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Representation of information eg numbers, characters, strings. Bitlevel manipulations Boolean Algebra. Numeric encodings 2's complement method. Basic operations addition, multiplication etc... How are information and operations simulated in computer hardware ? 1 Overview of Logic Design
Fundamental Hardware Requirements
Communication
How to get values from one place to another Computation Storage Bits are Our Friends
Everything expressed in terms of values 0 and 1 Communication
Low or high voltage on wire Computation
Compute Boolean functions Storage
Store bits of information
2 Communication  Digital Signals
0 1 0 Voltage Time Use voltage thresholds to extract discrete values from continuous signal Simplest version: 1bit signal
Either high range (1) or low range (0) With guard range between them Not strongly affected by noise or low quality circuit elements
Can make circuits simple, small, and fast
3 Transistor: Building Block of Computers
Microprocessors contain millions of transistors
Intel Pentium II: 7 million Compaq Alpha 21264: 15 million Intel Pentium III: 28 million Logically, each transistor acts as a switch Combined to implement logic functions
AND, OR, NOT Combined to build higherlevel structures
Adder, multiplexer, decoder, register, ... Combined to build processor 4 Simple Switch Circuit
Switch open:
No current through circuit Light is off Vout is +2.9V Switch closed:
Short circuit across switch Current flows Light is on Vout is 0V Switchbased circuits can easily represent two states: on/off, open/closed, voltage/no voltage.
5 Ntype MOS Transistor
MOS = Metal Oxide Semiconductor
two types: Ntype and Ptype Ntype
when circuit between #1voltage, short Gate has positive and #2 (switch closed) when Gate has zero voltage, open circuit between #1 and #2 (switch open) Gate = 1 Terminal #2 must be connected to GND (0V).
6 Gate = 0 Ptype MOS Transistor
Ptype is complementary to Ntype
when Gate has positive voltage, open circuit between #1 and #2 (switch open) when Gate has zero voltage, short circuit between #1 and #2 (switch closed) Gate = 1 Terminal #1 must be connected to +2.9V.
7 Gate = 0 Complementary MOS
Complementary devices work in pairs open when voltage at G is low closes when: voltage(G) > voltage (S) + nchannel closed when voltage at G is low opens when: voltage(G) < voltage (S) pchannel 8 Logic Gates
Use switch behavior of MOS transistors to implement logical functions : AND, OR, NOT. Digital symbols:
recall that we assign a range of analog voltages to each digital (logic) symbol assignment of voltage ranges depends on electrical properties of transistors being used
typical values for "1": +5V, +3.3V, +2.9V from now on we'll use +2.9V
9 Logic Gates 10 Logic Gates from Transistors
NOT Truth table
In 2.9 V 11 Out 0V In 0 1 Out 1 0 0 V 2.9 V Logic Gates from Transistors
NOR A 0 0 1 1 12 B 0 1 0 1 C 1 0 0 0 Note: Serial structure on top, parallel on bottom. Combinational Circuits
Acyclic Network Primary Inputs Primary Outputs Acyclic Network of Logic Gates
Continously responds to changes on primary inputs Primary outputs become (after some delay) Boolean functions of primary inputs 13 Truth Table
Shows all possible inputs and outputs of a function. Each input variable is either 1 or 0 A function with n variables has 2n possible combinations of inputs. Inputs are listed in binary order eg from 000 to 111
f(x,y,z) = (x + y')z + x'
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(x,y,z) 1 1 1 1 0 1 0 1 14 f(0,0,0) f(0,0,1) f(0,1,0) f(0,1,1) f(1,0,0) f(1,0,1) f(1,1,0) f(1,1,1) = (0 + 1)0 + 1 = (0 + 1)1 + 1 = (0 + 0)0 + 1 = (0 + 0)1 + 1 = (1 + 1)0 + 0 = (1 + 1)1 + 0 = (1 + 0)0 + 0 = (1 + 0)1 + 0 =1 =1 =1 =1 =0 =1 =0 =1 NAND Gate is complete in logic
We can build all other 2 input gates with 2 input nands. X = A' A.A = A (Idempotent law) (A.A)' = A' X = A.B; (A')' = A; ((A.B)')' = A.B X = A+B; 15 ((A+B)')' = (A+B); (A'.B')' = A+B (DeMorgan's Law) Boolean fns with Digital Circuits
a 0 0 0 0 1 1 1 1 16 b 0 0 1 1 0 0 1 1 c 0 1 0 1 0 1 0 1 f(a,b,c) 0 0 0 1 0 1 1 1
or f(a,b,c) = (a&b)  (b&c)  (a&c) 3 input majority function Boolean fns with digital circuits
a 0 0 0 0 1 1 1 1 17 b 0 0 1 1 0 0 1 1 c 0 1 0 1 0 1 0 1 f(a,b,c) 0 1 1 0 1 0 0 1
f(a,b,c) = (a^b)^c Parity function : true iff # inputs that are true is odd XOR is a 2input Parity function Boolean fns with digital circuits
Full Adder = Adds 2 bits and carryin, produces 1bit sum and carryout.
Sum = Parity Carryout = Majority a b 0 0 0 0 0 1 carry Carry sum in out 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 s c 0 1 1 0 1 0 1 1 1 1 18 Boolean fns with Digital Circuits
RippleCarry Adder Add 2, kbit numbers. 19 ...
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 Spring '08
 Chakraborty

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