Lecture 5: K-Maps in higher dimensions, K-map to product of sum minimization CSE 140: Components and Design Techniques for Digital Systems Spring 2014 CK Cheng, Diba Mirza Dept. of Computer Science and Engineering University of California, 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 a proper subset of 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 a proper subset of 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
K-Map to Minimized Product of Sum • Sometimes easier to reduce the K-map by considering the offset • Usually when number of zero outputs is less than number of outputs that evaluate to one OR offset is smaller than onset ab 11 00 01 10 cd 1 1 1 1 00 0 1 1 1 01 11 0 1 1 1 1 1 1 1 10 4
Minimum Sum of Product Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 1 3 7 5 1 Prime Implicants: Essential Prime Implicants: Min SOP exp: f(a,b,c)= 5
Minimum Sum of Product Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 Prime Implicant: Σ m (3), Σ m (4, 5) Essential Prime Implicant: Σ m (3), Σ m (4, 5) Min SOP exp: f(a,b,c) = a ’ bc + ab ’ 6
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 7
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 F (a,b,c) = Σ m (1, 2, 6,7)+ Σ d (0, 4) ab 00 01 11 10 c 0 2 6 4 0 1 3 7 5 1 8
Minimum Product of Sum: Boolean Algebra Rationale F (a,b,c) = Σ m (1, 2, 6,7)+ Σ d (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 1 1 X 1 3 7 5 1 1 0 1 0 9
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 10
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 PI Q: The adjacent cells grouped in red can be minimized to the following max term: A. a+b B. (a+b)’ C. a’+b’ 11
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 Prime Implicates: Essential Primes Implicates: Min exp: f(a,b,c) = 12
Minimum Product of Sum Given F (a,b,c) = Σ m (3, 5), D = Σ m (0, 4) ab 11 10 00 01 c 0 2 6 4 0 X 0 0 X 1 3 7 5 1 0 1 0 1 Prime Implicates: Π M (0, 1), Π M (0, 2, 4, 6), Π M (6, 7) Essential Primes Implicates: Π M (0, 1), Π M (0, 2, 4, 6), Π M(6, 7) Min exp: f(a,b,c) = (a+b)(c )(a ’ +b ’ ) 13
Corresponding Circuit a b f(a,b,c,d) a ’ b ’ c 14
Another min product of sums example Given R (a,b,c,d) = Σ m (3, 11, 12, 13, 14) D (a,b,c,d)= Σ m (4, 8, 10) K-map ab 00 01 11 10 cd 0 4 12 8 00 1 5 13 9 01 11 3 7 15 11 2 6 14 10 10 15
Another min product of sums example Given R (a,b,c,d) = Σ m (3, 11, 12, 13, 14) D (a,b,c,d)= Σ m (4, 8, 10) ab 00 01 11 10 cd 0 4 12 8 00 1 X 0 X 1 5 13 9 01 1 1 0 1 d 11 3 7 15 11 0 1 1 0 2 6 14 10 10 1 1 0 X a 16
Prime Implicates: Π M (3,11), Π M (12,13), Π M(10,11), Π M (4,12), Π M (8,10,12,14) PI Q: Which of the following is a non-essential prime implicate? A. Π M(3,11) ab B. Π M(12,13) 00 01 11 10 cd C. Π M(10,11) 0 4 12 8 00 1 X 0 X D. Π M(8,10,12,14) 1 5 13 9 01 1 1 0 1 d 3 7 15 11 11 0 1 1 0 2 6 14 10 10 1 1 0 X 17 a
Five variable K-map a=0 a=1 bc 11 10 00 01 11 10 00 01 de c c 0 4 12 8 16 20 28 24 00 1 5 13 9 17 21 29 25 01 e e 3 7 15 11 19 23 31 27 11 d d 2 6 14 10 18 22 30 26 10 b b 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 18
Reading a Five variable K-map a=0 a=1 bc 11 10 00 01 11 10 00 01 de c c 0 4 12 8 16 20 28 24 00 1 1 1 1 1 1 1 1 5 13 9 17 21 29 25 01 e e 3 7 15 11 19 23 31 27 11 1 1 1 1 1 d d 1 1 1 1 1 1 2 6 14 10 18 22 30 26 10 b b a 19
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 20
Reading [Harris] Chapter 3, 3.1 21
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