock and KeyboardData are at a 1. When a key is pressed (or released), the keyboard
sends a unique code for that key to the controller serially over the KeyboardData line. The serial data on the
KeyboardData line is synchronized between the keyboard and the controller by clock pulses that the keyboard sends
over the KeyboardClock line.
The data for each key that is sent over the KeyboardData line consists of eleven bits. These eleven bits are: a 0
for the start bit, eight data bits for the key code starting with the least significant bit to the most significant bit, an
odd parity bit, and lastly, a 1 for a stop bit. Figure 10.13 lists some of the key codes generated by the keyboard when
the corresponding key is pressed. When a key is released, a different code is generated. The odd parity bit is set such
that the total number of 1 bits in the eight data bits plus the parity bit is an odd number.
Figure 10.14 shows a sample timing diagram for the data transmission of the key code 4E (01001110 in binary)
for the hyphen key. Starting from the inactive state, where both the KeyboardData and KeyboardClock lines are at a
1, the transmission begins by setting the KeyboardData line to a 0 for the start bit. The keyboard then sends out the
data and parity bit on the KeyboardData line at a rate of one bit per clock cycle on the KeyboardClock line. The
clock pulses on the KeyboardClock line are generated by the keyboard. The parity bit for the key code 4E is a 1,
since the eight data bits consist of an even number of 1 bits, therefore, to make the parity odd, the parity bit must be
a 1.
Key
1
2
3
4
5
6
7
8
9
0 Key Code
16
1E
26
25
2E
36
3D
3E
46
45 Key
A
B
C
D
E
F
G
H
I
J Key Code
1C
32
21
23
24
2B
34
33
43
3B Key
K
L
M
N
O
P
Q
R
S
T Key Code
42
4B
3A
31
44
4D
15
2B
1B
2C Key
U
V
W
X
Y
Z
Esc
BS
CR
Ctrl Key Code
3C
2A
1D
22
35
1A
76
66
5A
14 Figure 10.13 A partial list of key codes generated by the keyboard.
start d0 d1 d2 d3 d4 d5 d6 d7 parity stop 0 0 1 1 1 0 0 1 0 1 1 KeyboardClock
KeyboardData Figure 10.14 Sample timing diagram for the data transmission of the key code 4E.
The state diagram for our keyboard controller shown in Figure 10.15(a) is derived by following the timing
diagram shown in Figure 10.14. In each of the eight data states, d0, d1, …, d7, we will get one corresponding data
bit from the KeyboardData input line. For example, suppose we use an 8-bit register named Keycode for storing the 352 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units eight data bits. Then in state d0, we will assign KeyboardData to Keycode0 (i.e., the 0th bit of Keycode), and in state
d1, we will assign KeyboardData to Keycode1, and so on for all eight data bits. This is possible because the
transition of the FSM from one state to another is synchronized by the keyboard clock signal KeyboardClock. For
simplicity, we will not check for the start bit, parity bit, nor the stop bit.
The next-state table and the implementation table using four D flip-flops to encode the eleven states are both
shown in Figure 10.15(b). The excitation equations derived from the implementation table are shown in Figure
10.15(c).
This controller circuit actually does not control the keyboard because it does not generate control signals for the
operation of the keyboard. Instead, it receives the serial data signals from the keyboard, and packages it into data
bytes. The output of this controller is simply the data bytes, which represents the key code of the keys being pressed
on the keyboard. The output table is shown in Figure 10.15(d) and the corresponding output equations in (e). In state
d0, the bit on the KeyboardData line is loaded into bit 0 of the Keycode register; in state d1, the bit on the
KeyboardData line is loaded into bit 1 of the Keycode register; and so on. Each bit of the Keycode register must,
therefore, be able to load in the KeyboardData independently, and each load enable line is asserted by the
corresponding state encoding. Hence, each of the output equations shown in Figure 10.15(e) is not implemented
simply as a 5-input AND gate. Instead, a D flip-flop with enable is used for each output signal. The D input to the
flip-flop is connected to the KeyboardData line. The load enable, E, on the flip-flop is asserted by ANDing the four
Q values for that state.
Using the excitation equations for the next-state circuit, four D flip-flops for the state memory, and the output
equations for the output circuit, we obtain the complete controller circuit, as shown in Figure 10.15(f). The
implementation of the controller circuit using the FLEX chip is shown in Figure 10.15(g) and is available on the
accompanying CD-ROM in the file UP2FLEX.GDF.
In practice, controllers are usually not design manually like what we have done. The purpose of this exercise is
only to provide an understanding of how controllers are designed. Section 10.5 shows how controllers can be
automatically synthesized using behavioral VHDL code.
♦ start d0 stop d1 parity d7 d6 d2 d3 d4 d5 (a) 353 Digital Logic and Microprocessor Design with VHDL Current State
Q3Q2Q1Q0
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010 Chapter 10 - Control Units Next State (Implementation)
Q3next Q2next Q1next Q0next (D3 D2 D1 D0) start
d0
d1
d2
d3
d4
d5
d6
d7
parity
stop 0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
0000
(b)
D3
Q1Q0 Q3Q2
00 01 11 10 00 1 01 1 11 1 10 D3 = Q3Q2'Q1' + Q3'Q2Q1Q0
D2
Q1Q0 Q3Q2
00 01 00
01 11 10 1
1 11 1 10 1 D2 = Q3'Q2Q1' + Q3'Q2Q0' + Q3'Q2'Q1Q0
D1
Q1Q0 Q3Q2
00 01 11 10 00
01 1 1 1 1 1 11
10 D1 = Q3'Q1'Q0 + Q2'Q1'Q0 + Q3'Q1Q0' 354 Digital Logic and Microprocessor Design with VHDL D0
Q1Q0 Chapter 10 - Control Units Q3Q2
00
00 01 1 1 1 11 10
1 1 01
11
10 D0 = Q3'Q0' + Q2'Q1'Q0'
(c)
Current State
Q3Q2Q1Q0
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010 start
d0
d1
d2
d3
d4
d5
d6
d7
parity
stop Output
Keycode7-0
– – – – – – – KeyboardData
– – – – – – KeyboardData –
– – – – – KeyboardData – –
– – – – KeyboardData – – –
– – – KeyboardData – – – –
– – KeyboardData – – – – –
– KeyboardData – – – – – –
KeyboardData – – – – – – – Where a dash (–) means no change to that Keycode bit.
(d)
Keycode7 = Q3Q2'Q1'Q0' KeyboardData
Keycode6 = Q3'Q2Q1Q0 KeyboardData
Keycode5 = Q3'Q2Q1Q0' KeyboardData
Keycode4 = Q3'Q2Q1'Q0 KeyboardData
Keycode3 = Q3'Q2Q1'Q0' KeyboardData
Keycode2 = Q3'Q2'Q1Q0 KeyboardData
Keycode1 = Q3'Q2'Q1Q0' KeyboardData
Keycode0 = Q3'Q2'Q1'Q0 KeyboardData
(e) 355 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units KeyboardData D
Q
E Clk
D3 D
Q
E Clk Clear D2 D
Q
E Clk D
Q
E Clk Keycode1 D
Q
E Clk Keycode0 Q'1 Q0 Clk
Clear Keycode2 D
Q
E Clk Clk D0 Keycode3 Q'2 Q1 Clear Keycode4 D
Q
E Clk Q2 D1 Keycode5 Q'3 Clk
Clear Keycode6 D
Q
E Clk Q3 Clk Keycode7 Q'0 KeyboardClock
Reset
(f) PS/2 port
KeyboardData
KeyboardClock BCD
KeyboardData
KeyboardClock a
f Keyboard
Controller
PB1
Reset e Keycode7-4
Reset g Keycode3-0 bf g ce
d 4 a
b
c
d BCD 4 (g)
Figure 10.15 Controller for Example 10.7: (a) state diagram; (b) next-state (implementation) table; (c) K-maps and
excitation equations; (d) output table; (e) output equations; (f) controller circuit; (g) interface circuit. 10.3.3 VGA Monitor Controller
Most people will think that it is way beyond the beginner’s level to design a VGA monitor controller. However,
to the contrary, a simple VGA monitor controller can be constructed using a couple of standard sequential 356 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units components. In Example 10.8, we will design and implement a VGA monitor controller using an FSM for the
control unit. However, it turns out that the VGA monitor controller is so simple that it can be constructed without
the need of a FSM. In Example 10.9, we will show how the VGA controller can be implemented without the FSM.
The UP2 development board has a 15-pin D-sub connector for connecting a standard VGA monitor to it. The signals
from the D-sub connector are connected directly to the FLEX FPGA chip on the board, so we will implement our
controller circuits on the FLEX chip. In order to design the VGA controller, we need to first understand how the
VGA monitor works.
The monitor screen for a standard VGA format contains 640 columns × 480 rows of picture elements called
pixels, as shown in Figure 10.16. An image is displayed on the screen by turning on or off individual pixels.
Although turning on just one pixel doesn’t represent much, when many pixels are turned on, the combined pixels
portray an image. The monitor continuously scans through the entire screen turning on or off one pixel at a time at a
very fast speed. Although only one pixel is turned on at any one time, but since the monitor is scanning through the
screen so fast, you get the impression that all of the pixels are on at the same time.
Figure 10.16 shows that the scanning starts from row 0, column 0 at the top left corner and moves to the right
until it reaches the last column in the row. When the scan reaches the end of a row, it retraces to the beginning of the
next row. When the scan reaches the last pixel at the bottom-right corner of the screen, it retraces back to the top-left
corner of the screen and repeats the scanning process again. In order to reduce flicker on the screen, the entire screen
must be scanned 60 times per second or higher. During the horizontal and the vertical retraces, all of the pixels are
turned off.
640 Pixels per Row
Column 0 Column 639 Row 0 Horizontal
Retrace Horizontal
Scan
480 Pixels
per Column VGA
Monitor
Screen
Vertical
Retrace Row 479 Figure 10.16 VGA monitor with 640 columns × 480 rows. The scan starts from row 0, column 0 and moves to the
right and down until row 479, column 639.
The VGA monitor is controlled by five signals: red, green, blue, horizontal synchronization, and vertical
synchronization. The three color signals, referred to collectively as the RGB signal, are used to control the color of a
pixel at a given location on the screen. These three color signals on the UP2 board are connected such that they can
individually be turned on or off, hence each pixel can display only one of eight colors. In order to produce more
colors, each analog color signal must be supplied with a voltage between 0 to 0.7 volts for varying the intensities of
the colors. The horizontal and vertical synchronization signals are used to control the timing of the scan rate. The
horizontal synchronization signal determines the time to scan a row, while the vertical synchronization signal
determines the time to scan the entire screen. By manipulating these five signals, images are formed on the monitor
screen.
The horizontal and vertical synchronization signals timing diagram is shown in Figure 10.17. When inactive,
both synchronization signals are at a 1. The start of a row scan begins with the horizontal synchronization signal
going low for 3.77 µsec, as shown by region B in Figure 10.17. This is followed by a 1.79 µsec high on the signal, 357 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units as shown by region C. Next, the data for the three color signals are sent (one pixel at a time) for the 640 columns, as
shown in region D for 25.42 µsec. Finally, after the last column pixel, there is another 0.79 µsec of inactivity on the
RGB signal lines, as shown in region E, before the horizontal synchronization signal goes low again for the next row
scan. The total time to complete one row scan is 31.77 µsec.
The timing for the vertical synchronization signal is analogous to the horizontal synchronization signal. The 64
µsec active-low vertical synchronization signal resets the scan to the top-left corner of the screen, as shown in region
P, followed by a 1020 µsec high on the signal as shown by region Q. Next, there are the 480 row scans of 31.77
µsec each, giving a total of 15250 µsec, as shown in region R. Finally, after the last row scan, there is another 450
µsec, as shown in region S, before the vertical synchronization signal goes low again to start another complete
screen scan starting at the top-left corner. The total time to complete one complete scan of the screen is 16784 µsec.
In order to get the monitor to operate properly, we simply have to get the horizontal and vertical
synchronization signals timing correct, and then send out the RGB data for each pixel at the right column and row
position. It turns out that it is fairly simple to get the correct timing for the two synchronization signals. The built-in
clock crystal on the UP2 board has a frequency of 25.175 MHz. The clock period is then 1 / 25.175 × 106, which is
about 0.0397 µsec per clock cycle. For region B in the horizontal synchronization signal, we need 3.77 µsec, which
is approximately 3.77 / 0.0397 = 95 clock cycles. For region C, we need 1.79 µsec, which is approximately 45 clock
cycles. Similarly, we need 640 clock cycles for region D for the 640 columns of pixels and 20 clock cycles for
region E. The total number of clock cycles needed for each row scan is, therefore, 800 clock cycles. Notice that with
a 25.175 MHz clock, region D requires exactly 640 cycles, giving us the 640 columns per row. Hence, a different
clock speed will produce a different screen resolution.
The vertical timings are multiples of the horizontal cycles. For example, region P is 64 µsec, which is
approximately two horizontal cycles (2 × 31.77). The calculation for region R is 15250 µsec / 31.77 µsec = 480. Of
course, it has to be exactly 480 times, since we need to have 480 rows per screen.
In addition to generating the correct horizontal and vertical synchronization signals, the circuit needs to keep
track of the current column within the D region and the current row within the R region of the scan in order to know
when to turn on or off a specific pixel. To make a particular pixel green for example, you need to test the values of
the column and row counts. If they are equal to the location of the pixel that you want to turn on, then you assert the
green signal, and that pixel will be green.
Red, Green, Blue
640 Column Pixels Horizontal Synchronization
Signal
Time and Number of
25.175 MHz Clock Cycles B
3.77 µs
95 cycles C
1.79 µs
45 cycles D
25.42 µs
640 cycles E
0.79 µs
20 cycles 31.77 µs
800 cycles Red, Green, Blue
480 Horizontal Cycles Vertical Synchronization
Signal
Time and Number of
Horizontal Cycles P
64 µs
2 cycles Q
1020 µs
32 cycles R
15250 µs
480 cycles
16784 µs
528 cycles Figure 10.17 Horizontal and vertical synchronization signals timing diagram.
Example 10.8: A VGA controller—version 1 358 S
450 µs
14 cycles Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units To get the horizontal and vertical synchronization timing correct, we can design an FSM with 800 states
running at a clock speed of 25.175 MHz. For the first 95 states, we will output a 0 for the horizontal synchronization
signal H_Sync. For the next 45 + 640 + 20 = 705 states, we will output a 1 for H_Sync. The problem with this,
however, is that it is difficult to manually derive the circuit for an 800-state FSM. A simple solution around this
difficulty is to use just two states: one for when H_Sync is a 0 in region B and one for when it is a 1 in regions C, D,
and E. We will then use a counter that runs at the same clock speed as the FSM to keep count of how many times we
have been in a state. For the first state, we will stay there for 95 counts before going to the next state, and for the
second state, we will stay there for 705 counts before going back to the first state. In the first state, we will output a
0 for H_Sync, and in the second state, we will output a 1 for H_Sync.
To help keep track of when the three color signals can be enabled, we will generate an additional H_Data_on
signal the same way we generated the H_Sync signal. The H_Data_on signal is asserted in region D, and de-asserted
in regions B, C, and E. Thus, for 640 counts of repeating in one state for region D, we will set H_Data_on to a 1 and
to a 0 in a second state for the remaining 160 counts for regions B, C, and E.
Combining the two states for H_Sync and two states for H_Data_on together results in the final state diagram
for the horizontal timing, as shown in Figure 10.18(a). This FSM has four states corresponding to the four regions,
B, C, D, and E. The counter initially is set to 0 and increments by 1 at every clock cycle. In state H_B for region B,
the FSM outputs a 0 for both H_Data_on and H_Sync. The FSM will stay in state H_B for 95 counts. The condition
(H_cnt = B) checks to see whether the counter is equal to B, where B is equal to 95. When the count is equal to 95,
the FSM goes to state H_C, which corresponds to region C. In state H_C, the FSM outputs a 0 for H_Data_on and a
1 for H_Sync for 45 counts (i.e., until H_cnt is B + C = 95 + 45 = 140). When H_cnt reaches 140, the FSM goes to
state H_D, and outputs a 1 for both H_Data_on and H_Sync. When H_cnt reaches B + C + D = 95 + 45 + 640 = 780,
the FSM goes to state H_E, and outputs a 0 for H_Data_on and a 1 for H_Sync. The FSM stays in state H_E for 20
more counts until H_cnt = B + C + D + E = 95 + 45 + 640 + 20 = 800, and then it goes back to state H_B. When the
FSM goes back to state H_B, H_cnt is reset back to 0, and the process starts all over again for the next row scan.
The vertical synchronization timing is analogous to the horizontal synchronization timing, so we can do the
same thing using a second counter and a second FSM. This second vertical FSM is identical to the horizontal FSM.
The only difference is in the timing. Looking at the times for each region in the vertical synchronization signal in
Figure 10.17, we see that the 64 µsec for region P is approximately two times the total horizontal scan time of 31.77
µsec each. The 1020 µsec for region Q is approximately 32 horizontal scan time (1020 / 31.77 ≈ 32). For region R, it
is 480 horizontal cycles, and for region S, it is approximately 14 horizontal cycles. Hence, the clock for both the
vertical counter and the vertical FSM can be derived from the horizontal counter. The vertical clock ticks once for
every 800 counts of the horizontal clock.
The next-state tables for the horizontal FSM and the vertical FSM are shown in Figure 10.18(c) and (d),
respectively. Notice that besides the four count conditions, the two tables are the same. Hence, we can use the same
circuit for both FSMs. The only difference is that their status signal inputs for the four count conditions come from
different counter comparators. The output table is also the same for both FSMs and is shown in Figure 10.18(e).
There are only two output signals to be generated: H_Data_on and H_Sync for the horizontal FSM, and V_Data_on
and V_Sync for the vertical FSM. Finally, the FSM circuit is shown in Figure 10.18(f).
We will need to use two instances of this FSM circuit: one for the horizontal FSM and one for the vertical FSM.
The clock for the horizontal FSM is the 25.175 MHz clock, while the clock for the vertical FSM is derived from the
rollover signal from the horizontal counter. The four status signals for the four counter conditions are generated
from two counters: a horizontal counter and a vertical counter. 359 Digital Logic and Microprocessor Design with VHDL H_Data_on = 0
H_Sync = 0 H_Data_on = 0
H_Sync = 1 H_Data_on = 1
H_Sync = 1 H_Data_on = 0
H_Sync = 1
(H_ H_E (H_ H_D c nt
+C
+D +D +D + E) +C ) +C )'
+C ' +D =B +C c nt =B (H_ =B =B =B = B) )
+C c nt =B (H_ c nt c nt = B) c nt (H_ c nt (H_ c nt (H_ H_C (H_ H_B Chapter 10 - Control Units + E) )' ' (a) V_Data_on = 0
V_Sync = 0 V_Data_on = 1
V_Sync = 1 V_Data_on = 0
V_Sync = 1
V_S =P
+Q nt = +R ' R+ + S) Q+ R) + R) Q )' +Q P+ Q+ ) P+ P )' =P +Q P+ (V_c nt = cn t
(V _ =P nt = nt = P) (V_c nt = (V_c cn t
(V _ V_R (V_c V_Q cn t
(V _ (V_c V_P V_Data_on = 0
V_Sync = 1 S) ' (b) Current
State
Q1Q0
00
01
10
11 H_B
H_C
H_D
H_E (H_cnt = B)
0
1
00
01 Next State (Implementation)
Q1next Q0next (D1 D0)
(H_cnt = B + C)
(H_cnt = B + C + D)
0
1
0
1 01
10 10
11 (H_cnt = B + C + D + E)
0
1 11
00 D1 = Q1'Q0 (H_cnt = B + C) + Q1Q0' + Q1Q0 (H_cnt = B + C + D + E)'
D0 = Q1'Q0' (H_cnt = B) + Q1'Q0 (H_cnt = B + C)' + Q1Q0' (H_cnt = B + C + D) + Q1Q0 (H_cnt = B + C + D + E)'
(c) 360 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units Next State (Implementation)
Q1next Q0next (D1 D0) Current
State
Q1Q0
00 V_P
01 V_Q
10 V_R
11 V_S 0(V_cnt = P)
0
01 0
1 (V_cnt = P + )
Q 01
10 0
1 (V_cnt = P + Q + R) 10
11
0
1 (V_cnt = P + Q + R + S) 0
1
11
00 D1 = Q1'Q0 (V_cnt = P + Q) + Q1Q0' + Q1Q0 (V_cnt = P + Q + R + S)'
D0 = Q1'Q0' (V_cnt = P) + Q1'Q0 (V_cnt = P + Q)' + Q1Q0' (V_cnt = P + Q + R) + Q1Q0 (V_cnt = P + Q + R + S)'
(d) Current State
Q1Q0 H_Data_on
or
V_Data_on H_Sync
or
V_Sync 00
01
10
11 0
0
1
0 0
1
1
1 BCDEorPQRS BCDorPQR BCorPQ BorP (e) H_Sync or V_Sync
D1 Q1 Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset
(f) 361 Q'0 H_Data_on or V_Data_on Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units "0000000000"
'1'
Clear
Clock Count
D9-0
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock 95 = 0001011111 10 (H_cnt = B) 140 = 0010001100 (H_cnt = B + C) 780 = 1100001100 (H_cnt = B + C + D) 800 = 1100100000 (H_cnt = B + C + D + E)
Rollover (g)
"0000000000"
'1'
Clear
Clock Count
D9-0
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock
10 2 = 0000000010
34 = 0000100010 (V_cnt = P)
(V_cnt = P + Q) 514 = 1000000010 (V_cnt = P + Q + R) 528 = 1000010000 (V_cnt = P + Q + R + S) (h) 362 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units HCount
Horizontal FSM
H_Sync_out BorP
H_Sync
BCorPQ H_Data_on
BCDorPQR
BCDEorPQRS H_Data_on Clock
Clear (H_cnt = B)
(H_cnt = B + C)
(H_cnt = B + C + D)
(H_cnt = B + C + D + E)
Rollover Clock
Reset 25.175MHzClock
VCount Vertical FSM
BorP
V_Sync
BCorPQ V_Data_on
BCDorPQR
BCDEorPQRS
Clock
Reset Reset V_Sync_out V_Data_on Clock
Clear (V_cnt = P)
(V_cnt = P + Q)
(V_cnt = P + Q + R)
(V_cnt = P + Q + R + S) Red_out Red Green_out Green Blue_out Blue
"0000000000" Column
D9-0
Count
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock Column_out
"0000000000" 10 Row
D9-0
Count
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock Row_out
10 (i)
Figure 10.18 Controller for the VGA monitor: (a) state diagram for horizontal synchronization; (b) state diagram for
vertical synchronization; (c) next-state table for horizontal synchronization; (d) next-state table for vertical
synchronization; (e) output table; (f) FSM circuit for both the horizontal and vertical synchronization; (g) horizontal
synchronization counter; (h) vertical synchronization counter; (i) complete circuit for the VGA controller.
The horizontal counter, HCount, with the four comparators for (H_cnt = B), (H_cnt = B + C), (H_cnt = B + C +
D), and (H_cnt = B + C + D + E) is shown in Figure 10.18(g). A 10-bit counter is needed for counting from 0 up to
800. A 10-input AND gate is used for each of the four comparators. The inputs to each AND gate is set to the
equivalent binary value for B = 95, B + C = 95 + 45 = 140, B + C + D = 95 + 45 + 640 = 780, and B + C + D + E =
95 + 45 + 640 + 20 = 800, respectively. The counter cycles back to 0 by asserting the Load line when the count
reaches 800 and loading in the value 0. The output from the comparator (H_cnt = 800) is the counter rollover signal
Roll_over, and is used as the vertical clock signal for the vertical counter, the vertical FSM, and the row counter.
The vertical counter, VCount, with the four comparators for (V_cnt = P), (V_cnt = P + Q), (V_cnt = P + Q + R),
and (V_cnt = P + Q + R + S), is shown in Figure 10.18(h). The circuit for this counter is just like the horizontal
counter, except that the comparator values are different, and we do not need to output a rollover clock signal. The
clock for this counter is the vertical clock signal Rollover from the horizontal counter.
The complete VGA monitor controller circuit is shown in Figure 10.18(i). To make sure that the three RGB
signals to the monitor are valid, they have to be turned on (if needed) only in regions D and R. Hence, the three color 363 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units output signals, Red_out, Green_out, and Blue_out, are ANDed with H_Data_on and V_Data_on. For example, if the
input Red signal is a 1, the output Red_out signal is a 1 only when the scan is within the regions D and R.
Finally, in order to turn on a specific pixel, the circuit needs to keep track of the current column within the D
region and the current row within the R region of the scan. Two additional counters, Column and Row, are used for
this purpose. Since they need to count only when the scan is in regions D and R, respectively, therefore, the Count
input for the Column counter is asserted by the H_Data_on signal, while the Count input for the Row counter is
asserted by the V_Data_on signal. Once the counts reach 640 and 480, respectively, they will have gone passed
these two regions, and so the counters will not count. They will have to be reset to 0 anytime before the scan reaches
the beginning of these two regions again. In the circuit, the two Load lines are asserted when the two respective
counters roll back to 0. Finally, the Column counter clock is from the 25.175 MHz source, and the Row counter
clock is from the vertical clock source Rollover derived from the horizontal counter.
To display something on the screen, you simply have to check the current column and row that the scan is at,
and then assert the RGB signal if you want the pixel at that location to be turned on. For example, if you simply
assert the Red signal continuously, all of the pixels will be red, and you will see the entire screen being red. On the
other hand, if you just want the first row of pixels to be red, then you need to assert the Red signal only when the
Row counter is 0. To get a red border around the screen, you would assert the Red signal when Row = 0, or Row =
639, or Column = 0, or Column = 479. Figure 10.19 shows the circuit to draw a red border around the entire screen
using the VGA controller circuit from Figure 10.18(i).
♦
Reset
25.175 MHz Clock
'0'
'0' Reset
V_Sync_out
25.175MHzClock
H_Sync_out
Red
Red_out
VGA Monitor
Green
Green_out
Controller
Blue
Blue_out
Column_out9-0
Row_out9-0 Vert_Sync (pin 14)
Horiz_Sync (pin 13)
Red (pin 1)
Green (pin 2)
Blue (pin 3) To
VGA Monitor
Connector Row = 0
Row = 479
= 0111011111
Column = 0
Column = 639
= 1001111111
Figure 10.19 Circuit using the VGA controller to generate a red border around the entire screen.
Example 10.9: A VGA controller—version 2
If you understood the construction of the VGA controller in Example 10.8, you probably will have noticed that
the controller circuit can be made much simpler. There are two reductions that we can make: (1) eliminate the
column and row counters, and (2) eliminate the horizontal and vertical FSMs.
