Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps

Transcription

1 Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology 16 September 2016 rev Egemen K. Çetinkaya

2 Introduction Karnaugh maps Karnaugh Maps Outline Incomplete Boolean functions Summary 2

3 Equations Boolean Representations Mappings allows simplification e.g. F = x Circuits using logic gates, represents actual physical implementation e.g. Truth tables represents all possible combinations of input values one function has only one truth table representation e.g. x x F F 3

4 Commutativity Distributivity Associativity Identity Complementery Null elements Idempotent law Involution law DeMorgan s law Absorption law Boolean Algebra Theorems 4

5 Boolean Representations Terminology Example F(a,b,c)=ab+c+ac What are the variables? What are the literals? What are the product terms? 5

6 Boolean Representations Terminology Example F(a,b,c)=ab+c+ac Variables: a, b, c Literals: a, b, c, a, c Product terms: ab, c, ac 6

7 Boolean Representations Sum-of-Products Form Equation written as OR of product terms Examples: Following are in sum-of-products form ab + a c a + b + cde Following are not in sum-of-products form (a+b)c (a ) + b variable should be complemented or uncomplemented form 7

8 Boolean Representations Product-of-Sums Form Equation written as AND of sum terms Examples: Following are in product-of-sums form a (a + c) (a + b) a b (c + d + e) Following are not in sum-of-products form (a b)+c (a ) b variable should be complemented or uncomplemented form 8

9 Boolean Representations Normal Term A product or a sum term no variable appears more than once A nonnormal term can be simplified to a constant or a normal term Nonnormal terms example: abbc, a+a+c, aa b Normal terms example: abc, a +b+c 9

10 Boolean Representations Minterm A product term in which literals include every variable only once, either true or complemented form Example: What are the minterms of the function: F(a,b,c)=ab+abc +ac +c 10

11 Boolean Representations Minterm A product term in which literals include every variable only once, either true or complemented form Example: What are the minterms of the function: F(a,b,c)=ab+abc +ac +c Answer: abc An equation is sum-of-minterms form if: every product term is a minterm Example: F(a,b,c)=abc+ab c+a b c 11

12 Boolean Representations Maxterm A sum term in which literals include every variable only once, either true or complemented form Example: What are the maxterms of the function: F(a,b,c)=(a+b) (a+b+c ) (a+c ) c 12

13 Boolean Representations Maxterm A sum term in which literals include every variable only once, either true or complemented form Example: What are the maxterms of the function: F(a,b,c)=(a+b) (a+b+c ) (a+c ) c Answer: a+b+c An equation is product-of-maxterms form if: every sum term is a maxterm Example: F(a,b,c)=(a+b+c) (a+b +c) (a +b +c ) 13

14 Boolean Representations Minterms and Maxterms Minterm is a product term that is 1 Maxterm is a sum term that is 0 row x y z F minterm maxterm F(0,0,0)?? F(0,0,1)?? F(0,1,0)?? F(0,1,1)?? F(1,0,0)?? F(1,0,1)?? F(1,1,0)?? F(1,1,1)?? 14

15 Boolean Representations Minterms and Maxterms Minterm is a product term that is 1 Maxterm is a sum term that is 0 row x y z F minterm maxterm F(0,0,0) x y z x+y+z F(0,0,1) x y z x+y+z F(0,1,0) x yz x+y +z F(0,1,1) x yz x+y +z F(1,0,0) xy z x +y+z F(1,0,1) xy z x +y+z F(1,1,0) xyz x +y +z F(1,1,1) xyz x +y +z 15

16 Boolean Representations Minterms and Maxterms Example Minterm is a product term that is 1 Maxterm is a sum term that is 0 What is the canonical sum? What is the canonical product? What is the minterm list? What is the maxterm list? row x y F

17 Boolean Representations Minterms and Maxterms Example Minterm is a product term that is 1 Maxterm is a sum term that is 0 Canonical sum: F=x y +x y Canonical product: F=(x +y) (x +y ) Minterm list: x,y (0,1) row x y F note that book shows m Maxterm list: x,y (2,3) 17

