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Quine-McClusky Minimization Method

Quine-McClusky Minimization Method. Discussion D3.2. Quine-McCluskey Method. Tabular Representations Prime Implicants Essential Prime Implicants. W'YZ' 0-10. W'X 01--. XY -11-. WY'Z 1-01. Tabular Representations. YZ. 00. 01. 11. 10. WX. 00. 1. 01. 1. 1. 1. 1.

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Quine-McClusky Minimization Method

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  1. Quine-McCluskyMinimization Method Discussion D3.2

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

  3. W'YZ' 0-10 W'X 01-- XY -11- WY'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 = XY + W'YZ' + WY'Z + W'X

  4. Prime Implicants Each product term is an implicant F = XY'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 = YZ' + 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 0 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 = YZ' + 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 YZ' 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 = YZ' + 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- (1,3,5,7) 0--1 (2,10) -010 (1,5,3,7) 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 + WXY + 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' WXY -0-0

  14. 00-- -111 1-10 3 Prime Implicants F = W'Z + WXY + 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' WXY -0-0

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