1 COMP541 Combinational Logic - II Montek Singh Jan 18, 2012.

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COMP541COMP541

Combinational Logic - IICombinational Logic - II

Montek SinghMontek Singh

Jan 18, 2012Jan 18, 2012

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TodayToday Basics of Boolean Algebra (review)Basics of Boolean Algebra (review)

Identities and SimplificationIdentities and Simplification

Basics of Logic ImplementationBasics of Logic Implementation Minterms and maxtermsMinterms and maxterms Going from truth table to logic implementationGoing from truth table to logic implementation

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IdentitiesIdentities Use identities to manipulate functionsUse identities to manipulate functions You can use distributive law … You can use distributive law …

… … to transform fromto transform from

ZY X F

))(( ZXY X F

))(( ZX YX YZ X

toto

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Table of IdentitiesTable of Identities

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DualsDuals Left and right columns are Left and right columns are dualsduals Replace AND and OR, 0s and 1sReplace AND and OR, 0s and 1s

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Single Variable IdentitiesSingle Variable Identities

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CommutativityCommutativity Operation is independent of order of variablesOperation is independent of order of variables

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AssociativityAssociativity Independent of order in which we groupIndependent of order in which we group

So can also be written asSo can also be written as

andandZYX

XYZ

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DistributivityDistributivity

Can substitute arbitrarily large algebraic Can substitute arbitrarily large algebraic expressions for the variablesexpressions for the variables Distribute an operation over the entire expressionDistribute an operation over the entire expression

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DeMorganDeMorgan’’s Theorems Theorem Used a lotUsed a lot NOR NOR invert, then AND invert, then AND

NAND NAND invert, then OR invert, then OR

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Truth Tables for DeMorganTruth Tables for DeMorgan’’ss

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Algebraic ManipulationAlgebraic Manipulation Consider functionConsider function

XZZYX YZ X F

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Simplify FunctionSimplify Function

XZZYX YZ X F

XZZ Z YX F )(

XZYX F 1

XZYX F

Apply

Apply

Apply

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Fewer GatesFewer Gates

XZYX F

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Consensus TheoremConsensus Theorem

The third term is redundantThe third term is redundant Can just dropCan just drop

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

ZXXYYZZXXY

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

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

Complement each literalComplement each literal

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Mechanically Go From Truth Table Mechanically Go From Truth Table to Functionto Function

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From Truth Table to FuncFrom Truth Table to Func Consider a truth tableConsider a truth table Can implement F by taking OR of all terms that Can implement F by taking OR of all terms that

are 1are 1

XYZZXYZYXZY XZ YX F

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Standard FormsStandard Forms Not necessarily simplest FNot necessarily simplest F But itBut it’’s a mechanical way to go from truth s a mechanical way to go from truth

table to functiontable to function

Definitions:Definitions: Product terms – AND Product terms – AND ĀBZ ĀBZ Sum terms – OR Sum terms – OR X + Ā X + Ā This is logical product and sum, not arithmeticThis is logical product and sum, not arithmetic

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Definition: MintermDefinition: Minterm Product term in which all variables appear Product term in which all variables appear

once (complemented or not)once (complemented or not)

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Number of MintermsNumber of Minterms For For nn variables, there will be variables, there will be 22nn minterms minterms Like binary numbers from 0 to 2Like binary numbers from 0 to 2nn-1-1 Often numbered same way (with decimal Often numbered same way (with decimal

conversion)conversion)

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MaxtermsMaxterms Sum term in which all variables appear once Sum term in which all variables appear once

(complemented or not)(complemented or not)

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Minterm related to MaxtermMinterm related to Maxterm Minterm and maxterm with same subscripts Minterm and maxterm with same subscripts

are complementsare complements

ExampleExample

33 MZYXYZXm

Mjm j

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Sum of MintermsSum of Minterms Like Slide 18Like Slide 18 OR all of the minterms of truth table row with OR all of the minterms of truth table row with

a 1a 1 ““ON-set mintermsON-set minterms””

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Sum of ProductsSum of Products Simplifying sum-of-minterms can yield a sum Simplifying sum-of-minterms can yield a sum

of productsof products Difference is each term need not be a mintermDifference is each term need not be a minterm

i.e., terms do not need to have all variablesi.e., terms do not need to have all variables

A bunch of ANDs and one ORA bunch of ANDs and one OR

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Two-Level ImplementationTwo-Level Implementation Sum of products has 2 levels of gatesSum of products has 2 levels of gates

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More Levels of Gates?More Levels of Gates? WhatWhat’’s best?s best?

Hard to answerHard to answer More gate delays (more on this later)More gate delays (more on this later) But maybe we only have 2-input gatesBut maybe we only have 2-input gates

So multi-input ANDs and ORs have to be decomposedSo multi-input ANDs and ORs have to be decomposed

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

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

Complement each literalComplement each literal

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Complement of FComplement of F Not surprisingly, just sum of the other Not surprisingly, just sum of the other

mintermsminterms ““OFF-set mintermsOFF-set minterms””

In this caseIn this case

mm11 + m + m33 + m + m44 + m + m66

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Product of MaxtermsProduct of Maxterms Recall that maxterm is true except for its own Recall that maxterm is true except for its own

casecase So M1 is only false for 001So M1 is only false for 001

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Product of MaxtermsProduct of Maxterms Can express F as AND of all rows that should Can express F as AND of all rows that should

evaluate to 0evaluate to 0

6431 MMMMF

))(( ZYXZYXF ))(( ZYXZYX

oror

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Product of SumsProduct of Sums Result: another standard formResult: another standard form ORs followed by ANDORs followed by AND

Terms do not have to be maxtermsTerms do not have to be maxterms

))(( ZYXZYXF

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RecapRecap Working (so far) with AND, OR, and NOTWorking (so far) with AND, OR, and NOT Algebraic identitiesAlgebraic identities Algebraic simplificationAlgebraic simplification Minterms and maxtermsMinterms and maxterms Can now synthesize function (and gates) from Can now synthesize function (and gates) from

truth tabletruth table

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Next TimeNext Time Lab on Fri, 1/20:Lab on Fri, 1/20:

Demo lab softwareDemo lab software Do 1Do 1stst lab assignment lab assignment

download software: see website for linkdownload software: see website for linkcouple of simple Verilog programming problemscouple of simple Verilog programming problems

Next Week: More on combinational logicNext Week: More on combinational logic