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Quine-McClusky Minimization Method

This lecture delves into the Quine-McCluskey method, a systematic approach for minimizing logic functions in digital design. We explore tabular representations, the identification of prime implicants, and essential prime implicants. Through step-by-step explanations, you will learn how to identify and form product terms that cover specified minterms efficiently. This method facilitates clear simplification of Boolean functions, assisting in digital circuit design and optimization.

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Quine-McClusky Minimization Method

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  1. Quine-McCluskyMinimization Method Lecture L5.3 Section 5.3

  2. Quine-McCluskey Method • Tabular Representations • Prime Implicants • Essential Prime Implicants

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

  4. 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.

  5. YZ 00 01 11 10 X 0 1 Prime Implicants -10 1 1 1 1 1 1-- F = Y & !Z # X

  6. YZ 00 01 11 10 X 0 1 1 1 1 1 1 Prime Implicants -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

  7. Finding Prime Implicants Step 3 Step 1 Step 2 (2,6) - 1 0 (4,5,6,7) 1 - - 2 0 1 0 4 1 O 0 5 1 0 1 6 1 1 0 7 1 1 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

  8. YZ 00 01 11 10 X 0 1 1 1 1 1 1 Prime Implicants -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

  9. 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

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

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

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

  13. 3 Prime Implicants F = !W & Z # W & X & Y # !X & !Z YZ 00 01 11 10 WX 00 1 1 1 1 0--1 01 1 1 !W & Z 11 1 1 111- 10 1 1 !X & !Z W & X & Y -0-0

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