Computer ArchitectureFall 2008 © August 20 th, 2008 Introduction to Computer Architecture Lecture 2 – Digital Logic Design

Computer Architecture Fall 2008 ©

August 20th, 2008

www.qatar.cmu.edu

Introduction to Computer Architecture

Lecture 2 – Digital Logic Design


Digital Systems

Digital vs. Analog Waveforms

Analog: values vary over a range continuously

Digital: only assumes discrete values

+5

V

–5

1 0 1

T ime

+5

V

–5

T ime


Digital Hardware Systems

Algebra: variables, values, operations

In Boolean algebra, the values are the symbols 0 and 1 If a logic statement is false, it has value 0 If a logic statement is true, it has value 1

Operations: AND, OR, NOT

Boolean Algebra and Logical Operators

0 0 1 1

X Y X AND Y

0 1 0 1

0 0 0 1

X Y X OR Y

0 0 1 1

0 1 0 1

0 1 1 1

X NOT X

0 1

1 0


L1

L6

L2

L3

L7

L4

L5

Example: Seven Segment Display

°Chip to drive digital display


B3 B2 B1 B0 Val L1 L2 L3 L4 L5 L6 L7

0 0 0 0 0 1 0 1 1 1 1 1

0 0 0 1 1 0 0 0 0 0 1 1

0 0 1 0 2 1 1 1 0 1 1 0

0 0 1 1 3 1 1 1 0 0 1 1

0 1 0 0 4 0 1 0 1 0 1 1

0 1 0 1 5 1 1 1 1 0 0 1

0 1 1 0 6 1 1 1 1 1 0 1

0 1 1 1 7 1 0 0 0 0 1 1

1 0 0 0 8 1 1 1 1 1 1 1

1 0 0 1 9 1 1 1 1 0 1 1

L1

L6

L2

L3

L7

L4

L5

Example (cont.)


Example (cont.)

° Implement L4:

Some gate level implementationof the Boolean function for L4


Representations of Digital Design: Switches

A switch connects two points under control signal.

when the control signal is 0 (false), the switch is open

when it is 1 (true), the switch is closed

when control is 1 (true), switch is open

when control is 0 (false), switch is closed

Normally Closed

Normally Open

Open Switch

Control

Normally Open Switch

Closed Switch

T rue

False

Open Switch

Control

Normally Closed Switch

Closed Switch

T rue

False


Switch RepresentationsExamples: routing inputs to outputs through a maze

Floating nodes: what happens if the car is not running? outputs are floating rather than forced to be false

Under all possible control signal settings (1) all outputs must be connected to some input through a path (2) no output is connected to more than one input through any path

EXAMPLE: IF car in driveway OR (car in garage AND NOT garage door closed) AND car running THEN can back out car

Car in garage Car

running

True

True

Car can back out

Garage door closed

Car in driveway

EXAMPLE: IF car in garage AND garage door open AND car running THEN back out car

T rue Car can back out

Garage door open

Car running

Car in garage


Steering Logic: Switches

Voltage Controlled Switches

Gate

Oxide

Source DrainSilicon Bulk

Channel Region

Metal Gate, Oxide, Silicon Sandwich

Diffusion regions: negatively charged ions driven into Si surface

Si Bulk: positively charged ions

By "pulling" electrons to the surface, a conducting channel is formed

"n-Channel MOS"

n-type Si

p-type Si


Switching or Steering Logic

Voltage Controlled Switches

Logic 0 on gate,Source and Drain connected

Gate

Source Drain

Gate

Source Drain

nMOS Transistor

pMOS Transistor

Logic 1 on gate,Source and Drain connected


Using Switches to implement Logic

Inverter Operation

+5V

"1" "0"

+5V

"0" "1"

Input is 1Pull-up does not conductPull-down conductsOutput connected to GND

Input is 0Pull-up conductsPull-down does not conductOutput connected to VDD


NAND Gate

NAND Gate Operation

A = 1, B = 1Pull-up network does not conductPull-down network conductsOutput node connected to GND

A = 0, B = 1Pull-up network has path to VDDPull-down network path brokenOutput node connected to VDD

+5V"1" "1"

"0"

+5V"0" "1"

"1"


NOR Gate

NOR Gate Operation

+5V"0" "0"

"1"

+5V"1" "0"

"0"

