advertisement

CpE 71 INTRODUCTION TO LOGIC DESIGN Σ Canonical/Standard Forms Π Product Term : A function that is defined to be the logical AND of a set of variables that are either primed(not) or unprimed, i.e. if f(A,B,C,D)is a function, the product terms of the function can be A . B , B . C ’ . D , A . B . C ’ . D , and etc. Minterm : A product term in which all variables (primed or unprimed) appear once and only n once, i.e. if a function has n variable, then there are 2 possible unique minterms. Each n minterms has a value of “1” for only one of the 2 possible values and “Zero” for all other value. A symbol for each minterm is of the form m k where k denotes the decimal equivalent of the binary. Example: The truth table below illustrates the form of all possible minterms for 3-variables. W 0 0 0 0 1 1 1 1 X 0 0 1 1 0 0 1 1 Y 0 1 0 1 0 1 0 1 TERM W’.X’.Y’ W’.X’.Y W’.X.Y’ W’.X.Y W.X’.Y’ W.X’.Y W.X.Y’ W.X.Y DESIGNATION m0 m1 m2 m3 m4 m5 m6 m7 Sum Term : A function that is defined to be the logical OR of a set of variables that are either primed(not) or unprimed, i.e. if f(A,B,C,D)is a function, the sum terms of the function can be (A+B), (B+C’+D) , (A+B+C’+D) , and etc. Maxterm : A sum term in which all variables (primed or unprimed) appear once and only n once, i.e. if a function has n variable, then there are 2 possible unique maxterms. Each n maxterms has a value of “0” for only one of the 2 possible values and “one” for all other value. A symbol for each maxterm is of the form Mk where k denotes the decimal equivalent of the binary. Example: The truth table below illustrates the form of all possible maxterms for 3-variables. W 0 0 0 0 1 1 1 1 X 0 0 1 1 0 0 1 1 Y 0 1 0 1 0 1 0 1 TERM DESIGNATION W+X+Y M0 W+X+Y’ M1 W+X’+Y M2 W+X’+Y’ M3 W’+X+Y M4 W’+X+Y’ M5 W’+X’+Y M6 W’+X’+Y’ M7 Klinkhachorn:Cpe 71:9/25/1999 CANONICAL REPRESENTATION OF LOGIC FUNCTION Σ Π SUM OF PRODUCTS: A boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables which produces a “1” in the function, and then taking OR (sum) of all those terms. Example: Decimal 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C F(A,B,C) 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 F(A,B,C) = Σm(0,3,4) = m0 +m3 +m4 = A’.B’.C’+A’.B.C+A.B’.C’ The function is said to be represented in the canonical sum of products form. PRODUCT OF SUMS: A boolean function can be expressed algebraically from a given truth table by forming a maxterm for each combination of the variables which produces a “0” in the function, and then taking AND (product) of all those terms. Example 1: Decimal 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C F(A,B,C) 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 F(A,B,C) = ΠM(1,2,5,6,7) = M1 .M2 .M5 .M6 .M7 =(A+B+C’) (A+B’+C) (A’+B+C’) (A'+B'+C) (A'+B'+C') The function is said to be represented in the canonical product of sums form. CONVERSION BETWEEN CANONICAL FORM Interchane the symbols Σm and ΠM and list those number missing from the original form Examples: F(W,X,Y) = Σm(0,1,3,5,7) = W’X’Y’ + W’X’Y + W’XY + WX’Y + WXY = ΠM (2,4,6) = (W+X’+Y) (W’+X+Y) (W’+X’+Y) F(A,B) = ΠM (0,1,2) = Σm(3) = (A+B) (A+B’) (A’+B) = A.B Klinkhachorn:Cpe 71:9/25/1999 EXPRESS THE BOOLEAN FUNCTION IN A CANONICAL SUM OF PRODUCT (MINTERMS) AND/OR PRODUCT OF SUMS(MAXTERMS) Σ Π Expand each individual term to incorporate all variables, i.e. use X+X’ = 1or X.X’=0. Example 1: Express F(A,B,C,D) = A’B + ACD into the canonical sum of products (Σm) and product of sum (ΠM) F(A,B,C,D) = A’B + ACD = A’B(C+C’) + ACD(B+B’) = A’BC+A’BC’+ABCD+AB’CD = A’BC(D+D’)+A’BC’(D+D’)+ ABCD+AB’CD = A’BCD+A’BCD’+A’BC’D+A’BC’D’+ ABCD+AB’CD = Σm(7,6,5,4,15,11) = ΠM(0,1,2,3,8,9,10,12,13,14) = (A+B+C+D).(A+B+C+D’).(A+B+C’+D).(A+B+C’+D’).(A’+B+C+D) .(A’+B+C+D’).(A’+B+C’+D). (A’+B’+C+D). (A’+B’+C+D’). (A’+B’+C’+D) Example 2: Express F(A,B,C,D) = (A’+B’+C).(A+C’+D).(B+C+D’) into the canonical product of sum (ΠM) and sum of products (Σm). F(A,B,C,D) = (A’+B’+C).(A+C’+D).(B+C+D’) = (A’+B’+C+D.D’).(A+B.B’+C’+D).(A.A’+B+C+D’) = (A’+B’+C+D)(A’+B’+C+D’).(A+B+C’+D) (A+B’+C’+D) .(A+B+C+D’) (A’+B+C+D’) = ΠM(12,13,2,6,1,9) = Σm(0,3,4,5,7,8,10,11,14,15) = A’B’C’D’+ A’B’CD+ A’BC’D’+ A’BC’D+ A’BCD+ AB’C’D’+ AB’CD’+ AB’CD+ ABCD’+ ABCD Klinkhachorn:Cpe 71:9/25/1999