03-Karnaugh-maps

03-Karnaugh-maps - Karnaugh maps So far this week weve used...

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June 18, 2003 ©2000-2003 Howard Huang 1 Karnaugh maps ± So far this week we’ve used Boolean algebra to design hardware circuits. — The basic Boolean operators are AND, OR and NOT. — Primitive logic gates implement these operations in hardware. — Boolean algebra helps us simplify expressions and circuits. ± Today we present Karnaugh maps , an alternative simplification method that we’ll use throughout the summer.
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June 18, 2003 Karnaugh maps 2 Minimal sums of products ± When used properly, Karnaugh maps can reduce expressions to a minimal sum of products , or MSP ,form. — There are a minimal number of product terms. — Each product has a minimal number of literals. ± For example, both expressions below (from the last lecture) are sums of products, but only the right one is a minimal sum of products. x’y’ + xyz + x’y = x’ + yz ± Minimal sum of products expressions lead to minimal two-level circuits. ± A minimal sum of products might not be “minimal” by other definitions! For example, the MSP xy + xz can be reduced to x(y + z) , which has fewer literals and operators—but it is no longer a sum of products.
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June 18, 2003 Karnaugh maps 3 Organizing the minterms ± We’ll rearrange these minterms into a Karnaugh map , or K-map . ± You can show either the actual minterms or just the minterm numbers. ± Notice the minterms are almost, but not quite, in numeric order. ± Recall that an n -variable function has up to 2 n minterms, one for each possible input combination. ± A function with inputs x , y and z includes up to eight minterms, as shown below. x y z’ x y z x y’z x y’z’ x’y z’ x’y z x’y’z x’y’z’ m 6 m 7 m 5 m 4 m 2 m 3 m 1 m 0 (m 7 ) x y z 1 1 1 (m 6 ) x y z’ 0 1 1 (m 5 ) x y’z 1 0 1 (m 4 ) x y’z’ 0 0 1 (m 3 ) x’y z 1 1 0 (m 2 ) x’y z’ 0 1 0 (m 1 ) x’y’z 1 0 0 (m 0 ) x’y’z’ 0 0 0 Minterm z y x
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June 18, 2003 Karnaugh maps 4 Reducing two minterms ± In this layout, any two adjacent minterms contain at least one common literal. This is useful in simplifying the sum of those two minterms. ± For instance, the minterms x’y’z’ and x’y’z both contain x’ and y’ , and we can use Boolean algebra to show that their sum is x’y’ . x’y’z’ + x’y’z = x’y’(z’ + z) = x’y’ • 1 = x’y’ ± You can also “wrap around” the sides of the K-map—minterms in the first and fourth columns are considered to be next to each other. x y’z’ + x y z’ = xz’(y’ + y) = xz’ • 1 = xz’ x y z’ x y z x y’z x y’z’ x’y z’ x’y z x’y’z x’y’z’ x y z’ x y z x y’z x y’z’ x’y z’ x’y z x’y’z x’y’z’
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June 18, 2003 Karnaugh maps 5 Reducing four minterms ± Similarly, rectangular groups of four minterms can be reduced as well. You can think of them as two adjacent groups of two minterms each.
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03-Karnaugh-maps - Karnaugh maps So far this week weve used...

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