电子科大数字电路完整4-1定理 chenyuppt课件

.,1,chapter4,CombinationalLogicDesignPrinciples,.,2,chapter4SwitchingAlgebraCombinationalCircuitAnalysisCombinationalCircuitSynthesisTimingHazards,.,3,BasicConcept(基本概念),Logiccircuitsareclassifiedintotwotypes（逻辑电路分为两大类）combinationallogiccircuit（组合逻辑电路）Acombinationallogiccircuitisonewhoseoutputsdependonlyonitscurrentinputs.（任何时刻的输出仅取决与当时的输入）characteristic：nofeedbackcircuitsequentiallogiccircuit（时序逻辑电路）Theoutputsofasequentiallogiccircuitdependnotonlyonthecurrentinputs,butalsoonthepastsequenceofinputs,possiblyarbitrarilyfarbackintime.（任一时刻的输出不仅取决于当时的输入，还取决于过去的输入顺序）,.,4,4.1SwitchingAlgebra,4.1.1Axioms（公理）P185（A1）X=0ifX1,（A1）X=1ifX0（A2）IfX=0,thenX=1（A2）IfX=1,thenX=0（A3）00=0（A3）1+1=1（A4）11=1（A4）0+0=0（A5）01=10=0（A5）1+0=0+1=1,Westatedtheseaxiomsasapair,withtheonlydifferencebetweenA1andA1beingtheinterchangeofthesymbols0and1.Thisisacharacteristicofalltheaxiomsofswitchingalgebra.P（185）,逻辑乘logicalmultiplicationdot乘点multiplicationdot,.,5,4.1.2Single-VariableTheorems(单变量开关代数定理)P188,Identities(自等律)：(T1)X+0=X(T1)X1=XNullElements(0-1律)：(T2)X+1=1(T2)X0=0Idempotency(同一律):(T3)X+X=X(T3)XX=XInvolution(还原律)：(T4)(X)=XComplements(互补律):(T5)X+X=1(T5)XX=0,变量和常量的关系,变量和其自身的关系,.,6,4.1.3Two-andThree-VariableTheorems(1),Commutativity(交换律)(T6)X+Y=Y+X(T6)XY=YXAssociativity(结合律)(T7)X(YZ)=(XY)Z(T7)X+(Y+Z)=(X+Y)+ZDistributivity(分配律)(T8)X(Y+Z)=XY+XZ(T8)X+YZ=(X+Y)(X+Z),Eachofthesetheoremsiseasilyprovedbyperfectinduction.(可以利用完备归纳法证明公式和定理)P188SimilarRelationshipwithGeneralAlgebra(与普通代数相似的关系),.,7,4.1.3Two-andThree-VariableTheorems(2),Covering(吸收律)(T9)X+XY=X(T9)X(X+Y)=XCombining(合并律)(T10)XY+XY=X(T10)(X+Y)(X+Y)=XConsensus（添加律（一致性定理）(T11)XY+XZ+YZ=XY+XZ(T11)(X+Y)(X+Z)(Y+Z)=(X+Y)(X+Z),.,8,Notes,nopowerofnumber(没有变量的乘方)AAAA3commonfactor(允许提取公因子)AB+AC=A(B+C)nodivision(没有定义除法)ifAB=BCA=C?,Nosubtracting(没有定义减法)ifA+B=A+CB=C?,A=1,B=0,C=0AB=BC=0,AC,A=1,B=0,C=1,错！,错！,.,9,4.1.4n-VariableTheorems(n变量定理),Generalizedidempotency(广义同一律)(T12)X+X+X=X(T12)XXX=XDeMorgansTheorems(德.摩根定理)(T13)(X1X2Xn)=X1+X2+Xn(T13)(X1+X2+Xn)=X1X2XnGeneralizedDeMorgansTheorems(广义德.