数字逻辑设计及应用教学课件：4-2函数表达

1、class-exercises1 、F=(A+B+D) (A+B+D) (A+B+D) (A+C+D) (A+C+D) FD = ? F= ? Review of the last class反演定理：对偶定理： FD(X1 , X2 , , Xn , + , , ) = F(X1 , X2 , ,Xn , , + , ) G1ABFA B FL L LL H LH L LH H HElectrical FunctionTable (电气功能表)A B F0 0 00 1 01 0 01 1 1A B F1 1 11 0 10 1 10 0 0Positive-Logic (正逻辑)： F =

2、 ABNegative-Logic (负逻辑)： F = A+BThe relationship of Positive-Logic Convention and Negative-Logic Convention are DualityABFA=A B=BF = FF(X1,X2,Xn，+，.,) = FD(X1,X2,Xn,+,.,)F(X1,X2,Xn，+，.,) = FD(X1,X2,Xn,+,.,)Example :F=(A+BC)+D(E+F)F=?FD=?Review of the last classF=(A+BC)+D(E+F)F= A (B+C) D+(E F)FD= A

3、(B+C) D+(E F)F= A (B+C) D+(E F) Example: Review of the last classF(X1,X2,Xn，+，.,) =FD(X1,X2,Xn,+,.,)Truth table 真值表product term 乘积项 sum term 求和项 sum-of-products expression “积之和”表达式product-of-sums expression “和之积”表达式n-variable minterm n 变量最小项n-variable maxterm n 变量最大项normal terms 标准项canonical sum 标准和

4、canonical product 标准积 最小项之和 最大项之积4.1.6 Standard Representations of Logic Functions (P196)A minterm can be defined as a product term that is 1 in exactly one row of the truth table.An n-variable minterm can be represented by an n-bit integer, the minterm number. (最小项编号) (P198)ABCABCABCABCABCABCABCABC

5、0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABC乘积项(Product Term )Minterms (最小项) (P197)Minterms (最小项) An n-variable Minterm is a normal product term with n literals (n个因子的标准乘积项)There are 2n such product terms (n变量函数具有2n个最小项)0 is the product of Any two different minterms. (任意两个不同最小项的乘积为0)Plus of all minte

6、rms is 1. (全体最小项之和为1)ABCABCABCABCABCABCABCABC0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCProduct Term(乘积项)最大项 ( Maxterms )a maxterm can be defined as a sum term that is 0 in exactly one row of the truth table. A+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+C0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCSum Term

7、(求和项)Maxterms (最大项) (P197)Maxterms (最大项) An n-variable maxterm is a normal sum term with n literals. (n变量最大项是具有n个因子的标准求和项)There are 2n such maxterms. (n变量函数具有2n个最大项)Any two different sum terms produce 1. (任意两个最大项的和为1) Product of all maxterms is 0. (全体最大项之积为0)A+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+C0

8、 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABC求和项(Sum Term)ABCABCABCABCABCABCABCABCm0m1m2m3m4m5m6m7 minterm0 0 0 00 0 1 10 1 0 20 1 1 31 0 0 41 0 1 51 1 0 61 1 1 7ABCrowA+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+CA+B+CM0M1M2M3M4M5M6M7 maxtermMinterms and MaxtermsStandard Representations of Logic Functions 逻辑函数的的

9、标准形式1、canonical sum (标准和) The canonical sum of a logic function is a sum of the minterms corresponding totruth-table rows (input combinations) for which the function produces a 1 output.F =X,Y,Z(0,3,4,6,7) =X.Y. Z + X. Y . Z + X . Y. Z + X . Y .Z + X .Y . ZX,Y,Z(0,3,4,6,7): is a minterm list . (最小项列

10、表)The minterm list is also knownas the on-set for the logic function. (开集)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 01 1 0 11 1 1 1ABCF真值表00010000F1= + +00000010F200000001F3Why the canonical sum of a logic is a sum of the minterms?Standard Representations of Logic Functions 逻辑函数的的标准形式2、canonical prod

11、uct(标准积) The canonical product of a logic function is a product of the maxterms corresponding to input combinations for which the function produces a 0 output. F =X,Y,Z(1,2,5) =(X+Y+Z). (X+Y+Z). (X+Y+Z) X,Y,Z(1,2,5) is a maxterm list . (最大项列表)The maxterm list is also knownas the off-set for the logi

12、c function. (閉集)0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 11 1 0 11 1 1 0ABCF真值表01111111F1= 11101111F211111110F3Why the canonical product of a logic is a product of the maxterms?Relationship of Minterm and Maxterm11101001G0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0ABCF(ABC) = A+B+C(ABC)

13、= A+B+C(ABC) = A+B+CMi = mimi = Mi标号互补exampleWhat is the duality of F1? F1D = (A,B,C) ( 0, 2, 6 )( m i )D = M (2 n -1) - i F1 = (A,B,C) ( 1, 5, 7) Mi = mi ； mi = Mi ；一个 n 变量函数，既可用最小项之和表示，也可用最大项之积表示，两者下标互补。 某逻辑函数 F，若用 P 项最小项之和表示，则其反函数 F 可用 P 项最大项之积表示，两者标号完全一致。Relationship of Minterm and Maxterm一个 n 变

