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CS 140 Lecture 5

CS 140 Lecture 5. Professor CK Cheng CSE Dept. UC San Diego. Part I. Combinational Logic. Specification Implementation K-map: Sum of products Product of sums. Implicant : A product term that has non-empty intersection with on-set F and does not intersect with off-set R .

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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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