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## Karnaugh Map Method

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**Karnaugh Map Technique**• K-Maps, like truth tables, are a way to show the relationship between logic inputs and desired outputs. • K-Maps are a graphical technique used to simplify a logic equation. • K-Maps are very procedural and much cleaner than Boolean simplification. • K-Maps can be used for any number of input variables, BUT are only practical for fewer than six.**K-Map Format**• Each minterm in a truth table corresponds to a cell in the K-Map. • K-Map cells are labeled so that both horizontal and vertical movement differ only in one variable. • Once a K-Map is filled (0’s & 1’s) the sum-of-products expression for the function can be obtained by OR-ing together the cells that contain 1’s. • Since the adjacent cells differ by only one variable, they can be grouped to create simpler terms in the sum-of-product expression.**X**0 0 1 1 Y 0 1 0 1 Z 1 0 1 1 X X 1 1 Y 0 1 Y Truth Table -TO- K-Map minterm 0 minterm 1 minterm 2 minterm 3 2 0 1 3**X Y**X Y X X X X X X X X 0 0 1 0 1 0 0 0 Y Y Y Y 0 0 0 1 1 0 0 0 Y Y Y Y X Y 2 Variable K-Map : Groups of One X Y**X Y**X Y X X X X 1 1 1 1 Y Y 1 Z = X Y + X Y = Y ( X + X ) = Y 0 0 0 0 Y Y Y = Z Adjacent Cells**Groupings**• Grouping a pair of adjacent 1’s eliminates the variable that appears in complemented and uncomplemented form. • Grouping a quad of 1’s eliminates the two variables that appear in both complemented and uncomplemented form. • Grouping an octet of 1’s eliminates the three variables that appear in both complemented and uncomplemented form, etc…..**X**Y X X X X X X X X 1 0 1 0 0 1 1 0 Y Y Y Y 0 1 1 0 0 1 0 1 Y Y Y Y 2 Variable K-Map : Groups of Two X Y**X**X 1 Y 1 1 1 1 Y 2 Variable K-Map : Group of Four**R**0 0 1 1 S 0 1 0 1 T 1 0 1 0 S R R 2 0 S 1 3 S 1 1 T = F(R,S) = S 0 0 Two Variable Design Example**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 1 0 0 0 A B C 1 0 1 1 3 Variable K-Map : Vertical 0 4 1 5 3 7 2 6**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 1 1 1 0 0 1 0 0 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**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**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**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**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.**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 J K J K J K J K J L 0 2 6 4 L 1 3 7 5 L J K L J K 1 1 0 0 0 1 0 1 M = F(J,K,L) = J L + J K + J K L Three Variable Design Example #1**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 A B A B A B A B B C 0 2 6 4 C 1 3 7 5 C 1 0 0 1 Z = F(A,B,C) = A C + B C 0 0 1 1 Three Variable Design Example #2 A C**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 B C B C B C B C A A B B C A 1 0 1 0 F2 = F(A,B,C) = B C + B C + A B F2 = F(A,B,C) = B C + B C + A C 1 1 1 0 Three Variable Design Example #3 A C 0 1 3 2 5 7 4 6 B C**W**0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 W X Y Z W X Y Z W X Y Z W X 1 1 0 1 0 1 1 1 0 1 1 0 Y Z 0 0 1 0 Four Variable K-Map minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 minterm 8 minterm 9 minterm 10 minterm 11 minterm 12 minterm 13 minterm 14 minterm 15 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10**W X**Y Z X Z W X Y Z W X Y Z W X X Z 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 X Z 0 0 1 0 0 0 0 0 0 0 1 0 Y Z 0 1 0 1 0 0 0 1 0 0 0 1 Four Variable K-Map : Groups of Four**W X**W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F1 1 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Y Z W X W Z Y Z W X Y Z W X 0 4 12 8 1 1 1 0 1 5 13 9 0 0 1 0 Y Z 3 7 15 11 0 0 0 0 2 6 14 10 1 1 0 1 F1 = F(w,x,y,z) = W X Y + W Z + X Y Z W X Y X Y Z Four Variable Design Example #1 min 0 min 15 **W X**W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F2 1 x 1 0 0 x 0 x x 1 0 1 x 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 X Y Z Y Z W X Y Z W X W X Y Z 1 X 0 1 0 4 12 8 0 X X 0 1 5 13 9 X 1 1 1 Y Z 3 7 15 11 X 1 1 0 2 6 14 10 F2 = F(w,x,y,z) = X Y Z + Y Z + X Y Four Variable Design Example #2 min 0 min 15 Y Z X Y