1. ECE 301 – Digital Electronics Karnaugh Maps and Determining a Minimal Cover (Lecture #8) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. Spring 2011ECE 301 - Digital Electronics2 Four-variable K-map row #ABCDminterm 00000m0m0 10001m1m1 20010m2m2 30011m3m3 40100m4m4 50101m5m5 …… 111011m 11 121100m 12 131101m 13 141110m 14 151111m 15

3. Spring 2011ECE 301 - Digital Electronics3 Four-variable K-map A B C D 0 0 1 1 1 0 0 0 1 1 1 0 Gray code m0 m0 m4m4 m 12 m8m8 m1 m1 m5m5 m 13 m9m9 m3 m3 m7m7 m 14 m 11 m2 m2 m6m6 m 15 m 10

4. Spring 2011ECE 301 - Digital Electronics4 Minimization: Example #7 Use a Karnaugh map to determine the minimum POS expression For the following logic function: F(A,B,C,D) =  m(0,1,3,4,5,7,8,11,14) Specify the equivalent maxterm expansion.

5. Spring 2011ECE 301 - Digital Electronics5 Minimization: Example #8 Use a Karnaugh map to determine the minimum SOP expression For the following logic function: F(A,B,C,D) =  M(0,2,5,7,8,11,13,15) Specify the equivalent minterm expansion.

6. Spring 2011ECE 301 - Digital Electronics6 Minimization: Example #9 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C,D) =  M(0,1,2,3,6,11,14) What is the cost of each logic circuit?

7. Spring 2011ECE 301 - Digital Electronics7 Karnaugh Maps Karnaugh maps can also be used to minimize incompletely specified functions.

8. Spring 2011ECE 301 - Digital Electronics8 Minimization: Example #10 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C) =  m(4,7) +  d(1,3)

9. Spring 2011ECE 301 - Digital Electronics9 Minimization: Example #11 Use a Karnaugh map to determine the minimum SOP expression For the following logic function: F(A,B,C,D) =  M(0,2,5,6,8,13,15).  D(3,4,10)

10. Spring 2011ECE 301 - Digital Electronics10 Minimization: Example #12 Use a Karnaugh map to determine the minimum POS expression For the following logic function: F(A,B,C,D) =  m(0,1,2,4,6,8,9,10) +  d(3,7,11,13,14)

11. Spring 2011ECE 301 - Digital Electronics11 Determining a Minimal Cover

12. Spring 2011ECE 301 - Digital Electronics12 Literals and Implicants Literal  Each occurrence of a variable or its complement in an expression Implicant (SOP)← represents a product term  A single 1 in the K-map  A group of adjacent 1's in the K-map Implicant (POS)← represents a sum term  A single 0 in the K-map  A group of adjacent 0's in the K-map

13. Spring 2011ECE 301 - Digital Electronics13 Prime Implicants Prime Implicant (SOP)  A product term implicant that cannot be combined with another product term implicant to eliminate a literal. Prime Implicant (POS)  A sum term implicant that cannot be combined with another sum term implicant to eliminate a literal.

14. Spring 2011ECE 301 - Digital Electronics14 Implicant Prime Implicant Implicant Prime Implicant Implicants and Prime Implicants Additional Prime Implicants?

15. Spring 2011ECE 301 - Digital Electronics15 Identifying Prime Implicants

16. Spring 2011ECE 301 - Digital Electronics16 Identifying Required Terms Is this term required?

17. Spring 2011ECE 301 - Digital Electronics17 If a minterm is covered by only one prime implicant, that prime implicant is said to be essential, and must be included in the minimum sum of products (SOP). Essential Prime Implicants Prime Implicants Implicants Essential Prime Implicants

18. Spring 2011ECE 301 - Digital Electronics18 Note: 1’s shaded in blue are covered by only one prime implicant. All other 1’s are covered by at least two prime implicants. Identifying Essential Prime Implicants

19. Spring 2011ECE 301 - Digital Electronics19 Determining a Minimal Cover Identify all prime implicants Select all essential prime implicants Select prime implicant(s) to cover remaining terms by considering all possibilities  Sometimes selection is obvious  Sometimes “guess” next prime implicant Continue, perhaps recursively Try all possible “guesses” Determine the Boolean expression  May not be unique

20. Spring 2011ECE 301 - Digital Electronics20 Shaded 1’s are covered by only one prime implicant. Essential prime implicants: A′B, AB′D′ Then AC′D covers the remaining 1’s. Determining a Minimal Cover

21. Spring 2011ECE 301 - Digital Electronics21 A Minimal Cover Thus … A minimal cover is an expression that consists of the fewest product terms (for a SOP expression) or sum terms (for a POS expression) and the fewest literals in each term.

22. Spring 2011ECE 301 - Digital Electronics22 Questions?