1. ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) 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 Simplification of Logic Functions Logic functions can generally be simplified using Boolean algebra. However, two problems arise: – It is difficult to apply to Boolean algebra laws and theorems in a systematic way. – It is difficult to determine when a minimum solution has been achieved. Using a Karnaugh map is generally faster and easier than using Boolean algebra.

3. Spring 2011ECE 301 - Digital Electronics3 Simplification using Boolean Algebra Given: F(A,B,C) =  m(0, 1, 2, 5, 6, 7) Find: minimum SOP expression Combining terms in one way: Combining terms in a different way:

4. Spring 2011ECE 301 - Digital Electronics4 Karnaugh Maps Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the input variables.

5. Spring 2011ECE 301 - Digital Electronics5 Two-variable K-map 0 1 01 m 0 m 2 m 3 m 1 B A row #ABminterm 000m0m0 101m1m1 210m2m2 311m3m3

6. Spring 2011ECE 301 - Digital Electronics6 Two-variable K-map: Example 02 13 Minterm expansion: F(A,B) =  m(0, 1) = A'B' + A'B Maxterm expansion: F(A,B) =  (2, 3) = (A'+B).(A'+B') numericalgebraic row #ABF 0001 1011 2100 3110

7. Spring 2011ECE 301 - Digital Electronics7 Three-variable K-map row #AB C minterm 000 0 m0m0 100 1 m1m1 201 0 m2m2 3011m3m3 4100m4m4 5101m5m5 6110m6m6 711 1 m7m7 m 0 m 4 m 5 m 1 BC A m 3 m 7 m 6 m 2 0 0 1 1 1 0 01 Gray Code

8. Spring 2011ECE 301 - Digital Electronics8 Three-variable K-map: Example 37 2 6 0 4 15 Minterm expansion: F(A,B,C) =  m(2, 3, 4, 6) Maxterm expansion: F(A,B,C) =  (0, 1, 5, 7) row #ABCF 00000 10010 20101 30111 41001 51010 61101 71110

9. Spring 2011ECE 301 - Digital Electronics9 Minimization using K-maps K-maps can be used to derive the  Minimum Sum of Products (SOP) expression  Minimum Product of Sums (POS) expression Procedure:  Enter functional values in the K-map  Identify adjacent cells with same logical value Adjacent cells differ in only one bit  Use adjacency to minimize logic function Horizontal and Vertical adjacency K-map wraps from top to bottom and left to right

10. Spring 2011ECE 301 - Digital Electronics10 Minimization using K-maps Logical Adjacency is used to  Reduce the number number of literals in a term  Reduce the number of terms in a Boolean expression. The adjacent cells  Form a rectangle  Must be a power of 2 (e.g. 1, 2, 4, 8, …) The greater the number of adjacent cells that can be grouped together (i.e. the larger the rectangle), the more the function can be reduced.

11. Spring 2011ECE 301 - Digital Electronics11 K-maps – Logical Adjacency Gray code

12. Spring 2011ECE 301 - Digital Electronics12 Minimization: Example #1 Minimize the following logic function using a Karnaugh map: F(A,B,C) =  m(2, 6, 7) Specify the equivalent maxterm expansion.

13. Spring 2011ECE 301 - Digital Electronics13 Minimization: Example #2 Minimize the following logic function using a Karnaugh map: F(A,B,C) =  M(1, 3, 5, 6, 7) Specify the equivalent minterm expansion.

14. Spring 2011ECE 301 - Digital Electronics14 Minimization: Example #3 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(0, 1, 5, 7) Specify the equivalent maxterm expansion.

15. Spring 2011ECE 301 - Digital Electronics15 Minimization: Example #4 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(0, 1, 5, 7) Specify the equivalent minterm expansion.

16. Spring 2011ECE 301 - Digital Electronics16 Minimization: Example #5 For the following truth table: #ABC F 0000 0 1001 1 2010 0 3011 1 4100 1 5101 0 6110 0 7111 1

17. Spring 2011ECE 301 - Digital Electronics17 Example #5 Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression

18. Spring 2011ECE 301 - Digital Electronics18 Minimization: Example #6 For the following truth table: #ABC F 0000 0 1001 1 2010 1 3011 1 4100 0 5101 1 6110 0 7111 0

19. Spring 2011ECE 301 - Digital Electronics19 Example #6 Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression

20. Spring 2011ECE 301 - Digital Electronics20 Minimal Forms Can a logic function have more than one minimum SOP expression? Can a logic function have more than one minimum POS expression?

21. Spring 2011ECE 301 - Digital Electronics21 K-maps – Two minimal forms F(A,B,C) =  m(0,1,2,5,6,7) =  M(3,4)

22. Spring 2011ECE 301 - Digital Electronics22 Questions?