摩根定理)(T14)F(X1,X2,Xn,+,)=F(X1,X2,Xn,+),Mostofthesetheoremscanbeprovedusingatwo-stepmethodcalledfiniteinductionfirstprovingthatthetheoremistrueforn=2(thebasisstep)andthenprovingthatifthetheoremistrueforn=i,thenitisalsotrueforn=i+1(theinductionstep).P(190),.,10,finiteinduction(P190),X+X+X+X=X+(X+X+X)(i+1Xsoneitherside)=X+(X)(ifT12istrueforn=i)=X(accordingtoT3),.,11,DemorgansTheorems(摩根定理)(P191),.,12,反演规则(ComplementRules)：swapping+and.andcomplementingallvariables.+，01，变量取反遵循原来的运算优先（Priority）次序不属于单个变量上的反号应保留不变,complementofalogicexpression(F)(反演定理)(P192),.,13,例1：写出下面函数的反函数（Complementfunction)F1=A(B+C)+CDF2=(AB)+CDE,例2：证明(AB+AC)=AB+AC,合理地运用反演定理能够将一些问题简化,.,14,合理地运用反演定理能够将一些问题简化,(AB+AC),AB+AC+BC=AB+AC,(A+B)(A+C),AA+AC+AB+BC,AC+AB,AC+AB+BC,Example2：prove(AB+AC)=AB+AC,.,15,4.1.5duality(对偶定理)(P193),FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),PrincipleofDualityAnytheoremoridentityinswitchingalgebraremainstrueif0and1areswappedand.and+areswappedthroughout.,thedualofalogicexpression,F(X1,X2,Xn)=FD(X1,X2,Xn),.,16,4.1.5duality(对偶定理)(P193),对偶规则+；01变换时不能破坏原来的运算顺序（优先级）对偶原理(PrincipleofDuality)若两逻辑式相等，则它们的对偶式也相等,例：写出下面函数的对偶函数F1=A+B(C+D)F2=(A(B+C)+(C+D),X+XY=XXX+Y=X(错),X(X+Y)=X,FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),.,17,对偶定理（DualityTheorems）,证明公式：A+BC=(A+B)(A+C),.,18,DualityandComplement(对偶和反演)(P194.P195),对偶(Duality)：FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),反演(Complement)：F(X1,X2,Xn,+,)=F(X1,X2,Xn,+),F(X1,X2,Xn)=FD(X1,X2,Xn),正逻辑约定和负逻辑约定互为对偶关系,.,19,ElectricalFunctionTable(电气功能表),Positive-LogicConvention,Negative-LogicConvention,Positive-Logic(正逻辑)：F=AB,Negative-Logic(负逻辑)：F=A+B,TherelationshipofPositive-LogicConventionandNegative-LogicConventionareDuality(正逻辑约定和负逻辑约定互为对偶关系),.,20,.,21,Shannonsexpansiontheorems(香农展开定理),香农展开定理主要用于证明等式或展开函数,将函数展开一次可以使函数内部的变量数从n个减少到n-1个.,.,22,Shannonsexpansiontheorems,F=X.Y+Y.Z=X(1.Y+Y.Z)+X(0.Y+Y.Z)+Y(1.X+1.Z)+Y(0.X+0.Z)+Z(X.Y+Y.1)+Z(X.Y+Y.0)=X.Y.Z+X.Y.Z+X.Y.Z,.,23,举重裁判电路,Y=F(A,B,C)=A(B+C),开关ABC1表闭合指示灯1表亮,4.1.6StandardRepresentationsofLogicFunctions(逻辑函数的标准表示方法),00000111,LogicFunctions,.,24,波形图(WaveForm),将输出与输入信号变化的时间关系用波形的形式描述，就得到了波形图,.,25,Truthtable真值表productterm乘积项sumterm求和项Asum-of-productsexpression“积之和”表达式product-of-sumsexpression“和之积”表达式n-variablemintermn变量最小项n-variablemaxtermn变量最