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Karnaugh Map Method

Karnaugh Map Method. X 0 0 1 1. Y 0 1 0 1. Z 1 0 1 1. Truth Table -TO- K-Map. minterm 0  minterm 1  minterm 2  minterm 3 . 2. 0. 1. 3. X. X. X. X. X. X. X. X. 1. 0. 0. 0. 1. 0. 0. 0. Y. Y. Y. Y. 0. 0. 0. 1. 0. 1. 0. 0. Y. Y. Y. Y.

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Karnaugh Map Method

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  1. Karnaugh Map Method

  2. X 0 0 1 1 Y 0 1 0 1 Z 1 0 1 1 Truth Table -TO- K-Map minterm 0  minterm 1  minterm 2  minterm 3  2 0 1 3

  3. X X X X X X X X 1 0 0 0 1 0 0 0 Y Y Y Y 0 0 0 1 0 1 0 0 Y Y Y Y 2 Variable K-Map : Groups of One

  4. X X X X 1 1 1 1 Y Y 0 0 0 0 Y Y Adjacent Cells Z =

  5. X X X X X X X X 1 0 0 1 1 1 0 0 Y Y Y Y 1 1 0 0 1 0 0 1 Y Y Y Y 2 Variable K-Map : Groups of Two

  6. X X Y 1 1 1 1 Y 2 Variable K-Map : Group of Four

  7. R 0 0 1 1 S 0 1 0 1 T 1 0 1 0 R R 2 0 S 1 3 S Two Variable Design Example T = =

  8. B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Y 1 0 1 1 0 0 1 0 A 0 0 0 0 1 1 1 1 A B C B C B C minterm 0  minterm 1  minterm 2  minterm 3  minterm 4  minterm 5  minterm 6  minterm 7  A B C 3 Variable K-Map : Vertical 0 4 1 5 3 7 2 6

  9. B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Y 1 0 1 1 0 0 1 0 A 0 0 0 0 1 1 1 1 A B A B A B A B C C 3 Variable K-Map : Horizontal minterm 0  minterm 1  minterm 2  minterm 3  minterm 4  minterm 5  minterm 6  minterm 7  0 2 6 4 1 3 7 5

  10. B C B C B C A B A B A B A C A B A B A B A B C B C A C A C C 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 A B A C 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 3 Variable K-Map : Groups of Two

  11. A B A B A B B B A B C C C C 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 A A 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 3 Variable K-Map : Groups of Four

  12. A B A B A B A B C 1 C 1 1 1 1 1 1 1 1 3 Variable K-Map : Group of Eight

  13. Simplification Process • Construct the K-Map and place 1’s in cells corresponding to the 1’s in the truth table. Place 0’s in the other cells. • Examine the map for adjacent 1’s and group those 1’s which are NOT adjacent to any others. These are called isolated 1’s. • Group any hex. • Group any octet, even if it contains some 1’s already grouped, but are not enclosed in a hex. • Group any quad, even if it contains some 1’s already grouped, but are not enclosed in a hex or octet. • Group any pair, even if it contains some 1’s already grouped, but are not enclosed in a hex, octet or quad. • Group any single cells remaining. • Form the OR sum of all the terms grouped.

  14. K 0 0 1 1 0 0 1 1 L 0 1 0 1 0 1 0 1 M 1 0 1 1 0 1 0 0 J 0 0 0 0 1 1 1 1 Three Variable Design Example #1 M = F(J,K,L) =

  15. B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Z 1 0 0 0 1 1 0 1 A 0 0 0 0 1 1 1 1 Three Variable Design Example #2 Z = F(A,B,C) =

  16. B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F2 1 0 0 1 1 1 0 1 A 0 0 0 0 1 1 1 1 Three Variable Design Example #3 F2 = F(A,B,C) = F2 = F(A,B,C) =

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