CS 140 Lecture 5 Professor CK Cheng 10/10/02. Part I. Combinational Logic 1.Spec 2.Implementation K-map: Sum of products Product of sums.

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CS 140 Lecture 5 Professor CK Cheng 10/10/02

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

Implicant: A product term tat covers at least an element in F and has no intersect with R. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes.

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

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

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

Min product of sums Given F =  m (3, 5), D =  m (0, 4) c a b 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’)

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

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

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)