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 17 @ 4 pm
It is posted on the class web page
I also sent you an with the link.

4. Administrative Stuff Sample homework solutions are posted on Canvas
Look under ‘Files’

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

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

51. 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

52. Gray Code & K-map
s x1 x2

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

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

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

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

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

58. differ only in the LAST bit
Gray Code & K-map
s x1 x2
000
010
110
100
001
011
111
101
These two neighbors
differ only in the LAST bit

59. differ only in the LAST bit
Gray Code & K-map
s x1 x2
000
010
110
100
001
011
111
101
These two neighbors
differ only in the LAST bit

60. differ only in the FIRST bit
Gray Code & K-map
s x1 x2
000
010
110
100
001
011
111
101
These two neighbors
differ only in the FIRST bit

61. Gray Code & K-map s x1 x2 These four neighbors
000
010
110
100
001
011
111
101
These four neighbors
differ in the FIRST and LAST bit
They are similar in their MIDDLE bit

62. Adjacency Rules
adjacent
columns

63. differ only in the FIRST bit
Gray Code & K-map
s x1 x2
000
010
110
100
001
011
111
101
These two neighbors
differ only in the FIRST bit

64. Gray Code & K-map s x1 x2 These four neighbors
000
010
110
100
001
011
111
101
These four neighbors
differ in the FIRST and LAST bit
They are similar in their MIDDLE bit

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

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

67. These are valid groupings

68. These are valid groupings

69. These are valid groupings

70. These are valid groupings

71. These are valid groupings

72. This is a valid grouping

73. Some invalid groupings

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

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

76. Four-Variable K-Map

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

78. 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

79. Adjacency Rules
adjacent
rows
adjacent
columns
adjacent
columns

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

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

82. Some Valid Groupings

83. Some Valid Groupings

84. Some Valid Groupings

85. Some Valid Groupings

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

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

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

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

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

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

92. Five-Variable K-Map

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

94. Questions?

95. THE END