First, the column counter is always 140 (regions B + C = 95 + 45 = 140) plus the horizontal counter when the
H_Data_on signal is asserted, and the row counter is always 34 (regions P + Q = 2 + 32 = 34) plus the vertical
counter when the V_Data_on signal is asserted. Therefore, if we shift the horizontal count so that count 0 is at the
beginning of region D, and the last count 800 is at the end of region C, then we do not need the extra column
counter. Similarly, we can get rid of the row counter by shifting the vertical count so that count 0 is at the beginning
of region R, and the last count 528 is at the end of region Q. 364 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units To eliminate the two FSMs, notice that the FSMs are used simply to assert the two pairs of signals: H_Sync,
H_Data_on and V_Sync, V_Data_on. The assertion of these four signals is dependent only on the horizontal and
vertical counters. Therefore, we can use just the horizontal and vertical counts to assert these signals directly.
However, we need to keep these signals unchanged until after a certain count. An SR flip-flop can be used for each
signal to set the signal to a 0 or a 1 at the right time. The SR flip-flop works just like the SR latch, except that the
output changes only at the active (rising) edge of the clock. When S is asserted, the flip-flop is set, and the output Q
= 1. When R is asserted, the flip-flop is reset, and the output Q = 0. For example, the H_Sync_out signal is a 1 when
the count is at 95 and onward, and it is a 0 when the count is at 800, wraps back to 0, and onward. Hence, we
connect the output of the comparator for (H_cnt = 95) to the S input of an SR flip-flop and the output of another
comparator for (H_cnt = 800) to the R input. The output Q of the flip-flop is the H_Sync_out signal. As a result, the
signal is set to a 1 when the count reaches 95, and it will remain at a 1 until it is reset to a 0 at count 800. The other
three signals: H_Data_on, V_Sync_out, and V_Data_on are done similarly.
The complete version 2 of the VGA monitor controller circuit is shown in Figure 10.20. The modified
horizontal counter, HCount, and the vertical counter, VCount, are shown in Figure 10.20(a) and (b) respectively.
Notice that the four condition outputs for each of the two counters are different because we have shifted the counts
so that the counting starts at the beginning of region D and R, for the horizontal and vertical counters respectively.
The complete monitor circuit is shown in Figure 10.20(c).
We can implement and test out this second version of the monitor controller by using the same circuit shown in
Figure 10.19, but with the controller replaced by this new version.
♦ "0000000000"
'1'
Clear
Clock Count
D9-0
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock
10 640 = 1010000000 (H_cnt = D) 660 = 1010010100 (H_cnt = D + E) 755 = 1011110011 (H_cnt = D + E + B) 800 = 1100100000 (H_cnt = D + E + B + C)
Rollover
Column_out 10 (a) 365 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units "0000000000"
'1'
Clear
Clock Count
D9-0
Load 10-bit Up Counter
with Load
Clear
Q9-0
Clock 480 = 0111100000 (V_cnt = R) 494 = 0111101110 (V_cnt = R + S) 496 = 0111110000 (V_cnt = R + S + P) 528 = 1000010000 10 (V_cnt = R + S + P + Q) Row_out
10 (b)
HCount
S
Clock
Clear (H_cnt = D)
(H_cnt = D + E)
(H_cnt = D + E + B)
(H_cnt = D + E + B + C)
Rollover
Q H_Sync_out Q Clk
R
S 25 MHz Clock Q H_data_on 9-0 Clear Clk
R VCount Clear S
Clock
Clear (V_cnt = R)
(V_cnt = R + S)
(V_cnt = R + S + P)
(V_cnt = R + S + P + Q) V_Sync_out Q Clk
R
S Q Clk
R Reset
Red Clear V_data_on Q9-0 Clear Red_out
Green_out Green Blue_out Blue Column_out
10 Row_out
10 (c)
Figure 10.20 Second version of the VGA monitor controller: (a) horizontal counter HCount; (b) vertical counter 366 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units VCount; (c) complete controller circuit. 10.4 * ASM Charts and State Action Tables
The drawback to using a state diagram to describe the behavior of a sequential circuit is that it does not portray
precisely the timing information, which is very important to the correct operation of the circuit. Algorithmic state
machine (ASM) charts and state action tables are two alternate methods for describing sequential circuits more
accurately and concisely. The state action table uses a tabular format, while the ASM chart uses a graphical
flowcharting format. 10.4.1 ASM Charts
Algorithmic state machine (ASM) charts are used to graphically portray the operations of a FSM more
accurately. They are similar to flowcharts used in computer programming but use different shaped boxes, as shown
in Figure 10.21, to describe a sequence of actions. The ASM chart portrays similar information as that in the state
diagram. However, in addition to just describing a sequence of actions as in a state diagram, ASM charts also
describe the timing relationships between the states.
The rectangle shown in Figure 10.21(a) is the state box for representing a state in an FSM. Each state box,
therefore, is executed in one clock cycle and is similar to a node in a state diagram. A state in an FSM is for
performing data manipulation and input/output actions. Each state box contains unconditional (Moore type) registertransfer operations that are to be performed in that state. There is one path going into the state box and one path
leading out of the state box. Outside the rectangle at the top-left corner is labeled with the symbolic state name,
while the top-right corner is labeled with the binary state encoding (if known). For example, Figure 10.21(b) shows
a state box with the register-transfer operation count = 0. The symbolic name given to the state is s2, and the binary
state encoding given to it is 010. When the FSM is in state s2, the register for storing the variable count is reset to 0.
When a register-transfer operation is not specified for a specific register in a state box, it is assumed that that
register content remains unchanged. A register changes its value in a state only when a register-transfer operation
writes to that register in that state box.
The diamond-shaped box shown in Figure 10.21(c) is the decision box. The purpose of the decision box is to
test whether the given condition written in the box is true or false. There is one path going into the decision box and
two paths leading out of the box. The two paths coming out of the decision box are labeled 1 and 0. If the given
condition is true, then the path labeled 1 is taken; otherwise, the path labeled 0 is taken. The decision box by itself
does not represent a state, and no actions are written inside the box. Instead, it is used to determine the next state to
go to. This is like the condition that is labeled on the edges in the state diagram.
The decision box is also used to describe a Mealy FSM where actions are performed depending on a condition
of an input. In this case, the decision box is used in conjunction with the condition box (explained next), for
performing conditional actions.
The oval-shaped condition box shown in Figure 10.21(d) is also used for performing register-transfer
operations like the state box. However, unlike the state box, the condition box is used for conditional data
manipulations in Mealy FSMs. The actual testing of the condition is not done within the condition box but rather in
a decision box. The condition box by itself is not equivalent to one state. It must be used together with a decision
box within an ASM block.
The ASM block shown in Figure 10.21(e), allows a state box and zero or more decision and condition boxes to
be grouped together to form one state. All of the actions specified inside an ASM block are executed within one
state or one clock cycle. Like the state box, the ASM block is labeled with the state name at the outside top-left
corner and the state encoding at the top-right corner. The ASM block must start with one state box containing zero
or more unconditional register-transfer operations. After the state box, there can be zero or more decision and
condition boxes. The ASM block will have one entry point, and one or more exit paths leading to other states.
Figure 10.21(f) shows an example of an ASM block. The symbolic state name is s0 using the binary encoding
00. When the FSM enters this state, the variable sum is initialized to 0 and the Start signal is tested. If there is no
Start signal, that is, Start = 0, the FSM re-enters this state at the next clock cycle. If there is a Start signal, an input is
performed to load the variable n. The actual writing of the registers sum and n with the new values occurs at the next 367 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units active clock edge. In other words, the register sum is not zeroed until the next clock cycle, and the input value for n
is not available until the next clock cycle.
State
Name State
Encoding
Unconditional
Variable Assignment
and Output 010 s2
count = 0 (a) (b) Condition Conditional
Variable Assignment
and Output 1 0 (c)
State
Name (d)
State
Encoding 00 s0
sum = 0
0 Start 1 INPUT n (e) (f) Figure 10.21 Symbols used in an ASM chart: (a) state box; (b) example of a state box; (c) decision box; (d)
condition box; (e) ASM block; (f) example of an ASM block.
Example 10.10: Moore ASM chart
In this example, we will derive an ASM chart for a Moore type FSM based on the algorithm and datapath
shown in Figure 10.3(a) and (b) for the IF-THEN-ELSE problem. Line 1 in the algorithm is put in a state box with the
label s_input and encoding 000, as shown in Figure 10.22(a). A decision box is used for the conditional test in line 2
and must be put inside an ASM block. The test is dependent on the value from state s_input; therefore, it cannot be
performed in the same state with line 1 but is assigned to state s_extra. In state s_extra, there is no action that can be
performed unconditionally during this clock cycle; hence, the initial state box is empty. If we put both lines 1 and 2
in the same ASM block, as in Figure 10.22(b), then the timing will be wrong because we are again trying to read
from the register for a value that has not been updated yet.
For a Moore FSM, actions are performed unconditionally; therefore, lines 3 and 5 in the algorithm are put into
two separate state boxes. The two exit paths from the decision box go to these two state boxes: one for when the
condition is true and the other for when the condition is false. Finally, both states, s_notequal and s_equal, go to
state s_output to do the output. If we do the output in states s_notequal and s_equal, as shown in Figure 10.22(c),
then again, the output will be wrong because we will be reading from register B before it is updated. If we want to 368 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units do the output together with the assignment of B in the same state, then what we can do is output the constant value
that is to be written into B rather than outputting the value from B.
♦
s_input 000 s_extra 001 INPUT A
INPUT A
s_extra 001 0 0 s_notequal A=5 1 1
(b)
s_equal 010 A=5 B = 13 011 B=8 s_notequal s_equal 010 B = 13
OUTPUT B
s_output 011 B=8
OUTPUT B 100 OUTPUT B
(a) (c) Figure 10.22 Moore ASM chart for Example 10.10: (a) Correct ASM chart; (b) Wrong ASM block; (c) Wrong
output.
Example 10.11: Mealy ASM chart
Let us derive the ASM chart for the same IF-THEN-ELSE algorithm and datapath used in Example 10.10, except
for a Mealy FSM. The resulting ASM chart is shown in Figure 10.23. The main difference between this and the one
in the previous example shown in Figure 10.22(a) is that the conditional assignments (lines 3 and 5 in the algorithm)
are performed in the same state as the test in s_extra. The two conditional assignments are put in two separate oval
condition boxes. The paths going to them are the same as before. Since the two condition boxes are put in the same
ASM block, the ASM block now has only one exit path to state s_output to perform the output.
The advantage of this Mealy machine is not only the fact that it requires fewer states, thus resulting in a smaller
control unit, but it also requires fewer clock cycles to execute the algorithm.
♦ 369 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units s_input 00 INPUT A
s_extra 01 0 A=5 1 B = 13 B=8 s_output 10 OUTPUT B Figure 10.23 Mealy ASM chart for Example 10.11. 10.4.2 State Action Tables
The state action table is another way to describe the behavior of a sequential circuit. Whereas the ASM chart is
like the state diagram, the state action table is similar to the next-state and output tables combined together. It has
three columns: current state, next state, and datapath actions. The next state and datapath action entries can be with
or without a condition. However, instead of enumerating all possible conditions and listing them out in columns, as
with the next-state table, the condition is written in front of the next-state entries whenever it is needed. If no
condition is written in front of a next-state entry, then it is an unconditional next state (i.e., the FSM will go to this
next state unconditionally). If a next state is qualified with a condition, then the FSM will go to that state only if the
condition is true. A branch in the state diagram corresponds to a next state with a condition, so the entry for that next
state will have a condition.
The datapath actions are like the control word table information. Just like the next states, these datapath actions
can be either conditional or unconditional. If an action is not qualified with a condition in front, then it is an
unconditional action. Otherwise, it is a conditional action, and the action is performed only if the condition is true.
All datapath actions for a Moore FSM will not have a condition, since Moore datapath actions are only dependent on
the current state. A Mealy FSM will have conditional actions for some of its datapath action entries.
Figure 10.24 shows the Moore-type state action table for the corresponding ASM chart from Example 10.10.
From current state s_input, the FSM unconditionally performs the action INPUT A, and then unconditionally goes to
state s_extra. In state s_extra, the FSM will go to either state s_notequal or s_equal, depending on the condition
(A = 5). No action is performed in this state. In state s_notequal, the action B = 13 is performed, and then the FSM
unconditionally transitions to state s_output. In state s_equal, the action B = 8 is performed, and then the FSM
unconditionally transitions to state s_output. In state s_output, the FSM performs the action OUTPUT B and halts in
that state by unconditionally going back to itself. All of the actions in this state action table are performed
unconditionally; therefore, it is for a Moore type FSM. When the FSM enters a state, it always performs the
unconditional action(s) assigned to that state first, and then determines what the next state should be. This is
equivalent to the ASM block where the unconditional state box is always written before a decision box, if any.
Figure 10.25 shows the Mealy-type state action table for the corresponding ASM chart from Example 10.10. In
current state s_extra, either one of the two conditional actions is performed, depending on the result of the
conditional test. After performing the action, the FSM goes to state s_output unconditionally. 370 Digital Logic and Microprocessor Design with VHDL Current
State
s_input
s_extra
s_notequal
s_equal
s_output Chapter 10 - Control Units Next State
Condition, State
s_extra
(A = 5) s_equal
(A = 5)' s_notequal
s_output
s_output
s_output Datapath Actions
Condition, Action
INPUT A B = 13
B=8
OUTPUT B Figure 10.24 A sample Moore-type state action table.
Current
State
s_input
s_extra Next State
Condition, State
s_extra
s_output s_output s_output Datapath Actions
Condition, Action
INPUT A
(A = 5) B = 8
(A = 5)' B = 13
OUTPUT B Figure 10.25 A sample Mealy-type state action table. 10.5 VHDL for Control Units
Figure 10.26 shows the behavioral VHDL code for the PS/2 keyboard controller from Example 10.7. The
format follows exactly the VHDL code for Moore FSMs discussed in Section 7.6. Notice how closely the code for
the FSM process follows the state diagram. Because of its simplicity, control units usually are synthesized this way
rather than constructed manually, as in Example 10.7. For this example, the output process does not generate output
signals to control the keyboard; rather, it reads in data signals from the keyboard and packages the eight data bits as
a byte for output.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY KeyboardCtrl IS PORT (
Reset: IN STD_LOGIC;
KeyboardClock: IN STD_LOGIC;
KeyboardData: IN STD_LOGIC;
keycode: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END KeyboardCtrl ;
ARCHITECTURE Structural OF KeyboardCtrl IS
TYPE state_type IS (
s_start,s_d0,s_d1,s_d2,s_d3,s_d4,s_d5,s_d6,s_d7,s_parity,s_stop);
SIGNAL state: state_type;
BEGIN
FSM: PROCESS(KeyboardClock, Reset)
BEGIN
IF (Reset = '1') THEN
state <= s_start;
-- this FSM is driven by the keyboard clock signal
ELSIF (KeyboardClock'EVENT AND KeyboardClock = '1') THEN
CASE state is
WHEN s_start =>
state <= s_d0; 371 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units WHEN s_d0 =>
state <= s_d1;
WHEN s_d1 =>
state <= s_d2;
WHEN s_d2 =>
state <= s_d3;
WHEN s_d3 =>
state <= s_d4;
WHEN s_d4 =>
state <= s_d5;
WHEN s_d5 =>
state <= s_d6;
WHEN s_d6 =>
state <= s_d7;
WHEN s_d7 =>
state <= s_parity;
WHEN s_parity =>
state <= s_stop;
WHEN s_stop =>
state <= s_start;
WHEN OTHERS =>
END CASE;
END IF;
END PROCESS;
output_logic: PROCESS (state)
BEGIN
CASE state IS
WHEN s_d0 =>
keycode(0) <= KeyboardData;
WHEN s_d1 =>
keycode(1) <= KeyboardData;
WHEN s_d2 =>
keycode(2) <= KeyboardData;
WHEN s_d3 =>
keycode(3) <= KeyboardData;
WHEN s_d4 =>
keycode(4) <= KeyboardData;
WHEN s_d5 =>
keycode(5) <= KeyboardData;
WHEN s_d6 =>
keycode(6) <= KeyboardData;
WHEN s_d7 =>
keycode(7) <= KeyboardData;
WHEN OTHERS =>
END CASE;
END PROCESS;
END Structural; -- read in data bit 0 from the keyboard
-- read in data bit 1 from the keyboard
-- read in data bit 2 from the keyboard
-- read in data bit 3 from the keyboard
-- read in data bit 4 from the keyboard
-- read in data bit 5 from the keyboard
-- read in data bit 6 from the keyboard
-- read in data bit 7 from the keyboard Figure 10.26 VHDL code for the PS/2 keyboard controller. 10.6 Summary Checklist
Control unit
Datapath
Control signals 372 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units Status signals
Constructing a control unit
Generating control signals
Generating status signals
Timing issues for a control unit
ASM charts
State action tables
Be able to derive a state diagram from an algorithm
Be able to derive output signals for a control unit from control words
Be able to derive status signals from the conditions in an algorithm
Be able to determine how a comparator for generating a status signal connects to the datapath
Be able to derive a control unit (FSM) from a state diagram 373 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units 10.7 Problems
10.1 Derive the control unit for the counting problem from Example 10.1, except use the general datapath from
Section 9.4.
Answer
Next State
Current State
control word 1 s0
Q1next Q0next
Q1Q0
(i ≠ 10)' (i ≠ 10)
(i ≠ 10)
(i ≠ 10)'
s0 00
s3 11
s1 01
s1 01
s2 10
s2 10
s2 10
s3 11
s1 01
control word 2 s1
s3 11
s3 11
s3 11
(i ≠ 10) (b)
s2 control word 3
(i ≠ 10)'
halt Implementation
D1 D0
(i ≠ 10)' (i ≠ 10)
11
01
10
10
11
01
11
11 Current State
Q1Q0
00
01
10
11 s3 (c)
(a)
Q1Q0
00
01
10
11 D 0 = Q 1 + Q 0'
D1 = (i ≠ 10)' + Q0 Clear
1
0
0
× Clear = Q1'Q0'
(e) (d) Q 1Q 0
00
01
10
11 IE
×
0
×
× ALU2
×
1
×
× ALU1
×
0
×
× ALU0
×
0
×
× Load
0
1
0
× Clear
1
0
0
× OE
0
0
1
0 IE
ALU2
ALU1
ALU0
Load
Clear
OE =0
= Q1'Q0
=0
=0
= Q1'Q0
= Q1'Q0'
= Q 1Q 0'
(g) (f) 374 Digital Logic and Microprocessor Design with VHDL status signal from
the datapath next-state
logic Chapter 10 - Control Units state
memory output logic and
control signals to the datapath (i≠10) ' 0'
D1 ALU2 Q1 Clk IE ' 0'
Q'1 ALU1 ' 0' ALU0
Load D0 Q0 OE Clk
Clk Clear Q'0 (h) 10.2 Use schematic entry to implement the control unit shown in Figure 10.2(g). Simulate and test the circuit.
10.3 Implement and test the control unit for Problem 10.2 on the UP2 board.
10.4 Use schematic entry to implement the control unit shown in Figure 10.6(g). Simulate and test the circuit.
10.5 Implement and test the control unit for Problem 10.4 on the UP2 board.
10.6 Use schematic entry to implement the control unit shown in Figure 10.11(g). Simulate and test the circuit.
10.7 Implement and test the control unit for Problem 10.6 on the UP2 board.
10.8 Derive the control unit for:
(a)
(b)
(c)
(d)
(e)
(f) Problem 9.23
Problem 9.24
Problem 9.25
Problem 9.26
Problem 9.27
Problem 9.28 10.9 Use schematic entry to implement the control units for Problem 10.8. Simulate and test the circuit.
10.10 Derive the control unit for:
(a)
(b)
(c)
(d)
(e)
(f) Problem 9.29
Problem 9.30
Problem 9.31
Problem 9.32
Problem 9.33
Problem 9.34 375 Digital Logic and Microprocessor Design with VHDL (g)
(h) Chapter 10 - Control Units Problem 9.35.
Problem 9.36. 10.11 Use schematic entry to implement the control units for Problem 10.10. Simulate and test the circuit.
10.12 Instead of using an FSM circuit for the keyboard controller, as shown in Section 10.3.2, design and
implement a keyboard controller circuit using just a serial-to-parallel shift register.
10.13 Modify the keyboard controller circuit from Section 10.3.2 so that it will test for the correct parity bit.
10.14 Modify the keyboard controller circuit from Problem 10.12 so that it will test for the correct parity bit.
10.15 Construct and implement a circuit using the VGA controller from Section 10.3.3 to draw a blue triangle on
the screen.
10.16 Construct and implement a circuit using the VGA controller from Section 10.3.3 to draw a white square in the
middle of the screen.
10.17 Construct and implement a circuit using the VGA controller from Section 10.3.3 to draw a white square and a
blue triangle on the screen.
10.18 Using the keyboard controller from Section 10.3.2 and the VGA controller from Section 10.3.3, construct and
implement a circuit that reads in a letter from the keyboard and displays it on the screen.
10.19 Write VHDL code for the control units for Problem 10.8. Simulate and test the circuit.
10.20 Write VHDL code for the control units for Problem 10.10. Simulate and test the circuit.
10.21 Derive the ASM charts for the control units for Problem 10.8.
10.22 Derive the ASM charts for the control units for Problem 10.10.
10.23 Derive the state action tables for the control units for Problem 10.8.
10.24 Derive the state action tables for the control units for Problem 10.10. 376 Digital Logic and Microprocessor Design with VHDL Chapter 10 - Control Units 00 s0
x=0
y=1
Start
1 0 s1 01
x=1
y=1
10 s2
x=1
y=1
n=9
1
s3 0 11
x=1
y=0 377 Chapter 11 Dedicated Microprocessors Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors All microprocessors can be divided into two main categories: general-purpose microprocessors and
dedicated microprocessors. General-purpose microprocessors are capable of performing a variety of computations.
In order to achieve this goal, each computation is not hardwired into the processor, but rather is represented by a
sequence of instructions in the form of a program that is stored in the memory, and executed by the microprocessor.
The program in the memory can be easily changed so that another computation can be performed. Because of the
general nature of the processor, it is likely that in performing a specific computation, not all of the resources
available inside the general-purpose microprocessor are used.
Dedicated microprocessors, also known as application specific integrated circuits (ASICs), on the other hand,
are dedicated to performing only one task. The instructions for performing that one task are, therefore, hardwired
into the processor itself, and once manufactured, cannot be modified again. In other words, no memory is required to
store the program because the program is built right into the circuit itself. If the ASIC is completely customized,
then only those resources that are required by the computation are included in the ASIC, and so no resources are
wasted. Another advantage of building the program instructions directly into the microprocessor circuit itself is that
the execution speed of the program is many times faster than if the instructions are stored in memory.
The design of a microprocessor, whether it be a general-purpose microprocessor or a dedicated microprocessor,
can be divided into two main parts, the datapath and the control unit as shown in Figure 11.1. The datapath is
responsible for all the operations perform on the data. It includes (1) functional units such as adders, shifters,
multipliers, ALUs, and comparators, (2) registers and other memory elements for the temporary storage of data, and
(3) buses and multiplexers for the transfer of data between the different components in the datapath. External data
can enter the datapath through the data input lines. Results from the computation can be returned through the data
output lines.
Control
Inputs Data
Inputs
Control unit 8 '0' Datapath mux
ff
Nextstate
Logic State
Memory Output
Logic register Control
Signals Status
Signals
Control
Outputs ALU 8 ff register 8
Data
Outputs Figure 11.1. Schematic of a microprocessor.
The control unit (or controller) is responsible for controlling all the operations of the datapath by providing
appropriate control signals to the datapath at the appropriate times. At any one time, the control unit is said to be in
a certain state as determined by the content of the state memory. The state memory is simply a register with one or
more (D) flip-flops. The control unit operates by transitioning from one state to another – one state per clock cycle,
and because of this behavior, the control unit is also referred to as a finite-state machine (FSM). The next-state
logic in the control unit will determine what state to go to next in the next clock cycle depending on the current state
that the FSM is in, the control inputs, and the status signals. In every state, the output logic that is in the control
unit generates all the appropriate control signals for controlling the datapath. The datapath, in return, provides status
signals for the next-state logic. Status signals are usually from the output of comparators for testing branch
conditions. Upon completion of the computation, the control output line is asserted to notify external devices that
the value on the data output lines is valid.
In chapters 9 and 10, you have learned how to design the datapath and the control unit separately. In this
chapter, you will learn how to put them together to form a dedicated microprocessor. There are several abstraction
levels at which a microprocessor can be designed. At the lowest level, you manually construct the circuit for both
the control unit and the datapath separately, and then connect them together using the control and status signals. This
manual process of constructing a microprocessor ties together everything that you have learned so far in this book. 379 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors However, in a real situation, this method is not too practical because the microprocessor will usually require many
more states, input, and output signals. As a result, the manual construction process becomes much more complex.
Furthermore, there are tools for automating this process. Nevertheless, being able to construct a microprocessor
manually shows that you fully understand the theories and concepts of how a microprocessor is designed.
Section 11.1 shows this complete dedicated microprocessor construction process. Section 11.2 provides several
complete manual construction examples. All the circuits are on the accompanying CD-ROM, and can be
downloaded onto the UP2 development board for testing.
After manually designing a microprocessor, we can actually implement the circuit either by manually drawing
the circuit using the Schematic Editor, or using VHDL to describe the circuit connections at the structural level.
Constructing a microprocessor manually this way uses the FSM+D (FSM plus datapath) model. In this model, both
the FSM and the datapath circuits are manually constructed as separate units. The FSM and the datapath are
connected together in an enclosing unit using the control and status signals.
The next abstraction level of microprocessor design also uses the FSM+D model. As before, you would
manually construct the datapath. However, instead of manually constructing the FSM, you would use behavioral
VHDL code to describe the operation of the FSM as discussed in Sections 7.6 and 10.7. There will be a next-state
process and an output process in the behavioral code. The next-state process will generate the next-state logic, and
the output process will generate all the control signals for driving the datapath. The final circuit for the FSM can
then be automatically synthesized. The automatically synthesized FSM and the manually constructed datapath are
then connected together like before in an enclosing unit using the control and status signals. In practice, this is
probably the lowest level in which you would want to design a dedicated microprocessor. The advantage of using
this FSM+D model is that you have full control as to how the datapath is built. Section 11.3.1 illustrates this process.
The third level of microprocessor design uses the FSMD (FSM with datapath) model. Using this model, you
would design the FSM using behavioral VHDL code just like in the previous level. However, instead of constructing
the datapath manually as a separate unit, all the datapath operations are embedded within the FSM entity using the
built-in VHDL operators. During the synthesis process, the synthesizer will automatically generate a separate FSM
and datapath units, but these two units will be automatically connected together as one microprocessor by the
synthesizer. The advantage of this model is that you do not have to manually design the datapath, but you still have
full control as to what operation is executed in what state or in what clock cycle. In other words, you have control
over the timing of the FSM circuit. Section 11.3.2 illustrates this process.
Finally, a microprocessor can be described algorithmically at the behavioral level using VHDL. This process
synthesizes the full microprocessor with its control unit and datapath automatically. The advantage of designing
microprocessors this way is that you do not need to know how to manually design a microprocessor. In other words,
you do not need to know most of the materials presented in this book. Instead, you only need to know how to write
VHDL codes. The disadvantage is that you do not have control over the timing of the circuit. You can no longer
specify what register-transfer operation is executed in what clock cycle. Section 11.3.3 illustrates this process. 11.1 Manual Construction of a Dedicated Microprocessor
In Chapter 9, we described how a datapath is designed, and how it is used to execute a particular algorithm by
specifying the control words to manipulate the datapath at each clock cycle. In that chapter, we tested the datapath
by setting the control word signals manually. However, to actually have the datapath automatically operate
according to the control words, a control unit is needed that will generate the control signals that correspond to the
control words at each clock cycle.
