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

2. Incompletely Specified Functions & Multiple-Output Circuits
CprE 281: Digital Logic
Iowa State University, Ames, IA
Copyright © Alexander Stoytchev

3. Administrative Stuff
HW4 is due today.

4. Administrative Stuff HW5 is out It is due on Monday Feb 17 @ 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

5. Administrative Stuff Midterm Exam #1 When: Monday Feb 24.
Where: This classroom
What: Chapter 1 and Chapter 2 plus number systems
The exam will be open book and open notes (you can bring up to 3 pages of handwritten notes).
More details to follow.

6. Quick Review

7. The Combining Theorems of Boolean Algebra

8. 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 ]

9. Two-Variable K-map 1
[ Figure 2.50 from the textbook ]

10. These are all valid groupings

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

12. Why are these two not valid?

13. Three-Variable K-map
[ Figure 2.51 from the textbook ]

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

15. Adjacency Rules
adjacent
columns

16. These are valid groupings

17. These are valid groupings

18. These are valid groupings

19. These are valid groupings

20. These are valid groupings

21. This is a valid grouping

22. Some invalid groupings

23. Three-Variable K-map
[ Figure 2.52 from the textbook ]

24. Two Different Ways to Draw the K-map
x2x3 x1 00 01 11 10 m0 m1 m3 m2 1 m4 m5 m7 m6

25. Another Way to Draw 3-variable K-mapx1
x2x3 1 00 m0 m4 01 m1 m5 11 m3 m7 10 m2 m6

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

27. 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

28. Adjacency Rules
adjacent
rows
adjacent
columns
adjacent
columns

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

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

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

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

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

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

35. Other Four-Variable K-map Examples
[ Figure 2.54 from the textbook ]

36. Strategy For Minimization

37. 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).

38. Terminology _ Literal: a variable, complemented or uncomplemented
Some Examples: X1 X2 _

39. Terminology
Implicant: product term that indicates the input combinations for which the function output is 1
Example
x indicates that x1x2 and x1x2 yield output of 1 _ _ _ _ x 2 1

40. Terminology Prime Implicant Prime Not prime
Implicant that cannot be combined into another implicant with fewer literals
Some Examples x 1 2 3 00 01 11 10 x 1 2 3 00 01 11 10
Prime
Not prime

41. Terminology Essential Prime Implicant
Prime implicant that includes a minterm not covered by any other prime implicant
Some Examples x 1 2 3 00 01 11 10

42. Terminology
Cover
Collection of implicants that account for all possible input valuations where output is 1
Ex. x1’x2x3 + x1x2x3’ + x1x2’x3’
Ex. x1’x2x3 + x1x3’ x 1 2 3 00 01 11 10

43. Example Give the Number of Implicants? Prime Implicants?
Essential Prime Implicants? x 1 2 3 00 01 11 10

44. Why concerned with minimization?
Simplified function
Reduce the cost of the circuit
Cost: Gates + Inputs
Transistors

45. Three-variable function f (x1, x2, x3) =  m(0, 1, 2, 3, 7)
[ Figure 2.56 from the textbook ]

46. 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

47. 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

48. 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

49. Example: Another Solution1 2 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
[ Figure 2.59 from the textbook ]

50. Example: Incompletely Specified Function

51. Three Ways to Specify the Function
f(x1, x2, x3, x4) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)

52. Three Ways to Specify the Function
f(x1, x2, x3, x4) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15) x 1 2 3 4 00 01 11 10 d

53. SOP implementation x 00 01 11 10 d (a) SOP implementation 1 2 3 400 01 11 10 d
(a) SOP implementation
[ Figure 2.62 from the textbook ]

54. POS implementation x ( ) 00 01 11 10 d + (b) POS implementation 1 2 34 00 01 11 10 d + ( )
(b) POS implementation
[ Figure 2.62 from the textbook ]

