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CSE20 Lecture 15 Karnaugh Maps Professor CK Cheng CSE Dept. UC San Diego 1 Example Given F = Sm (3, 5), D = Sm (0, 4) b 0 2 1 c 6 0 3 0 4 0 7 1 5 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’ 2 Boolean Expression K-Map Variable xi and its compliment xi’ Two half planes Rxi, and Rxi’ Product term P (Pxi* e.g. b’c’) Intersect of Rxi* for all i in P e.g. Rb’ intersect Rc’ Each minterm One element cell Two minterms are adjacent iff they differ by one and only one variable, eg: abc’d, abc’d’ The two cells are neighbors Each minterm has n adjacent minterms Each cell has n neighbors 3 Procedure Input: Two sets of F R D 1) Draw K-map. 2) Expand all terms in F to their largest sizes (prime implicants). 3) Choose the essential prime implicants. 4) Try all combinations to find the minimal sum of products. (This is the most difficult step) 4 Example Given F = Sm (0, 1, 2, 8, 14) D = Sm (9, 10) 1. Draw K-map b 0 4 1 1 0 5 1 3 c 12 0 2 1 13 0 6 1 0 0 7 8 9 0 15 0 14 0 11 d 0 10 1 a 5 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. Sm (0, 1, 8, 9), Sm (0, 2, 8, 10), Sm (10, 14) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. Sm (0, 1, 8, 9), Sm (0, 2, 8, 10), Sm (10, 14) 4. Min exp: Sm (0, 1, 8, 9) + Sm (0, 2, 8, 10) + Sm (10, 14) f(a,b,c,d) = b’c’ + b’d’+ acd’ 6 Another example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) 1. Draw K-map b 0 4 1 1 1 5 3 c 2 12 1 0 0 6 0 13 0 7 8 0 0 9 15 14 1 1 0 11 10 a d 0 7 2. 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) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g. Sm (0, 4), Sm (14, 15) 4. Min exp: Sm (0, 4), Sm (14, 15), ( Sm (3, 11) or Sm (1,3) ) f(a,b,c,d) = a’c’d’+ abc+ b’cd (or a’b’d) 8 Five variable K-map c d c 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 3 7 15 11 19 23 31 27 2 6 14 10 18 22 30 26 e d b e b a Neighbors of m5 are: minterms 1, 4, 7, 13, and 21 Neighbors of m10 are: minterms 2, 8, 11, 14, and 26 9 Six variable K-map d e d 0 4 12 8 16 20 28 24 1 5 13 9 17 21 29 25 3 7 15 11 19 23 31 27 2 6 14 10 18 22 30 26 e c d e f c d 32 36 44 40 48 52 60 56 33 37 45 41 49 53 61 57 35 39 47 43 51 55 63 59 34 38 46 42 50 54 62 58 f e c f f a c b 10 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. 11 Min product of sums Given F = Sm (3, 5), D = Sm (0, 4) b 0 - 1 c 2 0 3 0 6 0 7 1 4 - 5 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’) 12 Corresponding Circuit a b f(a,b,c,d) a’ b’ c 13 Quiz Given F = Sm (0, 6), D = Sm (2, 7), 1. Fill the Karnaugh map. 2. Identify all prime implicates 3. Identify all essential primes. 4. Find a minimal expression in product of sums format. 14 Another min product of sums example Given R = Sm (3, 11, 12, 13, 14) D = Sm (4, 8, 10) b K-map 0 4 1 1 5 1 3 c 2 12 0 1 0 6 - 13 1 7 8 1 1 9 0 15 14 1 11 1 10 0 a d 0 15 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. 16 Summary •Karnaugh Maps: Two dimensional truth table which mimics an n-variable cube with imaginary adjacency. •Theme: Relation between Boolean algebra and Karnaugh maps. •Key words: Primes, Essential Primes •Goal: Minimal expression in the format of sum-of-products or product-of-sums. 17