1 COMP541 Combinational Logic - II Montek Singh Aug 27, 2014.

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COMP541

Combinational Logic - II

Montek Singh

Aug 27, 2014

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Today Digital Circuits (review)

Basics of Boolean Algebra (review) Identities and Simplification

Basics of Logic ImplementationMinterms and maxtermsGoing from truth table to logic implementation

Digital Circuits Digital Circuit = network that processes binary

variablesone or more binary inputsone or more binary outputs

inputs and outputs are called “terminals”a functional specification

relationship between inputs and outputsa timing specification

describes delay from inputs changing to outputs responding

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Digital Circuits Inside the black box

subcircuits or components or elements

connected by wireswires and terminals often

called “nodes”each node has a binary

valueeach node is an input, an

output, or “internal”

Example:E1, E2, E3 are elementsA, B, C are input nodesY, Z are output nodesn1 is an internal node 4

Types of circuits Two types: with memory and without

Combinational Circuitoutput depends only on the current values of the inputs

– provided enough time is given for output to respondoutput does not depend on past inputs or outputscalled “memoryless”example: AND gate

Sequential Circuitanything not combinational is sequentialoutput depends on not only current inputs, but also past

behavior– previous inputs and/or outputs affect behavior

has “memory”, or is “stateful”example: counter

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Combinational Circuits: Examples

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Adder

OR

Multi-output exampleSlash notation

Combinational Circuits Theorem: A circuit is combinational if:

every element is itself combinationalevery node is either designated as an input, or

connects to exactly one output terminal of an elementoutputs of two elements are never “shorted together”ensures that each node has a unique/unambiguous value

contains no cyclic pathsevery path through the circuit visits each node at most

onceno “feedback”

Conditions above ensure that output is only a function of inputsProof: By induction

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Combinational Circuits: Examples Which meet the conditions for combinational

logic?

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Identities in Boolean Algebra Use identities to manipulate functions You can use distributive law …

… to transform

ZY X F

))(( ZXY X F

))(( ZX YX YZ X

to

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Table of Identities

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Duals Left and right columns are duals Replace ANDs and ORs, 0s and 1s

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Single Variable Identities

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Commutativity Operation is independent of order of variables

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Associativity Independent of order in which we group

So can also be written asand

ZYX XYZ

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Distributivity

Substitution 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)

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DeMorgan’s Theorem Used a lot

NOR invert, then AND

NAND invert, then OR

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Truth Tables for DeMorgan’s

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Algebraic/Boolean Manipulation Apply algebraic and Boolean identities to

simplify expressionexample: XZZYX YZ X F

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Simplification Example

XZZYX YZ X F

XZZ Z YX F )(

XZYX F 1

XZYX F

Apply

Apply

Apply

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Fewer Gates

XZYX F

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Consensus Theorem

The third term is redundantCan just drop third term (consensus term)

Proof summary (for first version):For third term to be true, Y & Z both must be 1Then one of the first two terms is already 1!

Exercise: Provide a similar proof for the 2nd version

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Complement of a Function Definition: 1s & 0s swapped in truth table Mechanical way to derive algebraic form

Take the dualRecall: Interchange AND and OR, and 1s & 0s

Complement each literalx becomes x’ (x’ means complement of x)

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Next Lecture Next Class: More on combinational logic

Commonly-used combinational building blocks