The construction of the control unit was described in Chapter 10. Given the algorithm and the control words, we
were able to derive the state diagram, from which we get the next-state table, and finally the next-state logic circuit
for the control unit. The control words also serve as the output table; from this we get the output logic circuit.
Combining the next-state logic circuit, the state memory, and the output logic circuit together produces the complete
control unit circuit.
To form a complete dedicated microprocessor, we simply have to join the control unit and the datapath together
using the control signals and the status signals. Recall that the control signals are generated by the control unit to
control the operations of the datapath, while the status signals are generated by the datapath to inform the next-state
logic in the control unit as to what the next state should be in the execution of the algorithm. 380 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors This method of manually constructing a dedicated microprocessor is referred to as the FSM+D (FSM plus
datapath) model because the FSM and the datapath are constructed separately, and then they are joined together
using the control and status signals. Example 11.1 shows this manual construction of a dedicated microprocessor.
Example 11.1
In this example, we will manually construct the dedicated microprocessor for the summation algorithm of
Example 9.10. Recall that the problem is to generate and add the numbers from n down to 1, where n is an input
number. In Example 9.10, we derived the control words for solving this algorithm using the general datapath shown
in Figure 9.32??? and repeated here in Figure 11.2(a). Example 10.6??? manually constructed the control unit circuit
based on the state diagram and control words for the algorithm. The complete control unit circuit from Figure 10.4 is
repeated here in Figure 11.2(b).
The datapath circuit shown in Figure 11.2(a) uses 16 control signals. Correspondingly, the control unit circuit
shown in Figure 11.2(b) generates the values for these 16 control signals. To combine the control unit and the
datapath together, we simply connect the 16 control signals from the control unit to the corresponding 16 control
signals on the datapath as shown in Figure 11.2(c).
control signals from
the control unit Data Input
status signal to the
control unit 8
IE 15 (n ≠ 0) 8
1 mux 0 8
8
WE
WA
RAE
RAA 14
13-12
11
10-9
Clock 4x8
RF RBE
RBA A B 8 8
7-6 8 A
B
ALU2
ALU1 ALU
ALU0 5
4
3 8
SH1
SH0 2
1 Shifter
8 OE 0 8 Data Output (a) 381 8 Digital Logic and Microprocessor Design with VHDL next-state
logic Start Chapter 11 - Dedicated Microprocessors state
memory output logic and
control signals to the datapath control
input RBE IE
D2 Q2 0 Clk RBA1 ALU2 WA0 ALU1
ALU0 Q1 Clk
Clear D0 Q'1 Q0 0 Clk
Clear Clk
Reset WA1 Q'2 0
D1 RBA0 RAE Clear WE 0 RAA1 Q'0 RAA0 OE (n ≠ 0 )
Done status signal from
the datapath control
output (b)
Data Input Start 8
Start Clock
Reset IE
WE
WA1-0
RAE
RAA1-0
Control RBE
Unit RBA1-0
ALU2-0
SH1-0
OE
Clock
Reset (n ≠ 0 ) Control
Signals Status
Signal Done Input
IE
WE
WA1-0
RAE
RAA1-0
RBE
RBA1-0 Datapath
ALU2-0
SH1-0
OE
Clock
(n ≠ 0)
Reset
Output 8
Done Data Output 382 SH1
SH0 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors (c)
Figure 11.2. Microprocessor for the summation algorithm of Example 11.1: (a) datapath; (b) control unit; (c)
complete microprocessor circuit.
In addition to the 16 control signals, the control unit requires one status signal from the datapath. This status
signal, (n ≠ 0), is generated by the OR gate comparator in the datapath, and goes to the input of the next-state logic in
the control unit.
The microprocessor has two types of inputs, and two types of outputs. There is a control input signal Start that
goes into the control unit. A control output signal Done that the control unit generates. Then there is the data input
going into the datapath. For this example, this is where the value for n is entered in. Finally, there is the data output
from which the datapath outputs the result of the computation. This complete dedicated microprocessor, as shown in
Figure 11.2(c), of course does nothing more then to sum the numbers from n down to 1, and outputs the result.
♦ 383 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors 11.2 Examples
This section provides several complete examples of manually designing and constructing dedicated
microprocessors. The complete schematic drawings are available on the accompanying CD-ROM in the respective
directories. All the examples’ microprocessor top-level schematic design files are named MP.GDF, and can be used
for simulation. The microprocessor circuits can also be downloaded onto the UP2 development board for testing. To
download the design onto the UP2 development board, use either the top-level file named UP2MAX.GDF for the
MAX chip, or UP2FLEX.GDF for the FLEX chip. In each respective directory on the CD-ROM, there is a
README.TXT file that describes how to use the microprocessor. Furthermore, each example also has the VHDL
codes for describing it using the FSMD model on the CD-ROM. The corresponding top level VHDL files on the
CD-ROM are MP.VHD, UP2MAX.VHD, and UP2FLEX.VHD. 11.2.1 Greatest Common Divisor
Example 11.2
In this example, we will manually design the complete dedicated microprocessor for evaluating the greatest
common divisor (GCD) of two 8-bit unsigned numbers X and Y. For example, GCD(3,5) = 1, GCD(10,4) = 2, and
GCD(12,60) = 12. The algorithm for solving the GCD problem is listed in Figure 11.3.
We will first design a dedicated datapath for the algorithm. Next, we will design the control unit for the
datapath. We will use schematic entry to implement the complete dedicated microprocessor. Finally, we will test it
using simulation, and on the UP2 board.
1
2
3
4
5
6
7
8
9
10 input
input
while
if X
Y
(X ≠ Y){
(X > Y) then
X=X-Y
else
Y=Y-X
end if
}
output X Figure 11.3. Algorithm for solving the GCD problem.
The algorithm shown in Figure 11.3 has five data manipulation statements in lines 1, 2, 5, 7, and 10. There are
two conditional tests in lines 3 and 4. We can conclude that the datapath requires two 8-bit registers, one for variable
X, and one for variable Y, and a subtractor. The dedicated datapath is shown in Figure 11.4.
We need a 2-to-1 mux for the input of each register because for each register, we need to initially load it with an
input number, and subsequently load it with the result from the subtractor. The two control signals, In_X and In_Y,
select which of the two sources are to be loaded into the registers X and Y respectively. The two control signals,
Load_X and Load_Y, load a value into the respective register.
The bottom two muxes, selected by the same XY signal, determine the source to the two operands for the
subtractor. When XY is asserted, then the value from register X goes to the left operand, and register Y goes to the
right operand of the subtractor. When XY is de-asserted, then Y goes to the left operand, and X goes to the right
operand. Thus, this allows the selection of one of the two subtraction operations, X – Y or Y – X, to perform. Finally,
a tri-state buffer is used for outputting the result from register X. The Out control signal is used to enable the tri-state
buffer.
A greater-than and equal-to comparator is used to generate the two needed conditional status signals. The
comparator inputs are directly from the two X and Y registers. There are two output signals, XeqY and XgtY, from the 384 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors comparator. XeqY is asserted if X is equal to Y, and XgtY is asserted if X is greater than Y. The operation table and
circuit for this comparator was discussed in Section 4.10, and shown in Figure 4.29.
This dedicated datapath for solving the GCD problem requires six control signals, In_X, In_Y, Load_X, Load_Y,
XY, and Out, and two status signals, XeqY, and XgtY.
Input_X Input_Y 8 8
1 In_X 0 1 0 In_Y
Load_X D7-0
Load 8-bit register
Clear
X
Q7-0 Load_Y
Clear
Clock
XeqY
XqtY D7-0
Load 8-bit register
Clear
Y
Q7-0 8 8 Comparator
(X>Y), (X=Y)
1 0 1 0 8 XY
8 8
Subtractor 8
Out
Output
Figure 11.4. Dedicated datapath for solving the GCD problem. The state diagram for the GCD algorithm requires five states as shown in Figure 11.5 (a). Four states are used
for the five data manipulation statements, since only one state is used for performing both inputs. One “no-op” state
is used for the conditional testing of the updated values of X and Y. The “no-operation” state 001 is needed since we
need to test the conditions on the updated values of X and Y. From state 001, we test for the two conditions, (X=Y)
and (X>Y). If (X=Y) is true, then the next state is 100. If (X=Y) is false, then the next state is either 010 or 011,
depending on whether the condition (X>Y) is true of false respectively.
This state diagram does not have a Start signal, so in order for the resulting microprocessor to read the inputs
correctly, we must first set up the input numbers, and then assert the reset signal to clear the state memory flip-flops
to 0. This way, when the FSM starts executing from state 000, the two input numbers are ready to be read in.
The next-state table as derived from the state diagram is shown in Figure 11.5 (b). The table requires five
variables; three to encode the six states, Q2, Q1, and Q0, and two status signals, (X=Y) and (X>Y). There are three
unused state encodings, 101, 110 and 111. We have assumed that the next states from these three unused states are
unconditionally back to state 000.
Using D flip-flops to implement the state memory, the implementation table is the same as the next-state table
except that the values in the table entries are the inputs to the flip-flops, D2, D1, and D0, instead of the flip-flops
outputs, Q2next, Q1next, and Q0next. The k-maps and the excitation equations for D2, D1, and D0 are shown in Figure
11.5 (c).
The control words and output truth table, having the six control signals, are shown in Figure 11.5 (d). State 000
performs both inputs of X and Y. The two mux select lines, In_X and In_Y must be asserted so that the data comes 385 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors from the two primary inputs. The two numbers are loaded into the two corresponding registers by asserting the
Load_X and Load_Y lines. State 001 is for testing the two conditions, so no operations are performed. The no-op is
accomplished by not loading the two registers, and not outputting a value. For states 010 and 011, the XY mux select
line is used to select which of the two subtraction operations is to be performed. Asserting XY performs the
operation X – Y, whereas, de-asserting XY performs the operation Y – X. The corresponding In_X or In_Y line is deasserted to route the result from the subtractor back to the input of the register. The corresponding Load_X or
Load_Y line is asserted to store the result of the subtraction into the correct register. State 100 outputs the result from
X by asserting the Out line.
The output equations as derived from the output table are shown in Figure 11.5 (e). There is one equation for
each of the six control signals. Each equation is only dependent on the current state, Q2, Q1, and Q0. We have
assumed that the control signals have “don’t care” values in all the unused states.
The complete control unit circuit is shown in Figure 11.5 (f). The state memory consists of three D flip-flops.
The inputs to the flip-flops are the next-state circuits derived from the three excitation equations. The output circuits
for the six control signals are derived from the six output equations. The two status signals, XeqY and XgtY, come
from the comparator in the datapath.
000 input X
input Y (X = Y ) 001
( X = Y) ' &
( X > Y)
X = X - Y 010 100 output X ( X = Y) ' &
( X > Y) '
011 Y=Y-X (a)
Current State
Q2Q1Q0
000
001
010
011
100 Next State Q2next Q1next Q0next
(X=Y), (X>Y)
00
01
10
11
001
001
001
001
011
010
100
100
001
001
001
001
001
001
001
001
100
100
100
100
(b) 386 Digital Logic and Microprocessor Design with VHDL D2 Q2 = 0 (X=Y),(X>Y) Q1Q0 00 Chapter 11 - Dedicated Microprocessors 01 11 Q2 = 1
10 01 1 00 01 11 10 1 00 1 1 1 Q2Q1'Q0'
Q2'Q1'Q0 (X=Y) 1 11
10 D2 = Q2Q1'Q0' + Q2'Q1'Q0 (X=Y)
D1 (X=Y),(X>Y) Q1Q0 00 01 1 Q2 = 0
11 Q2 = 1
10 00 01 11 10 1 00
01 Q2'Q1'Q0 (X=Y)' 11
10 D1 = Q2'Q1'Q0 (X=Y)'
D0 (X=Y),(X>Y) Q1Q0 Q2 = 0 Q2 = 1 00 01 11 10 00 1 1 1 1 01 1 11 1 1 1 1 10 1 1 1 00 01 11 10 1 Q2'Q0'
Q2' (X=Y)' (X>Y)'
Q2'Q1 D0 = Q2'Q0' + Q2' (X=Y)' (X>Y)' + Q2'Q1
(c)
Control
Word
0
1
2
3
4 State
Q2 Q1 Q0
000
001
010
011
100 Instruction
Input X, Input Y
none
X=X–Y
Y=Y–X
Output X In_X
1
×
0
×
×
(d) 387 In_Y
1
×
×
0
× Load_X
1
0
1
0
0 Load_Y
1
0
0
1
0 XY
×
×
1
0
× Out
0
0
0
0
1 Digital Logic and Microprocessor Design with VHDL In_X = Q1'
In_Y = Q0'
X Y = Q 0' Chapter 11 - Dedicated Microprocessors Load_X = Q2'Q0'
Load_Y = Q2'Q1'Q0' + Q2'Q1Q0
Out = Q2
(e) In_X
D2 In_Y Q2 Clk
Clr Q'2 D1 Load_X Q1 Load_Y Clk
Clr Q'1 XY
D0 Q0 Out Clk
Clr Clock
Reset Q'0 Done
XeqY
XgtY
(f) Figure 11.5. Control unit for solving the GCD problem: (a) state diagram; (b) next-state table; (c) k-maps and
excitation equations; (d) control word/output truth table; (e) output equations; (f) circuit.
The final microprocessor can now be easily formed by connecting the control unit and the datapath together
using the designated control and status signals as shown in Figure 11.6. A sample simulation is shown in Figure
11.7. This complete microprocessor circuit is on the accompanying CD-ROM in the file MP.GDF, and can be
synthesized and simulated. 388 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Input X
8
In_X
In_Y
Load_X
Load_Y
XY
CU
Out
Clock
Reset Clock
Reset XeqY
XgtY Control
Signals Status
Signals Done Input Y
8 Input_X
Input_Y
In_X
In_Y
Load_X
Load_Y
XY
DP
Out
Clock
Reset XeqY
XgtY
Output 8
Done Output Figure 11.6. Microprocessor for solving the GCD problem. Figure 11.7. Sample simulation for the GCD problem for the two input numbers 4 and 12. The GCD of these two
numbers is 4.
There is another top level file called UP2MAX.GDF in the same directory for downloading and testing this
microprocessor circuit on the UP2 development board using the MAX EPM7128S PLD.
You need to look at the Floorplan Editor to see how the I/O’s are mapped to the EPM7128S pins. If you prefer,
you can re-map these pins. Note that the Reset signal must be mapped to pin 1 on the MAX chip, otherwise you
need to remove the “Automatic Global Clear” selection checkmark from the Compiler menu Assign / Global Project
Synthesis in order to compile this design file correctly.
Jumper wires between the PLD pins and the appropriate switches and LED’s must be connected according to
this floor plan. There are two sets of eight wires for the two input DIP switches. Another eight wires for the eight
output LED’s, and one wire for the reset button. The original Done signal is mapped to segment a on digit 2. You
may want to refer to Appendix C.9 for making these connections.
To test this circuit, you need to first set up the two sets of eight DIP switches for the two binary unsigned input
numbers. Then press the button connected to the Reset line. You should see the answer displayed as an 8-bit binary
number on the eight LED’s. You should also see segment a on digit 2 light up for the Done signal.
♦ 389 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors 11.2.2 Summing Input Numbers
Example 11.3
In this example, we will manually design the complete dedicated microprocessor for inputting many 8-bit
unsigned numbers through one input port, and then output the sum of these numbers. The algorithm continues to
input numbers as long as the number entered is not a zero. Each number entered is also outputted. When the number
entered is a zero, the algorithm stops and outputs the sum of all the numbers entered. The algorithm for solving this
problem is listed in Figure 11.8.
We will first design a dedicated datapath for the algorithm. Next, we will design the control unit for the
datapath. We will use schematic entry to implement the complete dedicated microprocessor. Finally, we will test it
using simulation, and on the UP2 board.
1
2
3
4
5
6
7
8
9
10 Sum = 0
begin loop
input X
Sum = Sum + X
if (X = 0) then
exit loop
end if
output X
end loop
output Sum Figure 11.8. Algorithm for summing input numbers.
The algorithm shown in Figure 11.8 has five data manipulation statements in lines 1, 3, 4, 8, and 10. There is
one conditional test in line 5. The algorithm requires two 8-bit registers, one for variable X, and one for variable
Sum, and an adder. The dedicated datapath is shown in Figure 11.9.
Line 1 in the algorithm is performed by asserting the Clear signal, line 3 is performed by asserting the Load_X
signal, and line 4 is performed by asserting the Load_Sum signal. The algorithm continuously outputs either X or
Sum. The Out signal selects the 2-to-1 multiplexer for one of the two sources, register X or register Sum. The output
of the mux is always available at the output. The conditional test (X = 0) is generated by the 8-input NOR gate.
Input_X
8 Load_Sum
D7-0
Load 8-bit register
Clear
X
Q7-0 Load_X D7-0
Load 8-bit register
Clear Sum
Q7-0 Clear
Clock
Xeq0 8 8
8 8 Out 0 1 8 Output
Figure 11.9. Dedicated datapath for the summing input numbers problem. 390 8 + Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors At first glance, this algorithm is very similar to the GCD algorithm in Example 11.2. However, because of the
requirements of this problem, the actual hardware implementation of this microprocessor is slightly more difficult.
Specifically, the requirement that many different numbers be inputted through one input port requires careful timing
considerations, and an understanding of how mechanical switches behave.
As a first try, we begin with the state diagram shown in Figure 11.10 (a). Line 1 of the algorithm is performed
by the asynchronous reset, so it does not require a state to execute. Line 3 is performed in state 00, which is
unconditionally followed by line 4 in state 01. The condition (X=0) is then tested. If the condition is true, the loop is
exited, and the FSM goes to state 11 to output the value for Sum, and stays in that state until reset. If the condition is
false, the FSM goes to state 10 to output X, and the loop repeats back to state 00.
Enter' 00 input X
output X 00 input X
output X Enter 01
(X = 0)'
10 Sum = Sum + X
output X 01 (X = 0)
output X (X = 0)' 11 output Sum 10 (a) Sum = Sum + X
output X
(X = 0)
output X 11 output Sum (b) Figure 11.10. Incorrect state diagrams for the summing input numbers problem.
One thing to notice is that in all the states except for state 11, we need to output X. This is because in the
datapath, we do not have a tri-state buffer to disable the output. Instead, with the mux, either one of two numbers is
always being outputted. The behavior of this, nevertheless, is correct according to the problem specifications.
As an exercise, you can derive the FSM circuit based on this state diagram shown in Figure 11.10 (a), and
perform a simulation to test its operation. In the simulation, the input numbers are manually assigned, so you should
be able to obtain the correct simulation result.
However, if you implement this circuit in hardware, it will not work correctly. The reason is that the FSM
cycles through the three loop states, 00, 01, and 10, very fast because of the fast clock speed. As a result, the FSM
will have gone through state 00 to input a number many times before you can even change the input to another
number. Hence, the same number will be summed many times.
To correct this problem, we need to add another input signal that acts like the Enter key. This way, the FSM will
stay in state 00, waiting for the Enter signal to be asserted. This will give the user time to set up the input number
before pressing the enter switch. When the Enter signal is asserted, the FSM will exit state 00 with the new number
to be processed. This modified state diagram is shown in Figure 11.10 (b).
There is still a slight timing problem with this modified state diagram because of the fast clock speed. After
pressing the Enter switch, and before you have time to release it, the FSM will have cycled through the complete
loop, and is back at state 00. But since you have not yet released the Enter switch, the FSM will continue on another
loop with the same input number. What we need to do is to break the loop by waiting for the Enter switch to be
released. This is shown in the state diagram in Figure 11.11 (a). State 10 will wait for the Enter switch to be released
before continuing on, and looping back to state 00.
This last state diagram is correct. However, there might be a problem with the operation of the mechanical
switch used for the enter signal. When a mechanical switch is pressed, it usually goes on and off several times
before settling down in the on position. This is referred to as the “debounce” problem. When the switch is 391 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors fluctuating between the on and the off positions, the FSM can again be able to go through the loop many times.
What we need to do is to debounce the switch. This however, is not done in the FSM circuit itself, but in the
interface circuit between the FSM and the switch. We will come back to address this problem when building the
interface circuit.
We will now construct the control unit circuit based on the state diagram shown in Figure 11.11 (a). Four states
are used for the five data manipulation statements. All the states except for 11 will output X. State 00 inputs X and
waits for the Enter signal. This allows the user to set up the input number, and then press the Enter key. When the
Enter key is pressed, the FSM goes to state 01 to sum X, and test for (X = 0). If the condition is true, the FSM
terminates in state 11, and outputs X, otherwise, it goes to state 10 to wait for the Enter signal to be de-asserted by
the user releasing the Enter switch. After exiting state 10, the FSM continues on to repeat the loop in state 00.
The next-state table as derived from the state diagram is shown in Figure 11.11 (b). The table requires four
variables; two to encode the four states, Q1 and Q0, and two status signals, Enter and (X = 0).
Using D flip-flops to implement the state memory, the implementation table is the same as the next-state table
except that the values in the table entries are the inputs to the flip-flops, D1 and D0, instead of the flip-flops outputs,
Q1next and Q0next. The k-maps and the excitation equations for D1 and D0 are shown in Figure 11.11 (c).
The control words and output truth table for the three control signals, are shown in Figure 11.11 (d). State 00
performs Line 3 of the algorithm in Figure 11.8 by asserting Load_X, and Line 8 by de-asserting Out. When Out is
de-asserted, X is passed to the output. State 01 performs Line 4 and Line 8. Line 4 is executed by asserting
Load_Sum, and Line 8 is executed by de-asserting Out. State 10 again performs Line 8 by de-asserting Out. Finally,
State 11 performs Line 10 by asserting Out.
The output equations as derived from the output table are shown in Figure 11.11 (e). There is one equation for
each of the three control signals. Each equation is only dependent on the current state, Q1 and Q0.
The complete control unit circuit is shown in Figure 11.11 (f). The state memory consists of two D flip-flops.
The inputs to the flip-flops are the next-state circuits derived from the two excitation equations. The output circuits
for the three control signals are derived from the three output equations. The status signal, (Xeq0) come from the
comparator in the datapath. Enter'
00 input X
output X Enter
01
(X = 0)'
Enter' 10 Sum = Sum + X
output X
(X = 0)
output X 11 output Sum Enter (a)
Current State
Q 1Q 0
00
01
10 Next State Q2next Q1next Q0next
Enter, (X=0)
00
01
10
11
00
00
01
01
10
11
10
11
00
00
10
10 392 Digital Logic and Microprocessor Design with VHDL 11 Chapter 11 - Dedicated Microprocessors 11 11 11 11 (b)
D1 Enter,(X=0) Q1Q0 00 01 11 10 01 1 1 1 1 Q0 11 1 1 1 1 Q1 Enter 1 1 00 10 D1 = Q0 + Q1 Enter
D0 Enter,(X=0) Q1Q0 00 01 01
11 10 1 00 11 1 1
1 1 Q0 (X=0) 1 1 Q1'Q0' Enter Q1Q0 1 10 D0 = Q1Q0 + Q0 (X=0) + Q1'Q0' Enter
(c)
Control
Word
0
1
2
3 State
Q1 Q0
00
01
10
11 Instruction
input X, output X
Sum = Sum – X, output X
output X
output Sum Load_X
1
0
0
0 (d)
Load_X = Q1'Q0'
Load_Sum = Q1'Q0
Out = Q1Q0
(e) 393 Load_Sum
0
1
0
0 Out
0
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Enter D1 Q1 Load_X Q'1 Load_Sum Clk
Clr D0 Out Q0 Clk
Clr Clock
Reset Q'0 Done
Xeq0 (f)
Figure 11.11. Control unit for the summing input numbers problem: (a) state diagram; (b) next-state table; (c) kmaps and excitation equations; (d) control word/output truth table; (e) output equations; (f) circuit.
The final microprocessor can now be easily formed by connecting the control unit and the datapath together
using the designated control and status signals as shown in Figure 11.12. A sample simulation is shown in Figure
11.13. This complete microprocessor circuit is on the accompanying CD-ROM in the file MP.GDF, and can be
synthesized and simulated.
Input X Enter 8
Input_X Enter
Load_X
Load_Sum
Out Control
Signals Load_X
Load_Sum
Out CU
Clock
Reset Clock
Reset DP
Xeq0 Status
Signal Done Xeq0 Clock
Reset
Output 8
Done Output Figure 11.12. Microprocessor for solving the summing input numbers problem. 394 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Figure 11.13. Sample simulation for the summing input numbers problem.
In order to implement the circuit on the UP2 development board, we need to connect the microprocessor’s I/Os
to the switches, LEDs, and clock source. The most important interface circuit for this problem is to debounce the
Enter switch. A simple circuit to debounce a switch is to use a D flip-flop as shown in Figure 11.14. The clock
frequency for the D flip-flop must be slow enough for the switch bounce to settle so that the flip-flop will latch in a
single value. The exact clock frequency is not too critical. The clock divider circuit used in the example slows down
the on-board 25MHz clock to approximately 4Hz.
8 DIP switches
8
Reset push button switch
Enter push button switch D Q Enter
Clock 25MHz Clock Input_X Reset 25MHz
4Hz
Clock
Divider Clk MP
Done Q1
Q0 BCD
decoder 7-segment
LED Output
8 LED 8 LEDs Figure 11.14. Interface to the summing input numbers microprocessor for the UP2 development board.
The top-level interface circuit is in the file UP2MAX.GDF for the MAX chip on the accompanying CD-ROM.
This circuit also displays the current state value, Q1 and Q0, on the 7-segment LED. The FLEX version of this circuit
is in the file UP2FLEX.GDF. It displays the sum output on the two 7-segment displays instead of using the eight
discrete LEDs.
♦ 11.2.3 High-Low Guessing Game
Example 11.4
In this example, we will manually design the complete dedicated microprocessor for playing the high-low
guessing game. The user picks a number between 0 and 99, and the computer will use the binary search algorithm to
guess the number. After each guess, the user tells the computer whether the guess is high or low compared to the
picked number. Two push buttons, hi_button and lo_button, are used for the user to tell the computer whether the
guess is too high, too low, or correct. The hi_button is pressed if the guess is too high, and the lo_button is pressed if
the guess is too low. If the guess is correct, both buttons are pressed at the same time.
The algorithm for simulating this high-low guessing game is listed in Figure 11.15. The two range boundary
variables, Low and High, are initialized to 0 and 100 respectively. The loop between lines 3 to 11 will keep repeating
until both buttons, hi_button and lo_button, are pressed. Inside the loop, line 4 calculates the next guess by finding
the middle number between the lower and upper boundaries, and then outputs this new guess in line 5. Lines 6 to 10
checks which button is pressed. If the lo_button is pressed, that means the guess is too low, so line 7 changes the 395 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Low boundary to the current Guess. Otherwise if the hi_button is pressed, that means the guess is too high, and line
9 changes the High boundary to the current Guess. The loop is then repeated with the calculation of the new Guess
in line 4.
