Quine-McClusky Minimization Method

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Presentation transcript:

Quine-McClusky Minimization Method Lecture L5.3 Section 5.3

Quine-McCluskey Method Tabular Representations Prime Implicants Essential Prime Implicants

Tabular Representations YZ 00 01 11 10 !W & Y & !Z 0-10 WX 00 !W & X 01-- 1 01 1 1 1 1 11 1 1 1 X & Y -11- 10 1 W & !Y & Z 1-01 F = X & Y # !W & Y & !Z # W & !Y & Z # !W & X

Prime Implicants Each product term is an implicant F = X & !Y & Z # !X & !Z # !X & Y A product term that cannot have any of its variables removed and still imply the logic function is called a prime implicant.

Prime Implicants X YZ 00 01 11 10 1 -10 1 1 1 1 1 1-- F = Y & !Z # X

Prime Implicants X YZ 00 01 11 10 1 -10 Minterm X Y Z F 0 O O O 0 1 -10 Minterm X Y Z F 0 O O O 0 1 0 0 1 0 2 0 1 0 1 3 1 1 1 0 4 1 O O 1 5 1 0 1 1 6 1 1 0 1 7 1 1 1 1 1-- F = Y & !Z # X

Finding Prime Implicants Step 1 Step 2 Step 3 2 0 1 0 4 1 O 0 5 1 0 1 6 1 1 0 7 1 1 1 (2,6) - 1 0 (4,5,6,7) 1 - - (4,5) 1 0 - (4,6,5,7) 1 - - (4,6) 1 - 0 (5,7) 1 - 1 (6,7) 1 1 - All unchecked entries are Prime Implicants -10 Y & !Z 1-- X

Prime Implicants X YZ 00 01 11 10 1 -10 Minterm X Y Z F 0 O O O 0 1 -10 Minterm X Y Z F 0 O O O 0 1 0 0 1 0 2 0 1 0 1 3 1 1 1 0 4 1 O O 1 5 1 0 1 1 6 1 1 0 1 7 1 1 1 1 1-- F = Y & !Z # X

Essential Prime Implicants YZ 00 01 11 10 WX Find the essential prime implicants using the Q-M method. 00 1 1 1 1 01 1 1 11 1 1 10 1 1

Essential Prime Implicants minterms YZ 00 01 11 10 WX 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111 00 1 1 1 1 01 1 1 11 1 1 10 1 1

Finding Prime Implicants Step 1 Step 2 Step 3 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111 (0,1) 000- (0,1,2,3) 00-- (0,2) 00-0 (0,2,1,3) 00-- (0,8) -000 (0,2,8,10) -0-0 (1,3) 00-1 (1,5) 0-01 (0,8,2,10) -0-0 (2,3) 001- (1,5,3,7) 0--1 (2,10) -010 (1,3,5,7) 0--1 (8,10) 10-0 (3,7) 0-11 6 Prime Implicants (5,7) 01-1 1-10 -111 111- 00-- -0-0 0--1 (10,14) 1-10 (7,15) -111 (14,15) 111-

Find Essential Prime Implicants Covered minterms Minterms 0 1 2 3 5 7 8 10 14 15 1-10 -111 111- 00-- -0-0 0--1 10,14 7,15 14,15 0,1,2,3 0,2,8,10 1,3,5,7 X X X X * X X X X X X X X X X X X X X

3 Prime Implicants F = !W & Z # W & X & Y # !X & !Z YZ 00 01 11 10 WX 0--1 01 1 1 !W & Z 11 1 1 111- 10 1 1 !X & !Z W & X & Y -0-0