1. Instructor: Alexander Stoytchev
CprE 281: Digital Logic
Instructor: Alexander Stoytchev

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

3. Administrative Stuff HW4 is out It is due on Monday Sep 18 @ 4 pm
It is posted on the class web page
I also sent you an with the link.

4. Administrative Stuff
Homework Solutions are posted on BlackBoard

5. Quick Review

6. Do You Still Remember This Boolean Algebra Theorem?

7. Let’s prove 14.a

8. Let’s prove 14.a 1

9. Let’s prove 14.a 1 1

10. Let’s prove 14.a 1 1 1

11. Let’s prove 14.a 1 1 1 1

12. Let’s prove 14.a 1 1 1 1
They are equal.

14. Motivation
An approach for simplifying logic expressions How do we guarantee we have reached minimum SOP/POS representation?

15. Two-Variable K-Map

16. Karnaugh Map (K-map) View the function in a visual 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
[ Figure 2.49 from the textbook ]

17. 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

18. 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

19. 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

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

21. Grouping Example + = m0 + m1 = m0 + m1 x x x x x x 1 1 1 1 1 1 1 1 1 12 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1

22. Grouping Example + = m0 + m1 = m0 + m1 x x x x x x 1 1 1 1 1 1 1 1 1 12 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1

23. Grouping Example + = m0 + m1 = m0 + m1 _ _ _ _ + =2 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m1 =
m0 + m1 _ _ _ _
x1x2 +
x1x2 = x1
Property 14a (Combining)

24. Grouping Rules Group “1”s with rectangles Both sides a power of 2:
1x1, 1x2, 2x1, 2x2, 1x4, 4x1, 2x4, 4x2, 4x4
Can use the same minterm more than once
Can wrap around the edges of the map
Some rules in selecting groups:
Try to use as few groups as possible to cover all “1”s.
For each group, try to make it as large as you can (i.e., if you can use a 2x2, don’t use a 2x1 even if that is enough).

25. 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
[ Figure 2.49 from the textbook ]

26. Step-By-Step Example x x 1 2 1 1 1 1 1 1 1

27. 1. Draw The Map x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1

28. 2. Fill The Map x x 1 2 x 1 x 2 m 1 3 2 1 1 1 1 m m 2 1 1 m m 1 3 1 1 1

29. 2. Fill The Map x x 1 2 x 1 x 2 m 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1

30. 3. Group x x 1 2 x 1 x 2 m 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1

31. 3. Group x x 1 2 x 1 x 2 m 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1

32. 3. Group x x 1 2 x 1 x 2 m 1 3 2 1 1 1 1 1 1 1 1 1 1 1 1

33. 4. Write The Expression x x 1 2 x 1 x 2 1 1 1 1 1 1 1 1 1 1 1 1

34. 4. Write The Expression _ x1 + x2 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 1 1 _
x1 + x2

35. Writing The Expression
Find which variable is constant x 1 x 2 _ 1
x1 is constant 1 1 1

36. Writing The Expression
Find which variable is constant x 1 x 2 1
x1 is constant 1 1 1

38. These are all valid groupings

39. These are also valid
But try to use larger rectangles if possible.

40. This one is valid too
In this case the result is the constant function 1.

41. Why are these two not valid?

42. Let’s Find Out x 1 x 1 x 2 x 2 1 1 1 1 1 1 m0 m3

43. Let’s Find Out + = m0 + m3 = m0 + m3 x x x x x x 1 1 1 1 1 1 1 1 1 1 12 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m3 =
m0 + m3

44. Let’s Find Out + = m0 + m3 = m0 + m3 x x x x x x 1 1 1 1 1 1 1 1 1 1 12 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m3 =
m0 + m3

45. Let’s Find Out + = m0 + m3 = m0 + m3 _ _ = + _ _ +x 1 x 1 x 1 x 2 x 2 x 2 1 1 1 1 1 + = 1 1 1 1 1 m0 + m3 =
m0 + m3 _ _ =
x1x2 + _ _
x1x2 +
x1x2
We can’t use Property 14a here. This can’t be simplified.

46. Three-Variable K-Map

47. Location of three-variable minterms
[ Figure 2.51 from the textbook ]

48. Location of three-variable minterms
Notice the placement of
Variables
Binary pair values
Minterms

49. Gray Code Sequence of binary codes
Two neighboring lines vary by only 1 bit
000
001
011
010
110
111
101
100 00 01 11 10

50. Gray Code & K-map
s x1 x2

51. Gray Code & K-map
s x1 x2
000
010
110
100
001
011
111
101

52. Adjacency Rules
adjacent
columns

53. Adjacency Rules As if the K-map were adjacent drawn on a cylinder
columns

54. Adjacency Rules m0 m2 m4 m1 m3 m5 As if the K-map were adjacent
drawn on a cylinder
adjacent
columns

55. These are valid groupings

56. These are valid groupings

57. These are valid groupings

58. These are valid groupings

59. These are valid groupings

60. This is a valid grouping

61. Some invalid groupings

62. Examples of three-variable Karnaugh maps
[ Figure 2.52 from the textbook ]

63. Four-Variable K-Map

64. A four-variable Karnaugh map
[ Figure 2.53 from the textbook ]

65. A four-variable Karnaugh mapx1 x2 x3 x4 m0 1 m1 m2 m3 m4 m5 m6 m7 m8 m9
m10
m11
m12
m13
m14
m15

66. Adjacency Rules
adjacent
rows
adjacent
columns
adjacent
columns

67. Adjacency Rules adjacent rows As if the K-map were drawn on a torus
columns

68. Adjacency Rules m8 m10 m0 m2 m14 m12 m6 m4 adjacent rows
As if the K-map were
drawn on a torus
adjacent
columns

69. Some Valid Groupings

70. Some Valid Groupings

71. Some Valid Groupings

72. Some Valid Groupings

73. Some Invalid Groupings
All sides must be powers of 2.

74. Some valid Groupings
All sides must be powers of 2.

75. Example of a four-variable Karnaugh map
[ Figure 2.54 from the textbook ]

76. Example of a four-variable Karnaugh map
[ Figure 2.54 from the textbook ]

77. Example of a four-variable Karnaugh map
[ Figure 2.54 from the textbook ]

78. Example of a four-variable Karnaugh map
[ Figure 2.54 from the textbook ]

79. Five-Variable K-Map

80. A five-variable Karnaugh map
[ Figure 2.55 from the textbook ]

81. Questions?

82. THE END