說明案例數(shù)電

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(還原律): ( X ) = XIdempotency(同一律): X+X=XXX=XComplements(互補(bǔ)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 (交換律) AB = BA

7、A+B = B+AAssociativity (結(jié)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允許提取公因子 AB + AC = A(B+C)不存在變量的指數(shù) AAA A3沒有定義除法 if AB=BC A=C ? 沒有定義減法 if A+B=A+C B=C ?A=1, B=0, C=0AB=AC=0, ACA=1, B=0, C=1錯！錯！Jin. UESTC104.1.3 Two-

8、and Three-Variable TheoremsCovering (吸收律)X + XY = X X(X+Y) = XCombining (組合律)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 對偶 Jin. UESTC11對上述的公式、定理要熟記，做到舉一反三 (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： 在含有變量 X 的邏輯等式中，如果將式中所有出現(xiàn) X 的地方都用另一個表達(dá)式 F 來代替，則等式仍然成立。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 廣義同一律X + X + + X = X X X X = XShannon

10、s expansion theorem 香農(nóng)展開定理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回顧線與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 合理地運(yùn)用反演定理能夠?qū)⒁恍﹩栴}簡化 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 , , + , ) 回顧公理、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舉重裁判電路Y = F (A,B,C ) = A(B+C)ABYC&1ABCYLogic Circuit開關(guān)ABC:1-閉合

19、指示燈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“積之和”表達(dá)式“與-或”式literalproduct term乘積項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求和項Product-of-Sums expression“和之積”表達(dá)式“或-與”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真值表中使某行為1的乘積項Example: Truth table logic functionmintermF = ABCJin. UESTC29Canonical Sum: a sum of mintermsOn-Set開集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 (最小項) An n-variable minterm is a normal product term with n literals.There are 2n such pro