55. Example: A circuit with multiple outputs

56. Seven-Segment Indicator

57. Seven-Segment Indicator

58. Seven-Segment Indicator

59. Seven-Segment Indicator

60. Seven-Segment Indicator
d d d d d d d
d d d d d d d d d d d d d d

61. Seven-Segment Indicator
d d d d d d d
d d d d d d d d d d d d d d

62. Seven-Segment Indicatorx 3 2 1 00 01 11 10
d d d d d d d
d d d d d d d d d d d d d d

63. Seven-Segment Indicatorx 3 2 1 00 01 11 10 d
d d d d d d d
d d d d d d d d d d d d d d

64. Seven-Segment Indicatorx 3 2 1 00 01 11 10 d
d d d d d d d
d d d d d d d d d d d d d d

65. Seven-Segment Indicatorx 3 2 1 00 01 11 10
1 d d d d d d
1 d d d d d d
In this case all d's were treated as 1's.

66. Seven-Segment Indicator
1 d d d d d d
1 d d d d d d d d d d d d

67. Seven-Segment Indicatorx 3 2 1 00 01 11 10
1 d d d d d d
1 d d d d d d d d d d d d

68. Seven-Segment Indicatorx 3 2 1 00 01 11 10 d
1 d d d d d d
1 d d d d d d d d d d d d

69. Seven-Segment Indicatorx 3 2 1 00 01 11 10 d
1 d d d d d d
1 d d d d d d d d d d d d

70. Seven-Segment Indicatorx 3 2 1 00 01 11 10
1 d d d d d
1 d d d d d d d d d d
1 d d d d d d d d d d
1 d d d d d
In this case some d's were treated as 1's, others as 0's.

71. Seven-Segment Indicator

72. Another Example

73. [ Figure 2.64 from the textbook ]1 2 3 4 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
[ Figure 2.64 from the textbook ]

74. x 00 01 11 10 (a) Function f x 00 01 11 10 (b) Function f 1 2 3 4 1 2

75. _ x1 x3 x 00 01 11 10 (a) Function f x 00 01 11 10 (b) Function f 1 24 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
x1 x3 _

76. _ x1 x3 x 00 01 11 10 (a) Function f x 00 01 11 10 (b) Function f 1 24 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
x1 x3 _

77. _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function f x 00 01 11 102 3 4 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
x1 x3 _
x1 x3 _

78. _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function f x 00 01 11 102 3 4 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
x1 x3 _
x1 x3 _

79. _ x1 x3 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function f x 00 01 11 102 3 4 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f
x1 x3 _
x1 x3 _

80. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4
x1 x3 _
x1 x3 _

81. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4 f 1 2 x 3 4
(c) Combined circuit for
and
x1 x3 _
x1 x3 _

82. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4 f 1 2 x 3 4
(c) Combined circuit for
and
x1 x3 _
x1 x3 _

83. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4 f 1 2 x 3 4
(c) Combined circuit for
and
x1 x3 _
x1 x3 _

84. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4 f 1 2 x 3 4
(c) Combined circuit for
and
x1 x3 _
x1 x3 _

85. _ x1 x3 _ x2 x3 x4 x2 x3 x4 _ _ x1 x3 x1 x3 x 00 01 11 10 (a) Function
(b) Function f
x2 x3 x4 _
x2 x3 x4 f 1 2 x 3 4
(c) Combined circuit for
and
x1 x3 _
x1 x3 _

86. (c) Combined circuit for and [ Figure 2.64 from the textbook ]1 2 3 4 00 01 11 10
(a) Function f x 1 2 3 4 00 01 11 10
(b) Function f f 1 2 x 3 4
(c) Combined circuit for
and
[ Figure 2.64 from the textbook ]

87. Yet Another Example

88. Individual vs Joint Optimizationx x x x 1 2 1 2 x x x x 3 4 00 01 11 10 3 4 00 01 11 10 00 00 01 1 1 1 01 1 1 1 11 1 1 1 11 1 1 1 10 1 10 1
(a) Optimal realization of f
(b) Optimal realization of f 3 4 x x x x 1 2 1 2 x x x x 3 4 00 01 11 10 3 4 00 01 11 10 00 00 01 1 1 1 01 1 1 1 11 1 1 1 11 1 1 1 10 1 10 1
(c) Optimal realization of f
and f
together 3 4
[ Figure 2.65 from the textbook ]

89. Individual vs Joint Optimizationx x x x 1 2 1 2 x x x x 3 4 00 01 11 10 3 4 00 01 11 10 00 00 01 1 1 1 01 1 1 1 11 1 1 1 11 1 1 1 10 1 10 1
(c) Optimal realization of f
and f
together 3 4 f 3 4 x 1 2
(d) Combined circuit for
and
[ Figure 2.65 from the textbook ]

90. Individual vs Joint Optimizationx x x x 1 2 1 2 x x x x 3 4 00 01 11 10 3 4 00 01 11 10 00 00 01 1 1 1 01 1 1 1 11 1 1 1 11 1 1 1 10 1 10 1
(c) Optimal realization of f
and f
together 3 4 f 3 4 x 1 2
(d) Combined circuit for
and
[ Figure 2.65 from the textbook ]

91. Individual vs Joint Optimizationx x x x 1 2 1 2 x x x x 3 4 00 01 11 10 3 4 00 01 11 10 00 00 01 1 1 1 01 1 1 1 11 1 1 1 11 1 1 1 10 1 10 1
(c) Optimal realization of f
and f
together 3 4 f 3 4 x 1 2
(d) Combined circuit for
and
[ Figure 2.65 from the textbook ]

92. Questions?

93. THE END