Problem 9A (5 points): Represent the following signed-magnitude binary numbers
in 5-bit, 2s complement format and multiply them using the right-shift algorithm.
Show your work in the form of Figure 9.6.
1) a=+1001 and x = - 0101
2) a=+.1001 and x = -

Implementation Problem (30 points):
1. Use VHDL to implement an n-bit Ripple Carry Adder. Instantiate
and simulate your design using Modelsim for 8 bits.
2. Use VHDL to implement the 64-bit multi-level carry-lookahead
adder shown in Figure 6.4 of our

17 Floating-Point Representations
Chapter Goals
Study a representation method offering both
wide range (e.g., astronomical distances)
and high precision (e.g., atomic distances)
Chapter Highlights
Floating-point formats and related tradeoffs
The need for

14 High-Radix Dividers
Chapter Goals
Study techniques that allow us to obtain
more than one quotient bit in each cycle
(two bits in radix 4, three in radix 8, . . .)
Chapter Highlights
Radix > 2 quotient digit selection harder
Remedy: redundant quotient r

18 Floating-Point Operations
Chapter Goals
See how adders, multipliers, and dividers
are designed for floating-point operands
(square-rooting postponed to Chapter 21)
Chapter Highlights
Floating-point operation = preprocessing +
exponent and significand a

13 Basic Division Schemes
Chapter Goals
Study shift/subtract or bit-at-a-time dividers
and set the stage for faster methods and
variations to be covered in Chapters 14-16
Chapter Highlights
Shift/subtract divide vs shift/add multiply
Hardware, firmware, s

11 Tree and Array Multipliers
Chapter Goals
Study the design of multipliers for highest
possible performance (speed, throughput)
Chapter Highlights
Tree multiplier = reduction tree
+ redundant-to-binary converter
Avoiding full sign extension in multiplyin

10 High-Radix Multipliers
Chapter Goals
Study techniques that allow us to handle
more than one multiplier bit in each cycle
(two bits in radix 4, three in radix 8, . . .)
Chapter Highlights
High radix gives rise to difficult multiples
Recoding (change of

8
MAIN FACTORS AFFECTING COST AND EXECUTION TIME
RADIX r
QUOTIENT-DIGIT SET
1. CANONICAL: 0 qj r 1
2. REDUNDANT: qj Da = cfw_a, a + 1, . . . , 1, 0, 1, . . . , a 1, a
REDUNDANCY FACTOR: =
a
,
r1
>
1
2
REPRESENTATION OF RESIDUAL:
1. NONREDUNDANT (e.g.,

9 Basic Multiplication Schemes
Chapter Goals
Study shift/add or bit-at-a-time multipliers
and set the stage for faster methods and
variations to be covered in Chapters 10-12
Chapter Highlights
Multiplication = multioperand addition
Hardware, firmware, sof

8 Multioperand Addition
Chapter Goals
Learn methods for speeding up the
addition of several numbers (needed
for multiplication or inner-product)
Chapter Highlights
Running total kept in redundant form
Current total + Next number New total
Deferred carry a

7 Variations in Fast Adders
Chapter Goals
Study alternatives to the carry-lookahead
method for designing fast adders
Chapter Highlights
Many methods besides CLA are available
(both competing and complementary)
Best design is technology-dependent
(often hy

3 Redundant Number Systems
Chapter Goals
Explore the advantages and drawbacks
of using more than r digit values in radix r
Chapter Highlights
Redundancy eliminates long carry chains
Redundancy takes many forms: trade-offs
Redundant/nonredundant conversion

4 Residue Number Systems
Chapter Goals
Study a way of encoding large numbers
as a collection of smaller numbers
to simplify and speed up some operations
Chapter Highlights
Moduli, range, arithmetic operations
Many sets of moduli possible: tradeoffs
Conver

6 Carry-Lookahead Adders
Chapter Goals
Understand the carry-lookahead method
and its many variations
used in the design of fast adders
Chapter Highlights
Single- and multilevel carry lookahead
Various designs for log-time adders
Relating the carry determi

2 Representing Signed Numbers
Chapter Goals
Learn different encodings of the sign info
Discuss implications for arithmetic design
Chapter Highlights
Using sign bit, biasing, complementation
Properties of 2s-complement numbers
Signed vs unsigned arithmetic

Advanced Digital Hardware
Spring 2015
Dr. L. S. DeBrunner
NAME: _
Midterm Exam #1
Closed Book & Closed Notes
You may use a calculator for calculations only
Show all work
Corrections and general comments will be written on the board, as