# CprE 281: Digital Logic

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1 CprE 28: Digital Logic Instructor: Alexander Stoytchev

2 Minimization CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev

3 Administrative Stuff HW4 is out It is due on Monday Sep 4 pm It is posted on the class web page I also sent you an with the link.

4 Example: K-Map for the 2- Multiplexer

5 2- Multiplexer (Definition) Has two inputs: x and Also has another input line s If s=, then the output is equal to x If s=, then the output is equal to

6 Circuit for 2- Multiplexer x s s f x f (b) Circuit (c) Graphical symbol [ Figure 2.33b-c from the textbook ]

7 Truth Table for a 2- Multiplexer [ Figure 2.33a from the textbook ]

8 [ Figure 2.33a from the textbook ] Let s Draw the K-map

9 Let s Draw the K-map

10 Let s Draw the K-map

11 Let s Draw the K-map

12 Let s Draw the K-map

13 Let s Draw the K-map f (s, x, ) = x + s x

14 Let s Draw the K-map f (s, x, ) = x + s x Something is wrong!

15 Compare this with the SOP derivation

16 Let s Derive the SOP form

17 Let s Derive the SOP form

18 Let s Derive the SOP form Where should we put the negation signs? s x s x s x s x

19 Let s Derive the SOP form s x s x s x s x

20 Let s Derive the SOP form s x s x s x s x f (s, x, ) = s x + s x + s x + s x

21 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x

22 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x f (s, x, ) = s x ( + ) + s (x +x )

23 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x f (s, x, ) = s x ( + ) + s (x +x ) f (s, x, ) = s x + s

24 Let s Draw the K-map again [ Figure 2.33a from the textbook ]

25 Let s Draw the K-map again

26 Let s Draw the K-map again

27 Let s Draw the K-map again

28 Let s Draw the K-map again s x The order of the labeling matters.

29 Let s Draw the K-map again s x

30 Let s Draw the K-map again s x

31 Let s Draw the K-map again s x

32 Let s Draw the K-map again s x f (s, x, ) = s x + s This is correct!

33 Two Different Ways to Draw the K-map x x 3 m x x 3 m m 2 m m 2 m 6 m 4 m 3 m 4 m m 3 m 7 m 5 m 5 (b) Karnaugh map m 6 m 7 x 3 x (a) Truth table m m m 3 m 2 m 4 m 5 m 7 m 6

34 Another Way to Draw 3-variable K-map x x 3 m m m 2 m 3 m 4 m 5 m 6 m 7 x x 3 m m m 3 m 2 m 6 m 7 (b) Karnaugh map x x 3 m 4 m 5 (a) Truth table m m 4 m m 5 m 3 m 7 m 2 m 6

35 Gray Code Sequence of binary codes Consecutive lines vary by only bit

36 Gray Code & K-map s x

37 Gray Code & K-map s x

38 Gray Code & K-map s x These two neighbors differ only in the LAST bit

39 Gray Code & K-map s x These two neighbors differ only in the LAST bit

40 Gray Code & K-map s x These two neighbors differ only in the FIRST bit

41 Gray Code & K-map s x These two neighbors differ only in the FIRST bit

43 Gray Code & K-map s x These four neighbors differ in the FIRST and LAST bit They are similar in their MIDDLE bit

44 A four-variable Karnaugh map x x x 3 x 4 m m 4 m 2 m 8 m m 5 m 3 m 9 x 3 m 3 m 2 m 6 m 7 m 5 m 4 m m x 4 [ Figure 2.53 from the textbook ]

45 A four-variable Karnaugh map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 3 x 4 x 3 m m m 5 m 3 m 2 m 6 m 4 m 2 m 3 m 7 m 5 m 4 x m 8 m 9 m m x 4

47 Gray Code & K-map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 3 x 4 x 3 m m m 5 m 3 m 2 m 6 m 4 m 2 m 3 m 7 m 5 m 4 x m 8 m 9 m m x 4

48 Gray Code & K-map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 3 x 4 x 3 m m m 5 m 3 m 2 m 6 m 4 m 2 m 3 m 7 m 5 m 4 x m 8 m 9 m m x 4

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

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

51 Strategy For Minimization

52 Grouping Rules Group s with rectangles Both sides a power of 2: x, x2, 2x, 2x2, x4, 4x, 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 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 2x even if that is enough).

53 Terminology Literal: a variable, complemented or uncomplemented Some Examples: _ X X 2

54 Terminology Implicant: product term that indicates the input combinations for which function output is Example _ x - indicates that x and x yield output of _ x

55 Terminology Prime Implicant Implicant that cannot be combined into another implicant with fewer literals Some Examples x x x 3 x 3 Not prime Prime

56 Terminology Essential Prime Implicant Prime implicant that includes a minterm not covered by any other prime implicant Some Examples x 3 x

57 Terminology Cover Collection of implicants that account for all possible input valuations where output is Ex. Ex. x x 3 + x x 3 + x x 3 x x 3 + x x 3 x x 3

58 Give the Number of Implicants? Prime Implicants? Example Essential Prime Implicants? x x 3

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

60 Three-variable function f (x,, x 3 ) = Σ m(,, 2, 3, 7) x 3 x x x 3 [ Figure 2.56 from the textbook ]

61 Example x x 3 x 4

62 Example x x 3 x 4

63 Example x x 3 x 4 x x 3 x 4 x 3 x 4 x x 3 x 4 x 3 x 4

64 Example: Another Solution x x 3 x 4 x x 3 x 4 x 3 x 4 x x 3 x 4 x 3 x 4 x x 4 x x 4 x x 3 x x 3 [ Figure 2.59 from the textbook ]

65 f ( x,, x 4 ) = Σ m(2, 3, 5, 6, 7,,, 3, 4) x x 3 x 4 x x 4 x 3 x 4 x 3 x 4 x x 3 x 3 [ Figure 2.57 from the textbook ]

66 x f ( x,, x 4 ) = Σ m(, 4, 8,,, 2, 3, 5) x 3 x 4 x 3 x 4 x x 3 x x 4 x x 3 x 4 x x 3 x x 4 [ Figure 2.58 from the textbook ]

67 Minimization of Product-of-Sums Forms

68 Do You Still Remember This Boolean Algebra Theorem?

69 Let s prove 4.b

70 Let s prove 4.b

71 Let s prove 4.b

72 Let s prove 4.b

73 Let s prove 4.b

74 Let s prove 4.b They are equal.

75 Grouping Example x x 2 x M M 2

76 Grouping Example x x 2 x x * = M * M 2 = M * M 2

77 Grouping Example x x 2 x x * = M * M 2 = M * M 2

78 Grouping Example x x 2 x x * = M * M 2 = M * M 2

79 Grouping Example x x 2 x x * = M * M 2 = M * M 2 _ ( x + ) * ( x + ) = Property 4b (Combining)

80 POS minimization of f (x,, x 3 ) = Π M(4, 5, 6) x 3 x ( x + x 3 ) ( x + ) [ Figure 2.6 from the textbook ]

81 POS minimization of f ( x,, x 4 ) = Π M(,, 4, 8, 9, 2, 5) x x 3 x 4 ( x 3 + x 4 ) ( + x 3 ) ( x + + x 3 + x 4 ) [ Figure 2.6 from the textbook ]

82 Questions?

83 THE END

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