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1、1Digital Logic Design and ApplicationJin YanhuaLecture #7Basic Logic AlgebraUESTC, Spring 2012Jin. UESTC2Review of Chapter 2Logic Signals and GatesPositive Logic and Negative LogicBasic building blocks AND, OR, NOTCMOS LogicInverter, NAND, NOR, AND-OR-INVERTFan-in, non-inverting GatesSteady-State El

2、ectrical BehaviorLogic levels and noise marginsEffects of loading, Nonideal inputs, Unused InputsJin. UESTC3Review of Chapter 2Steady-State Electrical BehaviorCurrent Driving CapabilityDynamic Electrical BehaviorSpeed and Power ConsumptionOther CMOS Input and Output StructuresTransmission Gates, Sch

3、mitt-Trigger InputsThree-State Outputs, Open-Drain OutputsLogic Family: CMOS and TTLResistive LoadsGate Loads, Fanout4Jin. UESTCChapter 4 Combinational Logic Design PrinciplesBasic Logic AlgebraCombinational-Circuit AnalysisCombinational-Circuit SynthesisDigital Logic Design and ApplicationJin. UEST

4、C5Basic ConceptsTwo types of logic circuits:combinational logic circuitsequential logic circuitOutputs depend only on its current inputs.Outputs depends not only on the current inputs but also on the past sequence of inputs.A combinational circuit dont contain feedback loops which generally create s

5、equential circuit behavior.Jin. UESTC64.1 Switching Algebra4.1.1 AxiomsX = 0, if X 1X = 1, if X 00 = 11 = 000 = 01+1 = 111 = 10+0 = 001 = 10 = 01+0 = 0+1 = 1F = 0 + 1 ( 0 + 1 0 ) = 0 + 1 1= 0a.k.a. “Boolean algebra”Jin. UESTC74.1.2 Single-Variable TheoremsIdentities(自等律): X+0=XX1=XNull Elements(0-1律

6、): X+1=1X0=0Involution(还原律): ( X ) = XIdempotency(同一律): X+X=XXX=XComplements(互补律): X+X=1XX=0The relationship between variable and constantThe relationship between variable and itselfJin. UESTC84.1.3 Two- and Three-Variable TheoremsSimilar relationships with general algebraCommutativity (交换律) AB = BA

7、A+B = B+AAssociativity (结合律) A(BC) = (AB)CA+(B+C) = (A+B)+CDistributivity (分配律) A(B+C) = AB+BCA+BC = (A+B)(A+C) Proved by truth table.Jin. UESTC9Notices允许提取公因子 AB + AC = A(B+C)不存在变量的指数 AAA A3没有定义除法 if AB=BC A=C ? 没有定义减法 if A+B=A+C B=C ?A=1, B=0, C=0AB=AC=0, ACA=1, B=0, C=1错！错！Jin. UESTC104.1.3 Two-

8、and Three-Variable TheoremsCovering (吸收律)X + XY = X X(X+Y) = XCombining (组合律)XY + XY = X (X+Y)(X+Y) = XConsensus (添加律/一致性定理)XY + XZ + YZ = XY + XZ(X+Y)(X+Z)(Y+Z) = (X+Y)(X+Z)Some Special Relationships 对偶 Jin. UESTC11对上述的公式、定理要熟记，做到举一反三 (X+Y) + (X+Y) = 1A + A = 1XY + XY = X(A+B)(A(B+C) + (A+B)(A(B+C)

9、 = (A+B)代入定理： 在含有变量 X 的逻辑等式中，如果将式中所有出现 X 的地方都用另一个表达式 F 来代替，则等式仍然成立。Jin. UESTC12To prove: XY + XZ + YZ = XY + XZYZ = 1YZ = (X+X)YZXY + XZ + (X+X)YZ= XY + XZ + XYZ +XYZ= XY(1+Z) + XZ(1+Y)= XY + XZJin. UESTC134.1.4 n-Variable TheoremsGeneralized idempotency theorem 广义同一律X + X + + X = X X X X = XShannon

10、s expansion theorem 香农展开定理F(X1, X2, , Xn)= X1 F(1,X2,Xn) + X1 F(0,X2,Xn)= X1 + F(0,X2,Xn) X1 + F(1,X2,Xn) Jin. UESTC14To prove: AD + AC + CD + ABCD = AD + AC= A ( 1D + 1C + CD + 1BCD ) + A ( 0D + 0C + CD + 0BCD )= A ( D + CD + BCD ) + A ( C + CD )= AD( 1 + C + BC ) + AC( 1 + D )= AD + ACJin. UESTC15

