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

2. Using AND, OR, and NOT Gates
Synthesis
Using AND, OR, and NOT Gates
CprE 281: Digital Logic
Iowa State University, Ames, IA
Copyright © Alexander Stoytchev

3. Administrative Stuff HW2 is due on Monday Sep 3 @ 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

4. Administrative Stuff Next week we will start with Lab2
It will be graded!
Print the answer sheet for that lab and do the prelab at home. Otherwise you’ll lose 10% of your grade for that lab.

5. Quick Review

6. The Three Basic Logic Gatesx 1 2 × x 1 2 + x
NOT gate
AND gate
OR gate
[ Figure 2.8 from the textbook ]

7. Truth Table for NOT x 1 x

8. Truth Table for AND x 1 2 ×

9. Truth Table for OR x 1 2 +

10. Truth Tables for AND and OR
[ Figure 2.6b from the textbook ]

11. Operator Precedence
In regular arithmetic and algebra multiplication takes precedence over addition
This is also true in Boolean algebra

12. Operator Precedence (three different ways to write the same)

13. DeMorgan’s Theorem

14. Function Synthesis

15. A function to be synthesized
[ Figure 2.19 from the textbook ]

16. Let’s look at it row by row. How can we express the last row?

17. Let’s look at it row by row. How can we express the last row?

18. Let’s look at it row by row. How can we express the last row?

19. What about this row? x1 x2

20. What about this row? x1 x2

21. What about this row? x2 x1 x1 x2

22. What about the first row?x2 x1 x1 x2

23. What about the first row?x2 x1 x1 x2

24. What about the first row?x2 x1 x2 x1 x1 x2

25. Finally, what about the zero?x2 x1 x2 x1 x1 x2

26. Putting it all togetherf x1 x2

27. Let’s verify that this circuit implements correctly the target truth tablex1 x2

28. Putting it all togetherf x1 x2

29. Putting it all togetherf x1 x2

30. Canonical Sum-Of-Products (SOP)x 1 2
[ Figure 2.20a from the textbook ]

31. Summary of This Procedure
Get the truth table of the function
Form a product term (AND gate) for each row of the table for which the function is 1
Each product term contains all input variables
In each row, if xi =1 enter it at xi , otherwise use xi
Sum all of these products (OR gate) to get the function

32. Two implementations for the same function
(a) Canonical sum-of-products
(b) Minimal-cost realization x 2 1
[ Figure 2.20 from the textbook ]

33. Simplification Steps

34. Simplification Steps
replicate
this term

35. Simplification Steps
group
these terms

36. Simplification Steps
These two terms are trivially equal to 1

37. Simplification Steps
Drop the 1’s

38. Minimal-cost realizationf x 2 1
[ Figure 2.20b from the textbook ]

39. Let’s look at another problem
[ Figure 2.21 from the textbook ]

40. Let’s look at another problem
[ Figure 2.21b from the textbook ]

41. Let’s look at another problem

42. Let’s look at another problem

43. Let’s look at another problem

44. Let’s look at another problem (minimization)

45. Let’s look at another problem (minimization)f s3 s2 s1

46. Minterms and Maxterms

47. Sum-of-Products Form
f (x1, x2)

48. Sum-of-Products Form
f (x1, x2)

50. Product-of-Sums Form
f (x1, x2)

51. Product-of-Sums Form
f (x1, x2)

52. Product-of-Sums Form
f (x1, x2)

53. Product-of-Sums Form
f (x1, x2)

54. Product-of-Sums Form
f (x1, x2)

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

56. A three-variable function
[ Figure 2.23 from the textbook ]

57. Sum-of-Products Form

58. Sum-of-Products Form

59. Sum-of-Products Form

60. A three-variable function
[ Figure 2.23 from the textbook ]

61. Product-of-Sums Form

62. Product-of-Sums Form

63. Product-of-Sums Form

64. Shorthand Notation
Sum-of-Products or
Product-of-sums

65. Two realizations of that functionx 2 f x 3 x 1
(a) A minimal sum-of-products realization x 1 x 3 f x 2
(b) A minimal product-of-sums realization
[ Figure 2.24 from the textbook ]

66. The Cost of a Circuit Count all gates
Count all inputs/wires to the gates

67. What is the cost of each circuit?x 2 f x 3 x 1
(a) A minimal sum-of-products realization x 1 x 3 f x 2
(b) A minimal product-of-sums realization
[ Figure 2.24 from the textbook ]

68. What is the cost of this circuit?x 1 2

69. Questions?

70. THE END