Combinational Logic Implementation using Karnaugh Maps: Minimizing Sum of Products
This lecture focuses on implementing combinational logic through Karnaugh Maps (K-Maps). It demonstrates how to minimize the sum of products for a given function F using prime implicants and essential prime implicants. Through examples, the process for deriving a minimum expression is outlined, including finding the largest rectangles that represent on-set cells and ensuring coverage of all required outputs. The lecture also explores prime implicants for various functions, providing essential insights into logical circuit design.
Combinational Logic Implementation using Karnaugh Maps: Minimizing Sum of Products
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Presentation Transcript
CS 140 Lecture 4 Professor CK Cheng 4/11/02
Part I. Combinational Logic Implementation K-Map Given F R D Obj: Minimize sum of products Proc: Draw K-Map Derive prime implicants Derive the essential prime implicants Derive minimum expression
Example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) K-map b 0 4 12 8 1 1 0 0 1 5 13 9 - 0 - 0 d 3 7 15 11 1 0 1 - c 2 6 14 10 1 0 1 0 a
Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. E.g. Sm (0, 4), Sm (0, 1), Sm (1, 3), Sm (3, 11), Sm (14, 15), Sm (11, 15), Sm (13, 15) Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g. Sm (0, 4), Sm (14, 15) Min exp: Sm (0, 4), Sm (14, 15), ( Sm (3, 11) or Sm (1,3) ) f(a,b,c,d) = a’b’c’ + abc’ + b’cd (or a’b’d)
Corresponding circuit a’ c’ d’ f(a,b,c,d) a b c b’ c d
Another 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 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 5 are: 1, 4, 13, 7, and 21 Neighbors of 10 are: 2, 8, 10 ,14, and 26
6 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 27 45 41 49 53 61 57 b f f 35 39 47 43 51 53 63 59 e e 34 38 46 42 50 54 62 58 c c a
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’)
Corresponding Circuit a b f(a,b,c,d) a’ b’ c
Another min product of sums example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) K-map b 0 4 12 8 1 1 0 0 1 5 13 9 - 0 - 0 d 3 7 15 11 1 0 1 - c 2 6 14 10 0 0 1 0 a
Prime Implicates: PM (2,6), PM (2,10), PM (1,5,9,13), PM (5,7), PM (6,7), PM (8,9,10,11), PM (8,9,12,13) Essential Primes: PM (8,9,12,13) Min exp: PM (8,9,12,13) PM (5,7), PM (2,6), PM (8,9,10,11) or PM (6,7), PM (1,5,9,13), PM (2,10) f(a,b,c,d) = (a+b’+d’)(a’+c’+d)(a’+b)(a’+c)
