1. Guest Lecture by Kyle Tietz
CprE 281: Digital Logic
Guest Lecture by Kyle Tietz

2. Karnaugh Maps CprE 281: Digital Logic Iowa State University, Ames, IA
Copyright © 2013

3. Administrative Stuff HW4 is out It is due on Monday Sep 23 @ 4pm.
Please write clearly on the first page (in block capital letters) the following three things:
Your First and Last Name
Your Student ID Number
Your Lab Section Letter

4. Administrative Stuff Exam 1 on Monday Sep 30. Details to follow.
Homework Office Hours
Pratik Mishra
TLA
M 5:30-7:30pm
F 2:00-4:00pm

5. Motivation
Best approach for simplified logic expression? How do we guarantee we have reached minimum SOP/POS representation?

6. Karnaugh Map (K-map) View function in pictoral form
Same information as truth table
Easier to group minterms x x 1 2 x 1 x 2 m 1 1 m m m 1 2 1 m 2 1 m m 1 3 1 1 m 3
(a) Truth table
(b) Karnaugh map

7. Minterms x x x x m m m m m 1 1 m 1 1 1 m 1 1 1 1 m 1 1 1 1 2 1 2 1 2 31 2 3 m 1 1 m 1 1 1 1 m 1 1 2 1 1 m 1 1 1 3

8. Minterm Example x x x x m m m m m + m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 2 m m m m m
+ m 1 2 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

9. Minterm Example _ x1x2 + x1x2 = x2 x x x x m m m m m + m 1 1 1 1 1 1 11 2 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _
x1x2 + x1x2 = x2

10. Two-Variable K-map (a) Truth table (b) Karnaugh map x x x x m 1 1 m m2 x 1 x 2 m 1 1 m m m 1 2 1 m 2 1 m m 1 3 1 1 m 3
(a) Truth table
(b) Karnaugh map

11. Example x x 1 2 1 1 1 1 1 1 1

12. 1. Draw Map x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1

13. 2. Fill Map x x 1 2 x 1 x 2 1 1 1 1 m m 2 1 1 m m 1 3 1 1 1

14. 2. Fill Map x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

15. 3. Group x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

16. 3. Group x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

17. 3. Group x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

18. 4. Write Expression x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

19. 4. Write Expression x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1 _
x1 + x2

20. Grouping x 1 x 1 x 2 x 2 1 1 1 1 1 1 m0 m1

21. Grouping x 1 x 1 x 1 x 2 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1

22. Grouping + = m0 + m1 = m0 + m1 x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1

23. Grouping + = m0 + m1 = m0 + m1 x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 1 21 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1

24. Grouping + = m0 + m1 = m0 + m1 _ _ _ _ + = x1x2 x1x2 x1 x x x x x x 11 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1 _ _ _ _
x1x2 +
x1x2 = x1

25. Grouping Group with rectangles Both sides a power of 2:
1x1, 1x2, 2x1, 2x2, 1x4, 4x1, 2x4, 4x2, 4x4
Can use same minterm more than once
Can wrap around edges of map

26. Writing Expression Examine group and see which variables are constant1 x 2 _ 1
x1 is constant 1 1 1

27. Three-variable K-map

28. Three-variable K-map Notice placement of Variables Binary pair values
Minterms

29. Gray Code Sequence of binary codes Vary by only 1 bit 000 001 011 010
110
111
101
100 00 01 11 10

30. Three-variable K-map

31. Example x 1 2 3 00 01 11 10 f x 1 3 2 + =

32. Tips
Label rows / columns with variable names
Write out gray codes

33. Four-variable K-map

34. Example x x 1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1

35. Example x x 1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1

36. Example x x x x 00 01 11 10 00 1 1 x x x 01 1 1 x x x 11 1 1 x x x 102 x x 3 4 00 01 11 10 00 1 1 x x x 1 3 4 01 1 1 x x x 2 3 4 11 1 1 x x x 1 3 4 10 1 1 x x x 2 3 4

37. Example x x x x 00 01 11 10 00 1 1 x x x 01 1 1 x x x 11 1 1 x x x 102 x x 3 4 00 01 11 10 00 1 1 x x x 1 3 4 01 1 1 x x x 2 3 4 11 1 1 x x x 1 3 4 10 1 1 x x x 2 3 4 x x x x x x 1 2 4 1 2 4 x x x x x x 1 2 3 1 2 3

38. Example POS minimization of f =  M(0, 1, 4, 8, 9, 12, 15) ( ) ( ) ( )3 4 00 01 11 10 x 3 4 + ( ) x 2 3 + ( ) x 1 2 3 4 + ( )
POS minimization of f =  M(0, 1, 4, 8, 9, 12, 15)

39. More Examples

40. Next Time…
Minimization

41. Questions?

42. THE END