When both buttons are pressed, the condition in line 11 is true, and the loop is exited. Lines 12 to 15 simply
cause the display to blink the correct guess by turning it on and off.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 Low = 0
// initialize Low
High = 100
// initialize High
repeat {
Guess = (Low + High) / 2 // calculate guess using binary search
output Guess
if (lo_button = '1' and hi_button = '0')
// low button pressed
Low = Guess
else if (lo_button = '0' and hi_button = '1') // hi button pressed
High = Guess
end if
} until (lo_button = '1' and hi_button = '1') // repeat until both
// buttons are pressed
while (lo_button = '0' and hi_button = '0')
// blink correct guess
output Guess
turn off display
end while Figure 11.15. Algorithm for the high-low guessing game.
We will first design a dedicated datapath for the algorithm. Next, we will design the control unit for the
datapath. We will use schematic entry to implement the complete dedicated microprocessor. Finally, we will test it
using simulation, and on the UP2 board.
The algorithm shown in Figure 11.15 has eight data manipulation operations in lines 1, 2, 4, 5, 7, 9, 13 and 14.
The dedicated datapath for realizing this algorithm is shown in Figure 11.16. It requires three 8-bit registers, Low,
High, and Guess, for storing the low and high range boundary values, and the guess respectively. Two 2-to-1 muxes
are used for the inputs to the Low and High registers to select between the initialization values for lines 1 and 2, and
setting the new Guess value for lines 7 and 9.
The only arithmetic operations needed are the addition and division by 2 in line 4. Hence, the outputs from the
two registers, Low and High, go to the inputs of an adder for the addition, and the output of the adder goes to a
shifter. The division by 2 is performed by doing a right shift of 1 bit. The result from the shifter is stored in the
register Guess. Depending on the condition in line 6, the value in Guess is loaded into either the Low or the High
register by asserting the corresponding load signal for that register.
A 2-input by 8-bit AND gate is used to control the output of the Guess number. One 8-bit set of input is
connected to the output of the Guess register. The other 8-bit set of input is connected together in common to the
output enable OutE signal. By asserting OutE, the data from Guess is passed to the output port. To blink the output
display in lines 13 and 14, we just toggle the OutE line.
The datapath shown in Figure 11.16 requires five control signals, Init, LOload, HIload, GUESSload, and OutE.
The Init signal controls the two muxes to determine whether to load in the initialization values or the new guess. The
three load signals, LOload, HIload, and GUESSload control the writing of the three respective registers. Finally,
OutE controls the output of the guess value. 396 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors "00000000" "01100100" 8 8 8
1 0 1 0 Init
8
LOload 8 D7-0
Load 8-bit register
Clear Low
Q7-0 D7-0
Load 8-bit register
Clear High
Q7-0 HIload
Clear
Clock
8 8 8 + >> 1 8
D7-0
Load 8-bit register
Clear Guess
Q7-0 GUESSload 8
OutE
8
Output Figure 11.16. Dedicated datapath for the high-low guessing game.
The state diagram for this algorithm requires six states as shown in Figure 11.17 (a). State 000 is the starting
initialization state. State 001 executes lines 4 and 5 by calculating the new guess, and outputting it. State 001 also
waits for the user key press. If only the lo_button is pressed, then the FSM goes to state 010 to assign the guess as
the new low value. If only the hi_button is pressed, then the FSM goes to state 011 to assign the guess as the new
high value. If both buttons are pressed, then the FSM goes to state 100 go output the guess. From state 100, the FSM
turns on and off the output by cycling between states 100 and 101 until a button is pressed. When a button is pressed
from either state 100 or 101, the FSM goes back to the initialization state for a new game.
The output table showing the five output signals, Init, LOload, HIload, GUESSload, and OutE, to be generated
in each state are shown in Figure 11.17 (d). The corresponding output equations derived from the output table are
shown in (e).
Again, we will use D flip-flops to implement the state memory. Having six states, three flip-flops are needed,
with two unused states. Both the next-state table and the implementation table are shown in Figure 11.17 (b). Recall
that when D flip-flops are used for the implementation, the next-state table and the implementation table are the
same because the characteristic equation for the D flip-flop is Qnext = D.
The implementation table is simply the truth table for the three variables D2, D1, and D0. Hence, the excitation
equations for the three flip-flops, D2, D1, and D0, are derived directly from the implementation table. The K-maps
and equations for these three variables are shown in Figure 11.17 (c). 397 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Using the three excitation equations for deriving the next-state logic circuit, the three D flip-flops for the state
memory, and the five output equations for deriving the output logic circuit, we get the complete control unit circuit
for the high-low guessing game as shown in Figure 11.17 (f). 000 Low = 0
High = 100
Guess = (Low + High) / 2
output Guess hi' & lo'
001 hi' & lo hi & lo'
hi & lo 010 011 Low = Guess High = Guess hi or lo 100 hi' & lo' hi' & lo'
hi or lo Output Guess 101 Turn off lights
(a) Current State
Q2Q1Q0
000
001
010
011
100
101
110 (not used)
111 (not used) Next State Q2next Q1next Q0next (Implementation D2 D1 D0)
hi, lo
00
01
10
11
001
001
001
001
001
010
011
100
001
001
001
001
001
001
001
001
101
000
000
000
100
000
000
000
000
000
000
000
000
000
000
000
(b) D2 hi, lo
Q1Q0
00 Q2 = 0
01 11 00
01 Q2 = 1
10 00 01 11 10 1
1 Q2Q1' hi' lo' 1 Q2'Q1'Q0 hi lo 11
10 D2 = Q2Q1' hi' lo' + Q2'Q1'Q0 hi lo 398 Digital Logic and Microprocessor Design with VHDL D1 hi, lo
Q1Q0
00 Chapter 11 - Dedicated Microprocessors Q2 = 0
01 11 Q2 = 1
10 00 01 11 10 00
01 1 Q2'Q1'Q0 hi lo' 1 11
Q2'Q1'Q0 hi' lo 10 D1 = Q2'Q1'Q0 hi lo' + Q2'Q1'Q0 hi' lo D0 hi, lo
Q1Q0
00 Q2 = 0 Q2 = 1 01 11 10 00 1 1 1 1 00 1 01 1 11 1 1 1 1 1 1 11 1 10 Q1'Q0' hi' lo' 1 10 01 1 Q2'Q0'
Q2'Q1
Q2' lo' D0 = Q1'Q0' hi' lo' + Q2'Q0' + Q2'Q1 + Q2' lo'
(c)
Control
Word
0
1
2
3
4
5 State
Q2 Q1 Q0
000
001
010
011
100
101 Instruction
Initialize
Guess
Low = Guess
High = Guess
Correct
Flash output Init
1
0
0
0
0
0
(d) Init = Q2'Q1'Q0'
HIload = Q2'Q1'Q0' + Q2'Q1Q0
LOload = Q2'Q1'Q0' + Q2'Q1Q0'
GUESSload = Q2'Q1'Q0
OutE = Q2' + Q0'
(e) 399 HIload
1
0
0
1
0
0 LOload
1
0
1
0
0
0 GUESSload
0
1
0
0
0
0 OutE
1
1
1
1
1
0 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Init
D2 Q2 Clk HIload Clr Q'2 D1 Q1 LOload Clk
Clr Q'1 GUESSload
D0 Q0 Clk
Clr OutE
Q'0 Clock
Reset
lo
hi (f)
Figure 11.17. Control unit for the high-low guessing game: (a) state diagram; (b) next-state table; (c) k-maps and
excitation equations; (d) control word/output truth table; (e) output equations; (f) circuit.
Connecting the control unit circuit shown in Figure 11.17 (f) and the datapath circuit shown in Figure 11.16
together using the control and status signals produces the final microprocessor as shown in Figure 11.18. This
complete microprocessor circuit is on the accompanying CD-ROM in the file MP.GDF, and can be synthesized and
simulated. Init
HIload
LOload
GUESSload
CU OutE
Clock
Reset Clock
Reset lo
hi Control
Signals Status
Signals Init
HIload
LOload
GUESSload
OutE
DP
Clock
Reset lo
hi
Output 8
Output Figure 11.18. Microprocessor for the high-low guessing game. 400 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Figure 11.19 shows the interface needed to test the high-low guessing game dedicated microprocessor on the
UP2 development board. The top-level file UP2FLEX.GDF in the implementation directory can be used for
downloading and testing this circuit on the UP2 development board using the FLEX PLD.
♦
High push
button switch Low push
button switch In
Out
One-shot Reset
Clock
In
Out
One-shot Reset
Clock Hi
Lo
Reset
Clock MP Output binary
to
decimal two
7-segment
LEDs Reset dip
switch
25MHz Clock 25MHz
4Hz
Clock
Divider Figure 11.19. Interface to the high-low guessing game microprocessor for the UP2 development board. 11.2.4 Finding Largest Number
Example 11.5
In this example, we will implement the classic computer program of inputting several numbers, and then output
the largest of these numbers. We will assume that the numbers, entered through one input port, are 8-bit unsigned
numbers, and a zero will terminate the inputs. The current largest number is always displayed. The algorithm for
solving this problem is listed in Figure 11.20.
1
2
3
4
5
6
7
8
9 Largest = 0
// for storing the current largest number
input X
// enter first number
while (X ≠ 0){
if (X > Largest) then
// if new number greater?
Largest = X
// yes, remember new largest number
end if
output Largest
input X
// get next number
} Figure 11.20. Algorithm for the finding largest number problem.
We will first design a dedicated datapath for the algorithm. Next, we will design the control unit for the
datapath. We will use schematic entry to implement the complete dedicated microprocessor. Finally, we will test it
using simulation, and on the UP2 board.
The algorithm shown in Figure 11.20 has five data manipulation operations in lines 1, 2, 5, 7, and 8. It requires
two registers: an 8-bit register for storing X, and an 8-bit register for storing Largest. No functional unit for
performing arithmetic is needed. The dedicated datapath is shown in Figure 11.21. 401 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Input_X
8
D
Load 8-bit register
Clear
X Load_X
Clear
Clock 8
Xeq0
8 Load_Largest XgtLargest Load 8-bit register
Clear
Largest > 8 8
Output Figure 11.21. Dedicated datapath for the finding largest number problem.
Line 1 in the algorithm is performed by asserting the Clear signal, line 2 and 8 are performed by asserting the
Load_X signal, and line 5 is performed by asserting the Load_Largest signal. The algorithm continuously outputs
Largest. The conditional test (X = 0) is generated by the 8-input NOR gate, and the conditional test (X > Largest) is
generated by the greater-than comparator.
This algorithm is very similar to the summing input numbers problem in Example 11.3, especially the situation
for handling the Enter key for inputting each number through the same input port. The state diagram is shown in
Figure 11.22 (a). State 00 inputs X and waits for the Enter signal. This allows the user to set up the input number,
and then press the Enter key. When the Enter key is pressed, the FSM goes to state 01 to make two tests. If (X = 0) is
true, the FSM terminates in state 11. If (X > Largest) is true, then the FSM goes to state 10 to assign X as the new
largest number. If X is not 0 and not the largest, then the FSM will go back to state 00 to wait for another input
number. Before going back to state 00 from both states 01 and 10, we need to wait for the release of the Enter key as
explained in Example 11.3.
The next-state table as derived from the state diagram is shown in Figure 11.22 (b). The table requires five
variables; two to encode the four states, Q1 and Q0, and three status signals, Enter, (X = 0), and (X > Largest).
Using D flip-flops to implement the state memory, the implementation table is the same as the next-state table
except that the values in the table entries are for the inputs to the flip-flops, D1 and D0. The k-maps and the
excitation equations for D1 and D0 are shown in Figure 11.22 (c).
The control words and output truth table for the three control signals, are shown in Figure 11.22 (d). State 00
performs Lines 2 and 8 of the algorithm in Figure 11.20 by asserting the Load_X signal. All the states output
Largest, and this action does not require any control signals. State 10 performs Line 5 by asserting Load_Largest.
State 11 outputs a Done signal to inform the user that the FSM has stopped.
The output equations as derived from the output table are shown in Figure 11.22 (e). There is one equation for
each of the three control signals. Each equation is only dependent on the current state, Q1 and Q0.
The complete control unit circuit is shown in Figure 11.22 (f). The state memory consists of two D flip-flops.
The inputs to the flip-flops are the next-state circuits derived from the two excitation equations. The output circuits
for the three control signals are derived from the three output equations. The status signal (Xeq0) come from the NOR
gate comparator in the datapath, and the status signal (XgtLargest) come from the greater-than comparator in the
datapath. 402 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Enter'
00 input X
output Largest Enter
Enter
(X > Largest)'
& Enter' 01 output Largest
(X = 0) (X > Largest)
Enter' 11 output
Largest 10 output Largest Enter
(a)
Current State
Q1Q0
00
01
10
11 000
00
00
00
11 Next State Q2next Q1next Q0next
Enter, (X=0), (X>Largest)
001 010 011 100 101 110
00
00
00
01
01
01
10
11
11
01
10
11
00
00
00
10
10
10
11
11
11
11
11
11 111
01
11
10
11 (b)
D1 Enter = 0
(X=0),
(X>Largest)
Q1Q0
00 01 11 10 Enter = 1
00 01 11 10 1 1 1 1 1 1 1 1 1 1 1 00 Q0(X=0) 01
11 Q0(X>Largest) 1
1 1 1 1 1 1 10 Q1Enter Q1Q0 D1 = Q1Q0 + Q1Enter + Q0 (X=0) + Q0 (X>Largest)
D0 Enter = 0
(X=0),
(X>Largest)
Q1Q0
00 01 11 10 Q1'Enter(X>Largest)' 00 01 11 10 1 00 1 1 1 1 1 1 1 Q1'Q0'Enter 01
11 Enter = 1 1
1 1 1 1 1 1 1 10
Q1Q0 403 1 Q0(X=0) Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors D0 = Q1Q0 + Q0 (X=0) + Q1'Q0' Enter + Q1' Enter(X>Largest)'
(c)
Control
Word
0
1
2
3 State
Q1 Q0
00
01
10
11 Instruction
input X, output Largest
output Largest
Largest = X, output Largest
output Largest Load_X
1
0
0
0 Load_Largest
0
0
1
0 Done
0
0
0
1 (d)
Load_X = Q1'Q0'
Load_Largest = Q1Q0'
Done = Q1Q0
(e)
Enter Load_X
D1 Q1 Load_Largest Clk
Clr D0 Q'1 Done Q0 Clk Clock
Reset Clr Q'0 Xeq0
XqtLargest
(f)
Figure 11.22. Control unit for the finding largest number problem: (a) state diagram; (b) next-state table; (c) k-maps
and excitation equations; (d) control word/output truth table; (e) output equations; (f) circuit.
Connecting the control unit and the datapath together using the control and status signals produces the final
microprocessor as shown in Figure 11.23. A sample simulation is shown in Figure 11.24. This complete
microprocessor circuit is on the accompanying CD-ROM in the file MP.GDF, and can be synthesized and simulated.
The top-level interface circuit for testing this microprocessor on the UP2 development board using the MAX
chip is shown in Figure 11.25, and is in the file UP2MAX.GDF on the accompanying CD-ROM. Like the circuit for
Example 11.2, this circuit displays the largest number on the eight LEDs in binary, and the current state value, Q1
and Q0, on the 7-segment LED. Again, a debounce circuit is needed for the Enter switch. The 8-bit number inputs
are through the eight DIP switches. The FLEX version of this circuit is in the file UP2FLEX.GDF. Instead of
displaying the largest number on the eight discrete LEDs, it displays the number on the two 7-segment displays. ♦ 404 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Input X Enter 8
Enter Input_X Load_X
Load_Largest
Out Control
Signals Load_X
Load_Largest
Out CU
Clock
Reset DP Clock
Xeq0
Reset XgtLargest Status
Signal Clock
Reset Xeq0
XgtLargest Done Output 8
Done Output Figure 11.23. Microprocessor for solving the finding largest number problem. Figure 11.24. Sample simulation trace for the finding largest number problem. The last output number 15 when
Done is asserted is the largest of the six numbers entered 7, 4, 12, 8, 15, and 6.
8 DIP switches
8 Reset push button switch
Enter push button switch D Q Enter
Clock 25MHz Clock 25MHz
4Hz
Clock
Divider Input_X Reset Clk MP
Done Q1
Q0 BCD
decoder Output
8 LED 8 LEDs Figure 11.25. Interface to the finding largest number microprocessor for the UP2 development board. 405 7-segment
LED Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors 11.3 VHDL for Dedicated Microprocessors
In the introduction to this chapter, we mentioned that we can use VHDL based on different models to
automatically synthesize a microprocessor. The FSM+D model requires the manual construction of the datapath, but
the FSM is described using VHDL, and is synthesized automatically. The FSMD model also uses VHDL to describe
the FSM, but in addition, the datapath operations are imbedded within the FSM VHDL code using built-in VHDL
operators. Finally, the behavioral model uses VHDL to describe the behavior of the complete microprocessor
algorithmically, and so both the control unit and the datapath are synthesized automatically. 11.3.1 FSM + D Model
In Section 11.1, we constructed a dedicated microprocessor by manually constructing the circuits for a control
unit and a datapath. These two components are then connected together to form the microprocessor. Instead of
manually constructing the control unit circuit, an alternative way is to automatically synthesize it from behavioral
VHDL code. A separate datapath circuit is still manually constructed like before. The two components are then
joined together to form the complete microprocessor.
Writing the behavioral VHDL code for the control unit was discussed in Chapter 10. The method is again
illustrated in Example 11.6. The final dedicated microprocessor is constructed using an enclosing VHDL entity
written at the structural level that combines the VHDL code for both the control unit entity and the datapath entity.
Example 11.6
In this example, we will again construct the dedicated microprocessor for the summation algorithm of
Example 9.3 similar to Example 11.1. However, unlike Example 11.1 where the circuit for the control unit is
manually derived, we will use VHDL to automatically synthesize the control unit circuit. In order to write the
behavioral VHDL code for the control unit, we need to use the information from the state diagram and the control
words derived for the algorithm. The control words were manually derived in Example 9.3, and the state diagram
was manually derived in Example 10.2. The state diagram and control words from these two examples are repeated
here in Figure 11.26.
The complete behavioral VHDL code for the control unit is shown in Figure 11.27. In the entity section, the
output signals include the 16 control signals for controlling the datapath. There is one input signal neq0 in the entity
section, which is the status signal (n ≠ 0) from the datapath. In addition to the control and status signals, there is the
control input signal Start and the control output signal Done. Finally, there are the two global input signals Clock
and Reset. The Reset signal is to reset all the flip-flops in the state memory to zero.
The architecture section starts out with using the TYPE statement to define all the states used in the state
diagram. The SIGNAL statement declares the signal state to remember the current state of the FSM. There are two
processes in the architecture section, the next-state logic process and the output logic process that execute
concurrently. As the name suggests, the next-state process defines the next-state logic circuit that is inside the
control unit, and the output logic process defines the output logic circuit inside the control unit. The main statement
within these two processes is the CASE statement that determines what the current state is. For the next-state process,
the CASE statement is executed only at the rising clock edge because of the test (clock'EVENT AND clock = '1') in the
IF statement. Hence, the State signal is assigned a new state value only at the rising clock edge. The new state value
is, of course, dependent on the current state and input signals, if any. For example, in state s0, the next state is
dependent on the input signal Start, whereas, in state s1, the next state is unconditionally s2.
In the output process, all control signals are generated for every case, i.e., all the control signals must be
assigned a value in every case. Recall from Section 6.13.1 that VHDL synthesizes a signal using a memory element
if the signal is not assigned a value in all possible cases. However, the output circuit is a combinational circuit, and it
should not contain any memory elements. For each state in the CASE statement in the output process, the values
assigned to each of the output signal are taken directly from the control word/output table.
For the datapath, we are using the same general datapath as discussed in Section 8.5. The VHDL code for this
datapath is listed in Figure 8.12 and Figure 8.13. Figure 8.12 describes the individual components used in the
datapath and Figure 8.13 combines these components into the desired datapath. 406 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Finally, Figure 11.28 combines the datapath and the control unit together using structural VHDL coding to
produce the top-level entity sum for the microprocessor. The entity section specifies the primary I/O signals for the
microprocessor. In addition to the global Clock and Reset signals, there are the four I/O signals shown in Figure
11.2. These are the control input Start signal, the control output Done signal, the datapath input, and the datapath
output. The architecture section declares the two components, fsm and datapath, used in this module. These two
components are connected together using the 16 control signals and one status signal, and are declared as internal
signals using the SIGNAL declaration statement. Finally, two PORT MAP statements are used to actually connect the
control unit and the datapath together.
A sample simulation trace is shown in Figure 11.29 for the input n = 10. After asserting the Start signal in state
s0, the input value 10 is read in during state s2. The value 10 is written to RF1 at the next rising clock edge, which
also brings the FSM to state s3. RF1 is for storing n that counts from 10 down to 0. RF0 is for storing sum and is
updated after each count of n. When RF1 reaches zero, the sum 55 from RF0 is sent to the output, and the Done
♦
signal is asserted. At this point, the FSM stays in state s0 indefinitely waiting for another start signal. Start'
s0
Start
s1 s2 control word 2
input n s3 (n ≠ 0 ) control word 1
sum = 0 control word 3
sum = sum + n s4 control word 4
n=n-1 (n ≠ 0 )'
s5 (a) 407 control word 5
output sum Digital Logic and Microprocessor Design with VHDL Control
Word
1
2
3
4
5 Instruction
sum = 0
input n
sum = sum + n
n=n–1
output sum IE
15
0
1
0
0
× WE
14
1
1
1
1
0 WA1,0
13-12
00
01
00
01
×× Chapter 11 - Dedicated Microprocessors RAE
11
1
0
1
1
1 RAA1,0
10-9
00
××
00
01
00 RBE
8
1
0
1
0
0 RBA1,0
7-6
00
××
01
××
×× ALU2,1,0
5-3
101 (subtract)
×××
100 (add)
111 (decrement)
000 (pass) SH1,0
2-1
00
00
00
00
00 (b)
Figure 11.26. For Example 11.6: (a) state diagram; (b) control words.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY fsm IS PORT (
clock, reset, start: IN STD_LOGIC;
IE: OUT STD_LOGIC;
WE: OUT STD_LOGIC;
WA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
RAE: OUT STD_LOGIC;
RAA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
RBE: OUT STD_LOGIC;
RBA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
aluSel : OUT STD_LOGIC_VECTOR(2 DOWNTO 0);
shSel: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
OE: OUT STD_LOGIC;
done: OUT STD_LOGIC;
neq0: IN STD_LOGIC);
END fsm;
ARCHITECTURE fsm_arc OF fsm IS
TYPE state_type IS (s0, s1, s2, s3, s4, s5);
SIGNAL state: state_type;
BEGIN
next_state_logic: PROCESS(reset, clock)
BEGIN
IF(reset = '1') THEN
state <= s0;
ELSIF(clock'EVENT AND clock = '1') THEN
CASE state IS
WHEN s0 =>
IF(start = '1') THEN state <= s1; ELSE state <= s0; END IF;
WHEN s1 =>
state <= s2;
WHEN s2 =>
state <= s3;
WHEN s3 =>
state <= s4;
WHEN s4 =>
IF(neq0 = '1') THEN state <= s3; ELSE state <= s5; END IF;
WHEN s5 =>
state <= s0;
WHEN OTHERS =>
state <= s0;
END CASE; 408 OE
0
0
0
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors END IF;
END PROCESS;
output_logic: PROCESS(state)
BEGIN
CASE state IS
WHEN s1 =>
IE<='0'; WE<='1'; WA<="00"; RAE<='1'; RAA<="00"; RBE<='1';
RBA<="00"; aluSel<="101"; shSel<="00"; OE<='0'; done<='0';
WHEN s2 =>
IE<='1'; WE<='1'; WA<="01"; RAE<='0'; RAA<="00"; RBE<='0';
RBA<="00"; aluSel<="000"; shSel<="00"; OE<='0'; done<='0';
WHEN s3 =>
IE<='0'; WE<='1'; WA<="00"; RAE<='1'; RAA<="00"; RBE<='1';
RBA<="01"; aluSel<="100"; shSel<="00"; OE<='0'; done<='0';
WHEN s4 =>
IE<='0'; WE<='1'; WA<="01"; RAE<='1'; RAA<="01"; RBE<='0';
RBA<="00"; aluSel<="111"; shSel<="00"; OE<='0'; done<='0';
WHEN s5 =>
IE<='0'; WE<='0'; WA<="00"; RAE<='1'; RAA<="00"; RBE<='0';
RBA<="00"; aluSel<="000"; shSel<="00"; OE<='1'; done<='1';
WHEN others =>
IE<='0'; WE<='0'; WA<="00"; RAE<='0'; RAA<="00"; RBE<='0';
RBA<="00"; aluSel<="000"; shSel<="00"; OE<='0'; done<='0';
END CASE;
END PROCESS;
END fsm_arc;
Figure 11.27. FSM+D model of the control unit for the summation algorithm.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY sum IS PORT (
clock,
reset, start: IN STD_LOGIC;
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
done: OUT STD_LOGIC;
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END sum;
ARCHITECTURE Structural OF sum IS
COMPONENT fsm PORT (
clock, reset, start: IN STD_LOGIC;
IE: OUT STD_LOGIC;
WE: OUT STD_LOGIC;
WA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
RAE: OUT STD_LOGIC;
RAA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
RBE: OUT STD_LOGIC;
RBA: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
aluSel : OUT STD_LOGIC_VECTOR(2 DOWNTO 0);
shSel: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
OE: OUT STD_LOGIC;
done: OUT STD_LOGIC;
neq0: IN STD_LOGIC); 409 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors END COMPONENT;
COMPONENT datapath PORT (
clock: IN STD_LOGIC;
input: IN STD_LOGIC_VECTOR( 7 DOWNTO 0 );
IE, WE: IN STD_LOGIC;
WA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
RAE: IN STD_LOGIC;
RAA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
RBE: IN STD_LOGIC;
RBA: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
aluSel : IN STD_LOGIC_VECTOR(2 DOWNTO 0);
shSel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
OE: IN STD_LOGIC;
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0);
neq0: OUT STD_LOGIC);
END COMPONENT;
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL
SIGNAL sIE, sWE: STD_LOGIC;
sWA: STD_LOGIC_VECTOR(1 DOWNTO 0);
sRAE: STD_LOGIC;
sRAA: STD_LOGIC_VECTOR(1 DOWNTO 0);
sRBE: STD_LOGIC;
sRBA: STD_LOGIC_VECTOR(1 DOWNTO 0);
sAluSel: STD_LOGIC_VECTOR(2 DOWNTO 0);
sShSel: STD_LOGIC_VECTOR(1 DOWNTO 0);
sOE: STD_LOGIC;
sneq0: STD_LOGIC; BEGIN
-- doing structural modeling here
-- Control unit
U0: fsm PORT MAP(clock,reset,start,sIE,sWE,sWA,sRAE,sRAA,sRBE,sRBA,
sAluSel,sShSel,sOE,done,sneq0);
-- Datapath
U1: datapath PORT MAP (clock,input,sIE,sWE,sWA,sRAE,sRAA,sRBE,sRBA,
sAluSel,sShSel,sOE,output,sneq0);
END Structural;
Figure 11.28. FSM+D model of the microprocessor for the summation algorithm. . Figure 11.29. Simulation trace of the FSM+D summation algorithm with input n = 10. 410 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors 11.3.2 FSMD Model
When writing VHDL code using the FSM+D model, we need to manually construct the datapath module. The
FSM module can be constructed either manually or synthesize automatically. These two modules are then joined
together at the structural level using the control signals and status signals. Although we can automatically synthesize
the FSM, using the FSM+D method to build a microprocessor with a large datapath still requires a lot of work. We
need to first manually construct the datapath, and then we need to connect the numerous control and status signals
together between the two modules.
