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

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

3. Administrative Stuff HW3 is out It is due on Monday Sep 11 @ 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, please
Staple your pages

4. Administrative Stuff TA Office Hours:
5:00 pm - 6:00 pm on Tuesdays (Vahid Sanei-Mehri)
Location: 3125 Coover Hall
11:10 am-1:10 pm on Wednesdays (Siyuan Lu)
Location: TLA (Coover Hall - first floor)
5:00 pm - 6:00 pm on Thursdays (Vahid Sanei-Mehri)
10:00 am-12:00 pm on Fridays (Krishna Teja)
Location: 3214 Coover Hall

5. Administrative Stuff
Homework Solutions will be posted on BlackBoard

6. Quick Review

7. The Three Basic Logic Gatesx 1 2 • x 1 2 + x
NOT gate
AND gate
OR gate
You can build any circuit using only these three gates
[ Figure 2.8 from the textbook ]

8. Figure B.21. A 7400-series chip. (a) Dual-inline package V GndDD
Gnd
(b) Structure of 7404 chip
Figure B A 7400-series chip.

9. Figure B.22. An implementation of f = x1x2 + x2x3.V DD x 1 2 3 f
7404
7408
7432

10. NAND Gate x1 x2 f 1

11. NOR Gate x1 x2 f 1

12. Why do we need two more gates?
They can be implemented with fewer transistors.
(more about this later)

13. Building a NOT Gate with NANDx x x f 1 x 1
impossible
combinations
Thus, the two truth tables are equal!

14. Building an AND gate with NAND gates[

15. Building an OR gate with NAND gates[

16. Any Boolean function can be implemented
Implications
Any Boolean function can be implemented
with only NAND gates!

17. Implications Any Boolean function can be implemented
with only NAND gates!
The same is also true for NOR gates!

18. NAND-NAND Implementation of Sum-of-Products Expressions

19. Sum-Of-Products This circuit uses ANDs & ORx x 1 1 x x 2 2 x x 3 3 x x 4 4 x x 5 5
This circuit uses ANDs & OR x 1 x 2 x 3 x 4 x 5
This circuit uses only NANDs
[ Figure 2.27 from the textbook ]

20. NAND followed by NOT = ANDx 1 2 • x 1 2 • x 1 2 • x 1 x 2 x1 x2 f 1 f 1 x1 x2 f 1

21. DeMorgan’s Theorem

22. DeMorgan’s Theorem x x x y • =
x + y y y

23. Sum-Of-Products x1 x2 x3 x4

24. Sum-Of-Products x1 x2 x3 x4 x 1 2 • 3 4 +

25. Sum-Of-Products
AND x1 x2 x3 x4 OR
AND x 1 2 • 3 4 +

26. Sum-Of-Products
AND x1 x2 x3 x4 OR
AND
AND x 1 2 • 3 4 + OR
AND

27. Sum-Of-Products
AND x1 x2 x3 x4 OR
AND x 1 2 • 3 4 +

28. Sum-Of-Products
AND x1 x2 x3 x4 OR
AND
NAND x 1 2 • 3 4 +

29. Sum-Of-Products AND OR AND x1 x2 x3 x4 + • • • • x x x x x x x x 1 2 1

30. Sum-Of-Products This circuit uses only NANDs AND OR AND NAND NAND NANDx1 x2 x3 x4 OR
AND
NAND x 1 2 • x
NAND 1 x 2 x 1 2 • + x 3 4 •
NAND x 3 4 • x 3 x 4
This circuit uses only NANDs

31. Sum-Of-Products This circuit uses only NANDs x1 x2 x3 x4 + • • • • x x

32. NOR-NOR Implementation of Product-of-Sums Expressions

33. Product-Of-Sums This circuit uses ORs & ANDx 1 2 3 4 5
This circuit uses ORs & AND
This circuit uses only NORs
[ Figure 2.28 from the textbook ]

34. NOR followed by NOT = OR x 1 2 + x 1 2 + x1 x2 f 1 f 1 x1 x2 f 1

35. DeMorgan’s Theorem

36. DeMorgan’s Theorem x y
x • y x y + =

37. Product-Of-Sums x1 x2 x3 x4

38. Product-Of-Sums + + + + x1 x2 x3 x4 x x x x (x1 + x2) • (x3 + x4) x x

39. Product-Of-Sums AND OR OR + + + + x1 x2 x3 x4 x x x x

40. Product-Of-Sums AND AND OR OR OR OR + + + + x1 x2 x3 x4 x x x x

41. Product-Of-Sums AND OR OR + + + + x1 x2 x3 x4 x x x x

42. Product-Of-Sums AND OR OR NOR + + + + x1 x2 x3 x4 x x x x

43. Product-Of-Sums AND OR OR + + x1 x2 x3 x4 x x x (x1 + x2) • (x3 + x4)

44. Product-Of-Sums AND This circuit uses only NORs OR OR NOR NOR NOR + +x1 x2 x3 x4
AND OR
NOR x 1 2 + x 1
NOR x 2
(x1 + x2) • (x3 + x4)
NOR x 3 4 + x 3 x 4
This circuit uses only NORs