A = 0, B = 0Pull-up network conductsPull-down network brokenOutput node at VDD

A = 1, B = 0Pull-up network brokenPull-down network conductsOutput node at GND


Switch Representations

Implementation of AND and OR Functions with Switches

A

False

T rue

output

B A

False

T rue

output

B

AND functionSeries connection to TRUE

OR functionParallel connection to TRUE


Representations of a Digital Design

Truth Tables

tabulate all possible input combinations and their associated output values

Example: half adder adds two binary digits to form Sum and Carry

Example: full adder adds two binary digits and Carry in to form Sum and Carry Out

NOTE: 1 plus 1 is 0 with a carry of 1 in binary

A B

0 0 1 1

0 1 0 1

Sum Carry

0 1 1 0

0 0 0 1

A 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

C in 0 1 0 1 0 1 0 1

S um 0 1 1 0 1 0 0 1

C out 0 0 0 1 0 1 1 1


Representing Digital Design: Boolean Algebra

NOT X is written as XX AND Y is written as X & Y, or sometimes X YX OR Y is written as X + Y

values: 0, 1variables: A, B, C, . . ., X, Y, Zoperations: NOT, AND, OR, . . .

A

0011

B

0101

Sum

0110

Carry

0001

Sum = A B + A B

Carry = A B

OR'd together product terms for each truth table

row where the function is 1

if input variable is 0, it appears in complemented form;

if 1, it appears uncomplemented

Deriving Boolean equations from truth tables:


Representing Digital Logic: Boolean Algebra

A

00001111

B

00110011

Cin

01010101

Sum

01101001

Cout

00010111

Another example:

Sum = A B Cin + A B Cin + A B Cin + A B Cin

Cout = A B Cin + A B Cin + A B Cin + A B Cin


Gate Representations of a Digital Designmost widely used primitive building block in digital system design

StandardLogic Gate

RepresentationHalf Adder Schematic

Netlist: tabulation of gate inputs & outputs and the nets they are connected to

Net: electrically connected collection of wires

Inverter

AND

OR

A

B

CARR Y

SUM


Representations of a Digital Design: Gates

Full Adder Schematic

Fan-in: number of inputs to a gateFan-out: number of gate inputs an output is connected to

Technology "Rules of Composition" place limits on fan-in/fan-out

Cin B A \Cin \ B \ A

A

B

Cin SUM

Cout

A B

B C in

A C in

C out


Waveform Representationdynamic behavior of a circuitreal circuits have non-zero delays

Timing Diagram of the Half Adder

sumpropagation

delay

circuit hazard: 1 plus 0 is 1, not 0!

sumpropagation

delay

Output changes are delayed from input changes

The propagation delay is sensitive to paths in the circuit

Outputs may temporarily change from the correct value to the wrong value back again to the correct value: this is called a glitch or hazard

100 200

A B SUM CARR Y


Routing Stuff Around – Multiplexors and Demultiplexors

Multi-point connections

MUX MUX

DEMUX

A B

Sum

A0 A1 B0 B1

Sa Sb

Ss

S0 S1

Multiple input sources

Multiple output destinations


So far Combinatorial (or stateless) logic°Things don’t get really interesting until we have memory elements

• Think about a digital watch, or a vending machine (or a microprocessor!)

• This is called “Sequential Logic”

°Two questions:• How do we design memory elements?

- How do we create logic that remembers?

• How do we design sequential logic


Memory Elements°The trick is to use feedback

Cascaded Inverters: Static Memory Cell

"0"

"1"

°Example: D-Latch

°Latch vs. Flip Flop• Flip flop: two latches in a row

• Allows you to read the old value and write a new value without a race condition

D Q

D

Clk

Q

Clk


Sequential Logic Design

° State as part of the input

° Next state as part of the output

° State stored in flip flops (registers!)

° Clock controls everything (synchronous logic)

° Lets rethink our 8 segment display example to make it display a counter

inputs outputsCombinatorialLogic

State

Current state Next state


Take home points°Logic built using switches (transistors)

°Boolean algebra and truth tables

°Translation to gates: sum of products

°Multiplexors, demultiplexors, ALU … all designed this way

°Sequential logic• Memory elements• Old state part of inputs, next state part of outputs

• Counters, registers, control all designed this way

Documents

Computer ArchitectureFall 2008 © August 20 th, 2008 Introduction to Computer Architecture Lecture 2 – Digital Logic Design