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Lecture 4: Four Input K-Maps 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 Outlines Boolean Algebra vs.

  1. Lecture 4: Four Input K-Maps 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

  2. Outlines • Boolean Algebra vs. Karnaugh Maps – Algebra: variables, product terms, minterms, consensus theorem – Map: planes, rectangles, cells, adjacency • Definitions: implicants, prime implicants, essential prime implicants • Implementation Procedures 2

  3. 4-input K-map A B C D Y Y 0 AB 0 0 0 1 CD 00 01 11 10 0 0 0 1 0 0 0 1 0 1 00 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 01 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 11 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 10 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 1 1 1 1 0 3

  4. 4-input K-map A B C D Y 0 0 0 0 1 0 0 0 1 0 Y 0 0 1 0 1 AB 0 0 1 1 1 CD 00 01 11 10 0 1 0 0 0 0 1 0 1 1 00 1 0 0 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 01 0 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 11 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 10 1 1 0 1 1 1 1 1 0 4

  5. 4-input K-map • Identify adjacent cells containing 1’s • What happens when we combine these cells? Y AB CD 00 01 11 10 00 1 0 0 1 01 0 1 0 1 11 1 1 0 0 10 1 1 0 1 5

  6. Boolean Expression K-Map Variable x i and its Two half planes Rx i , ó compliment x i ’ and Rx i ’ Product term P ó Intersect of Rx i * for all i in ( P x i * e.g. b ’ c ’ ) P e.g. Rb ’ intersect Rc ’ Each minterm ó One element cell Two minterms are adjacent iff they differ by one and The two cells are only one variable, eg: ó neighbors abc ’ d, abc ’ d ’ Each minterm has n ó Each cell has n adjacent minterms neighbors 6

  7. Procedure for finding the minimal function via K-maps (layman terms) 1. Convert truth table to K-map 2. Group adjacent ones: In doing so include Y the largest number of adjacent ones (Prime AB CD 00 01 11 10 Implicants) 00 1 0 0 1 3. Create new groups to cover all ones in the map: create a new group only to include at 01 0 1 0 1 least once cell (of value 1 ) that is not covered by any other group 11 1 1 0 0 4. Select the groups that result in the minimal 10 1 1 0 1 sum of products (we will formalize this because its not straightforward) 7

  8. Reading the reduced K-map Y AB 00 01 11 10 CD 00 1 0 0 1 01 0 1 0 1 11 1 1 0 0 10 1 1 0 1 Y = AC + ABD + ABC + BD 8

  9. Definitions: implicant, prime implicant, essential prime implicant • 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 i.e. it is not completely covered by any single implicant • Essential Prime Implicant: A prime implicant with atleast one element that is not covered by one or more prime implicants. 9

  10. Definition: Prime Implicant • 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 i.e. it is not completely covered by any single implicant Q: Is this a prime implicant? Y AB CD 00 01 11 10 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 10

  11. Definition: Prime Implicant • 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 i.e. it is not completely covered by any single implicant Y Q: Is this a prime implicant? AB CD 00 01 11 10 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 11

  12. Definition: Prime Implicant • 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 i.e. it is not completely covered by any single implicant Q: How about this one? Is it a Y AB CD 00 01 11 10 prime implicant? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 12

  13. Definition: Prime Implicant • 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 i.e. it is not completely covered by any single implicant Q: Is the red group a prime Y AB CD 00 01 11 10 implicant? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No: Because it is covered by a larger 11 1 1 0 0 group 10 1 1 0 1 13

  14. Definition: Essential Prime • Essential Prime Implicant: A prime implicant with atleast one element that is not covered by one or more prime implicants Q: Is the blue group an Y AB CD 00 01 11 10 essential prime? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 14

  15. Definition: Essential Prime • Essential Prime Implicant: A prime implicant with atleast one element that is not covered by one or more prime implicants Q: Is the blue group an Y AB CD 00 01 11 10 essential prime? 00 1 0 0 1 A. Yes 01 0 1 0 1 B. No 11 1 1 0 0 10 1 1 0 1 15

