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

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

3. 4-Variable K-Maps: An example f(a,b,c,d) Id a b c d f (a,b,c,d) 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 3

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

5. Another example w/ 4 bits: 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 5

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

7. Boolean Expression K-Map Variable x i and its compliment x i ’  Two half planes Rx i, and Rx i ’ Product term P (  x i * e.g. b’c’)  Intersect of Rx i * 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 7

8. ProcedureInput: 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) 8

9. 4-input K-map 9

10. 10

11. K-maps with Don’t Cares 11

12. K-maps with Don’t Cares 12

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

14. 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms.  m (0, 1, 8, 9),  m (0, 2, 8, 10),  m (10, 14) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes.  m (0, 1, 8, 9),  m (0, 2, 8, 10),  m (10, 14) 4. Min exp:  m (0, 1, 8, 9) +  m (0, 2, 8, 10) +  m (10, 14) f(a,b,c,d) = b’c’ + b’d’+ acd’ 14

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

16. 2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms. E.g.  m (0, 4),  m (0, 1),  m (1, 3),  m (3, 11),  m (14, 15),  m (11, 15),  m (13, 15) 3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes. E.g.  m (0, 4),  m (14, 15) 4. Min exp:  m (0, 4),  m (14, 15), (  m (3, 11) or  m (1,3) ) f(a,b,c,d) = a’c’d’+ abc+ b’cd (or a’b’d) 16