数字电子技术(Floyd 第十版)课件Chapter 5.ppt

数字电子技术(Floyd 第十版)课件Chapter 5.ppt

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,DigitalFundamentalsTenthEditionFloyd,Chapter4,2008PearsonEducation,InBooleanalgebra,avariableisasymbolusedtorepresentanaction,acondition,ordata.Asinglevariablecanonlyhaveavalueof1or0.,Summary,BooleanAddition,Thecomplementrepresentstheinverseofavariableandisindicatedwithanoverbar.Thus,thecomplementofAisA.,Aliteralisavariableoritscomplement.,AdditionisequivalenttotheORoperation.Thesumtermis1ifoneormoreiftheliteralsare1.Thesumtermiszeroonlyifeachliteralis0.,Example,Solution,Eachliteralmust=0;thereforeA=1,B=0andC=1.,InBooleanalgebra,multiplicationisequivalenttotheANDoperation.Theproductofliteralsformsaproductterm.Theproducttermwillbe1onlyifalloftheliteralsare1.,Summary,BooleanMultiplication,Example,Solution,Eachliteralmust=1;thereforeA=1,B=0andC=0.,Summary,CommutativeLaws,Intermsoftheresult,theorderinwhichvariablesareORedmakesnodifference.,Thecommutativelawsareappliedtoadditionandmultiplication.Foraddition,thecommutativelawstates,A+B=B+A,Intermsoftheresult,theorderinwhichvariablesareANDedmakesnodifference.,Formultiplication,thecommutativelawstates,AB=BA,Summary,AssociativeLaws,WhenORingmorethantwovariables,theresultisthesameregardlessofthegroupingofthevariables.,Theassociativelawsarealsoappliedtoadditionandmultiplication.Foraddition,theassociativelawstates,A+（B+C）=（A+B）+C,Formultiplication,theassociativelawstates,WhenANDingmorethantwovariables,theresultisthesameregardlessofthegroupingofthevariables.,A（BC）=（AB）C,Summary,DistributiveLaw,Thedistributivelawisthefactoringlaw.Acommonvariablecanbefactoredfromanexpressionjustasinordinaryalgebra.Thatis,AB+AC=A（B+C）,Thedistributivelawcanbeillustratedwithequivalentcircuits:

AB+AC,A（B+C）,Summary,RulesofBooleanAlgebra,1.A+0=A,2.A+1=1,3.A.0=0,4.A.1=1,5.A+A=A,7.A.A=A,10.A+AB=A,12.（A+B）（A+C）=A+BC,Summary,RulesofBooleanAlgebra,RulesofBooleanalgebracanbeillustratedwithVenndiagrams.ThevariableAisshownasanarea.,TheruleA+AB=Acanbeillustratedeasilywithadiagram.AddanoverlappingareatorepresentthevariableB.,TheoverlapregionbetweenAandBrepresentsAB.,ThediagramvisuallyshowsthatA+AB=A.Otherrulescanbeillustratedwiththediagramsaswell.,=,Summary,RulesofBooleanAlgebra,Example,Solution,IllustratetherulewithaVenndiagram.,Summary,RulesofBooleanAlgebra,Rule12,whichstatesthat（A+B）（A+C）=A+BC,canbeprovenbyapplyingearlierrulesasfollows:

（A+B）（A+C）=AA+AC+AB+BC=A+AC+AB+BC=A（1+C+B）+BC=A.1+BC=A+BCThisruleisalittlemorecomplicated,butitcanalsobeshownwithaVenndiagram,asgivenonthefollowingslide,Summary,ThearearepresentingA+Bisshowninyellow.,ThearearepresentingA+Cisshowninred.,ThreeareasrepresentthevariablesA,B,andC.,Theoverlapofredandyellowisshowninorange.,ORingwithAgivesthesameareaasbefore.,=,TheoverlappingareabetweenBandCrepresentsBC.,Summary,DeMorgansTheorem,Thecomplementofaproductofvariablesisequaltothesumofthecomplementedvariables.,DeMorgans1stTheorem,ApplyingDeMorgansfirsttheoremtogates:

Summary,DeMorgansTheorem,DeMorgans2ndTheorem,Thecomplementofasumofvariablesisequaltotheproductofthecomplementedvariables.,A+B=A.B,ApplyingDeMorganssecondtheoremtogates:

Summary,ApplyDeMorganstheoremtoremovetheoverbarcoveringbothtermsfromtheexpressionX=C+D.,DeMorgansTheorem,Example,Solution,Summary,BooleanAnalysisofLogicCircuits,CombinationallogiccircuitscanbeanalyzedbywritingtheexpressionforeachgateandcombiningtheexpressionsaccordingtotherulesforBooleanalgebra.,ApplyBooleanalgebratoderivetheexpressionforX.,Example,Solution,Writetheexpressionforeachgate:

ApplyingDeMorganstheoremandthedistributionlaw:

X,Summary,BooleanAnalysisofLogicCircuits,UseMultisimtogeneratethetruthtableforthecircuitinthepreviousexample.,Example,Solution,SetupthecircuitusingtheLogicConverterasshown.（Notethatthelogicconverterhasno“real-world”counterpart.）,Double-clicktheLogicConvertertopopenit.Thenclickontheconversionbarontherightsidetoseethetruthtableforthecircuit（seenextslide）.,Summary,BooleanAnalysisofLogicCircuits,Thesimplifiedlogicexpressioncanbeviewedbyclicking,Summary,SOPandPOSforms,Booleanexpressionscanbewritteninthesum-of-productsform（SOP）orintheproduct-of-sumsform（POS）.Theseformscansimplifytheimplementationofcombinationallogic,particularlywithPLDs.Inbothforms,anoverbarcannotextendovermorethanonevariable.,AnexpressionisinSOPformwhentwoormoreproducttermsaresummedasinthefollowingexamples:

AnexpressionisinPOSformwhentwoormoresumtermsaremultipliedasinthefollowingexamples:

Summary,SOPStandardform,InSOPstandardfor