  16. Definition: Non-Essential Prime Non Essential Prime Implicant : Prime implicant that has no element that cannot be covered by other prime implicant Q: Is the blue group a non-essential prime implicant? ab 00 01 11 10 cd A. Yes 1 1 1 00 B. No 1 1 01 11 1 1 1 1 10 1 16

  17. Definition: Non-Essential Prime Non Essential Prime Implicant : Prime implicant that has no element that cannot be covered by other prime implicant Q: Is the blue group a non-essential prime implicant? ab 00 01 11 10 cd A. Yes 1 1 1 00 B. No 1 1 01 11 1 1 1 1 10 1 17

  18. Procedure for finding the minimal function via K-maps (formal terms) Y AB CD 00 01 11 10 1. Convert truth table to K-map 00 1 0 0 1 2. Include all essential primes 3. Include non essential primes as 01 0 1 0 1 needed to completely cover the onset 11 1 1 0 0 (all cells of value one) 10 1 1 0 1 18

  19. K-maps with Don ’ t Cares A D B C Y Y 0 0 0 0 1 AB 0 0 0 1 0 00 01 11 10 CD 0 0 1 0 1 0 0 1 1 1 00 0 1 0 0 0 0 1 0 1 X 0 1 1 0 1 01 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 11 1 X 0 1 0 1 X 0 1 1 1 X 1 0 0 10 1 X 1 0 1 1 X 1 1 0 1 1 1 1 X 19

  20. K-maps with Don ’ t Cares A D B C Y Y 0 0 0 0 1 AB 0 0 0 1 0 00 01 11 10 CD 0 0 1 0 1 0 0 1 1 1 00 1 0 X 1 0 1 0 0 0 0 1 0 1 X 0 1 1 0 1 01 0 X X 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 11 1 1 X X 1 X 0 1 0 1 X 0 1 1 1 X 1 0 0 10 1 1 X X 1 X 1 0 1 1 X 1 1 0 1 1 1 1 X 20

  21. K-maps with Don ’ t Cares Y AB A D B C Y 00 01 11 10 CD 0 0 0 0 1 0 0 0 1 0 00 1 0 X 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 01 0 X X 1 0 1 0 1 X 0 1 1 0 1 0 1 1 1 1 11 1 1 X X 1 1 0 0 0 1 1 0 0 1 1 X 0 1 0 10 1 1 X X 1 X 0 1 1 1 X 1 0 0 1 X 1 0 1 Y = A + BD + C 1 X 1 1 0 1 1 1 1 X 21

  22. Reducing Canonical expressions Given F(a,b,c,d) = Σ m (0, 1, 2, 8, 14) D(a,b,c,d) = Σ m (9, 10) 1. Draw K-map ab 11 00 01 10 cd 00 01 11 10 22

  23. Reducing Canonical Expressions Given F(a,b,c,d) = Σ m (0, 1, 2, 8, 14) D(a,b,c,d) = Σ m (9, 10) 1. Draw K-map ab 11 00 01 10 cd 0 4 12 8 00 1 5 13 9 01 3 7 15 11 11 2 6 14 10 10 23

  24. Reducing Canonical Expressions Given F(a,b,c,d) = Σ m (0, 1, 2, 8, 14) D(a,b,c,d) = Σ m (9, 10) 1. Draw K-map ab 11 00 01 10 cd 0 4 12 8 1 0 0 1 00 1 5 13 9 01 1 0 0 X 3 7 15 11 11 0 0 0 0 2 6 14 10 10 1 0 1 X 24

  25. Reducing Canonical Expressions 1. Draw K-map 2. Identify Prime implicants 3. Identify Essential Primes ab 11 00 01 10 cd 0 4 12 8 1 0 0 1 00 PI Q: How many primes (P) and essential primes (EP) 1 5 13 9 01 1 0 0 X are there? A. Four (P) and three (EP) 3 7 15 11 11 0 0 0 0 B. Three (P) and two (EP) C. Three (P) and three (EP) 2 6 14 10 10 1 0 1 X D. Four (P) and Four (EP) 25

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