14、量函数的最小项 mi,其对偶(Duality)为：( m i )D = M (2 n -1) - i five possible representations for a combinational logic function:1. A truth table.2. An algebraic sum of minterms, the canonical sum.3. A minterm list using the notation.4. An algebraic product of maxterms, the canonical product.5. A maxterm list us

15、ing the notation.1. Logic Expression to Truth Table 逻辑表达式真值表Y = (B+C) (A+B+C)0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCB+CA+B+CY001111110111111111110000product-of-sums expression (“和之积”表达式 )OR-AND expression(“或与”式)2、 Truth Table t o Logic Expression 真值表 逻辑表达式ABC0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01

16、0 1 11 1 0 11 1 1 0ABCF真值表ABCABCF = ABC + ABC + ABCsum-of-products expression (“积之和”表达式)AND-OR expression (“与-或”式)2、 Truth Table to Logic Expression 真值表 逻辑表达式0 0 0 10 0 1 10 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1ABCF真值表A+B+CA+B+CF = (A+B+C) (A+B+C)product-of-sums expression (“和之积”表达式 )OR-AND expre

17、ssion(“或与”式)2、 Truth Table t LogicExpression 真值表 逻辑表达式11101111G0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 01 1 0 01 1 1 0ABCF真值表(ABC) = A+B+CF = ABCG = (A+B+C)=FRepresent the logical function with canonical Sum: F(A,B,C) = AB +AC利用基本公式 A + A = 1F(A,B,C) = AB + AC = AB(C+C) + AC(B+B) = ABC + ABC + ABC

18、+ ABC1 1 11 1 00 1 10 0 1= A,B,C(1,3,6,7)G(A,B,C) = (A+B) (A+C) = (A+B+CC) (A+C+BB) = (A+B+C)(A+B+C)(A+B+C)(A+B+C)0 0 00 0 11 0 01 1 0= A,B,C(0,1,4,6)Represent the logical function with canonical productThinkingIf ， ,what is the relation between the fonction F and fonction G, complement or duality?B

19、asic operations of the logic functions相加（或）相乘（与）反演对偶 F1D = (A,B,C,D) ( 2, 6, 8, 10, 14 ) F1 = (A,B,C,D) ( 1, 5, 7, 9, 13 )F2 = (A,B,C,D) ( 2, 6, 9, 13, 15 )F = F1 + F2 = (A,B,C,D)(1,2,5,6,7,9,13,15)F = F1 F2 = (A,B,C,D) (9,13)F1 = (A,B,C,D) ( 1, 5, 7, 9, 13 )= (A,B,C,D) ( 0,2,3,4,6,8,10,11,12,14,15

20、)补充： XOR(异或) 、 XNOR(同或)Exclusive OR (异或) 当两个输入相异时, 结果为1。 同或(Exclusive NOR)当两个输入相同时， 结果为1。F = AB =AB+ABF = AB =AB+ABA B F0 0 00 1 11 0 11 1 0异 或A B F0 0 10 1 01 0 01 1 1同 或AB = (AB)Basic formula 异或(XOR)交换律：AB = BA结合律：A(BC) = (AB)C分配律：A(BC) = (AB)(AC) 因果互换关系 AB=C AC=B BC=A ABCD=0 0ABC=D c h a p t e r

21、4 4.1 Switching Algebra 4.2 Combinational Circuit Analysis4.3 Combinational Circuit Synthesis 4.4 Timing Hazards c h a p t e r 4 4.1 Switching Algebra 4.2 Combinational Circuit Analysis4.3 Combinational Circuit Synthesis 4.4 Timing Hazards 4.2 Combinational Circuit Analysis(分析) (P199) We analyze a c

22、ombinational logic circuit by obtaining a formal description of its logic function.ABFAB(AB)(AB)F = (AB) (AB) = AB + AB = ABProcess of the Analysis (P201)1.We build up a parenthesized logic expression corresponding to the logic operators and structure of the circuit. We start at the circuit inputs a

23、nd propagate expressions through gates toward the output.2.Using the theorems of switching algebra,we may simplify the expressions as we go, or we may defer all algebraic manipulations until an output expression is obtained. 4.2 Combinational Circuit Analysis分析步骤：由输入到输出逐级写出逻辑函数表达式对输出逻辑函数表达式进行化简（列真值表

24、或画波形图）判断逻辑功能例P149-图4-17 c h a p t e r 4 4.1 Switching Algebra 4.2 Combinational Circuit Analysis4.3 Combinational Circuit Synthesis 4.4 Timing Hazards c h a p t e r 4 4.1 Switching Algebra 4.2 Combinational Circuit Analysis4.3 Combinational Circuit Synthesis 4.4 Timing Hazards 4.3 Combinational Circ

25、uit Synthesis(综合) (P205)the description is a list of input combinations for which a signal should be on or off, the verbal equivalent of a truth table or the or notation introduced previously. write the corresponding logic expressions .Combinational Circuit Minimization Circuit Manipulations .4.3 Combinational Circuit Synthesis(综合) (P205)1、用真值表表示输入输出的逻辑关系；2、写出函数表达式并化简；3、选择合适的逻辑门电路，列出相应门器件 的表达式；4、画出逻辑电路图；5、安装调试，验证设计。4.3 Combinational Circuit Synthesis(综合) (P205)4.3.1 Circuit Descriptions and De