The FSMD (FSM with datapath) model is an abstraction used when we write VHDL code for a microprocessor.
Instead of separating the FSM and the datapath into two different modules, we combine the FSM and the datapath
together into the same entity. In other words, all the data operations that need to be performed by the datapath are
actually imbedded in the FSM coding itself as VHDL data manipulation statements. After all, there are built-in
VHDL operators to perform data manipulations. As a result, when writing VHDL code for the FSMD model, we do
not need to manually construct the datapath module, and therefore, there are no control and status signals to be
connected together. The FSM module is written using behavioral VHDL code just like with the FSM+D model.
Thus, the FSMD model is an abstraction for simplifying the VHDL code for a microprocessor.
Using the FSM+D model allows us to have full control as to what components are used in the datapath, and
how these components are used by the FSM. Whereas, using the FSMD model automates the datapath construction
process. The synthesizer now decides what components are needed by the datapath. However, whether you use the
FSM+D model or the FSMD model, the FSM is still constructed the same way using behavioral VHDL code. It is
just that for the FSMD model, the VHDL code for the FSM has extra data manipulation statements imbedded in it.
Because the FSM is still written manually, you still have full control as to what instructions are executed in what
state, and the number of states needed to execute the algorithm. Example 11.7 shows the process to construct a
microprocessor using the FSMD model.
Example 11.7
Figure 11.30 shows the complete microprocessor VHDL code using the FSMD model for the summation
algorithm of Example 9.3. Notice the simplicity of this code as compare to the code for the FSM+D model shown in
the previous section. Here, we have just one entity, which serves as both the top-level microprocessor entity and the
FSMD entity. The architecture section is written similar to a regular FSM with the different cases for each of the
states. Like before for the next-state process, the CASE statement selects the current state and determines the next
state. But in addition to setting the next state, each case (state) also contains data operation statements such as sum =
sum + n. The primary output signals are still generated in the output process. A sample simulation trace is shown in
Figure 11.31 for the input n = 10.
♦
LIBRARY ieee;
USE ieee.std_logic_1164.ALL;
USE ieee.std_logic_unsigned.ALL;
ENTITY sum IS PORT (
clock, reset, start: IN std_logic;
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
done: OUT STD_LOGIC;
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END sum;
ARCHITECTURE FSMD OF sum IS
TYPE state_type IS (s0, s1, s2, s3, s4, s5);
SIGNAL state: state_type;
SIGNAL sum: STD_LOGIC_VECTOR(7 DOWNTO 0);
SIGNAL n: STD_LOGIC_VECTOR(7 DOWNTO 0);
BEGIN
next_state_logic: PROCESS(reset, clock)
BEGIN 411 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors IF(reset = '1') THEN
state <= s0;
ELSIF(clock'EVENT AND clock = '1') THEN
CASE state IS
WHEN s0 =>
IF (start = '1') THEN state <= s1; ELSE state <= s0; END IF;
WHEN s1 =>
state <= s2;
sum <= (others => '0');
WHEN s2 =>
state <= s3;
n <= input;
WHEN s3 =>
state <= s4;
sum <= sum + n;
WHEN s4 =>
-- reading n in the following IF statement is BEFORE the
decrement
-- update, therefore, we need to compare with 1 and not 0
n <= n - 1;
IF (n /= 1) THEN state <= s3; ELSE state <= s5; END IF;
WHEN s5 =>
state <= s0;
WHEN OTHERS =>
state <= s0;
END CASE;
END IF;
END PROCESS;
output_logic: PROCESS(state)
BEGIN
CASE state IS
WHEN s5 =>
done <= '1';
output <= sum;
WHEN OTHERS =>
done <= '0';
output <= (others => 'Z');
END CASE;
END PROCESS;
END FSMD;
Figure 11.30. FSMD model of the microprocessor for the summation algorithm. 412 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Figure 11.31. Simulation trace of the FSMD summation algorithm with input n = 10. 11.3.3 Behavioral Model
The complete microprocessor can also be designed by writing VHDL code in a truly behavioral style so that
both the FSM and the datapath are synthesized automatically. Using the behavioral model to design a circuit is quite
similar to writing computer programs using a high-level language. The synthesizer, like the compiler, will translate
the VHDL behavioral description of the circuit automatically to a netlist. This netlist can then be programmed
directly onto a FPGA chip.
Since the synthesizer automatically constructs both the FSM and the datapath, therefore, you have no control
over what components are used in the datapath, and what control words are executed in what state of the FSM. Not
being able to decide what components are used in the datapath is not too big of a problem because the synthesizer
does do a good job in deciding that for you. The issue is with not being able to specify what control words are
executed in what state of the FSM. This is purely a timing issue. In some timing critical applications such as
communication protocols and real-time controls, we need to control exactly in what clock cycle a certain registertransfer operation is performed. In other words, we need to be able to assign a control word to a specific state of the
FSM.
Behavioral VHDL code offers all the basic language constructs that are available in most computer
programming languages such as variable assignments, FOR LOOPs and IF-THEN-ELSEs. These statements, written
within a process block, are executed sequentially. Besides the fact that we have no control over which clock cycle
each operation is performed in, using the behavioral model is very powerful and simple. If timing is not an issue for
your circuit, then between the different models, this should be the method of choice.
Example 11.8
Figure 11.32 shows the VHDL code using the behavioral model for the summation algorithm of Example 9.3.
Note that some VHDL synthesizers such as MaxPlus+II do not allow the use of loops that cannot be unrolled, i.e.,
loops with variables for their starting or ending value whose values are unknown at compile time. Hence, in the
code, the value for the variable n cannot be from a user input.
♦
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY sum IS PORT (
start: IN STD_LOGIC;
done: OUT STD_LOGIC;
output: OUT INTEGER);
END sum;
ARCHITECTURE Behavioral OF sum IS
BEGIN 413 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors PROCESS
VARIABLE n: integer;
VARIABLE sum: integer;
BEGIN
IF (start = '0') THEN
done <= '0';
output <= 0;
ELSE
sum := 0;
n := 10; -- cannot be user input
FOR i IN n DOWNTO 1 LOOP
sum := sum + i;
END LOOP;
done <= '1';
output <= sum;
END IF;
END PROCESS;
END Behavioral;
Figure 11.32. Behavioral model of the microprocessor for the summation algorithm. 11.4 Summary Checklist
Be able to manually construct the complete circuit for a dedicated microprocessor from a given algorithm
Be able to construct a dedicated microprocessor from a given algorithm using the FSM+D model with
VHDL
Be able to construct a dedicated microprocessor from a given algorithm using the FSMD model with
VHDL
Be able to construct a dedicated microprocessor from a given algorithm using the behavioral model with
VHDL 11.5 Problems
11.1 Manually construct a dedicated microprocessor to enter two 8-bit numbers, and output the larger of the two
numbers. The two numbers are entered through two separate input ports. Implement the circuit on the UP2
board.
11.2 Manually construct a dedicated microprocessor to enter two 8-bit numbers, and output the larger of the two
numbers. The two numbers are entered through one input port. Assume that the two input numbers are always
available at the input port whenever it is needed.
11.3 Implement the circuit from Problem 11.2 on the UP2 board. Notice that in order for the number to be inputted
correctly, you cannot assume that the two input numbers are always available. Instead, you need an extra
Enter signal to tell the FSM when the data is available.
11.4 Manually construct a dedicated microprocessor to enter one 8-bit number. Output a 1 if the number has five 1
bits, otherwise, output a 0. 414 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors 11.5 Manually construct a dedicated microprocessor to enter two 8-bit numbers. Output a 1 if the two numbers
together has five 1 bits, otherwise, output a 0.
11.6 Manually construct a dedicated microprocessor to enter two numbers, and output the product of these two
numbers.
11.7 Manually construct a dedicated microprocessor to enter two numbers. Output a 1 if the first number is
divisible by the second number, otherwise, output a 0.
11.8 Manually construct a dedicated microprocessor to enter three numbers, and output the larger of the three
numbers.
11.9 Manually construct a dedicated microprocessor to enter three numbers. Output the three numbers in
ascending order.
11.10 Manually construct a dedicated microprocessor to evaluate the factorial of n. The algorithm is shown below
product = 1
input n
while (n > 1){
product = product * n
n=n–1
}
output product
11.11 Manually construct a dedicated microprocessor to enter several numbers until a 0 is entered. Output the
largest and second largest of the numbers entered.
11.12 Manually construct a dedicated microprocessor to input one 8-bit value, and then determine whether the input
value has an equal number of 0 and 1 bits. The microprocessor outputs a 1 if the input value has the same
number of 0’s and 1’s, otherwise, it outputs a 0. For example, the number 10111011 will produce a 0 output,
whereas, the number 00110011 will produce a 1 output. The algorithm is as follows:
1
2
3
4
5
6
7
8
9 Count
input
while
if =0
N
(N ≠ 0){
(N(0) = 1) then
Count = Count +
end if
N = N >> 1 }
output (Count = 4) // for counting the number of 1 bits // least significant bit of N
1
// shift N right one bit
// output 1 if the test (Count = 4) is true Answer:
Custom datapath 415 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors Input_N
8 Shift_N
Load_N
Clear
Clock Shift
D
Load 8-bit shifter
Clear with load
N 8 Neq0
N0eq1
Count 8 N0 4-bit
Count
Clear up counter
Q3 Q2 Q1 Q0 Output
000 input N 001
(N = 0 )' &
(N(0) = 1)
Count = Count + 1 010
N = N >> 1 (N = 0) 100 output (Count = 4) (N = 0)' &
(N(0) = 1)'
011 N = N >> 1 (a)
Current State
Q2Q1Q0
000
001
010
011
100 Next State Q2next Q1next Q0next
(N=0), (N(0)=1)
00
01
10
11
001
001
001
001
011
010
100
100
001
001
001
001
001
001
001
001
100
100
100
100
(b) 416 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors D2 Q2 = 0
(N=0),(N(0)=1)
Q1Q0
00 01 11 10 Q2 = 1
00 01 11 10 1 1 1 1 00
01 1 Q2Q1'Q0'
Q2'Q1'Q0 (N=0) 1 11
10 D2 = Q2Q1'Q0' + Q2'Q1'Q0 (N=0)
D1 Q2 = 0
(N=0),(N(0)=1)
Q1Q0
00 01 11 10 Q2 = 1
00 01 11 10 00
01 1 Q2'Q1'Q0 (N=0)' 1 11
10 D1 = Q2'Q1'Q0 (N=0)'
D0 Q2 = 0
(N=0),(N(0)=1)
Q1Q0
00 01 11 10
00 1 01 1 1 1 1 1 1 01 11 10 1 10 00 1 11 Q2 = 1 1 1 1 1 Q2'Q0'
Q2' (N=0)' (N(0)=1)'
Q2'Q1 D0 = Q2'Q0' + Q2' (N=0)' (N(0)=1)' + Q2'Q1
(c)
Control
Word
0
1
2
3
4 State
Q2 Q1 Q0
000
001
010
011
100 Instruction
Input N
none
Count = Count + 1, N = N >> 1
N = N >> 1
Output (Count = 4)
(d) Load_N = Q2'Q1'Q0'
Shift_N = Q1
Count = Q2'Q1Q0'
Done = Q2
(e) 417 Load_N
1
0
0
0
0 Shift_N
0
0
1
1
0 Count
0
0
1
0
0 Done
0
0
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors D2 Load_N Q2 Shift_N Clk
Clr Q'2 Count
Done D1 Q1 Clk
Clr D0 Q'1 Q0 Clk
Clr Clock
Reset Q'0 Neq0
N0eq1
(f)
Input X
8
Input_N
Shift_N
Load_N
Count Control
Signals Shift_N
Load_N
Count CU
Clock
Reset Clock
Reset DP
Neq0
N0eq1 Status
Signals Neq0
N0eq1 Clock
Reset Done Output Done Output
(g) 11.13 Assume that the following control unit and datapath circuits are used to construct a dedicated microprocessor.
Determine the instructions being executed in each state of the FSM. Write out the complete algorithm that the
resulting microprocessor will execute. In other words, write out the pseudo-code for the algorithm. Briefly
describe what the algorithm does. 418 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors (Z≠0) D1 Q1 ClrX
LoadY
inZ
LoadX
stat1
LoadZ
subtract Clk
Clear D0 Q'1 Q0 Clk
Clear Clock
Reset Q'0 input '4'
8 8
1 inZ LoadX
ClrX D7-0
Load 8-bit register
Clear
X
Clock Q7-0 D7-0
Load 8-bit register
Clear
Y
Clock Q7-0 0 D7-0
Load 8-bit register
Clear
Z
Clock Q7-0 Clock
LoadY
LoadZ 8
8 '1' 8
1 0 1 0 stat1
+/- subtract 8 output Answer:
Excitation equations / Next-state equations:
D1 = Q1next = Q0 + (Z≠0)'
D0 = Q0next = Q1 + Q0'
Output equations:
ClrX = LoadY = inZ = Q1'Q0'
LoadX = stat1 = Q1'Q0
LoadZ = Q0'
subtract = Q1Q0'
Next-state table / Output table: 419 (Z≠0) 8 Digital Logic and Microprocessor Design with VHDL Current
State
Q1Q0
00
01
10
11 Next State
Q1next Q0next
(Z≠0) = 0
(Z≠0) = 1
11
01
10
10
11
01
11
11 Chapter 11 - Dedicated Microprocessors Outputs
ClrX
1
0
0
0 LoadX
0
1
0
0 LoadY
1
0
0
0 LoadZ
1
0
1
0 inZ
1
0
0
0 State diagram: 0
(Z≠0)
ClrX=0
LoadX=1
LoadY=0
LoadZ=0
inZ=0
stat1=1
subtract=0 ClrX=1
LoadX=0
LoadY=1
LoadZ=1
inZ=1
stat1=0
subtract=0
(Z≠0)' 1 3 (Z≠0)
(Z≠0)'
2
ClrX=0
LoadX=0
LoadY=0
LoadZ=1
inZ=0
stat1=0
subtract=1 State 0:
X=0
Y=4
input Z
State 1:
X=X+Y
State 2:
Z=Z–1
State 3:
none
Algorithm: Outputs 4 × Z, where Z is an input value.
X=0
Y=4
input Z
while (Z ≠ 0){ 420 ClrX=0
LoadX=0
LoadY=0
LoadZ=0
inZ=0
stat1=0
subtract=0 stat1
0
1
0
0 subtract
0
0
1
0 Digital Logic and Microprocessor Design with VHDL Chapter 11 - Dedicated Microprocessors X=X+Y
Z=Z–1
}
11.14 Write the VHDL code for Problems 11.1 to 11.9 using the FSM+D model.
11.15 Write the VHDL code for Problems 11.1 to 11.9 using the FSMD model. 421 Chapter 12 General-Purpose Microprocessors Control
Inputs Data
Inputs
Control Unit 8 '0' Datapath MUX
ff
Nextstate
Logic State
Memory Output
Logic Register Control
Signals Status
Signals
Control
Outputs ALU 8 ff Register 8
Data
Outputs Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors Unlike a dedicated or custom microprocessor that is capable of performing only one function, a general-purpose
microprocessor, such as the Pentium CPU, is capable of performing many different functions under the direction
of program instructions. Given a different instruction set or program, the general-purpose microprocessor will
perform a different function. However, a general-purpose microprocessor can also be viewed as a dedicated
microprocessor because it is made to perform only one function, and that is to execute the program instructions. In
this sense, we can design and construct a general-purpose microprocessor in the same way that we constructed the
dedicated microprocessors in the last chapter. 12.1 Overview of the CPU Design
A general-purpose microprocessor is often referred to as the central processing unit (CPU). The CPU is simply
a dedicated microprocessor that only executes software instructions. Figure 12.1 shows an overview of a generalpurpose microprocessor. The following discussion references this diagram.
Input
Control
Unit
fetch Datapath
+1
PC control
signals address
Memory decode execute
instruction1 execute
instructionn status
signals IR instruction Datapath
for
executing
all the
instructions control
signals Output
Figure 12.1. Overview of a general-purpose microprocessor.
In designing a CPU, we must first define its instruction set, and how the instructions are encoded and executed.
We need to answer questions such as how many instructions do we want? What are the instructions? What operation
code (opcode) do we assign to each of the instructions? How many bits do we use to encode an instruction?
Once we have decided on the instruction set, we can proceed to designing a datapath that can execute all the
instructions in the instruction set. In this step we are creating a custom datapath, so we need to answer questions
such as what functional units do we need? How many registers do we need? Do we use a single register file or
separate registers? How are the different units connected together?
Creating the datapath for a general-purpose microprocessor is exactly the same as creating the datapath for a
dedicated microprocessor. However, in addition to being able to perform all the instructions in the instruction set,
there are other data operations and registers that must be included in the datapath for the general-purpose
microprocessor. These data operations and registers deal with how the general-purpose microprocessor fetches the
instructions from memory and executes them. In particular, there is a program counter (PC) that contains the
memory location of where the next instruction is stored. There is an instruction register (IR) for storing the
instruction being fetched from the memory. Each time an instruction is fetched from a memory location pointed to
by the PC, normally the PC must be incremented to the next memory location for the next instruction. Alternatively,
if the instruction is a jump instruction, the PC must be loaded with a new memory address instead. 423 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors The control unit for a general-purpose microprocessor basically cycles through three main steps, usually
referred to as the instruction cycle: 1) fetches an instruction; 2) decodes the instruction; and 3) executes the
instruction. Each step is executed in one state of the finite-state machine. For step 3, each instruction is usually
executed in one clock cycle, although some memory access instructions may require two or more clock cycles to
complete, hence they may require several states for correct timing.
For fetching the instruction in step 1, the control unit simply reads the memory location specified by the PC,
and copies the content of that location into the instruction register (IR). The PC is then incremented by 1 (assuming
that each instruction occupies one memory location). For decoding the instruction in step 2, the control unit extracts
the opcode bits from the instruction register and determines what the current instruction is by jumping to the state
that is assigned for executing that instruction. Once in that particular state, the control unit performs step 3 by simply
asserting the appropriate control signals for controlling the datapath to execute that instruction.
Instructions for the program are usually stored in external memory, so in addition to the CPU, there is external
memory that is connected to the CPU via an address bus and a data bus. Hence, step 1 (fetch an instruction) usually
involves the control unit setting up a memory address on the address bus and telling the external memory to output
the instruction from that memory location onto the data bus. The control unit then reads the instruction from the data
bus. To keep our design simple, instead of having external memory, we will include the memory as part of the
datapath. This way, we do not have to worry about the handshaking and timing issues involved for accessing
external memory. 12.2 The EC-1 General-Purpose Microprocessor
This first version of the EC1 computer is extremely small and very limited as to what it can do, and therefore, its
general-purpose microprocessor is very “E-Cee” to design manually. In order to keep the manual design of the
microprocessor manageable, we have to keep the number of variables small. Since these variables determine the
number of states and input signals for the finite-state machine, therefore these factors have to be kept to the bare
minimum. Nevertheless, the building of this computer demonstrates how a general-purpose microprocessor is
designed, and how the different components are put together. After this exercise, you will quickly appreciate the
power of designing with VHDL at a higher abstraction level, and the use of an automatic synthesizer.
We will now design the general-purpose microprocessor for our EC-1 computer. After which, we will interface
this microprocessor to external I/O’s, and implement the complete computer using the FLEX FPGA chip on the UP2
development board to make it into a real working general-purpose computer. Using the few instructions available,
we will then write a program to execute on the EC-1, and see that it actually works! 12.2.1 Instruction Set
The instructions that our EC-1 general-purpose microprocessor can execute, and the corresponding encodings
are defined in Figure 12.2. The Instruction column shows the syntax and mnemonic to use for the instruction when
writing a program in assembly language. The Encoding column shows the binary encoding defined for the
instruction, and the Operation column shows the operation of the instruction.
As we can see from Figure 12.2, our little computer’s instruction set has only five instructions. To encode five
instructions, the operation code (opcode) will require three bits – giving us eight different combinations. As shown
in the Encoding column, the first three most significant bits are the opcode given to an instruction. For example, the
opcode for the IN A instruction is 011, and the opcode for OUT A is 100, and so on. The three encodings, 000, 001,
and 010, are not defined, and can be used as a no-operation (nop) instruction. Since the width of each instruction is
fixed at eight bits, the last five bits are not used by all the instructions except for the JNZ instruction. Normally, for a
more extensive instruction set, these extra bits are used as operand bits to specify what registers or other resources to
use. In our case, only the JNZ instruction uses the last four bits, designated as aaaa, to specify an address in the
memory to jump to.
The IN A instruction inputs an 8-bit value, and stores it into the accumulator (A) . The accumulator is an 8-bit
register for performing data operations. The OUT A instruction copies the content of the accumulator to the output
port. For the EC-1, the content of the accumulator is always available at the output port, so this OUT A instruction is
1 “EC” is the acronym for Enoch’s Computer. 424 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors not really necessary. It is only included just because a program should have an output instruction. The DEC A
instruction decrements the content of A by 1, and stores the result back into A. The JNZ (Jump Not Zero) instruction
tests to see if the value in A is equal to 0 or not. If A is equal to 0, then nothing is done. If A is not equal to 0, then
the last four bits of the instruction, designated as aaaa in the encoding, is loaded into the PC. The four bits, aaaa,
represent a memory address. When this value is loaded into the PC, we are essentially performing a jump to this new
memory address, since the value stored in the PC is the location for the next fetch operation. Finally, the HALT
instruction halts the CPU by having the control unit stay in the halt state indefinitely until reset.
Instruction
IN A
OUT A
DEC A
JNZ address
HALT Encoding
011 ×××××
100 ×××××
101 ×××××
110 ×aaaa
111 ××××× Operation
A ← input
output ← A
A←A–1
if (A != 0) then PC = aaaa
Halt Comment
Input to A
Output from A
Decrement A
Jump to address if A is not zero
Halt execution Notations:
A
= accumulator.
PC = program counter.
aaaa = four bits for specifying a memory address.
×
= don’t cares.
Figure 12.2. Instruction set for the EC-1. 12.2.2 Datapath
Having defined the instruction set for the EC-1 general-purpose microprocessor, we are now ready to design the
custom datapath that will execute all the operations as defined by all the instructions. The custom datapath for the
EC-1 is shown in Figure 12.3.
The datapath can be viewed as having three separate parts: 1) for performing the instruction cycle operations of
fetching an instruction, and incrementing or loading the PC, 2) the memory, and 3) for performing the data
operations for all the instructions in the instruction set.
The portion of the datapath for performing the instruction cycle operations basically contains the instruction
register (IR), and the program counter (PC). The bit width of the instructions determine the size of the IR, whereas,
the number of addressable memory locations determines the size of the PC. For this datapath, we want a memory
with 16 locations, each being 8-bits wide, so we need a 4-bit (24 = 16) address. Hence, the PC is 4-bits wide, and the
IR is 8-bits wide. A 4-bit increment unit is used to increment the PC by 1. The PC needs to be loaded with either the
result of the increment unit, or the address from the JNZ instruction. A 2-to-1 multiplexer is used for this purpose.
One input of the multiplexer is from the increment unit, and the other input is from the four least significant bits of
the IR, IR3-0.
As mentioned in Section 12.1, to keep our design simple, instead of having external memory, we have included
the memory as part of the datapath. In this design, the memory is a 16 locations × 8 bits wide read-only memory
(ROM). Since, the instruction set does not have an instruction that writes to memory, we only need a read-only
memory. The output of the PC is connected directly to the 4-bit memory address lines, since the memory location is
always determined by the content of the PC. The 8-bits memory output Q7..0 is connected to the input of the IR for
executing the instruction fetch operation.
The portion of the datapath for performing the instruction set operations includes the 8-bit accumulator A, and
an 8-bit decrement unit. A 2-to-1 multiplexer is used to select the input to the accumulator. For the IN A instruction,
the input to the accumulator is from the data input port, whereas for the DEC A instruction, the input is from the
output of the decrement unit, which performs the decrement of A. The output of the accumulator is directly
connected to the data output port, hence the OUT A instruction does not require any specific datapath actions.
Furthermore, with this direct connection, it is equivalent to always performing the OUT A instruction. The JNZ
instruction requires the 8-input OR gate connected to the output of the accumulator to test for the condition (A ≠ 0). 425 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors The actual operation required by the JNZ instruction is to load the PC with the four least significant bits of the IR.
The HALT instruction also does not require any specific datapath actions.
The control word for this custom datapath has five control signals, IRload, PCload, INmux, Aload, and JNZmux.
The datapath provides one status signal, Aneq0, to the control unit. The control words for executing the instruction
cycle operations and the instruction set operations are discussed in the next section.
Input
IRload IR7-5 D7-0
Load 8-bit register
Clear
IR
IR7-5 IR3-0 4 3 4 1 JNZmux 8
8 0 4
D3-0
Load 4-bit register
Clear PC
PC3-0 PCload 1 4-bit
increment 8 Q7-0 4 16 locations × 8bits
ROM
Address3-0 0 D7-0
Load 8-bit register
Clear
A
A7-0 INmux
Aload
Reset
Clock 8
8 8 8-bit
decrement Aneq0
Output
Instruction Cycle Operations Memory Instruction Set Operations Figure 12.3. Datapath for the EC-1. 12.2.3 Control Unit
The state diagram for the control unit is shown in Figure 12.4 (a), and the actions that are executed, specifically
the control signals that are asserted in each state are shown in (d). The first START state 000 serves as the initial
reset state. No action is performed in this state. In addition, this START state provides one extra clock cycle for
instructions that require an extra clock cycle to complete its operation. Although, for those instructions that do not
require this extra clock cycle to complete its operation should technically go back to the FETCH state instead, we
have made all instructions go back to the START state so that the next-state table and the excitation equations are
simpler.
Of the five instructions, only the JNZ instruction requires an extra clock cycle to complete its operation. This is
because if the condition is tested true, the PC is loaded with a new address value. This new address value is actually
loaded into the PC at the beginning of the next clock cycle. So if we have the FSM go to the FETCH state in the
next clock cycle, then the IR will be loaded with the memory from the old address, and not the new address.
However, by making the FSM go back to the START state, the PC will be updated with the new address in this
state, and the memory will be accessed during the next FETCH state from the new address.
