Transistors amp Logic II Montek Singh Nov 1 2017 Lecture 10 Todays Topics Basic gates Boolean algebra Synthesis using standard gates Truth tables Universal gates NAND and NOR Gates with more than 2 inputs ID: 759690 Download Presentation

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Transistors & Logic - II. Montek Singh. Nov 1, 2017. Lecture . 10. Today’s Topics. Basic gates. Boolean algebra. Synthesis using standard gates. Truth tables. Universal gates: NAND and NOR. Gates with more than 2 inputs.

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Slide1

Computer Organization and DesignTransistors & Logic - II

Montek Singh

Nov 1, 2017

Lecture

10

Slide2Today’s Topics

Basic gatesBoolean algebraSynthesis using standard gatesTruth tablesUniversal gates: NAND and NORGates with more than 2 inputsSum-of-ProductsDeMorgan’s Law

2

Slide3Single-Input Logic Gates

3

Slide4Two-Input Logic Gates

4

Slide5More Two-Input Logic Gates

5

Slide6Multiple-Input Logic Gates

6

Slide7Multiple-Input Logic Gates

7

Slide8Basic gates vs. single CMOS gates

Not all basic gates can be implemented using a single CMOS gate

Some can be

inverter, NAND, NOR

we have covered their CMOS implementation

Others need multiple CMOS gates connected together

AND: implemented as NAND + inverter

OR: implemented as NOR + inverter

buffer: implemented as inverter + inverter

XOR / XNOR: will see shortly

Slide9Boolean Algebra

Algebra of 1s and 0s

Slide1010

Table of Identities

Slide1111

Duals

Left and right columns are dualsReplace ANDs and ORs, 0s and 1s

Slide1212

Single Variable Identities

Slide1313

Commutativity

Operation is independent of order of variables

Slide1414

Associativity

Independent of order in which we groupSo can also be simply written as:X+Y+Z, andXYZ

Slide1515

Distributivity

Slide16Substitution

Can substitute arbitrarily large algebraic expressions for the variablesDistribute an operation over the entire expressionExample: X + YZ = (X+Y)(X+Z) Substitute ABC for X ABC + YZ = (ABC + Y)(ABC + Z)

16

Slide1717

DeMorgan’s Theorem

Used a lotNOR invert, then ANDNAND invert, then OR

Slide1818

Truth Tables for DeMorgan’s

Slide1919

DeMorgan’s Thm.: “Bubble Pushing”

Bubble pushing:imagine the bubble at the output is being pushed towards the inputsit becomes a bubble at every input, andthe shape of the gate changes from AND to OR, and vice versa

Slide2020

Algebraic/Boolean Manipulation

Apply algebraic and Boolean identities to simplify expressionExample:

Slide2121

Simplification Example

Apply

Apply

Apply

Slide2222

Fewer Gates

Slide23From Truth Table to Gate-Level Circuit

Slide24Start with Functional Spec

We need to start somewhereusually it’s the functional specification

A

B

Y

If C is 1 then

copy B to Y,

otherwise copy

A to Y

C

First step is to translate a verbal description into a tabular form. Any combinational function can be represented as a

“

truth table.

”A truth table lists the output(s) for each combination of inputs.

Truth Table

Slide25We Can Make Most Gates Out of Others

Example 1: B > AOutput Y is 1 if and only if B is 1 AND A is 0 Y = B AND (NOT(A))

B>A

A

B

y

Slide26We Can Make Most Gates Out of Others

Example 2: A XOR BOutput Y is 1 if and only if …B is 1 AND A is 0 --OR—B is 0 AND A is 1 Y = B AND (NOT(A)) OR A AND (NOT(B))

XOR

A

B

Y

A

B

Y

Symbol for XOR

Slide27How many gates do we really need?

Slide28One Will Do!

NANDs and NORs are universalone can make any circuit out of just NANDs, or out of just NORs!

=

=

=

=

=

=

Slide29Gates with more than two inputs

Sometimes can be directly created in CMOSe.g., 3-input NOR, 4-input NAND etc.Often constructed using smaller gates:e.g., N-input AND gate using several 2-input AND gates AND(A0, A1, A2 … AN-1) = AND … (AND(AND(A0, A1), A2) … AN-1)Delay in computing final output is linear in # of gates: O(N)can we do it faster?

A

0

A

1

A

2

A

3

A

N-1

Slide30Gate trees are faster

More parallelism: combine two at a time in parallelmuch like a tournament bracket!

A

1

A

0

A

3

A

2

A

N-3

A

N-4

A

N-1

A

N-2

Delay is now

logarithmic

: O(log

2

(N))

Slide31Systematic Approach

Given truth table:

Develop Boolean equation

Slide32Design Approach: Sum-of-Products

Three steps:Write functional spec as a truth tableWrite down a Boolean expression forevery row with ‘1’ in the outputan input that is ‘0’ becomes invertedAND all the inputs in each termWire up the gates!This approach give us expressions of the type:SUM-OF-PRODUCTS (“SOP”)Boolean “SUM” actually means ORBoolean “PRODUCT” actually means AND

Truth Table

Slide33Gate-level circuit

We can implement SUM-OF-PRODUCTS……with just three levels of logic:INVERTERS/AND/OR

A

B

C

A

B

C

A

B

C

A

B

C

Y

Slide34Notations

Symbols and Boolean operators:

Slide35An Interesting 3-Input Gate: Multiplexer

Based on C, select the A or B input to be copied to the output Y.

A

B

Y

C

If C is 1 then

copy B to Y,

otherwise copy

A to Y

2-input

Multiplexer (“mux”)

A

B

C

0

1

Gate

symbol

Truth Table

Slide36Multiplexer (MUX) Shortcuts

0

1

0

1

S

0

1

0

1

S

0

1

0

1

S

I

0

I

1

I

2

I

3

Y

S

0

S

1

A 4-input Mux

(implemented as a tree)

0

1

0

1

S

0

1

0

1

S

A

2

B

2

A

3

B

3

Y

0

S

0

1

0

1

S

0

1

0

1

S

A

0

B

0

A

1

B

1

Y

1

Y

2

Y

3

A 4-bit wide 2-input Mux

A

B

C

D

S

0

1

2

3

Y

A

0-3

B

0-3

S

Y

0-3

Slide37Next Class

Arithmetic

c

ircuits

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