CS 140 Lecture 5 Professor CK Cheng CSE Dept. UC San Diego 1.

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CS 140 Lecture 5 Professor CK Cheng CSE Dept. UC San Diego 1

Part I. Combinational Logic 1.Specification 2.Implementation K-map: Sum of products Product of sums 2

Implicant: A product term that has non-empty intersection with on-set F and does not intersect with off-set R. Prime Implicant: An implicant that is not covered by any other implicant. Essential Prime Implicant: A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants. Implicate: A sum term that has non-empty intersection with off-set R and does not intersect with on-set F. Prime Implicate: An implicate that is not covered by any other implicate. Essential Prime Implicate: A prime implicate that has an element in off-set R but this element is not covered by any other prime implicates. 3

Example Given F =  m (3, 5), D =  m (0, 4) b c a Primes:  m (3),  m (4, 5) Essential Primes:  m (3),  m (4, 5) Min exp: f(a,b,c) = a’bc + ab’ 4

Five variable K-map c d b e c d b e a Neighbors of m 5 are: minterms 1, 4, 7, 13, and 21 Neighbors of m 10 are: minterms 2, 8, 11, 14, and 26 5

Six variable K-map d e c f d e c d e c f d e c b a f f

Min product of sums Given F =  m (3, 5), D =  m (0, 4) b c a Prime Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Essential Primes Implicates:  M (0,1),  M (0,2,4,6),  M (6,7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’) 7

Corresponding Circuit a b a’ b’ c f(a,b,c,d) 8

Another min product of sums example Given R =  m (3, 11, 12, 13, 14) D =  m (4, 8, 10) K-map b c a d

Prime Implicates:  M (3,11),  M (12,13),  M(10,11),  M (4,12),  M (8,10,12,14) Essential Primes:  M (8,10,12,14),  M (3,11),  M(12,13) Exercise: Derive f(a,b,c,d) in minimal product of sums expression. 10