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CS 140 Lecture 4 Combinational Logic: K-Map

This lecture covers the implementation of combinational logic using K-Maps. It includes examples and step-by-step procedures for finding the minimal sum of products.

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CS 140 Lecture 4 Combinational Logic: K-Map

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  1. CS 140 Lecture 4Combinational Logic: K-Map Professor CK Cheng CSE Dept. UC San Diego

  2. Part I. Combinational Logic • Implementation • K-map

  3. Id a b c d f (a,b,c,d) 0 0 0 0 0 0 1 0 0 0 1 0 2 0 0 1 0 1 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 0 6 0 1 1 0 1 7 0 1 1 1 1 8 1 0 0 0 0 9 1 0 0 1 0 10 1 0 1 0 1 11 1 0 1 1 1 12 1 1 0 0 0 13 1 1 0 1 0 14 1 1 1 0 1 15 1 1 1 1 1 4-variable K-maps

  4. CorrespondingK-map b 0 4 12 8 0 0 0 0 1 5 13 9 0 0 0 0 d 3 7 15 11 1 1 1 1 c 2 6 14 10 1 1 1 1 a f (a, b, c, d) = c

  5. Id a b c d f (a,b,c,d) 0 0 0 0 0 1 1 0 0 0 1 1 2 0 0 1 0 1 3 0 0 1 1 0 4 0 1 0 0 0 5 0 1 0 1 0 6 0 1 1 0 0 7 0 1 1 1 0 8 1 0 0 0 1 9 1 0 0 1 - 10 1 0 1 0 - 11 1 0 1 1 0 12 1 1 0 0 0 13 1 1 0 1 0 14 1 1 1 0 1 15 1 1 1 1 0 Another example w/ 4 bits:

  6. Corresponding4-variable K-map b 0 4 12 8 1 0 0 1 1 5 13 9 1 0 0 - d 3 7 15 11 0 0 0 0 c 2 6 14 10 1 0 1 - a f (a, b, c, d) = b’c’ + b’d’ + acd’

  7. Boolean Expression K-Map Variable xi and its compliment xi’ Two half planes Rxi, Rxi’  Product term P (PXi* e.g. b’c’) Intersect of Rxi* for all i in P (Rb’ intersect Rc’)  U Each minterm  1-cell Two minterms are adjacent iff they differ by one and only one variable, eg:abc’d, abc’d’ The two 1-cells are neighbors  Each minterm has n adjacent minterms Each 1-cell has n neighbors 

  8. Procedure Input: Two sets of F R D • Draw K-map. • Expand all terms in F to their largest sizes (prime implicants). • Choose the essential prime implicants. • Try all combinations to find the minimal sum of products. (This is the most difficult step)

  9. 4-input K-map

  10. 4-input K-map

  11. K-maps with Don’t Cares

  12. K-maps with Don’t Cares

  13. Example Given F = Sm (0, 1, 2, 8, 14) D = Sm (9, 10) 1. Draw K-map b 0 4 12 8 1 0 0 1 1 5 13 9 1 0 0 - d 3 7 15 11 0 0 0 0 c 2 6 14 10 1 0 1 - a

  14. 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. Sm (0, 1, 2, 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’

  15. Another example Given F = Sm (0, 3, 4, 14, 15) D = Sm (1, 11, 13) 1. Draw 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

  16. 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)

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