From the START state, the control unit goes to the FETCH state unconditionally. In the FETCH state, the IR is
loaded with the memory content in the location specified by the PC by asserting the IRload signal. Furthermore, the
PC is incremented by 1, and the result is loaded back to the PC by asserting the PCload signal. The DECODE state
tests the three most significant bits of the IR, IR7-5, and goes to the corresponding state as encoded by the three bit
opcode for executing the instruction. 426 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors In the five execute states corresponding to the five instructions, the appropriate control signals for the datapath
are asserted to execute that instruction. For example, the IN A instruction requires setting the INmux signal to a 1 for
the input multiplexer, and setting the Aload signal to a 1 to load the input value into A. Notice that in order for the
input instruction to read in the correct value, the input value must be set up first before resetting the CPU.
Furthermore, since the INPUT state does not wait for an Enter key signal, therefore, only one value can be read in,
even if there are multiple input statements. The DEC A instruction requires setting INmux to 0 and Aload to 1 so that
the output from the decrement unit is routed back to the accumulator, and gets loaded in.
The JNZ instruction asserts the JNZmux signal to route the four address bits from the IR to the PC. Whether the
PC actually gets loaded with this new address depends on the condition of the Aneq0 status signal. Hence, the
PCload control signal is asserted only if Aneq0 is a 1. By asserting the PCload signal conditionally depending on the
status signal Aneq0, the state diagram will require one less state, thus making the finite-state machine smaller, and
making it into a Mealy FSM. Otherwise, the FSM will need two states for the JNZ instruction: one state for asserting
the PCload signal when Aneq0 is true, and one state for de-asserting the PCload signal when Aneq0 is false.
Once the FSM enters the HALT state, it unconditionally loops back to the HALT state, giving the impression
that the CPU has halted.
The next-state / implementation table for the state diagram, and the three excitation equations as derived from
the implementation table are shown in Figure 12.4 (b) and (c) respectively. With eight states, three D flip-flops are
used for the implementation of the control unit circuit. Notice that the derivation of the excitation equations is fairly
straight forward since most of the entries in the table contain 0’s. Only the decode state row contains different
values.
The output equations shown in Figure 12.4 (e) are derived directly from the output table in (d).
Finally, we can derive the circuit for the control unit based on the excitation equations and the output equations.
The complete control unit circuit for the EC-1 general-purpose microprocessor is shown in Figure 12.4 (f). start
000 fetch
001 decode
010 IR[7..5] =
011
input
011 IR[7..5] =
100
output
100 IR[7..5] =
101
dec
101 IR[7..5] = 000, 001, 010 IR[7..5] =
110
jnz
110 (a) 427 IR[7..5] =
111
halt
111 Digital Logic and Microprocessor Design with VHDL Current State
Q2Q1Q0
000 start
001 fetch
010 decode
011 input
100 output
101 dec
110 jnz
111 halt Chapter 12 - General-Purpose Microprocessors Next State Q2next Q1next Q0next (Implementation D2 D1 D0)
IR7, IR6, IR5
000 001 010 011
100
101 110 111
nop nop nop input output dec jnz halt
001 001 001 001
001
001 001 001
010 010 010 010
010
010 010 010
000 000 000 011
100
101 110 111
000 000 000 000
000
000 000 000
000 000 000 000
000
000 000 000
000 000 000 000
000
000 000 000
000 000 000 000
000
000 000 000
111 111 111 111
111
111 111 111
(b) D2 = Q2Q1Q0 + Q2'Q1Q0'IR7
D1 = Q2Q1Q0 + Q2'Q1Q0' (IR6IR5 + IR7IR6) + Q2'Q1'Q0
D0 = Q2Q1Q0 + Q2'Q1Q0' (IR6IR5 + IR7IR5) + Q2'Q1'Q0'
(c)
Control
Word
0
1
2
3
4
5
6
7 State
Q2Q1Q0
000 start
001 fetch
010 decode
011 input
100 output
101 dec
110 jnz
111 halt IRload
0
1
0
0
0
0
0
0 PCload
0
1
0
0
0
0
if (Aneq0 = 1) then 1 else 0
0 INmux
0
0
0
1
0
0
0
0 Aload
0
0
0
1
0
1
0
0 (d)
IRload = Q2'Q1'Q0
PCload = Q2'Q1'Q0 + Q2Q1Q0' (Aneq0)
INmux = Q2'Q1Q0
Aload = Q2'Q1Q0 + Q2Q1'Q0
JNZmux = Q2Q1Q0'
Halt
= Q2Q1Q0
(e) 428 JNZmux
0
0
0
0
0
0
1
0 Halt
0
0
0
0
0
0
0
1 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors IRload
D2 Q2 Clk
Clear PCload
Q'2 INmux
D1 Q1 Clk
Clear Aload
Q'1 JNZmux
Halt
D0 Q0 Clk
Clear Clock
Reset Q'0 Aneq0
IR7
IR6
IR5 (f)
Figure 12.4. Control unit for the EC-1: (a) state diagram; (b) next-state / implementation table; (c) excitation
equations; (d) control word/output table; (e) output equations; (f) circuit. 12.2.4 Complete Circuit
The complete circuit for the EC-1 general-purpose microprocessor is constructed by connecting the datapath
from Figure 12.3 and the control unit from Figure 12.4 (f) together using the designated control and status signals as
shown in Figure 12.5. 429 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors Input
8
Input
IRload
PCload
INmux
CU Aload
JNZmux
Clock
Reset Clock
Reset IR7-5
Aneq0 Control
Signals Status
Signals IRload
PCload
INmux
Aload
JNZmux DP Clock
Reset IR7-5
Aneq0 Halt Output 8
Halt Output Figure 12.5. Complete circuit for the EC-1 general-purpose microprocessor. 12.2.5 Sample Program
Dedicated microprocessors, as discussed in the previous chapter, have the algorithm built right into the circuit
of the microprocessor. General-purpose microprocessors, on the other hand, do not have an algorithm built into it. It
is designed only to execute program instructions fetched from the memory. Hence, in order to test out the EC-1
computer, we need to write a program using the instructions available in the instruction set.
There are only five instructions in the instruction set defined for the EC-1 as shown in Figure 12.2. For our
sample program, we will use these five instructions to write a program to input a number, and then count down from
this input number to 0. The program is shown in Figure 12.6 (a).
Since we do not have a compiler for the EC-1, we need to manually compile this program by hand. The binary
executable code for this program is shown in Figure 12.6 (b). The binary code is obtained by replacing each
instruction with its corresponding 3-bit opcode as listed in Figure 12.2, followed by five bits for the operand. All the
instructions, except for the JNZ instruction, do not use these five operand bits, so either a 0 or a 1 can be used. From
Figure 12.2, we find that the opcode for the IN A instruction is 011, therefore, the encoding for this first instruction
is 01100000. Similarly, the opcode for the OUT A instruction is 100, therefore, the encoding used is 10000000.
For the JNZ instruction, the least significant four bits represent the memory address to jump to if the condition
is true. In the example, we are assuming that the first instruction, IN A, is stored in memory location 0000. Since the
JNZ instruction jumps to the second instruction, OUT A, which will be stored in memory location 0001, therefore,
the four address bits for the JNZ instruction are 0001. The opcode for the JNZ instruction is 110, hence, the
encoding for the complete JNZ instruction is 11000001.
IN A
loop: OUT A
DEC A
JNZ loop
HALT ------ input a value into the A register
output the value from the A register
decrement A by one
go back to loop if A is not zero
halt
(a) 430 Digital Logic and Microprocessor Design with VHDL memory
address
0000
0001
0010
0011
0100 instruction
encoding
01100000;
10000000;
10100000;
11000001;
11111111; ------ Chapter 12 - General-Purpose Microprocessors IN A
OUT A
DEC A
JNZ loop
HALT
(b) Figure 12.6. Countdown program to run on the EC-1: (a) assembly code; (b) binary code.
Normally, the program instructions are stored in memory that is external to the CPU, and the computer (with the
help of the operating system) will provide means to independently load the instructions into the memory. However,
to keep our design simple, we have included the memory as part of the CPU inside the datapath. Furthermore, we do
not have an operating system for loading the instructions into the memory separately. Therefore, our program must
be “loaded” into the memory before the synthesis of the datapath and the microprocessor. The memory circuit that
we have used is from Altera’s LPM component library. Refer to the Altera documentation for more information on
the usage of this component. This memory is initialized with the content of the text file named PROGRAM.MIF.
Therefore, to load our memory with the program, we must enter the binary encoding of the program in this text file,
and then re-synthesize the microprocessor circuit.
The content of this PROGRAM.MIF file for the countdown program is shown in Figure 12.7. Texts after the
two hypens ( --) are comments, and all the capitalized words are keywords.
Remember to re-synthesize your computer every time you make changes to the PROGRAM.MIF file. For this
purpose, you may want to check the Processing | Smart Recompile option under the Compiler menu. With this
option checked, the compiler will not have to re-synthesize the entire computer circuit.
-- Content of the ROM memory in the file PROGRAM.MIF
DEPTH = 16; -- Number of memory locations: 4-bit address
WIDTH = 8;
-- Data Width of memory: 8-bit data
ADDRESS_RADIX = BIN;
DATA_RADIX = BIN; -- Specifies the address values are in binary
-- Other valid radixes are HEX, DEC, OCT, BIN
-- Specifies the data values are in binary -- Specify memory content.
-- Format of each memory location is
-address : data
CONTENT
BEGIN
[0000..1111] : 00000000; -- Initialize locations range 00-1F to 0 -- Program to countdown from an input to 0
0000
0001
0010
0011
0100 :
:
:
:
: 01100000;
10000000;
10100000;
11000001;
11111111; ------ input A
output A
decrement A
jnz 0001
halt END; 431 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors Figure 12.7. The PROGRAM.MIF file containing the Countdown program to run on the EC-1. 12.2.6 Simulation
Having typed in the countdown program into the PROGRAM.MIF file, and re-synthesize the computer, we can
now run a simulation of the microprocessor executing the program. A sample simulation of the microprocessor is
shown in Figure 12.8 showing the count down from the input 3 on the Output signal. This complete microprocessor
circuit is on the accompanying CD-ROM in the file MP.GDF, and can be synthesized and simulated. Figure 12.8. A sample simulation trace of the countdown program running on the EC-1 starting at the input 3. 12.2.7 Hardware Implementation
A complete computer, according to the Von Neumann model, includes not only the microprocessor, but also the
memory, input and output devices. We have so far only constructed the general-purpose microprocessor with the
built-in memory for the EC-1 as shown in Figure 12.5. The final step in building the complete EC-1 computer is to
add the input and output devices. Figure 12.9 shows the interface between the microprocessor, and the input and
output devices on the UP2 development board. The input simply consists of eight DIP switches, and the output is the
two 7-segment LED display. Since the microprocessor outputs an 8-bit binary number, we need an 8-bit binary to
two digit BCD (binary coded decimal) decoder so that we can see the 8-bit binary number as two decimal digits on
the two 7-segment LED display. A single LED is used to show when the microprocessor has halted. A push button
switch is used as the reset button. Finally, the on-board 25MHz clock is slowed down with a clock divider circuit.
The reason for using the slower clock speed is so that we can see some intermediate results on the display.
This complete hardware implementation circuit is in the file UP2FLEX.GDF in the appropriate directory on the
accompanying CD-ROM. You can use this file to download this complete EC-1 computer onto the UP2
development board, and see the count down on the 7-segment display from any given input provided by the eight
DIP switches. The FLEX_PB1 pushbutton switch is the Reset switch. The output is shown on the two 7-segment
LED as a decimal number. The decimal point on the right digit is lit if the number is in the 100’s. The decimal point
on the left digit is the Halt light, and is lit when the program terminates with the HALT instruction. 432 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors
8 DIP switches
8 Reset push button switch
FLEX_PB1
25MHz Clock Input Reset
Clock Divider
25MHz
4Hz EC-1
microprocessor
Clock Output
8 binary to
BCD
decoder two
7-segment
LEDs Halt LED
left digit decimal point Figure 12.9. Hardware implementation of the EC-1 computer. 12.3 The EC-2 General-Purpose Microprocessor
For our next example, we will design the general-purpose microprocessor for a second version of the EC
computer, the EC-2. 12.3.1 Instruction Set
The instruction set for the EC-2 general-purpose microprocessor has eight instructions as shown in Figure
12.10. The reason for keeping this number at eight is so that we can still use only three bits to encode them.
The LOAD instruction loads the content of the memory at the specified address into the accumulator A. The
address is specified by the five least significant bits of the instruction. The STORE instruction is similar to the
LOAD except that it stores the value in A to the memory at the specified address. The ADD and SUB instructions
respectively adds and subtracts the content of A with the content in a memory location, and stores the result back
into A. The IN instruction inputs a value and stores it into A. The JZ instruction loads the PC with the specified
address if A is zero. Loading the PC with a new address simply causes the CPU to jump to this new memory
location. The JPOS instruction loads the PC with the specified address if A is positive. The value in A is taken as a
2’s complement signed number, so a positive number is one where the most significant bit of the number is a 0.
Finally, the HALT instruction halts the CPU.
Instruction
LOAD A, address
STORE A, address
ADD A, address
SUB A, address
IN A
JZ address
JPOS address
HALT
Notations:
A
M
PC
aaaaa
× Encoding
000 aaaaa
001 aaaaa
010 aaaaa
011 aaaaa
100 ×××××
101 aaaaa
110 aaaaa
111 ××××× Operation
A ← M[aaaaa]
M[aaaaa] ← A
A ← A + M[aaaaa]
A ← A – M[aaaaa]
A ← input
if (A = 0) then PC = aaaaa
if (A > 0) then PC = aaaaa
Halt = accumulator.
= memory.
= program counter.
= four bits for specifying a memory address.
= don’t cares. Figure 12.10. Instruction set for the EC-2. 433 Comment
Load A with content of memory location aaaaa
Store A into memory location aaaaa
Add A with M[aaaaa] and store the result back into A
Subtract A with M[aaaaa] and store result back into A
Input to A
Jump to address if A is zero
Jump to address if A is positive
Halt execution Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors 12.3.2 Datapath
The custom datapath for the EC-2 is shown in Figure 12.11.
The portion of the datapath for performing the instruction cycle operations is very similar to that of the previous
EC-1 with the instruction register (IR), the program counter (PC), and the increment unit for incrementing the PC.
The minor differences between the two are in the size of the PC, and the increment unit. In this second design, we
want a memory with 32 locations, hence the address, and therefore, the size of the PC and the increment unit must
all be 5-bits wide.
The main modification to this portion of the datapath is in the addition of a second 2-to-1 multiplexer that is
connected between the output of the PC and the memory address input. One input of this multiplexer comes from
the PC, and the other input comes from the five least significant bits of the IR, IR4-0. The reason for this is because
there are now two different types of operations that can access memory. The first is still for the fetch operation
where the memory address is given by the content of the PC. The second type is for the four instructions, LOAD,
STORE, ADD, and SUB, where they use the memory as an operand. Hence, the memory address for these four
instructions is given by the five least significant bits of the IR. The select signal for this multiplexer is Meminst.
The memory size for the EC-2 is increased to 32 locations, thus requiring five address bits. The memory is still
included as part of the datapath rather then as an independent external unit to the CPU. In order to accommodate the
STORE instruction for storing the value of A into the memory, we need to use a RAM instead of the previous ROM.
To realize this operation, the output of the accumulator A is connected to the memory data input D7-0. The signal
MemWr, when asserted, causes the memory to write the value from register A into the location specified by the
address in the instruction.
The output of the memory at Q7-0 is connected to both the input of the IR and to the input of the accumulator A
through a 4-to-1 multiplexer. The connection to the IR is for the fetch operation just like in the EC-1 design. The
connection to the accumulator is for performing the LOAD instruction where the content of the memory is loaded
into A. Since the memory is only one source among two other sources that is loaded into A, therefore, the
multiplexer is needed.
The portion of the datapath for performing the instruction set operations includes the 8-bit accumulator A, an 8bit adder-subtractor unit, and a 4-to-1 multiplexer. The adder-subtractor unit performs the ADD and SUB
instructions. The Sub signal, when asserted selects the subtraction operation, and when de-asserted selects the
addition operation. The 4-to-1 multiplexer allows the accumulator A input to come from one of three sources. For
the ADD and SUB instructions, the A input comes from the output of the adder-subtractor unit; for the IN
instruction, the A input comes from the input port; and for the STORE instruction, the A input comes from the
output of the memory. The selection of this multiplexer is through the two signal lines Asel1-0. The fourth input of
the multiplexer is not used.
Similar to the previous EC-1 design, the output port is connected directly to the output of the accumulator A.
Therefore, the value of the accumulator is always available at the output port, and no specific output instruction is
necessary to output the value in A.
For the two conditional jump instructions, JZ and JPOS, the datapath provides the two status signals, Aeq0 and
Apos respectively, that are generated from two comparators. The Aeq0 status signal outputs a 1 if the value in A is a
0, hence an 8-input NOR gate is used. The Apos status signal outputs a 1 if the value in A, which is treated as a 2’s
complement signed number, is a positive number. Since for a 2’s complement signed number, a leading 0 means
positive, and a leading 1 means negative, hence, Apos is simply the negated value of A7, the most significant bit of
A.
The control word for this custom datapath has five control signals, IRload, JMPmux, PCload, Meminst,
MemWr, Asel1-0, Aload, and Sub. The datapath provides two status signals, Aeq0 and Apos, to the control unit The
control words for executing the instruction cycle operations and the instruction set operations are discussed in the
next section. 434 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors Input
IRload D7-0
Load 8-bit register
Clear
IR
IR7-5 IR4-0 3 IR7..5 5 8 8 5
1 JMPmux 0 5-bit
increment 5 32 10
s1,0 D4-0
Load 5-bit register
Clear PC
PC4-0 PCload 8 1 Meminst 0 5 32 locations × 8bits
RAM
Address4-0
WE MemWr D7-0
Load 8-bit register
Clear
A
A7-0 Q7-0 5 8 8 8 D7-0 Asel1-0
Aload
Reset
Clock
Sub
Aeq0
Apos add /
Subsubtract 8
1
A7 Output
Instruction Cycle Operations Memory Instruction Set Operations Figure 12.11. Datapath for the EC-2. 12.3.3 Control Unit
The state diagram for the control unit is shown in Figure 12.12 (a), and the actions that are executed,
specifically the control signals that are asserted in each state are shown in (d). The first three states, START,
FETCH, and DECODE, serve the same purpose as in the previous EC-1’s control unit. The DECODE state for this
second design, however, needs to decode eight opcodes by branching to eight different states for executing the
corresponding eight instructions. Like before, the decoding of the opcodes depends on the three most significant bits
of the IR.
A very important timing issue for this control unit has to do with the memory accesses of the three instructions,
LOAD, ADD, and SUB. The problem here is that only after fetching these instructions will the address of the
memory location for these instructions be available. Furthermore, only after decoding the instruction will the control
unit know that the memory needs to be read. If we change the memory address during the execute state, the memory
will not have enough time to output the value for the instruction to operate on.
Normally, for instructions requiring a memory access for one of its operands, an extra memory read state will be
inserted between the DECODE state and the execute state. This way, the memory will have one clock cycle to
output the data for the instruction to operate on in the following clock cycle. This of course, is assuming that the
memory requires only one clock cycle for a read operation. If the memory is slower, then more states must be
inserted in between.
In our design, however, again to minimize the number of states, we have used the decode state to also perform
the memory read. This way, when the control unit gets to the execute state, the memory will already have the data
ready. Whether the data from the memory is actually used or not will depend on the instruction being executed. If 435 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors the instruction does not require the data from the memory, it is simply ignored. On the other hand, if the instruction
needs the data, then the data is there and ready to be used. This solution works in this design because it does not
conflict with the operations for the rest of the instructions in our instruction set. The memory read operation
performed in the DECODE state is accomplished by asserting the Meminst signal from this state. Looking at the
Output table in Figure 12.12 (d), this is reflected by the 1 under the Meminst column for the DECODE state.
The actual execution of each instruction is accomplished by asserting the correct control signals to control the
operation of the datapath. This is shown by the assignments made for the respective rows in the Output table in
Figure 12.12 (d). At this point, you should be able to understand why each assignment is made by looking at the
operation of the datapath. For example, for the LOAD A, instruction, the Asel1 signal needs to be asserted and the
Asel0 signal needs to be de-asserted in order to select input-2 of the multiplexer so that the output from the memory
can pass to the input of the accumulator A. The actual loading of A is done by asserting the Aload signal. To
perform the STORE A instruction, the memory address is taken from the IR by asserting Meminst. The writing into
memory takes place when MemWr is asserted.
The INPUT state for this state diagram waits for the Enter key signal before looping back to the START state.
In so doing, we can read in several values correctly by having multiple input statements in the program. Notice that
after the Enter signal is asserted, there is no state that waits for the Enter signal to be de-asserted, i.e. for the Enter
key to be released. Hence, the input device must resolve this issue by outputting exactly one clock pulse each time
the Enter key is pressed. This is accomplished at the computer circuit level by using a one-shot circuit.
The next-state / implementation table for the state diagram, and the three excitation equations as derived from
the implementation table are shown in Figure 12.12 (b) and (c) respectively. To keep the table reasonable small, all
the possible combinations of the input signals are not listed out. All the states, except for the INPUT state, depends
only on the three IR bits, whereas, the INPUT state depends only on the Enter signal. The blank entries in the table,
therefore, can be viewed as having all zeros. With 11 states, four D flip-flops are used for the implementation of the
control unit circuit.
The output equations shown in Figure 12.12 (e) are derived directly from the output table in (d).
Finally, we can derive the circuit for the control unit based on the excitation equations and the output equations.
The complete control unit circuit for the EC-2 general-purpose microprocessor is shown in Figure 12.12 (f). start
0000 fetch
0001 decode
0010 IR[7..5] =
000
load
1000 IR[7..5] =
001
store
1001 IR[7..5] =
010
add
1010 IR[7..5] =
011 IR[7..5] =
100 IR[7..5] =
101 IR[7..5] =
110 IR[7..5] =
111 input
1100 jz
1101 jpos
1110 halt
1111 sub
1011 Enter (a) 436 Enter' Digital Logic and Microprocessor Design with VHDL Current State
Q3Q2Q1Q0
0000 start
0001 fetch
0010 decode
1000 load
1001 store
1010 add
1011 sub
1100 input
1101 jz
1110 jpos
1111 halt Chapter 12 - General-Purpose Microprocessors Next State Q3next Q2next Q1next Q0next (Implementation D3 D2 D1 D0)
IR7, IR6, IR5
Enter
000
001
010
011
100
101
110
111
0
1
load store add
sub input
jz
jpos
halt
0001 0001 0001 0001 0001 0001 0001 0001
0010 0010 0010 0010 0010 0010 0010 0010
1000 1001 1010 1011 1100 1101 1110 1111
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
1100 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
1111 1111 1111 1111 1111 1111 1111 1111
(b)
D3 = Q3'Q2'Q1Q0' + Q3Q2Q1'Q0'Enter' + Q3Q2Q1Q0
D2 = Q3'Q2'Q1Q0'IR7 + Q3Q2Q1'Q0'Enter' + Q3Q2Q1Q0
D1 = Q3'Q2'Q1'Q0 + Q3'Q2'Q1Q0'IR6 + Q3Q2Q1Q0
D0 = Q3'Q2'Q1'Q0' + Q3'Q2'Q1Q0'IR5 + Q3Q2Q1Q0
(c) State
Q3Q2Q1Q0
0000 start
0001 fetch
0010 decode
1000 load
1001 store
1010 add
1011 sub
1100 input
1101 jz
1110 jpos
1111 halt IRload
0
1
0
0
0
0
0
0
0
0
0 JMPmux
0
0
0
0
0
0
0
0
1
1
0 PCload
0
1
0
0
0
0
0
0
Aeq0
Apos
0 Meminst
0
0
1
0
1
0
0
0
0
0
0 MemWr
0
0
0
0
1
0
0
0
0
0
0 Asel1,0
00
00
00
10
00
00
00
01
00
00
00 Aload
0
0
0
1
0
1
1
1
0
0
0 Sub
0
0
0
0
0
0
1
0
0
0
0 Halt
0
0
0
0
0
0
0
0
0
0
1 (d)
IRload = Q3'Q2'Q1'Q0
JMPmux = Q3Q2Q1'Q0 + Q3Q2Q1Q0'
PCload = Q3'Q2'Q1'Q0 + Q3Q2Q1'Q0Aeq0 + Q3Q2Q1Q0' Apos
Meminst = Q3'Q2'Q1Q0' + Q3Q2'Q1'Q0
MemWr = Q3Q2'Q1'Q0
Asel1
= Q3Q2'Q1'Q0'
Asel0
= Q3Q2Q1'Q0'
Aload = Q3Q1'Q0' + Q3Q2'Q1
Sub
= Q3Q2'Q1Q0
Halt
= Q3Q2Q1Q0 437 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors (e) IRload
D3 Q3 JMPmux Clk
Clear Q'3 PCload Enter
D2 Q2 Clk
Clear Q'2 Meminst
MemWr D1 Q1 Clk
Clear D0 Q'1 Q0 Clear Asel0
Aload
Sub
Halt Clk Clock
Reset Asel1 Q'0 Aeq0
Apos
IR7
IR6
IR5 (f)
Figure 12.12. Control unit for the EC-2: (a) state diagram; (b) next-state / implementation table; (c) excitation
equations; (d) control word/output table; (e) output equations; (f) circuit. 12.3.4 Complete Circuit
The complete circuit for the EC-2 general-purpose microprocessor is constructed by connecting the datapath
from Figure 12.11 and the control unit from Figure 12.12 (f) together using the designated control and status signals
as shown in Figure 12.13. 438 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors Input Enter 8
Enter
IRload
JMPmux
PCload
Meminst
MemWr
Asel1-0
CU Aload
Sub Clock
Reset IR7-5
Aeq0
Apos Clock
Reset Control
Signals Status
Signals Input
IRload
JMPmux
PCload
Meminst
MemWr
Asel1-0
Aload
DP
Sub
IR7-5
Aeq0
Apos Clock
Reset
Output Halt 8
Halt Output Figure 12.13. Complete circuit for the EC-2 general-purpose microprocessor. 12.3.5 Sample Program
The memory for our computer is initialized with the content of the text file PROGRAM.MIF. A sample file is
shown in Figure 12.14. The file contains three programs: GCD calculates the greatest common divisor of two input
numbers; SUM evaluates the sum of all the numbers between an input n and 1; and COUNT displays the count from
input n down to 0. Only the last program (COUNT) listed in the file is executed. To try out another program, move
the code for that program to the end, then re-compile the entire microprocessor circuit, i.e. the UP2FLEX.GDF file
as explained in the next section, and download the new code to the UP2 board.
