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CS 140 Lecture 5

Learn about combinatorial logic, including specification, implementation, K-map analysis, implicants, and circuit design. Understand the concepts of sum of products and product of sums, as well as prime and essential prime implicants. Work through examples and exercises to gain practical knowledge.

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CS 140 Lecture 5

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

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

  3. Implicant: A product term that has non-empty intersection with on-setF 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.

  4. Example Given F = Sm (3, 5), D = Sm (0, 4) b 0 2 6 4 - 0 0 - 1 3 7 5 c 0 1 0 1 a Primes: Sm (3), Sm (4, 5) Essential Primes: Sm (3), Sm (4, 5) Min exp: f(a,b,c) = a’bc + ab’

  5. Five variable K-map c c 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 e e 3 7 15 11 19 23 31 27 d d 2 6 14 10 18 22 30 26 b b a Neighbors of m5 are: minterms 1, 4, 7, 13, and 21 Neighbors of m10 are: minterms 2, 8, 11, 14, and 26

  6. Six variable K-map d d 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 f f 3 7 15 11 19 23 31 27 e e 2 6 14 10 18 22 30 26 c c d d 32 36 44 40 48 52 60 56 33 37 45 41 49 53 61 57 a f f 35 39 47 43 51 55 63 59 e e 34 38 46 42 50 54 62 58 c c b

  7. Min product of sums Given F = Sm (3, 5), D = Sm (0, 4) b 0 2 6 4 - 0 0 - 1 3 7 5 c 0 1 0 1 a Prime Implicates: PM (0,1), PM (0,2,4,6), PM (6,7) Essential Primes Implicates: PM (0,1), PM (0,2,4,6), PM (6,7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’)

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

  9. Another min product of sums example Given R = Sm (3, 11, 12, 13, 14) D = Sm (4, 8, 10) K-map b 0 4 12 8 1 - 0 - 1 5 13 9 1 1 0 1 d 3 7 15 11 0 1 1 0 c 2 6 14 10 1 1 0 - a

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

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