11、4.1.4 n-Variable TheoremsDeMorgans Theorem 摩根定理 Complement Theorem 反演定理 (A B) = A + B(A + B) = A B回顾线与Jin. UESTC16DeMorgan SymbolsJin. UESTC174.1.4 n-Variable TheoremsComplement of a logic expression: , 0 1, Complementing all VariablesKeep the previous priorityNotice the out of parenthesesExample1:

12、Write the complement function for each of the following logic functions.F1 = A(B+C)+CDF2 = (AB)+CDE 合理地运用反演定理能够将一些问题简化 Example2: Prove that (AB + AC) = AB + ACJin. UESTC18Example1: Write the complement function for each of the following logic functions.F1 = A(B+C)+CDF2 = (AB)+CDEF1 = (A+BC)(C+D)F2 =

13、 (A+B)(C+D+E)F2 = AB(C+D+E)AB + AC + BC = AB + AC(A+B)(A+C)AA +AC + AB + BCAC + AB AC + AB + BCExample2: Prove (AB + AC) = AB + ACJin. UESTC194.1.5 DualityDuality Rule , 0 1 Keep the previous priorityExample: Write the Duality function for each of the following Logic functions. F1 = A+B(C+D) F2 = (

14、A(B+C) + (C+D) )X(X+Y) = X FD(X1 , X2 , , Xn , + , , ) = F(X1 , X2 , , Xn , , + , ) 回顾公理、定理Counterexample: X+XY = XXX+Y = X X+Y = XJin. UESTC204.1.5 DualityDuality Rule , 0 1Keep the previous priorityPrinciple of Duality Any logic equation remains true if the duals of it is true. To prove: A+BC = (A

15、+B)(A+C)A(B+C)AB+ACJin. UESTC21Example: Write the Duality function for each of the following Logic functions. F1 = A+B(C+D)F2 = ( A(B+C) + (C+D) )F1D = A(B+CD)F2D = ( (A+BC) (CD) )Jin. UESTC22Duality and ComplementDuality: FD(X1 , X2 , , Xn , + , , ) = F(X1 , X2 , , Xn , , + , ) Complement: F(X1 , X

16、2 , , Xn , + , ) = F(X1 , X2, , Xn , , + ) F(X1 , X2 , , Xn) = FD(X1 , X2, , Xn ) The relation between the positive-logic convention and the negative-logic convention is duality.Jin. UESTC23The relation between the positive-logic convention and the negative-logic convention is duality.G1ABFA B FL L

17、LL H LH L LH H Helectrical functionA B F0 0 00 1 01 0 01 1 1positive logicA B F1 1 11 0 10 1 10 0 0negative logicF = ABF = A+BJin. UESTC24More definitionsLiteral: a variable or its complement such as X, X, CS_LExpression: literals combined by AND, OR, parentheses, complementation( FREDZ + CS_LABC +

18、Q5 )RESET Product term: PQRSum term: X+Y+ZSum-of-products expression: A + BC + ABC Product-of-sums expression: (B+C) (A+B+C)Equation: Variable = expressionP = ( FREDZ + CS_LABC + Q5)RESET Jin. UESTC25Logic Function and its Representations举重裁判电路Y = F (A,B,C ) = A(B+C)ABYC&1ABCYLogic Circuit开关ABC:1-闭合

19、指示灯Y:1-亮000001110 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ABCYTruth TableLogic Equation Jin. UESTC26Logic Expression Truth TableY = A + BC + ABC0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCBCABCY110000000111111000000100Sum-of-Products expression“积之和”表达式“与-或”式literalproduct term乘积项1111Jin. UESTC2

20、7Logic Expression Truth TableY = (B+C)(A+B+C)0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCB+CA+B+CY001111110111111111110000sum term求和项Product-of-Sums expression“和之积”表达式“或-与”式Jin. UESTC284.1.6 Standard Representations of Logic FunctionsABC1 variable0 (variable)product terms: 0 0 0 00 0 1 00 1 0 00 1 1

21、11 0 0 01 0 1 01 1 0 01 1 1 0ABCFTruth TableA product term that is 1 in exactly one row of the truth table真值表中使某行为1的乘积项Example: Truth table logic functionmintermF = ABCJin. UESTC29Canonical Sum: a sum of mintermsOn-Set开集0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0ABCF00010000F1= + +00000100F200000010F3F = ABC + ABC + ABC= A,B,C(3,5,6)MintermListJin. UESTC304.1.6 Standard Representations of Logic FunctionsMinterm (最小项) An n-variable minterm is a normal product term with n literals.There are 2n such pro