-- Content of the RAM memory in the file PROGRAM.MIF
DEPTH = 32; -- Depth of memory: 5-bit address
WIDTH = 8;
-- Width of memory: 8-bit data
ADDRESS_RADIX = BIN;-- All values in binary (HEX, DEC, OCT, BIN)
DATA_RADIX = BIN;
---------- Opcodes for the EC-2
000 = load A,aaaaa
001 = store A,aaaaa
010 = add A,aaaaa
011 = sub A,aaaaa
100 = in A
101 = jz aaaaa
110 = jpos aaaaa
111 = halt -- Specify the memory content. 439 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors -- Format of each memory location is
-address : data
CONTENT
BEGIN
[00000..11111] : 00000000; -- Initialize locations range 00-1F to 0000
--------------------------------------------------------- There are three programs listed below: GCD, Sum, and Count
-- Only the progam listed last is ran.
-- To try out a different program, move the code for the program you
-- want to the end of the list, re-compile, and download to the FPGA
--------------------------------------------------------- GCD
-- Program to calculate the GCD of two numbers
00000
00001
00010
00011 :
:
:
: 10000000;
00111110;
10000000;
00111111; ----- input
store
input
store A
A,x
A
A,y 00100
00101
00110
00111 :
:
:
: 00011110;
01111111;
10110000;
11001100; ----- loop: load A,x
sub A,y
jz out
jp xgty 01000
01001
01010
01011 :
:
:
: 00011111;
01111110;
00111111;
11000100; ----- load A,y
sub A,x
store A,y
jp loop -- y>x
-- y-x 01100
01101
01110
01111 :
:
:
: 00011110;
01111111;
00111110;
11000100; ----- xgty: load A,x
sub A,y
store A,x
jp loop -- x>y
-- x-y -- x = y?
-- x=y
-- x>y 10000 : 00011110; -- load A,x
10001 : 11111111; -- halt
11110 : 00000000; -- x
11111 : 00000000; -- y
--------------------------------------------------------- SUM
-- Program to sum N downto 1
00000 : 00011101; -- load A,one
00001 : 01111101; -- sub A,one
00010 : 00111110; -- store A,sum -- zero sum by doing 1-1 00011 : 10000000; -- input A
00100 : 00111111; -- store A,n
00101 : 00011111; -- loop: load A,n
00110 : 01011110; -- add A,sum
00111 : 00111110; -- store A,sum 440 -- n + sum Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors 01000 : 00011111; -- load A,n
01001 : 01111101; -- sub A,one
01010 : 00111111; -- store A,n
01011
01100
01101
01110 :
:
:
: 10101101;
11000101;
00011110;
11111111; ----- -- decrement A jz out
jp loop
out: load A,sum
halt 11101 : 00000001; -- one
11110 : 00000000; -- sum
11111 : 00000000; -- n
--------------------------------------------------------- COUNT
-- Program to countdown from input N to 0
00000
00001
00010
00011
00100
11111 :
:
:
:
:
: 10000000;
01111111;
10100100;
11000001;
11111111;
00000001; ------- input
Sub A,11111
jz 00100
jp 00001
halt
constant 1 END;
Figure 12.14. The PROGRAM.MIF file containing three programs for the EC-2. Only the last program, COUNT, is
executed. 12.3.6 Hardware Implementation
Figure 12.15 shows the interface between the EC-2 microprocessor and the input and output devices on the UP2
development board. The input simply consists of eight DIP switches, and the output is the two 7-segment LED
display. Since the microprocessor outputs an 8-bit binary number, we need an 8-bit binary to two digit BCD (binary
coded decimal) decoder so that we can see the 8-bit binary number as two decimal digits on the two 7-segment LED
display. A single LED is used to show when the microprocessor has halted. A push button switch is used as the reset
button, and a second push button switch is used as the Enter key. A one-shot circuit is used to generate only one
clock pulse for each key press of the Enter key switch. Finally, the on-board 25MHz clock is slowed down with a
clock divider circuit. The reason for using the slower clock speed is so that we can see some intermediate results on
the display.
This complete hardware implementation circuit is in the file UP2FLEX.GDF in the appropriate directory on the
accompanying CD-ROM. You can use this file to download this complete EC-2 computer onto the UP2
development board, and see your very own microprocessor running a program. The COUNT program that is listed at
the end of the PROGRAM.MIF file is executed. The FLEX_PB1 pushbutton switch is the Reset switch. The
FLEX_PB2 pushbutton switch is the Enter switch. The eight dip switches are for the input port. The output is shown
on the two 7-segment LED as a decimal number. The decimal point on the right digit is lit if the number is in the
100’s. The decimal point on the left digit is the Halt light, and is lit when the program terminates with the HALT
instruction. 441 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors
8 DIP switches
8 Reset push button switch
FLEX_PB1
Enter push button switch
FLEX_PB2
25MHz Clock Reset
One-Shot
in
out
Clock Divider
25MHz
4Hz Input Enter EC-2
microprocessor Clock Output
8 binary to
BCD
decoder two
7-segment
LEDs Halt LED
left digit decimal point Figure 12.15. Hardware implementation of the EC-2. 12.4 VHDL for General-Purpose Microprocessors
This section presents the VHDL code for the EC-2 general-purpose microprocessor using both the structural
and the behavioral levels. First, in Section 12.4.1, we use the VHDL structural level to define the microprocessor
based on the FSM+D model as designed manually in Section 12.3. In order to use this method, the microprocessor
must first be designed manually. Remember that in practice, we really do not want to use this method. This exercise
is just to show how tedious and complicated it is to do it this way, especially for a more realistic microprocessor.
Section 12.4.2 uses the VHDL behavioral level to define the EC-2 microprocessor based on the FSMD model.
Using this method, we only need to manually derive the state diagram. Using the VHDL template for defining a
FSM (from Section 7.6), we can quickly translate the state diagram to the behavioral VHDL code. Comparing this
behavioral code with the structural code, we can see how much easier it is to construct a microprocessor circuit
using the behavioral method, and quickly appreciate the power of the VHDL synthesizer. 12.4.1 Structural FSM+D
Figure 12.16 shows the structural VHDL code of the datapath for the EC-2 general-purpose microprocessor.
This code is based on the datapath circuit as derived manually from Section 12.3.2. lpm memory Figure 12.17 shows
the code for the control unit as derived manually from Section 12.3.3. This control unit code follows very closely the
template for writing a FSM sequential circuit as discussed in Section 7.6. Figure 12.18 shows the structural code for
the complete EC-2 general-purpose microprocessor.
LIBRARY IEEE;
USE IEEE.std_logic_1164.ALL;
LIBRARY lpm;
-- for memory
USE lpm.lpm_components.ALL;
ENTITY dp IS PORT (
Clock, Clear: IN STD_LOGIC;
-- datapath input
Input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- control signals
IRload, JMPmux, PCload, MemInst, MemWr: IN STD_LOGIC;
ASel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
Aload, Sub: IN STD_LOGIC;
-- status signals
IR: OUT STD_LOGIC_VECTOR(7 DOWNTO 5);
Aeq0, Apos: OUT STD_LOGIC; 442 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors -- datapath output
Output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END dp;
ARCHITECTURE Structural OF dp IS
COMPONENT reg
GENERIC (size: INTEGER := 4); -- the actual size is defined in the
instantiation GENERIC MAP PORT (
Clock, Clear, Load: IN STD_LOGIC;
D: IN STD_LOGIC_VECTOR(size-1 DOWNTO 0);
Q: OUT STD_LOGIC_VECTOR(size-1 DOWNTO 0));
END COMPONENT;
COMPONENT increment
GENERIC (size: INTEGER := 8);
-- default number of bits
PORT (
A: IN STD_LOGIC_VECTOR(size-1 DOWNTO 0);
F: OUT STD_LOGIC_VECTOR(size-1 DOWNTO 0));
END COMPONENT;
COMPONENT mux2
GENERIC (size: INTEGER := 8);
PORT (
S: IN STD_LOGIC;
D1, D0: IN STD_LOGIC_VECTOR(size-1 DOWNTO 0);
Y: OUT STD_LOGIC_VECTOR(size-1 DOWNTO 0));
END COMPONENT;
COMPONENT mux4
GENERIC (size: INTEGER := 8);
PORT (
S: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
D3, D2, D1, D0: IN STD_LOGIC_VECTOR(size-1 DOWNTO
Y: OUT STD_LOGIC_VECTOR(size-1 DOWNTO 0));
END COMPONENT; -- default size
-- select line
-- data bus input
-- data bus output -- default size
-- select line
0);
-- data bus input
-- data bus output COMPONENT addsub
GENERIC(n: INTEGER :=4);
-- default number of bits = 4
PORT(S: IN std_logic;
-- select subtract signal
A: IN std_logic_vector(n-1 DOWNTO 0);
B: IN std_logic_vector(n-1 DOWNTO 0);
F: OUT std_logic_vector(n-1 DOWNTO 0);
unsigned_overflow: OUT std_logic;
signed_overflow: OUT std_logic);
END COMPONENT;
SIGNAL dp_IR, dp_RAMQ: STD_LOGIC_VECTOR(7 DOWNTO 0);
SIGNAL dp_JMPmux, dp_PC, dp_increment, dp_meminst
: STD_LOGIC_VECTOR(4 DOWNTO 0);
SIGNAL dp_Amux, dp_addsub, dp_A: STD_LOGIC_VECTOR(7 DOWNTO 0);
BEGIN
-- doing structural modeling for the datapath here
-- IR
U0: reg GENERIC MAP(8) PORT MAP(Clock, Clear, IRLoad, dp_RAMQ, dp_IR);
IR <= dp_IR(7 DOWNTO 5); 443 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors -- JMPmux
U1: mux2 GENERIC MAP(5) PORT MAP(JMPmux,dp_IR(4 DOWNTO 0),
dp_increment,dp_JMPmux);
-- PC
U2: reg GENERIC MAP(5) PORT MAP(Clock, Clear, PCLoad, dp_JMPmux, dp_PC);
-- Meminst
U3: mux2 GENERIC MAP(5) PORT MAP(Meminst,dp_IR(4 DOWNTO 0),
dp_PC,dp_meminst);
-- increment
U4: increment GENERIC MAP(5) PORT MAP(dp_PC,dp_increment);
-- memory
U5: lpm_ram_dq
GENERIC MAP (
lpm_widthad => 5,
lpm_outdata => "UNREGISTERED",
-lpm_indata => "UNREGISTERED",
-lpm_address_control => "UNREGISTERED",
lpm_file => "program.mif",-- fill ram with content of file program.mif
lpm_width => 8)
PORT MAP (
data => dp_A,
address => dp_meminst,
we
=> MemWr,
inclock => Clock,
q
=> dp_RAMQ);
-- A input mux
U6: mux4 GENERIC MAP(8) PORT MAP (
Asel,dp_RAMQ,dp_RAMQ,Input,dp_addsub,dp_Amux);
-- Accumulator
U7: reg GENERIC MAP(8) PORT MAP(Clock, Clear, ALoad, dp_Amux, dp_A);
-- Adder-subtractor
U8: addsub GENERIC MAP(8) PORT MAP(Sub,dp_A,dp_RAMQ,dp_addsub,open,open);
Aeq0 <= '1' WHEN dp_A = "00000000" ELSE '0';
Apos <= NOT dp_A(7);
Output <= dp_A;
END Structural;
Figure 12.16. VHDL code for the datapath of the EC-2.
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL;
ENTITY cu IS PORT (
Clock, Reset : IN STD_LOGIC;
-- control input
Enter: IN STD_LOGIC; 444 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors -- control signals
IRload, JMPmux, PCload, MemInst, MemWr: OUT STD_LOGIC;
Asel: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
Aload, Sub: OUT STD_LOGIC;
-- status signals
IR: IN STD_LOGIC_VECTOR(7 DOWNTO 5);
Aeq0, Apos: IN STD_LOGIC;
-- control outputs
Halt: OUT STD_LOGIC);
END cu;
ARCHITECTURE FSM OF cu IS
TYPE state_type IS (s_start,s_fetch,s_decode,
s_load,s_store,s_add,s_sub,s_in,s_jz,s_jpos,s_halt);
SIGNAL state: state_type;
BEGIN
next_state_logic: PROCESS(Reset, Clock)
BEGIN
IF(Reset = '1') THEN
state <= s_start;
ELSIF(Clock'EVENT AND Clock = '1') THEN
CASE state IS
WHEN s_start => -- reset
state <= s_fetch;
WHEN s_fetch =>
state <= s_decode;
WHEN s_decode =>
CASE IR IS
WHEN "000" => state <= s_load;
WHEN "001" => state <= s_store;
WHEN "010" => state <= s_add;
WHEN "011" => state <= s_sub;
WHEN "100" => state <= s_in;
WHEN "101" => state <= s_jz;
WHEN "110" => state <= s_jpos;
WHEN "111" => state <= s_halt;
WHEN OTHERS => state <= s_halt;
END CASE;
WHEN s_load =>
state <= s_start;
WHEN s_store =>
state <= s_start;
WHEN s_add =>
state <= s_start;
WHEN s_sub =>
state <= s_start;
WHEN s_in =>
IF (Enter = '0') THEN -- wait for the Enter key for inputs
state <= s_in;
ELSE
state <= s_start;
END IF;
WHEN s_jz =>
state <= s_start;
WHEN s_jpos => 445 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors state <= s_start;
WHEN s_halt =>
state <= s_halt;
WHEN OTHERS =>
state <= s_start;
END CASE;
END IF;
END PROCESS;
output_logic: PROCESS(state)
BEGIN
CASE state IS
WHEN s_fetch =>
IRload <= '1'; -- load IR
JMPmux <= '0';
PCload <= '1'; -- increment PC
Meminst <= '0';
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
WHEN s_decode => -- also set up for memory access
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '1';
-- pass IR address to memory
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
WHEN s_load =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '1';
MemWr <= '0';
Asel <= "10";-- pass memory to A
Aload <= '1'; -- load A
Sub
<= '0';
Halt <= '0';
WHEN s_store =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '1';
-- pass IR address to memory
MemWr <= '1'; -- store A to memory
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
WHEN s_add =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '1'; 446 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors MemWr <= '0';
Asel <= "00";-- pass add/sub unit to A
Aload <= '1'; -- load A
Sub
<= '0'; -- select add
Halt <= '0';
WHEN s_sub =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '1';
MemWr <= '0';
Asel <= "00";-- pass add/sub unit to A
Aload <= '1'; -- load A
Sub
<= '1'; -- select subtract
Halt <= '0';
WHEN s_in =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '0';
MemWr <= '0';
Asel <= "01";-- pass input to A
Aload <= '1'; -- load A
Sub
<= '0';
Halt <= '0';
WHEN s_jz =>
IRload <= '0';
JMPmux <= '1'; -- pass IR address to PC
PCload <= Aeq0;
-- load PC if condition is true
Meminst <= '0';
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
WHEN s_jpos =>
IRload <= '0';
JMPmux <= '1'; -- pass IR address to PC
PCload <= Apos;
-- load PC if condition is true
Meminst <= '0';
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
WHEN s_halt =>
IRload <= '0';
JMPmux <= '0';
PCload <= '0';
Meminst <= '0';
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '1';
WHEN OTHERS =>
IRload <= '0'; 447 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors JMPmux <= '0';
PCload <= '0';
Meminst <= '0';
MemWr <= '0';
Asel <= "00";
Aload <= '0';
Sub
<= '0';
Halt <= '0';
END CASE;
END PROCESS;
END FSM;
Figure 12.17. VHDL code for the control unit of the EC-2.
LIBRARY IEEE;
USE IEEE.std_logic_1164.all;
ENTITY mp IS PORT (
Clock, Reset: IN STD_LOGIC;
Enter: IN STD_LOGIC;
Input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
Output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0);
Halt: OUT STD_LOGIC);
END mp;
ARCHITECTURE Structural OF mp IS
COMPONENT cu PORT (
Clock, Reset : IN STD_LOGIC;
-- control input
Enter: IN STD_LOGIC;
-- control signals
IRload, JMPmux, PCload, MemInst, MemWr: OUT STD_LOGIC;
Asel: OUT STD_LOGIC_VECTOR(1 DOWNTO 0);
Aload, Sub: OUT STD_LOGIC;
-- status signals
IR: IN STD_LOGIC_VECTOR(7 DOWNTO 5);
Aeq0, Apos: IN STD_LOGIC;
-- control outputs
Halt: OUT STD_LOGIC);
END COMPONENT;
COMPONENT dp PORT (
Clock, Clear: IN STD_LOGIC;
-- datapath input
Input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- control signals
IRload, JMPmux, PCload, MemInst, MemWr: IN STD_LOGIC;
ASel: IN STD_LOGIC_VECTOR(1 DOWNTO 0);
Aload, Sub: IN STD_LOGIC;
-- status signals
IR: OUT STD_LOGIC_VECTOR(7 DOWNTO 5);
Aeq0, Apos: OUT STD_LOGIC;
-- datapath output
Output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0));
END COMPONENT; 448 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors -- control signals
SIGNAL mp_IRload, mp_JMPmux, mp_PCload, mp_MemInst, mp_MemWr: STD_LOGIC;
SIGNAL mp_Asel: STD_LOGIC_VECTOR(1 DOWNTO 0);
SIGNAL mp_Aload, mp_Sub: STD_LOGIC;
-- status signals
SIGNAL mp_IR: STD_LOGIC_VECTOR(7 DOWNTO 5);
SIGNAL mp_Aeq0, mp_Apos: STD_LOGIC;
BEGIN
-- doing structural modeling for the microprocessor here
U0: cu PORT MAP (
Clock, Reset,
-- control input
Enter,
-- control signals
mp_IRload, mp_JMPmux, mp_PCload, mp_MemInst, mp_MemWr,
mp_Asel,
mp_Aload, mp_Sub,
-- status signals
mp_IR,
mp_Aeq0, mp_Apos,
-- control outputs
Halt);
U1: dp PORT MAP (
Clock, Reset,
-- datapath input
Input,
-- control signals
mp_IRload, mp_JMPmux, mp_PCload, mp_MemInst, mp_MemWr,
mp_Asel,
mp_Aload, mp_Sub,
-- status signals
mp_IR,
mp_Aeq0, mp_Apos,
-- datapath output
Output);
END Structural;
Figure 12.18. Structural VHDL code for the complete EC-2 general-purpose microprocessor. 12.4.2 Behavioral FSMD
Figure 12.19 shows the complete behavioral FSMD VHDL code for the EC-2 general-purpose microprocessor.
The memory used is from the Altera’s LPM component library. The three registers, IR, PC, and A, are declared as
signals in the architecture section. Recall from Section 6.13 that if a signal is not assigned a value from all
conditional paths, then it is synthesized as a register. The process block is structured just like a regular FSM
following the state diagram from Figure 12.12 (a). The DECODE state uses a CASE statement to check the opcode,
which is the first three bits of the IR. From there, the FSM jumps to the state for executing the corresponding
instruction. For the STORE instruction, an extra state is needed to de-assert the MemWr signal before changing the
memory address back to the location of the next instruction.
-- EC-2 Behavioral FSMD description
LIBRARY IEEE;
USE IEEE.STD_LOGIC_1164.ALL; 449 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors USE IEEE.STD_LOGIC_ARITH.ALL;
USE IEEE.STD_LOGIC_UNSIGNED.ALL;
LIBRARY lpm;
USE lpm.lpm_components.ALL;
ENTITY mp IS PORT (
clock, reset: IN STD_LOGIC;
enter: IN STD_LOGIC;
-- data input
input: IN STD_LOGIC_VECTOR(7 DOWNTO 0);
-- data output
output: OUT STD_LOGIC_VECTOR(7 DOWNTO 0);
-- control outputs
halt: OUT STD_LOGIC);
END mp;
ARCHITECTURE FSMD OF mp IS
TYPE state_type IS(s_start,s_fetch,s_decode,s_load,s_store,s_store2,
s_add,s_sub,s_input,s_jz,s_jpos,s_halt);
SIGNAL state: state_type;
-- states
SIGNAL IR: STD_LOGIC_VECTOR(7 DOWNTO 0);
-- Instruction register
SIGNAL PC: STD_LOGIC_VECTOR(4 DOWNTO 0);
-- Program counter
SIGNAL A: STD_LOGIC_VECTOR(7 DOWNTO 0);
-- Accumulator
SIGNAL memory_address: STD_LOGIC_VECTOR(4 DOWNTO 0); -- memory address
SIGNAL memory_data: STD_LOGIC_VECTOR(7 DOWNTO 0); -- memory data input
SIGNAL MemWr: STD_LOGIC;
BEGIN
memory: lpm_ram_dq
-- 32 locations x 8 bits wide asynchronous memory
GENERIC MAP (
lpm_widthad => 5,
lpm_outdata => "UNREGISTERED",
lpm_indata => "UNREGISTERED",
lpm_address_control => "UNREGISTERED",
lpm_file => "program.mif", -- fill ram with content of file program.mif
lpm_width => 8)
PORT MAP (
data => A,
address => memory_address,
we
=> MemWr,
q
=> memory_data);
PROCESS(clock,reset)
BEGIN
IF(reset = '1') THEN
PC <= "00000";
IR <= "00000000";
A <= "00000000";
MemWr <= '0';
halt <= '0';
state <= s_start;
ELSIF(clock'EVENT AND clock = '1') THEN
CASE state IS
WHEN s_start => -- reset, start
memory_address <= PC;
state <= s_fetch; 450 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors WHEN s_fetch => -- fetch
IR <= memory_data;
PC <= PC + 1;
state <= s_decode;
WHEN s_decode => -- decode
-- memory access using last 5 bits of IR
memory_address <= IR(4 DOWNTO 0);
CASE IR(7 DOWNTO 5) IS
-- decode first 3 bits of IR as opcode
WHEN "000" => state <= s_load;
WHEN "001" => state <= s_store;
WHEN "010" => state <= s_add;
WHEN "011" => state <= s_sub;
WHEN "100" => state <= s_input;
WHEN "101" => state <= s_jz;
WHEN "110" => state <= s_jpos;
WHEN "111" => state <= s_halt;
WHEN OTHERS => state <= s_start;
END CASE;
WHEN s_load =>
-- load A from memory
A <= memory_data;
state <= s_start;
WHEN s_store =>
-- store A to memory
MemWr <= '1';
state <= s_store2;
WHEN s_store2 =>
-- need an extra state to de-assert MemWr
MemWr <= '0';
-- before changing the memory address
state <= s_start;
WHEN s_add =>
-- add
A <= A + memory_data;
state <= s_start;
WHEN s_sub =>
-- subtract
A <= A - memory_data;
state <= s_start;
WHEN s_input =>
A <= input;
IF (Enter = '0') THEN -- wait for Enter key
state <= s_input;
ELSE
state <= s_start;
END IF;
WHEN s_jz =>
IF (A = 0) THEN
-- jump if A is 0
PC <= IR(4 DOWNTO 0);
END IF;
state <= s_start;
WHEN s_jpos =>
IF (A(7) = '0') THEN
-- jump if MSB(A) is 0
PC <= IR(4 DOWNTO 0);
END IF;
state <= s_start;
WHEN s_halt =>
halt <= '1';
state <= s_halt;
WHEN OTHERS =>
state <= s_halt;
END CASE;
END IF; 451 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors END PROCESS;
output <= A; -- send value of Accumulator to the output
END FSMD;
Figure 12.19. Behavioral VHDL code for the complete EC-2 general-purpose microprocessor. 12.5 Summary Checklist
Instruction set
Opcode
Instruction register (IR)
Program counter (PC)
Instruction cycle
Be able to derive the datapath circuit for a given instruction set
Be able to derive the state diagram for a given instruction set
Be able to derive the control unit circuit for a given instruction set
Be able to derive the complete general-purpose microprocessor circuit
Be able to implement the complete computer using the general-purpose microprocessor developed
Be able to write programs in machine language using the implemented instruction set
Be able to run the programs on the implemented computer
Be able to write behavioral VHDL codes for general-purpose microprocessors 12.6 Problems
12.1 Manually re-design the EC-1 microprocessor to accommodate each of the following changes. The changes
are to be done separately.
a) Modify the OUTPUT instruction so that the output port outputs a value only when the OUTPUT
instruction is executed.
b) Modify the INPUT instruction so that it will wait for an external Enter key signal before continuing to
the next instruction.
c) Add an extra INC instruction, using the opcode 000, to the EC-1 instruction set. The INC instruction
increments the accumulator.
d) Add an extra LOAD instruction, using the opcode 001, to the EC-1 instruction set. The LOAD
instruction loads the accumulator with the content of memory location aaaa, where aaaa are the least
four significant bits of the instruction encoding.
12.2 Write the behavioral VHDL code for the EC-1 microprocessor.
12.3 Re-write the behavioral VHDL code for the EC-1 microprocessor with each of the changes from
Problem 12.1.
12.4 Write and run the following programs on the EC-2 computer:
e) Input two numbers, and output the sum of these two numbers.
f) Input two numbers, and output the larger of the two numbers.
g) Input two numbers, and output the product of these two numbers.
h) Keep inputting numbers until a 0. Output the number of numbers entered.
i) Keep inputting numbers until a 0. Output the sum of these numbers.
j) Keep inputting numbers until a 0. Output the largest of these numbers.
k) Keep inputting numbers until a 0. Output the largest and second largest of these numbers.
l) Input three numbers, and output these numbers in ascending order.
12.5 Manually re-design the EC-2 microprocessor to accommodate each of the following changes. The changes
are to be done separately. 452 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors a) Replace the SUB instruction in the EC-2 instruction set with a LSHIFT instruction. The LSHIFT
instruction shifts the content of the accumulator left by one bit. The result of the shift operation is
written back into the accumulator.
b) Replace the SUB instruction in the EC-2 instruction set with an OUTPUT instruction. The OUTPUT
instruction outputs the content of the accumulator to the output port. The output port should not show
anything when the OUTPUT instruction is not being executed.
c) Add an extra LSHIFT instruction to the EC-2 instruction set. The LSHIFT instruction is defined in a)
above.
d) Add an extra OUTPUT instruction to the EC-2 instruction set. The OUTPUT instruction is defined in
b) above.
e)
12.6 Re-write the behavioral VHDL code for the EC-2 microprocessor with each of the changes from
Problem 12.5.
