2 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.
3 K-Map FormatEach 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.
4 Truth Table -TO- K-Map Y 1 Z X Y X 1 1 1 minterm 0 minterm 1 1ZXYX1minterm 0 minterm 1 minterm 2 minterm 3 12113
5 2 Variable K-Map : Groups of One YX1YX1X YX YYX1YX1X YX Y
6 Adjacent Cells Y X 1 X Y X Y Z = X Y + X Y = Y ( X + X ) = Y 1 Y X 1 X YX YZ = X Y + X Y = Y ( X + X ) = Y1YX1Y = Z
7 GroupingsGrouping 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…..
8 2 Variable K-Map : Groups of Two YX1YX1YXYX1YX1XY
16 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.
17 Three Variable Design Example #1 1MKJLJ K12367451J LJ KJ K LM = F(J,K,L) = J L + J K + J K L
18 Three Variable Design Example #2 C1ZBAB CCA B12367451A CZ = F(A,B,C) = A C + B C
19 Three Variable Design Example #3 C1F2BAAB CA C1A B1324576B CB CF2 = F(A,B,C) = B C + B C + A BF2 = F(A,B,C) = B C + B C + A C
20 Four Variable K-Map W X Y Z 1 Z 1 F1 Y X W minterm 0 minterm 1 1F1YXWminterm 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 XY Z1412815139371511261410
21 Four Variable K-Map : Groups of Four W XY Z1X ZX Z11X Z
22 Four Variable Design Example #1 Z1F1YXW145121389327615141110W XY Zmin 0 min 15 W X Y1W ZX Y ZF1 = F(w,x,y,z) = W X Y + W Z + X Y Z
23 Four Variable Design Example #2 145121389327615141110W XY ZXZ1F2xYXWX Y Zmin 0 min 15 Y ZX YF2 = F(w,x,y,z) = X Y Z + Y Z + X Y