數(shù)字邏輯設(shè)計(jì)及應(yīng)用教學(xué)課件：4-2函數(shù)表達(dá)

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：對(duì)偶定理： FD(X1 , X2 , , Xn , + , , ) = F(X1 , X2 , ,Xn , , + , ) G1ABFA B FL L LL H LH L LH H HElectrical FunctionTable (電氣功能表)A B F0 0 00 1 01 0 01 1 1A B F1 1 11 0 10 1 10 0 0Positive-Logic (正邏輯)： F =

2、 ABNegative-Logic (負(fù)邏輯)： 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 乘積項(xiàng) sum term 求和項(xiàng) sum-of-products expression “積之和”表達(dá)式product-of-sums expression “和之積”表達(dá)式n-variable minterm n 變量最小項(xiàng)n-variable maxterm n 變量最大項(xiàng)normal terms 標(biāo)準(zhǔn)項(xiàng)canonical sum 標(biāo)準(zhǔn)和

4、canonical product 標(biāo)準(zhǔn)積 最小項(xiàng)之和 最大項(xiàng)之積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. (最小項(xiàng)編號(hào)) (P198)ABCABCABCABCABCABCABCABC

5、0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABC乘積項(xiàng)(Product Term )Minterms (最小項(xiàng)) (P197)Minterms (最小項(xiàng)) An n-variable Minterm is a normal product term with n literals (n個(gè)因子的標(biāo)準(zhǔn)乘積項(xiàng))There are 2n such product terms (n變量函數(shù)具有2n個(gè)最小項(xiàng))0 is the product of Any two different minterms. (任意兩個(gè)不同最小項(xiàng)的乘積為0)Plus of all minte

6、rms is 1. (全體最小項(xiàng)之和為1)ABCABCABCABCABCABCABCABC0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1ABCProduct Term(乘積項(xiàng))最大項(xiàng) ( 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、(求和項(xiàng))Maxterms (最大項(xiàng)) (P197)Maxterms (最大項(xiàng)) An n-variable maxterm is a normal sum term with n literals. (n變量最大項(xiàng)是具有n個(gè)因子的標(biāo)準(zhǔn)求和項(xiàng))There are 2n such maxterms. (n變量函數(shù)具有2n個(gè)最大項(xiàng))Any two different sum terms produce 1. (任意兩個(gè)最大項(xiàng)的和為1) Product of all maxterms is 0. (全體最大項(xiàng)之積為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求和項(xiàng)(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 邏輯函數(shù)的的

9、標(biāo)準(zhǔn)形式1、canonical sum (標(biāo)準(zhǔn)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 . (最小項(xiàng)列

10、表)The minterm list is also knownas the on-set for the logic function. (開集)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 邏輯函數(shù)的的標(biāo)準(zhǔn)形式2、canonical prod

11、uct(標(biāo)準(zhǔn)積) 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 . (最大項(xiàng)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標(biāo)號(hào)互補(bǔ)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 ；一個(gè) n 變量函數(shù)，既可用最小項(xiàng)之和表示，也可用最大項(xiàng)之積表示，兩者下標(biāo)互補(bǔ)。 某邏輯函數(shù) F，若用 P 項(xiàng)最小項(xiàng)之和表示，則其反函數(shù) F 可用 P 項(xiàng)最大項(xiàng)之積表示，兩者標(biāo)號(hào)完全一致。Relationship of Minterm and Maxterm一個(gè) n 變

14、量函數(shù)的最小項(xiàng) mi,其對(duì)偶(Duality)為：( 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 邏輯表達(dá)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 (“和之積”表達(dá)式 )OR-AND expression(“或與”式)2、 Truth Table t o Logic Expression 真值表 邏輯表達(dá)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 (“積之和”表達(dá)式)AND-OR expression (“與-或”式)2、 Truth Table to Logic Expression 真值表 邏輯表達(dá)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 (“和之積”表達(dá)式 )OR-AND expre

17、ssion(“或與”式)2、 Truth Table t LogicExpression 真值表 邏輯表達(dá)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（與）反演對(duì)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、)補(bǔ)充： XOR(異或) 、 XNOR(同或)Exclusive OR (異或) 當(dāng)兩個(gè)輸入相異時(shí), 結(jié)果為1。 同或(Exclusive NOR)當(dāng)兩個(gè)輸入相同時(shí)， 結(jié)果為1。F = AB =AB+ABF = AB =AB+ABA B F0 0 00 1 11 0 11 1 0異 或A B F0 0 10 1 01 0 01 1 1同 或AB = (AB)Basic formula 異或(XOR)交換律：AB = BA結(jié)合律：A(BC) = (AB)C分配律：A(BC) = (AB)(AC) 因果互換關(guān)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分析步驟：由輸入到輸出逐級(jí)寫出邏輯函數(shù)表達(dá)式對(duì)輸出邏輯函數(shù)表達(dá)式進(jìn)行化簡(jiǎn)（列真值表

24、或畫波形圖）判斷邏輯功能例P149-圖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(綜合) (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(綜合) (P205)1、用真值表表示輸入輸出的邏輯關(guān)系；2、寫出函數(shù)表達(dá)式并化簡(jiǎn)；3、選擇合適的邏輯門電路，列出相應(yīng)門器件 的表達(dá)式；4、畫出邏輯電路圖；5、安裝調(diào)試，驗(yàn)證設(shè)計(jì)。4.3 Combinational Circuit Synthesis(綜合) (P205)4.3.1 Circuit Descriptions and De