82Boolean Algebra and Logic Circuit

5.2BooleanAlgebraandLogicCircuit布尔代数与逻辑电路BooleanAlgebra布尔代数LogicCircuit逻辑电路LogicGate逻辑门mathematicallogic数理逻辑logicvariable逻辑变量logicoperation逻辑运算expression表达式ANDoperation与运算ANDgate与门functiontable真值表，函数表，功能表ORoperation或运算ORgate或门NOToperation非运算complement补码，反码Inversion取反，反向Inverter反相器DeMorgan’sTheorem摩根定理identity恒等式compositeoperations复合运算NAND与非NOR或非integratedcircuit集成电路logicflowdiagram逻辑流程图Flip-Flop（FF）触发器memorycharacteristic记忆特性SET置位CLEAR复位symmetrical对称的illegal非法的internal内部的circuitry电路,线路memoryelement存储元件,记忆元件burglaralarm防盗报警器deactivate使无效，使不活动photocell光电池，光电管illuminate照明，照亮saturated饱和的activate使动作，使活动synchronous同步的sequential顺序的synchronoussequentialsystem同步时序系统synchronizev.同步masterclock主时钟periodic周期的pulse脉冲periodicpulse周期脉冲squarewave方波dutycycle占空比interval时间间隔，间隔risingedge上升沿fallingedge下降沿positive-goingedge正向沿negative-goingedge负向沿clockedflip-flop时钟触发器frequency频率propagationdelays传播延迟trigger触发waveforms波形triggerinput触发脉冲输入controlinput控制输入edge-triggered边沿触发triangle三角形BooleanAlgebraandLogicGatesThesectionisconcernedwithdigitalsystemvariablesthattakeononlytwovalues(binaryvariables).Weconventionallydenotethesevaluesas“0”and”1”,andthenuseaspecialsetofrulescalledBooleanalgebratosummarizethevariouswaysinwhichdigitalvariablescanbecombined.Thisalgebraandmuchofthenotationareadopteddirectlyfrommathematicallogic.Thus,”logicvariable”or“logicoperation”arecommonlyusedinplaceof“digitalvariable”or“digitaloperation”.DefinitionoftheANDoperation:Giventwoinputvariables,AandB,andanoutputvariableC,theexpression.C=AandBmeansC=1ifA=1andB=1otherwiseC=0AcircuitthatperformstheANDoperationiscalledanANDgate.Thelogicsymbolforatwo-inputANDgateisshowninFigure5.3.AdotisusedasashorthandfortheANDoperation,sothatEq.5.2maybewrittenC=AB,thedotisoftenomittedsimplifyingfurtherC=AB.Onenicefeatureofdigitaloperationsisthatthecompletesetofinputoutputvariablevaluescanbewrittendown.Figure5.3(a)showssuchafunctiontable,correspondingtoequationC=AB,whichlistsallpossiblecombinationsofinputvariablesAandBtogetherwiththecorrespondingoutputvariableC.FromthisfunctiontableweseethatinalgebraictermstheANDoperationisaformofmultiplication,withthesemanipulationrules:DefinitionoftheORoperation:GiventwoinputDandE,andanoutputvariableF,theexpressionF=DOREMeansF=1ifD=1orE=1orbothD=1andE=1The+signisusedasashorthandforOR,andisneveromittedinalgebraicexpressions,Thus,Eq.5.5iswrittenalgebraicallyasF=D+EFigure5.3(b)showsthelogicsymbolusedforthetwo-inputORgatetogetherwiththecorrespondingfunctiontable.Algebraically,theORoperationisaspecialformofadditionperformedaccordingtotheserules:Notethatthelastmanipulation,1+1=1,differsfromtheordinaryarithmeticuseofthe+sign.Asinordinaryalgebra,parenthesesmaybeusedinbooleanexpressionstogrouptermsandgiveprecedencetooperations.Ifthesearenoparentheses,theANDfunctionsinanequationareevaluatedfirst.DefinitionoftheNOToperation:Insomesituations,theoppositevalueofaparticularvariableisrequired.InBooleanalgebra,theoppositevalueofavariableiscalledthecomplementofthatvariable,andisdenotedbyabardrawnoverthevariableinquestion.ThecomplementoperationissummarizedbelowusingvariableGasanexample.Thelogicoperationthatproducesthecomplementiscalledinversion,ortheNOToperation.ThelogicsymbolandfunctiontableforaninverterisshownisFigure5.3(c).DeMorgan’sTheorem.DeMorgan’stheoremisaBooleanalgebraidentityexpressionthatstatesorequivalentlyYXYX)(YXYX(Notethatthecompletealgebraicexpressionunderneaththecomplementbarmustfirstbeevaluated,thenthecomplementtaken.)Thistheoremiseasilyverifiedbyexaminingthefunctiontablesforthetwosidesofeachequation.Insummary,DeMorgan’stheoremstatesthatthecomplementoftheORoperationisequivalenttoperformingtheANDoperationonthecomplementvariables,andviceversa.DeMorgan’stheoremisofgreatuseinmanipulatingandsimplifyingBooleanalgebraicexpressionthatcontainmorethanonebasiclogicoperation.ThecompositeoperationsNANDandNOR.Twocombinationsofbasicoperationsarisesooftenthattheyaregivenindividualnamesandlogicsymbols.TheNORoperationisthecomplementoftheORoperation(thenameissimplyacontractionof“NOTOR”),andisdefinedbyor,byDeMorgan’stheorem,BACBACTwoequivalentsymbolsfortheNORgate,representingEq.5.9andEq5.10respectively,areshowninFigure5.4(a)alongwiththeNORfunctiontable.NotethatthesmallcircleadjacenttotheinputoroutputofthebasicgatesymbolsproducestheINVERSIONofthevariableineachcase.ThecomplementoftheANDoperationiscalledtheNANDoperation(from“NOTAND”),andisdefinedbythetwoequivalentformsorThetwoequivalentsymbolsforNANDgatesandthefunctiontableareshowninFigure5.4(b).EDFEDFTheprincipalimportanceofNORandNANDisthattheyarethesimplestlogicfunctionstoconstructinintegratedcircuitform.Thus,whileitmaybeeasierforthebeginnertolearnto“think”withORandAND,heshouldalsopracticethinkingwithNORandNANDasthesefunctionsarelikelytobeusedinthefinalcircuitrealization.Also,itispossibletosynthesizeallofthelogicfunctionsusingonlyNORgatesoronlyNANDgates.LetusformulateasimpleeverydaysituationintermsofdigitalvariablesandBooleanoperations.Supposeyouaredrivinghomeandbecomethirstyforahotdrink.Youseeadinerahe