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 out It is due on Monday Sep 18 @ 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
Also, staple all of your pages together

4. Administrative Stuff Midterm Exam #1 When: Friday Sep 22.
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).
Sample exams are posted on the class web page

5. Topics for the Midterm Exam
Binary Numbers
Octal Numbers
Hexadecimal Numbers
Conversion between the different number systems
Truth Tables
Boolean Algebra
Logic Gates
Circuit Synthesis with AND, OR, NOT
Circuit Synthesis with NAND, NOR
Converting an AND/OR/NOT circuit to NAND circuit
Converting an AND/OR/NOT circuit to NOR circuit
SOP and POS expressions

6. Topics for the Midterm Exam
Mapping a Circuit to Verilog code
Mapping Verilog code to a circuit
Multiplexers
Venn Diagrams
K-maps for 2, 3, and 4 variables
Minimization of Boolean expressions using theorems
Minimization of Boolean expressions with K-maps
Incompletely specified functions (with don’t cares)
Functions with multiple outputs

7. Quick Review

8. The Combining Theorems of Boolean Algebra

9. One-Variable K-map (a) Truth table (b) Karnaugh map x x m 1 1 m m m 1m 1 1 m m m 1 1
(a) Truth table
(b) Karnaugh map

10. One-Variable K-map x 1 x 1 1 1 1 1 x1
(a) Truth table
(b) Karnaugh map

11. 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 ]

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

13. These are all valid groupings

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

15. These two are not valid

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

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

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

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

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

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

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

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

24. Gray Code & K-map
s x1 x2

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

26. 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

27. 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

28. 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

29. 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

30. 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

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

32. 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

33. Gray Code & K-map x1 x2 x3 x4 m0 1 m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11m0 1 m1 m2 m3 m4 m5 m6 m7 m8 m9
m10
m11
m12
m13
m14
m15

34. Gray Code & K-map x1 x2 x3 x4 m0 1 m1 m2 m3 m4 m5 m6 m7 m8 m9
m10
m11
m12
m13
m14
m15
0000
0100
1100
1000
0001
0101
1101
1001
0011
0111
1111
1011
0010
0110
1110
1010

35. Adjacency Rules
adjacent
rows
adjacent
columns
adjacent
columns

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

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

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

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

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

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

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

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

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

45. Strategy For Minimization

46. 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).

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

48. 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

49. 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

50. 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

51. 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

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

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

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

55. 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

56. 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

57. 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

58. Example f = + x x x x 00 01 11 10 00 1 1 x x x 01 1 1 x x x 11 1 1 x x2 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 1 3 4 2
f = +

59. Example: Another Solution1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1

60. Example: Another Solution1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1
[ Figure 2.59 from the textbook ]

61. Example: Another Solution1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 x x x x x x 1 2 4 1 2 4 x x x x x x 1 2 3 1 2 3

62. Example: Another Solution1 2 x x 3 4 00 01 11 10 00 1 1 01 1 1 11 1 1 10 1 1 x x x x x x 1 2 4 1 2 4 x x x x x x 1 2 3 1 2 3 x 1 2 4 3 +
f =

63. Example: Both Are Valid Solutions1 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 ]

64. Example: Both Are Valid Solutions1 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 x 1 3 4 2
f = + x 1 2 4 3 +
f =

65. Example: Incompletely Specified Function

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

67. 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

68. 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 ]

69. 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 ]

70. Example: A circuit with multiple outputs

71. Seven-Segment Indicator

72. Seven-Segment Indicator

73. Seven-Segment Indicator

74. Seven-Segment Indicator
d d d d d d d
d d d d d d d d d d d d d d

75. Seven-Segment Indicator
d d d d d d d
d d d d d d d d d d d d d d

76. 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

77. 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

78. 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

79. 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.

80. Seven-Segment Indicator
1 d d d d d d
1 d d d d d d d d d d d d

81. 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

82. 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

83. 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

84. 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.

85. Seven-Segment Indicator

86. Another Example

87. [ 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 ]

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

89. _ 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 _

90. _ 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 _

91. _ _ 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 _

92. _ _ 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 _

93. _ 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 _

94. _ 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 _

95. _ 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 _

96. _ 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 _

97. _ 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 _

98. _ 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 _

99. _ 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 _

100. (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 ]

101. Yet Another Example

102. 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 ]

103. 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 ]

104. 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 ]

105. 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 ]

106. Questions?

107. THE END