45. Product-Of-Sums AND This circuit uses only NORs + + x1 x2 x3 x4 x x x

46. Another Synthesis Example

47. Truth table for a three-way light control
[ Figure 2.31 from the textbook ]

48. Minterms and Maxterms (with three variables)
[ Figure 2.22 from the textbook ]

49. Let’s Derive the SOP form

50. Let’s Derive the SOP form

51. Sum-of-products realizationx1 x3 x2
[ Figure 2.32a from the textbook ]

52. Let’s Derive the POS form
[ Figure 2.31 from the textbook ]

53. Let’s Derive the POS form

54. Product-of-sums realizationx3 x1 x2
[ Figure 2.32b from the textbook ]

55. Multiplexers

56. 2-1 Multiplexer (Definition)
Has two inputs: x1 and x2
Also has another input line s
If s=0, then the output is equal to x1
If s=1, then the output is equal to x2

57. Graphical Symbol for a 2-1 Multiplexer
[ Figure 2.33c from the textbook ]

58. Analogy: Railroad Switch

59. Analogy: Railroad Switchx2 x1
select f

60. Analogy: Railroad Switchx2 x1
select f
This is not a perfect analogy because the trains can go in either direction,
while the multiplexer would only allow them to go from top to bottom.

61. Truth Table for a 2-1 Multiplexer
[ Figure 2.33a from the textbook ]

62. Let’s Derive the SOP form

63. Let’s Derive the SOP form

64. Let’s Derive the SOP form
Where should we
put the negation signs?
s x1 x2
s x1 x2
s x1 x2
s x1 x2

65. Let’s Derive the SOP form
s x1 x2
s x1 x2
s x1 x2
s x1 x2

66. Let’s Derive the SOP form
s x1 x2
s x1 x2
s x1 x2
s x1 x2
f (s, x1, x2) =
s x1 x2 +

67. Let’s simplify this expression
f (s, x1, x2) =
s x1 x2 +

68. Let’s simplify this expression
f (s, x1, x2) =
s x1 x2 +
f (s, x1, x2) =
s x1 (x2 + x2)
s (x1 +x1 )x2 +

69. Let’s simplify this expression
f (s, x1, x2) =
s x1 x2 +
f (s, x1, x2) =
s x1 (x2 + x2)
s (x1 +x1 )x2 +
f (s, x1, x2) =
s x1
s x2 +

70. Circuit for 2-1 Multiplexers
(c) Graphical symbol
(b) Circuit
f (s, x1, x2) =
s x1
s x2 +
[ Figure 2.33b-c from the textbook ]

71. More Compact Truth-Table Representation1
(a)
Truth
table s x1 x2
f (s, x1, x2) 1
f (s, x1, x2) s x1 x2
[ Figure 2.33 from the textbook ]

72. 4-1 Multiplexer (Definition)
Has four inputs: w0 , w1, w2, w3
Also has two select lines: s1 and s0
If s1=0 and s0=0, then the output f is equal to w0
If s1=0 and s0=1, then the output f is equal to w1
If s1=1 and s0=0, then the output f is equal to w2
If s1=1 and s0=1, then the output f is equal to w3

73. 4-1 Multiplexer (Definition)
Has four inputs: w0 , w1, w2, w3
Also has two select lines: s1 and s0
If s1=0 and s0=0, then the output f is equal to w0
If s1=0 and s0=1, then the output f is equal to w1
If s1=1 and s0=0, then the output f is equal to w2
If s1=1 and s0=1, then the output f is equal to w3
We’ll talk more about this when we get
to chapter 4, but here is a quick preview.

74. Graphical Symbol and Truth Table
[ Figure 4.2a-b from the textbook ]

75. The long-form truth table

76. The long-form truth table[

77. The long-form truth table[

78. The long-form truth table[

79. The long-form truth table[

80. 4-1 Multiplexer (SOP circuit)
[ Figure 4.2c from the textbook ]

81. Using three 2-to-1 multiplexers to build one 4-to-1 multiplexerw w 1 1 f 1 w 2 w 3 1
[ Figure 4.3 from the textbook ]

82. Analogy: Railroad Switches

83. Analogy: Railroad Switchesf

84. Analogy: Railroad Switches
these two
switches are
controlled
together s1 f

85. Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

86. Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

87. Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

88. Using three 2-to-1 multiplexers to build one 4-to-1 multiplexerw0 w1 s1 f s0 w2 w3

89. That is different from the SOP form of the 4-1 multiplexer shown below, which uses fewer gates

90. 16-1 Multiplexer s f w [ Figure 4.4 from the textbook ] 1 3 4 2 7 8 113 4 7 12 15 2 f
[ Figure 4.4 from the textbook ]

91. [http://upload. wikimedia. org/wikipedia/commons/2/26/SunsetTracksCrop[

92. Questions?

93. THE END