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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.

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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.

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**Truth Table -TO- K-Map Y 1 Z X Y X 1 1 1 minterm 0 minterm 1 **

1 Z X Y X 1 minterm 0 minterm 1 minterm 2 minterm 3 1 2 1 1 3

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**2 Variable K-Map : Groups of One**

Y X 1 Y X 1 X Y X Y Y X 1 Y X 1 X Y X Y

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**Adjacent Cells Y X 1 X Y X Y Z = X Y + X Y = Y ( X + X ) = Y 1 Y X 1**

X Y X Y Z = X Y + X Y = Y ( X + X ) = Y 1 Y X 1 Y = Z

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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…..

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**2 Variable K-Map : Groups of Two**

Y X 1 Y X 1 Y X Y X 1 Y X 1 X Y

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**2 Variable K-Map : Group of Four**

Y X 1 1

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**Two Variable Design Example**

1 T R S R 1 2 3 1 S T = F(R,S) = S

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**3 Variable K-Map : Vertical**

B C minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 C 1 Y B A 1 4 1 5 3 7 2 6

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**3 Variable K-Map : Horizontal**

C 1 Y B A minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 C A B 1 2 6 4 1 3 7 5

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**3 Variable K-Map : Groups of Two**

C A B 1 B C 1 A B 1 A C 1 A B 1 B C 1 A B 1 A B 1 A C 1 B C 1 A C 1 B C 1 A C

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**3 Variable K-Map : Groups of Four**

C A B 1 B 1 A 1 C 1 B 1 A 1 C

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**3 Variable K-Map : Group of Eight**

C A B 1 1

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**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.

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**Three Variable Design Example #1**

1 M K J L J K 1 2 3 6 7 4 5 1 J L J K J K L M = F(J,K,L) = J L + J K + J K L

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**Three Variable Design Example #2**

C 1 Z B A B C C A B 1 2 3 6 7 4 5 1 A C Z = F(A,B,C) = A C + B C

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**Three Variable Design Example #3**

C 1 F2 B A A B C A C 1 A B 1 3 2 4 5 7 6 B C B C F2 = F(A,B,C) = B C + B C + A B F2 = F(A,B,C) = B C + B C + A C

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**Four Variable K-Map W X Y Z 1 Z 1 F1 Y X W minterm 0 minterm 1 **

1 F1 Y X W 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 W X Y Z 1 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10

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**Four Variable K-Map : Groups of Four**

W X Y Z 1 X Z X Z 1 1 X Z

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**Four Variable Design Example #1**

Z 1 F1 Y X W 1 4 5 12 13 8 9 3 2 7 6 15 14 11 10 W X Y Z min 0 min 15 W X Y 1 W Z X Y Z F1 = F(w,x,y,z) = W X Y + W Z + X Y Z

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**Four Variable Design Example #2**

1 4 5 12 13 8 9 3 2 7 6 15 14 11 10 W X Y Z X Z 1 F2 x Y X W X Y Z min 0 min 15 Y Z X Y F2 = F(w,x,y,z) = X Y Z + Y Z + X Y

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Techniques for Combinational Logic Optimization

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