12.7 Given the instruction set as defined below, manually design a datapath that can realize this instruction set.
Instruction Encoding Data movement instructions
LDA A,rrr
0001 0rrr
STA rrr,A
0010 0rrr
LDM A,aaaaaa 0011 0000
00aaaaaa
STM aaaaaa,A 0100 0000
00 aaaaaa
LDI A,iiiiiiii
0101 0000
iiiiiiii
Jump instructions
JMP absolute
0110 0000
00 aaaaaa
JMPR relative 0110 smmm JZ absolute
JZR relative JNZ absolute
JNZR relative JP absolute
JPR relative 0111 0000
00 aaaaaa
0111 smmm 1000 0000
00 aaaaaa
1000 smmm 1001 0000
00 aaaaaa
1001 smmm Operation Comment A ← R[rrr]
R[rrr] ← A
A ← M[aaaaaa] Load accumulator from register
Load register from accumulator
Load accumulator from memory M[aaaaaa] ← A Load memory from accumulator A ← iiiiiiii Load accumulator with immediate
value (iiiiiiii is a signed number) PC = aaaaaa Absolute unconditional jump if (smmm != 0) then
if (s == 0) then PC = PC + mmm
else PC = PC – mmm
if (A == 0) then PC = aaaaaa Relative unconditional jump (smmm
is in sign and magnitude format) if (A == 0 and smmm != 0) then
if (s == 0) then PC = PC + mmm
else PC = PC – mmm
if (A != 0) then PC = aaaaaa Relative jump if A is zero (smmm is
in sign and magnitude format)
Absolute jump if A is not zero if (A != 0 and smmm != 0) then
if (s == 0) then PC = PC + mmm
else PC = PC – mmm
if(A == positive) then PC = aaaaaa Relative jump if A is not zero
(smmm is in sign and magnitude
format)
Absolute jump if A is positive if(A == positive and smmm != 0) then
if (s == 0) then PC = PC + mmm
else PC = PC – mmm Relative jump if A is positive (smmm
is in sign and magnitude format) Arithmetic and logical instructions
AND A,rrr
1010 0rrr
A ← A AND R[rrr]
OR A,rrr
1011 0rrr
A ← A OR R[rrr] 453 Absolute jump if A is zero Accumulator AND register
Accumulator OR register Digital Logic and Microprocessor Design with VHDL ADD A,rrr
SUB A,rrr
NOT A
INC A
DEC A
SHFL A
SHFR A
ROTR A 1100 0rrr
1101 0rrr
1110 0000
1110 0001
1110 0010
1110 0011
1110 0100
1110 0101 Input / Output and Miscellaneous
In A
1111 0000
Out A
1111 0001
HALT
1111 0010
NOP
0000 0000 Chapter 12 - General-Purpose Microprocessors A ← A + R[rrr]
A ← A − R[rrr]
A ← NOT A
A←A+1
A←A−1
A ← A << 1
A ← A >> 1
A ← Rotate_right(A) Accumulator + register
Accumulator − register
Invert accumulator
Increment accumulator
Decrement accumulator
Shift accumulator left
Shift accumulator right
Rotate accumulator right A ← input
output ← A
Halt
no operation Input to accumulator
Output from accumulator
Halt execution
No operation Notations:
A = accumulator.
R = general register.
M = memory.
PC = program counter.
rrr = three bits for specifying the general register number (0 – 7).
aaaaaa = six bits for specifying the memory address.
iiiiiiii = an eight bit signed number.
smmm = four bits for specifying the relative jump displacement in sign and magnitude format. The most
significant bit (s) determines whether to jump forward or backward (0 = forward, 1 = backward). The last
three bits (mmm) specify the number of locations to increment or decrement from the current PC location.
Answer: 454 Digital Logic and Microprocessor Design with VHDL Chapter 12 - General-Purpose Microprocessors imm_dp input_dp
8
8 8 8 3
2
1
sel1,0 Mux
y muxsel_dp1,0
zero_dp
positive_dp
accwr_dp
rst_dp 0 C_muxout
MSB 8 wr Accumulator
clear
C_accout rfwr_dp wr rfaddr_dp2,1,0 8
Register File addr2,1,0
C_rfout A alusel_dp2,1,0 B
sel2,1,0 ALU
C_aluout shiftsel_dp1,0 sel1,0 Shifter
8 C_shiftout outen_dp output_dp 12.8 Write the behavioral VHDL code for a microprocessor that can execute the instructions in the instruction set
defined in Problem 12.7. 455 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 Appendix A Schematic Entry—Tutorial 1
The MAX+plus II software and the UP2 development board provide all of the necessary tools for implementing
and trying out all of the examples, including building the final general-purpose microprocessor, discussed in this
book. The MAX+plus II software offers a completely integrated development tool and easy-to-use graphical-user
interface for the design, and synthesis of digital logic circuits. Together with the UP2 development board, these
circuits can be implemented on a programmable logic device (PLD) chip. After downloading the circuit netlist to the
PLD, you can see the actual operation of these circuits in the hardware.
A Student Edition version of the MAX+plus II software is included on the accompanying CD-ROM and can
also be downloaded from the Altera website found at www.altera.com. The UP2 development board can be
purchased directly from Altera. The full User Guide for using the UP2 board is on the CD-ROM, and can also be
downloaded from the Altera website. This tutorial assumes that you are familiar with the Windows environment,
and that the MAX+plus II software has already been installed on your computer. Instructions for the installation of
the MAX+plus II software can be found on the CD-ROM. You also must obtain a license file from the Altera
website in order for the software to function correctly. Be careful that you obtain the Student Edition license, and not
the Baseline Edition license.
The MAX+plus II development software provides for both schematic and text entry of a circuit design. The
Schematic Editor is used to enter a schematic drawing of a circuit. Using the Schematic Editor, logic symbols for the
circuit can be inserted and connected together using the drawing tools. The Text Editor is used to enter VHDL or
Verilog code for describing a circuit.
This tutorial provides a step-by-step instruction for the schematic entry, synthesis, and simulation of an 8-bit 2to-1 multiplexer circuit. Tutorial 3 (Appendix C) will show how a circuit can be downloaded to the PLD on the UP2
development board so that you actually can see this circuit executed in the hardware. A.1 Getting Started
A.1.1 Preparing a Folder for the Project
Each circuit design in MAX+plus II is called a project. Each project should be placed in its own folder, since
the synthesizer creates many associated working files for a project. Using Windows File Manager, create a new
folder for your new project.
• For this tutorial, create a folder called 2x8mux in the root directory of the C drive. A.1.2 Starting MAX+plus II
After the successful installation of the MAX+plus II software, there should be a link for the program under the
Start button. Click on this link to start the program. You should see the MAX+plus II Manager window, as shown in Figure A.1. 456 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 Figure A.1 The MAX+plus II Manager window.
Figure A.2 shows the toolbar for accessing the main development tools. The buttons from left to right are:
•
•
•
•
•
•
•
•
• Hierarchy display—to show the design files used in the current project
Floorplan Editor—to map the I/O signals from the circuit to the pins on the PLD chip
Compiler (synthesizer)—to synthesize the circuit to its netlist
Simulator—to perform circuit simulation
Timing analyzer—to perform circuit timing analysis
Programmer—to program the circuit to the PLD chip
Open existing or new project—to select a new project
Change project name to current filename—to use the current file as the new project
Open top-level design file—to open the top-level design file for the current project Figure A.2 The MAX+plus II development software toolbar.
In the MAX+plus II software, different commands in the menus are available when different windows are
activated. This might cause some confusion at first. If you cannot find a particular command from the menu, make
sure that the correct window for that command is the active window. A.1.3 Starting the Graphic Editor
From the Manager window, select Max+plus II | Graphic Editor. You should see the Graphic Editor window
similar to the one shown in Figure A.3. Any circuit diagram can be drawn in this Graphic Editor window. 457 Digital Logic and Microprocessor Design with VHDL
• Appendix A - Schematic Entry. Tutorial 1 Alternatively, you can select File | New from the Manager window menu. Select Graphic Editor file using the
extension .gdf, and click OK.
Multifunction Pointer Tool
Text Tool Line/connection Tools Zooming
Fit Drawing in Window
Connection Dot
T urns On Rubberbanding
Turns Off Rubberbanding Figure A.3 The Graphic Editor window with the graphics toolbar on the left. A.2 Using the Graphic Editor
A.2.1 Drawing Tools
In Figure A.3, the tools for drawing a circuit in the Graphic Editor are shown in the toolbar on the left side.
There are the standard text writing tool, line drawing tools for making connections between logic symbols, and
zooming tools. The main tool that you will use is the multifunction pointer tool. This multifunction pointer allows
you to perform many different operations depending on the context in which it is used. Two main operations
performed by this multifunction pointer are selecting objects and making connections between logic symbols. The
connection dot tool either makes or deletes a connection point between two crossing lines. Finally, the two
rubberbanding buttons turn on or off the rubberbanding function. When rubberbanding is turned on, connection lines
are adjusted automatically when symbols are moved from one location to another. When rubberbanding is turned
off, moving a symbol will not affect the lines connected to it. A.2.2 Inserting Logic Symbols
1. To insert a logic symbol, first select the multifunction pointer tool, and then double-click the pointer on an
empty spot in the Graphic Editor window. You should see the Enter Symbol window, as shown in Figure A.4. Available symbol libraries are listed in the Symbol Libraries selection box. These libraries include the standard
primitive gates, standard combinational and sequential components, and your own logic symbols located in the
current project directory.
All of the primitive logic gates, latches, flip-flops, and input and output connectors that we need are in the
primitive library: …\prim. Your own circuits that you want to reuse in building larger circuits are in the directory
that they are stored in. 458 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 Figure A.4 The Enter Symbol selector window.
2. Double-click on the prim library to see a list of logic symbols available in that library. A list of logic symbols is
shown in the Symbol Files selection box. The logic symbols are sorted in alphabetical order. To narrow down
the list, you can type the first few letters of the symbol name followed by an asterisk in the Symbol Name text
box. You need to either press Enter or click on the OK button to update the list. For example, typing a*, and
pressing Enter will produce a list of all of the symbols whose name starts with the letter “a”. 3. Double-click on the logic symbol name that you want in order to insert that symbol into the Graphic Editor. If
you just select the symbol with a single-click, then you will also have to click on the OK button.
For this tutorial, insert the following symbols:
•
•
•
•
• A 2-input AND gate (and2)
A 2-input OR gate (or2)
A NOT gate (not)
An input signal connector (input)
An output signal connector (output) A unique number is given to each instance of a symbol and is written at the lower-left corner of the symbol.
This number is used only as a reference number in the output netlist and report files. These numbers may be
different from those in the examples. A.2.3 Selecting, Moving, Copying, and Deleting Logic Symbols
• To select a logic symbol in the Graphic Editor, simply single-click on the symbol using the multifunction
pointer tool. You can also select multiple symbols by holding down the Shift key while you select the symbols.
An alternative method is to trace a rectangle with the multifunction tool around the objects that you want to 459 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 select. All objects inside the rectangle will be selected.
• To move a symbol, simply drag the symbol. • To copy a symbol, first select it and then perform the Copy and Paste operations. An alternative method is to
hold down the Ctrl key while you drag the symbol. • To delete a symbol, first select it and then press the Delete key. • To rotate a symbol, right-click on the symbol, select Rotate from the drop-down menu, and select the angle to
rotate the symbol.
Perform the following operations for this tutorial:
1. Make a copy of the 2-input AND gate 2. Make two more copies of the input signal connector 3. Position the symbols to look like Figure A.5 Figure A.5 Symbol placements for the 2-to-1 multiplexer circuit. A.2.4 Making and Naming Connections
• To make a connection between two connection points, use the multifunction pointer tool and drag from one
connection point to the second connection point. Notice that, when you position the multifunction pointer tool
to a connection point, the arrow pointer changes to a crosshair. • To change the direction of a connection line while dragging the line, simply release and press the mouse button
again and then continue to drag the connection line. • You can also make a connection between two connection points by moving a symbol so that its connection
point touches the other connection point. • If you want to make a connection line that does not start from a connection point, you will need to use either the
straight line drawing tool or the vertical and horizontal line drawing tool instead of the multifunction pointer
tool. • Once a connection is made to a symbol, you can move the symbol to another location, and the connection line is
adjusted automatically if the rubberbanding function is turned on. However, if the rubberbanding function is
turned off, the connection will be broken if the symbol is moved. • To make a connection between two lines that cross each other, you need to use the multifunction pointer tool to
select the junction point (i.e., the point where the two lines cross) and then press the connection tool button, as 460 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 shown in Figure A.6. The connection tool button is enabled only after you have selected a junction point.
• To remove a connection point, select (single-click) the point, and then press the connection tool button. Figure A.6 Making or deleting a connection point with the connection tool button.
• To make a bus connection (for grouping two or more lines together), first draw a regular connection line, then
right-click on the line, select Line Style, and select the thicker, solid line style from the pop-up menu. • A bus must also have a name and a width associated with it. Select the bus line at the point where you want to
place the name and then type in the name and the width for the bus. For example, data[7..0] is an 8-bit bus with
the name data, as shown in Figure A.7. • To change the name, just double-click on the name and edit it. • To connect one line to a bus, connect a single line to the bus, and then give it the same name as the bus with the
line index appended to it. For example, data2, is bit two of the data bus, as shown in Figure A.7. Figure A.7 A single connection line connected to an 8-bit bus with the name data.
• To check whether a name is attached correctly to a line, select the line, and the name that is attached to the line
will also be selected. • To name an input or output connector, select its name label by double-clicking it, and then type in the new
name. Pressing the Enter key will move the text entry cursor to the name label for the symbol below the current
symbol. • A bus line connected to an input or output connector must have the same name as the connector.
Perform the following operations for this tutorial:
1. Name the three input connectors d0[7..0], s, and d1[7..0], as shown in Figure A.8 2. Name the output connector y[7..0], as shown in Figure A.8 3. Connect and name the five bus lines d0[7..0], d1[7..0], and0[7..0], and1[7..0], and y[7..0], as shown in
Figure A.8 4. Connect the single lines from the input connector s to the inverter and to the two AND gates, as shown in
Figure A.8 461 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 Figure A.8 Connections and names for the 2-to-1 multiplexer circuit. A.2.5 Selecting, Moving and Deleting Connection Lines
• To select a straight connection line, just single-click on it. • To select an entire connection line with horizontal and vertical segments, double-click on it. • To select a portion of a line, trace a rectangle with the multifunction tool around the segment. • After a line is selected, it can be moved by dragging. • After a line is selected, it can be deleted by pressing the Delete key. A.3 Specifying the Top-Level File and Project
A.3.1 Saving the Schematic Drawing
1. From the Graphic Editor menu, select File | Save. Select the 2x8mux directory that you created on the C drive in
Section A.1.1. Type in the filename 2x8mux. The extension should be .gdf (for graphic design file). 2. Click OK. A.3.2 Specifying the Project
1. To use the schematic drawing file saved in Section A.3.1 as the top-level project file, select File | Project | Set
Project to Current File from the Manager window menu, or simply click on the icon • . You can open any graphic design file (with the extension .gdf) using the Manager menu command File | Open,
and then select File | Project | Set Project to Current File, or click on the icon
top-level project file. to make that particular file the A.4 Synthesis for Functional Simulation
1. From the Manager window menu, select MAX+plus II | Compiler, or click on the icon
Compiler window. 2. From the Compiler window menu (that is, with the Compiler window selected as the active window), select
Processing | Functional SNF Extractor so that a check mark appears next to it. To actually see whether there is a
check mark or not, you need to select the Processing menu again. The Compiler window for functional 462 to bring up the Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 extraction is shown in Figure A.9.
3. Click on the Start button to start the synthesis. You will then see the progress of the synthesis. 4. At the end of the synthesis, if there are no syntax errors, you will see a message window saying that the
compilation was successful. Click OK to close the message window. Figure A.9 Compiler window for functional extraction. A.5 Circuit Simulation
A.5.1 Selecting Input Test Signals
1. Before you can simulate the design, you need to create test vectors for specifying what the input values are.
From the Manager window menu, select MAX+plus II | W aveform Editor. 2. From the W aveform Editor window menu, select Node | Enter Nodes from SNF. You can also right-click under
the Name section in the W aveform Editor window and select Enter Nodes from SNF from the pop-up menu. You
will see something similar to the Enter Nodes from SNF window shown in Figure A.10. Figure A.10 Window for adding signals for simulation. 463 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 3. Click on the List button in the Enter Nodes from SNF window, and a list of available nodes and groups will be
displayed in the Available Nodes & Groups box. 4. Select the signals that you want to see in the simulation trace. The signals that we want are: s (I), d1[7..0] (I),
d0[7..0] (I), and y[7..0] (O). The letters I and O in parenthesis next to each signal denote whether the signal is an
input or output signal, respectively. Note that the signal name such as y7 is bit seven of the bus named y, and
d16 is bit six of the bus named d1. Multiple nodes can be selected by holding down the Ctrl or Shift key while
clicking on the signal names. 5. After selecting the signals, click on the => button to move the selected signals to the Selected Nodes & Groups
box. 6. Repeat Steps 4 and 5 until all of the signals that you want to see in the simulation are moved to the Selected
Nodes & Groups box. 7. Click on OK when you are finished. The selected signals will now be inserted in the Waveform Editor window
similar to Figure A.11. Figure A.11 Waveform Editor window for simulation. A.5.2 Customizing the Waveform Editor
1. You can rearrange the signals in the Waveform Editor by dragging the signal icons like 2. To delete a signal in the Waveform Editor, just select the signal by clicking on its name and press the Delete
key. 3. For signals that are composed of a group of bits (such as the data input d1), you can separate them into
individual bits or change the radix for the displayed value by first selecting that signal and then right-click the
mouse. A drop-down menu appears. Select Ungroup to separate the bits. To regroup them, select the bits you
want to group and then right-click the mouse. A drop-down menu appears. Select Enter Group. Type in a group
name, and select the Decimal radix for the display. 4. We want to simulate for 500 nanoseconds. To change the simulation end time, select File | End Time from the
Waveform Editor window menu. 5. In the End Time window, type in 500ns, and click OK. 6. To fit the entire simulation time range inside the window, select View | Fit in Window from the W aveform Editor
window menu, or click on the icon
look like the one in Figure A.12. up or down. in the toolbar on the left. Your W aveform Editor window should now 464 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 Figure A.12 Waveform Editor window after changing the value radix and fitting the entire time range inside the
window. Notice also the toolbar buttons on the left. A.5.3 Assigning Values to the Input Signals
The next thing is to assign values to all of the input signals.
1. Drag from time 200ns to 400ns for the s signal only, as shown in Figure A.13. 2. Click on the icon 3. Drag from time 0ns to 100ns for the d0 signal only, as shown in Figure A.13. 4. Click on the icon
signal to decimal 5. 5. Repeat Steps 3 and 4 for the remaining input values for d0 and d1, as shown in Figure A.13. in the toolbar on the left to set the signal in the selected range to a logical 1 value. in the toolbar on the left. Type in the value 5 and click OK to set the value for the d0 bus Figure A.13 Changing the input signal values. A.5.4 Saving the Waveform File
1. Save the W aveform Editor window by selecting File | Save. The Save As window appears. Notice that the
default file name is the same as the top-level entity name, and the extension is .scf. For this example, the name
is 2x8mux.scf. 2. Click on OK to save the file. 465 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 A.5.5 Starting the Simulator
1. We are now ready to simulate the design. From the Manager window menu, select MAX+plus II | Simulator, or
click on the icon to bring up the Simulator window. • You can also save the waveform file and start the simulator in one step by selecting from the Manager window
menu File | Project | Save & Simulate. 2. The Simulator window as shown in Figure A.14 is displayed. Make sure that the Simulation Input filename is
2x8mux.scf. This is the same name as your top-level entity. 3. Click on the Start button and watch the progress of the simulation. Figure A.14 Simulator window for the multiplexer design.
4. At the end of the simulation, if there are no errors, you will see a message window saying that the simulation
was successful. Click OK to close the message window. 5. Click on the Open SCF button in the Simulator window to bring up the Waveform Editor with the resulting
simulation waveforms. The simulation result is shown in Figure A.15. The signal y is the multiplexer output.
Notice that when s is a 0, the y output follows the d0 input, and when s is a 1, the y output follows the d1 input. Figure A.15 Resulting waveform after the simulation.
6. You can change the input signal values of s, d0, and d1 to something different and run the simulation again. 466 Digital Logic and Microprocessor Design with VHDL Appendix A - Schematic Entry. Tutorial 1 A.6 Creating and Using the Logic Symbol
If you want to use this circuit as part of another circuit, you need to create a logic symbol for this circuit.
1. To create a logic symbol for the current active circuit diagram that is in the Graphic Editor window, from the
Graphic Editor menu, select File | Create Default Symbol. The name of this logic symbol is the same as the name
of the current active circuit diagram in the Graphic Editor, but with the extension .sym. 2. You can view and edit the logic symbol by selecting File | Edit Symbol. The placements of the input and output
signals can be moved to different locations by dragging them. The size of the symbol can also be changed by
dragging the edges of the symbol rectangle. 3. To use this circuit in another project, you need to copy this .sym file and the corresponding .gdf circuit file to
the other project’s directory. This new symbol name will now show up in the Enter Symbol window like the one
shown in Figure A.4. 467 Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 Appendix B VHDL Entry—Tutorial 2
This tutorial provides a step-by-step instruction for the VHDL entry, synthesis, and simulation of a 4-bit binary
counter circuit. However, no knowledge of VHDL is required to follow this tutorial. Tutorial 3 (Appendix C) will
show how this circuit can be downloaded to the PLD on the UP2 development board so that you actually can see this
circuit executing in the hardware.
This tutorial is very similar to Tutorial 1 (Appendix A) for schematic entry. The main difference is in using the
Text Editor rather than the Graphic Editor. The procedures for project creation, synthesis, and simulation are the
same in both cases. Even if you do not intend to write VHDL code, you should go through this tutorial so that you
can continue on to Tutorial 3 (Appendix C) and learn how to download a circuit to the PLD on the UP2 development
board. B.1 Getting Started
B.1.1 Preparing a Folder for the Project
1. Each circuit design in MAX+plus II is called a project. Each project should be placed in its own folder, since
the synthesizer creates many associated working files for a project. Using Windows File Manager, create a new
folder for your new project. This tutorial uses the folder called counter created in the root directory of the C
drive. 2. The VHDL source code for the counter circuit can be found on the accompanying CD-ROM in the file
counter.vhd located in the directory <CD-ROM drive>:\VHDL Examples\Appendix B\counter. Using
Windows File Manager, copy this file to the new folder c:\counter that you created in Step 1. B.1.2 Starting MAX+plus II
After the successful installation of the MAX+plus II software, there should be a link for the program under the
Start button. Click on this link to start the program. You should see the MAX+plus II Manager, window as shown in Figure B.1. 468 Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 Figure B.1. The MAX+plus II Manager window.
Figure B.2 shows the toolbar for accessing the main development tools. The buttons from left to right are:
•
•
•
•
•
•
•
•
• Hierarchy display—to show the design files used in the current project
Floorplan Editor—to map the I/O signals from the circuit to the pins on the PLD chip
Compiler (synthesizer)—to synthesize the circuit to its netlist
Simulator—to perform circuit simulation
Timing analyzer—to perform circuit timing analysis
Programmer—to program the circuit to the PLD chip
Open existing or new project—to select a new project
Change project name to current filename—to use the current file as the new project
Open top-level design file—to open the top-level design file for the current project Figure B.2 The MAX+plus II development software tool bar.
In the MAX+plus II software, different commands in the menus are available when different windows are
activated. This might cause some confusion at first. If you cannot find a particular command from the menu, make
sure that the correct window is the active window. B.1.3 Creating a Project
1. From the Manager window menu, select File | Project | Name, or simply click on the icon 469 . You should see Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 the Project Name window similar to the one shown in Figure B.3.
2. Select the C drive from the Drives dropdown list. 3. Move to the counter directory on the C drive and double-click on it. You should see the file counter.vhd listed
in the Files box. 4. Select the file counter.vhd. The filename will be copied to the Project Name text field. 5. Click OK. The MAX+plus II Manager window title should now show c:\counter\counter. Figure B.3 Project name window for creating a new project.
• Alternatively, you can open any VHDL entity source file (with the extension .vhd) using the Manager menu
command File | Open. With the VHDL entity source file in the active Text Editor window, select File | Project |
to make that particular file the top-level project file. Note
Set Project to Current File, or click on the icon
that the name of the file must be the same as the name of the entity that is in this file. B.1.4 Editing the VHDL Source Code
• From the Manager window menu, select File | Hierarchy Project Top, or click on the icon
to open the
VHDL source code for the counter. Notice that the entity name for this circuit is also counter. The top entity
name for the project must be the same as the project name, and the file name.
You can use this Text Editor to modify the code if necessary. For now, we will not make any modifications, so
you can close this Text Editor window. • If you need to create a new VHDL source file, select File | New from the Manager window menu. Select Text
Editor file, and click OK. B.2 Synthesis for Functional Simulation
1. From the Manager window menu, select MAX+plus II | Compiler, or click on the icon
Compiler window. 470 to bring up the Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 2. From the Compiler window menu, that is, with the Compiler window selected as the active window, select
Processing | Functional SNF Extractor so that a check mark appears next to it. To actually see whether there is a
check mark or not, you need to select the Processing menu again. The compiler window for functional
extraction is shown in Figure B.4. 3. Click on the Start button to start the synthesis. You will then see the progress of the synthesis. 4. At the end of the synthesis, if there are no syntax errors, you will see a message window saying that the
compilation was successful. Click OK to close the message window. Figure B.4. Compiler window for functional extraction. B.3 Circuit Simulation
B.3.1 Selecting Input Test Signals
1. Before you can simulate the design, you need to create test vectors for specifying what the input values are.
From the Manager window menu, select MAX+plus II | Waveform Editor. 2. From the W aveform Editor window menu, select Node | Enter Nodes from SNF. You can also right-click under
the Name section in the W aveform Editor window, and select Enter Nodes from SNF from the pop-up menu.
You will see something similar to the Enter Nodes from SNF window shown in Figure B.5. 471 Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 Figure B.5 Window for adding signals for simulation.
3. Click on the List button in the Enter Nodes from SNF window, and a list of available nodes and groups will be
displayed in the Available Nodes & Groups box. 4. Select the signals that you want to see in the simulation trace. The signals that we want are: ResetN (I), Clock
(I), and Q (O). Be careful that it is Q, and not Q0, Q1, Q2, or Q3. The letters I and O in parenthesis next to each
signal denote whether the signal is an input or output signal, respectively. Multiple nodes can be selected by
holding down the Ctrl or Shift key while clicking on the signal names. 5. After selecting the signals, click on the => button to move the selected signals to the Selected Nodes & Groups
box. 6. Repeat Steps 4 and 5 until all of the signals that you want to see in the simulation are moved to the Selected
Nodes & Groups box. 7. Click on OK when you are finished. The selected signals will now be inserted in the Waveform Editor window
similar to Figure B.6. Figure B.6 Waveform Editor window for simulation. 472 Digital Logic and Microprocessor Design with VHDL Appendix B - VHDL Entry. Tutorial 2 B.3.2 Customizing the Waveform Editor
1. You can rearrange the signals in the Waveform Editor by dragging the signal icons like
the Clock signal to the very top of the list. • To delete a signal in the Waveform Editor, just select the signal by clicking on its name and press the Delete
key. • For signals that are composed of a group of bits (such as the counter output Q), you can separate them into
individual bits or change the radix for the displayed value by first selecting that signal and then right-click the
mouse. A drop-down menu appears. Select Ungroup to separate the bits. To regroup them, select the bits you
want to group and then right-click the mouse. A drop-down menu appears. Select Enter Group. Type in a group
name, and select the radix you want for the display. 2. We want to simulate for 2 microseconds. To change the simulation end time, select File | End Time from the
Waveform Editor window menu. 3. In the End Time window, type in 2us, and click OK. 4. To fit the entire simulation time range inside the window, select View | Fit in Window from the W aveform Editor
window menu or click on the icon
look like the one in Figure B.7. up or down. Drag in the toolbar on the left. Your W aveform Editor window should now Figure B.7 Waveform Editor window after rearranging the Clock signal, and fitting the entire time range inside.
Notice also the toolbar buttons on the left. B.3.3 Assigning Values to the Input Signals
The next thing is to assign values to all of the input s