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# 05-Kmaps - Karnaugh Maps 1 Karnaugh maps Last time we saw...

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1 Karnaugh Maps

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2 Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can also be directly translated into a hardware circuit. Boolean algebra helps us simplify expressions and circuits. Today we’ll look at a graphical technique for simplifying an expression into a minimal sum of products (MSP) form: There are a minimal number of product terms in the expression. Each term has a minimal number of literals. Circuit-wise, this leads to a minimal two-level implementation.
3 Re-arranging the truth table A two-variable function has four possible minterms. We can re-arrange these minterms into a Karnaugh map . Now we can easily see which minterms contain common literals. Minterms on the left and right sides contain y’ and y respectively. Minterms in the top and bottom rows contain x’ and x respectively. x y minterm 0 0 x’y’ 0 1 x’y 1 0 xy’ 1 1 xy Y 0 1 0 x’y’ x’y X 1 xy’ xy Y 0 1 0 x’ y’ x’ y X 1 x y’ x y Y’ Y X’ x’ y’ x’ y X x y’ x y

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4 Karnaugh map simplifications Imagine a two-variable sum of minterms: x’y’ + x’y Both of these minterms appear in the top row of a Karnaugh map, which means that they both contain the literal x’ . What happens if you simplify this expression using Boolean algebra? x’y’ + x’y = x’(y’ + y) [ Distributive ] = x’ 1 [ y + y’ = 1 ] = x’ [ x 1 = x ] Y x’y’ x’y X xy’ xy
5 More two-variable examples Another example expression is x’y + xy . Both minterms appear in the right side, where y is uncomplemented. Thus, we can reduce x’y + xy to just y . How about x’y’ + x’y + xy ? We have x’y’ + x’y in the top row, corresponding to x’ . There’s also x’y + xy in the right side, corresponding to y . This whole expression can be reduced to x’ + y . Y x’y’ x’y X xy’ xy Y x’y’ x’y X xy’ xy

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6 A three-variable Karnaugh map For a three-variable expression with inputs x, y, z, the arrangement of minterms is more tricky: Another way to label the K-map (use whichever you like): Y x’y’z’ x’y’z x’yz x’yz’ X xy’z’ xy’z xyz xyz’ Z Y m 0 m 1 m 3 m 2 X m 4 m 5 m 7 m 6 Z YZ 00 01 11 10 0 x’y’z’ x’y’z x’yz x’yz’ X 1 xy’z’ xy’z xyz xyz’ YZ 00 01 11 10 0 m 0 m 1 m 3 m 2 X 1 m 4 m 5 m 7 m 6
7 Why the funny ordering? With this ordering, any group of 2, 4 or 8 adjacent squares on the map contains common literals that can be factored out. “Adjacency” includes wrapping around the left and right sides: We’ll use this property of adjacent squares to do our simplifications. x’y’z + x’yz = x’z(y’ + y) = x’z 1 = x’z x’y’z’ + xy’z’ + x’yz’ + xyz’ = z’(x’y’ + xy’ + x’y + xy) = z’(y’(x’ + x) + y(x’ + x)) = z’(y’+y) = z’ Y x’y’z’ x’y’z x’yz x’yz’ X xy’z’ xy’z xyz xyz’ Z Y x’y’z’ x’y’z x’yz x’yz’ X xy’z’ xy’z xyz xyz’ Z

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05-Kmaps - Karnaugh Maps 1 Karnaugh maps Last time we saw...

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