L6+K+Map - Lecture 6 Karnaugh Map Ve270 Introduction to...

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Ve270 Introduction to Logic Design 1 Lecture 6 Karnaugh Map
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Ve270 Introduction to Logic Design 2 Karnaugh Map (K-map) Technique K-map, like a true table, is a way to show the relationship between the logic inputs and desired output K-map is a graphical technique used to simplify a logic equation (Boolean function) Much cleaner and more procedural than algebraic simplification by theorems of Boolean algebra Theoretically, it can be used for any number of input variables, BUT is only practical for less than six, we will limit our discussion to logic equations with four or less variables
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Ve270 Introduction to Logic Design 3 Building a K-map K-map cells are labeled so that both horizontal and vertical movement differ only in one variable K-map can be filled up directly from a truth table, each minterm corresponds to a cell in the K-map Once a K-map is filled (with 0’s & 1’s), minterm expression for the output can be obtained by OR-ing together the cells that contain 1’s Since the adjacent cells differ in only one variable, they can be grouped to create simpler terms in the sum-of-product expression. X X’ Y Y’ X Y m1 m2 m3 m0 F Y’Z YZ X X’ YZ’ Y’Z’ X YZ m1 m2 m3 m0 m5 m6 m7 m4 F WX W’X Y’Z YZ WX’ W’X’ YZ’ Y’Z’ WX YZ m1 m2 m3 m0 m5 m6 m7 m4 m13 m14 m15 m12 m9 m10 m11 m8 F
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Ve270 Introduction to Logic Design 4 Two-Variable K-map There are four minterms – 2 by 2 square map X X’ Y Y’ m3 m2 m1 m0 1 1 0 1 F XY 1 1 XY’ 0 1 X’Y 1 0 X’Y’ 0 0 Y X X Y m1 m2 m3 m0 1 0 1 1 Show the variables along which the labels varying horizontally Show the variables along which the labels varying vertically F
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Ve270 Introduction to Logic Design 5 Three-Variable K-map There are 2 3 = 8 minterms – 2 by 4 rectangular map Y’Z YZ X X’ YZ’ Y’Z’ XYZ’ m6 0 0 1 1 XY’Z m5 0 1 0 1 XY’Z’ m4 0 0 0 1 X’YZ m3 1 1 1 0 1 0 0 0 X m7 m2 m1 m0 1 1 0 1 F XYZ 1 1 X’YZ’ 0 1 X’Y’Z 1 0 X’Y’Z’ 0 0 Z Y X YZ m1 m2 m3 m0 1 01 1 Show the variables along which the labels varying horizontally Show the variables along which the labels varying vertically m5 m6 m7 m4 1 0 0 0
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Ve270 Introduction to Logic Design 6 Four-Variable K-map There are 2 4 =16 minterms – 4 by 4 square map WX W’X Y’Z YZ WX’ W’X’ YZ’ Y’Z’ W’XYZ m7 1 1 1 1 0 W’XYZ’ m6 0 0 1 1 0 W’XY’Z m5 0 1 0 1 0 W’XY’Z’ m4 0 0 0 1 0 W’X’YZ m3 1 1 1 0 0 W’X’YZ’ m2 1 0 1 0 0 W’X’Y’Z m1 0 1 0 0 0 W’X’Y’Z’ m0 1 0 0 0 0 1 1 1 1 0 0 0 0 X WXYZ’ m14 0 0 1 1 WXY’Z m13 0 1 0 1 WXY’Z’ m12 1 0 0 1 WX’YZ m11 0 1 1 1 1 1 1 1 W m15 M10 m9 M8 1 0 1 1 F WXYZ 1 1 WX’YZ’ 0 1 WX’Y’Z 1 0 WX’Y’Z’ 0 0 Z Y WX YZ m1 m2 m3 m0 1 01 1 Show the variables along which the labels varying vertically or horizontally m5 m6 m7 m4 1 0 0 0 m13 m14 m15 m12 m9 m10 m11 m8 1 10 0 0 1
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Ve270 Introduction to Logic Design 7 Label the Rows and Columns by 0 and 1 X 1 X’ 0 Y 1 Y’ 0 X Y m1 m2 m3 m0 1 0 1 1 F Y’Z 01 YZ 11 X 1 X’ 0 YZ’ 10 Y’Z’ 00 X YZ m1 m2 m3 m0 1 01 1 m5 m6 m7 m4 1 0 0 0 WX 11 W’X 01 Y’Z 01 YZ 11 WX’ 10 W’X’ 00 YZ’ 10 Y’Z’ 00 WX YZ m1 m2 m3 m0 1 1 m5 m6 m7 m4 1 0 0 0 m13 m14
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L6+K+Map - Lecture 6 Karnaugh Map Ve270 Introduction to...

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