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Digital Logic Notes 4

Digital Logic Notes 4 - Karnaugh Maps Graphical...

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Karnaugh Maps Graphical representation of a logical function’s truth table. The map for an n -input logic function is an array with 2 n cells. Each cell is associated with a minterm.

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Two Variable Karnaugh Map x 2 (a) Truth table (b) Karnaugh map 0 1 0 1 m 0 m 2 m 3 m 1 x 1 x 2 0 0 0 1 1 0 1 1 m 0 m 1 m 3 m 2 x 1
Function Representation with Karnaugh Map To represent a logic function on a Karnaugh map, we simply copy the 0s and 1s from the truth table to the corresponding cell of the map. Notice that the cells are numbered in such a way that they differ by a single bit.

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Function Minimization with Karnaugh Map In general we can simplify a logic function by combining adjacent 1-cells whenever possible, and writing a sum of product term that covers all of the 1-cells. The number of adjacent 1-cells being simplified has to be a power of 2.
x1 x2 f 0 0 1 0 1 1 1 0 0 1 1 1 x 1 x 2 1 0 1 1 f x 2 x 1 + = 0 1 0 1 1

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Three Variable K-maps x 1 x 2 x 3 00 01 11 10 0 1 (b) Karnaugh map x 2 x 3 0 0 0 1 1 0 1 1 m 0 m 1 m 3 m 2 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 m 4 m 5 m 7 m 6 x 1 (a) Truth table m 0 m 1 m 3 m 2 m 6 m 7 m 4 m 5
Adjacent Cells in a K-map The cells wrap around when adjacency is considered. Therefore cells 1 and 5 are adjacent. Notice that the cells are numbered in such a way that they differ by a single bit. f x 1 x 3 x 2 x 3 + = x 1 x 2 x 3 0 0 1 0 1 1 0 1 00 01 11 10 0 1

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x 1 x 2 x 3 1 1 0 0 1 1 0 1 f x 3 x 1 x 2 + = 00 01 11 10 0 1 x1 x2 x3 f 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0
Four Variable Karnaugh Maps x 1 x 2 x 3 x 4 00 01 11 10 00 01 11 10 x 2 x 4 x 1 x 3 m 0 m 1 m 5 m 4 m 12 m 13 m 8 m 9 m 3 m 2 m 6 m 7 m 15 m 14 m 11 m 10

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x 1 x 2 x 3 x 4 1 00 01 11 10 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 00 01 11 10 x 1 x 2 x 3 x 4 1 00 01 11 10 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 00 01 11 10 x 1 x 2 x 3 x 4 0 00 01 11 10 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 00 01 11 10 x 1 x 2 x 3 x 4 0 00 01 11 10 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 00 01 11 10 f 1 x 2 x 3 x 1 x 3 x 4 + = f 2 x 3 x 1 x 4 + = f 3 x 2 x 4 x 1 x 3 x 2 x 3 x 4 + + = f 4 x 1 x 3 x 1 x 3 + + = x 1 x 2 x 2 x 3 or
Complete and Minimal Sums Complete sum is the sum of all prime implicants of a logic function. A minimal sum of a logic function F(X 1 , …,X n ) is a sum of products expression for F such that no sum of product expression for F has fewer product terms, and any sum of products expression with the same number of product terms has at least as many literals.

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Cover A logic function P(X 1 ,…,X n ) implies
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