lec14

# lec14 - CS 20 Lecture 14 Karnaugh Maps Professor CK Cheng...

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1 CS 20 Lecture 14 Karnaugh Maps Professor CK Cheng CSE Dept. UC San Diego

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2 1. Introduction 2. The Maps 3. Boolean Optimization Karnaugh Maps
3 Definitions: Literal: x i or x i Product Term: x 2 x 1 ’x 0 Sum Term: x 2 + x 1 ’ + x 0 Minterm of n variables: A product of n literals in which every variable appears exactly once. f(a,b,c,d): ab’cd’, a’bc’d’ Maxterm of n variables: A sum of n literals in which every variable appears exactly once. f(a,b,c,d): (a’+b+c+d), (a’+b’+c+d) Introduction

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4 Introduction Input: Boolean expression of n binary variables Goal: Simplification of the expression. E.g. we want to minimize # terms and # literals. Applications: Logic: rule reduction Hardware Design: cost and performance optimization. Cost (wires, gates): # literals, product terms, sum terms Performance: speed, reliability
5 Introduction ID A B f(A,B) minterm 0 0 0 0 1 0 1 1 A’B 2 1 0 1 AB’ 3 1 1 1 AB An example of 2-variable function f(A,B)=A+B

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6 Function can be represented by sum of minterms: f(A,B) = A’B+AB’+AB This is not minimal however! We want to minimize the number of literals and terms. We factor out common terms – A’B+AB’+AB= A’B+AB’+ AB+AB =(A’+ A )B+A(B’+ B )=B+A Hence, we have f(A,B) = A+B
7 K-Map: Truth Table in 2 Dimensions A = 0 A = 1 B = 0 B = 1 0 2 1 3 0 1 1 1 A’B AB’ AB f(A,B) = A + B

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8 ID A B f(A,B) minterm 0 0 0 0 1 0 1 1 A’B 2 1 0 0 3 1 1 1 AB Another Example f(A,B)=B f(A,B)=A’B+AB=(A’+A)B=B
9 On the K-map: A = 0 A= 1 B= 0 B = 1 0 2 1 3 0 0 1 1 A’B AB f(A,B)=B

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