03-Karnaugh-maps

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

This preview shows pages 1–6. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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’
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 29

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online