18 Karnaugh Maps Overview Graphical method to simplify Boolean equations Also called K-maps Originally published by Edward W. Veitch in 1952 Further improved by Maurice Karnaugh in 1953 by extending Veitch s work to 3-dimensional arrays Generally more common for 3-4 variable functions Not practical for functions with many variables 18

19 Gray Code Overview One bit changes between two successive integers Shown below up to decimal 7 Decimal Binary Gray Gray as Decimal

20 Karnaugh Maps Overview K-maps: diagrammatic representation of truth table i.e. Boolean function One-variable k-map Two-variable k-map Three-variable k-map Four-variable k-map Five-variable k-map Six-variable k-map 20

21 Karnaugh Maps One-Variable K-map One-variable k-map has 2 1 cells x f (x) 0 f (0) x f (1) f (0) f (1) 21

22 Karnaugh Maps Two-Variable K-map Two-variable k-map has 2 2 cells Neighborhood is based on 1 difference remember the gray code x y f (x, y) 0 0 f (0,0) 0 1 f (0,1) 1 0 f (1,0) 1 1 f (1,1) x 0 1 y 0 1 f (0,0) f (0,1) f (1,0) f (1,1) 22

23 Karnaugh Maps Two-Variable K-map Two-variable k-map has 2 2 cells Neighborhood is based on 1 difference remember the gray code Note that x and y can substitute for 2-variable case x y f (x, y) 0 0 f (0,0) 0 1 f (0,1) 1 0 f (1,0) 1 1 f (1,1) y 0 1 x 0 1 f (0,0) f (1,0) f (0,1) f (1,1) 23

24 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy 24

25 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y 25

26 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y x y f (x, y)

27 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y x y f (x, y) y 0 1 x

28 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y x y f (x, y) y 0 1 x Grouping of 1s: 2 0, 2 1,

29 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y x y f (x, y) y 0 1 x If we group 1s, what would be the minimized f? 29

30 Two-Variable K-map Example 1 Lets try to simplify f = x y + xy f = x y + xy = y (x + x) = y x y f (x, y) y 0 1 x If we group 1s, the minimized f would be y 30

31 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy 31

32 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y)

33 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y) x 0 1 y

34 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y) x 0 1 y Note that a 1 can belong more than one group What is the minimized function f? 34

35 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y) x 0 1 y Note that a 1 can belong more than one group The minimized function f = x + y 35

36 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y) x 0 1 y Can we write the function as: f = x + y x? 36

37 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy x y f (x, y) x 0 1 y Can we write the function as: f = x + y x? Logically, both is same, but it is not minimized 37

38 Two-Variable K-map Example 2 Compare the circuits: x + y vs. x + y x 38

39 Two-Variable K-map Example 2 Compare the circuits: x + y vs. x + y x x + y : has 1 OR gate (excluding inverters) x + y x : has 1 OR and 1 AND gate (excluding inverters) 39

40 Two-Variable K-map Example 2 Lets try to simplify f = x y + x y + xy Use Boolean algebra to simplify above equation 40

41 Two-Variable K-map Example 3 Lets try to simplify f = x y + xy 41

42 Two-Variable K-map Example 3 Lets try to simplify f = x y + xy x y f (x, y)

43 Two-Variable K-map Example 3 Lets try to simplify f = x y + xy x y f (x, y) x 0 1 y

44 Two-Variable K-map Example 3 Lets try to simplify f = x y + xy x y f (x, y) x 0 1 y

45 Two-Variable K-map Example 3 Lets try to simplify f = x y + xy x y f (x, y) x 0 1 y It is not possible to group 1s Then, leave as is 45

46 Karnaugh Maps Three-Variable K-map Three-variable k-map has 2 3 cells Neighborhood is based on 1 variable difference x y z f (x, y, z) f (0,0,0) f (0,0,1) f (0,1,0) f (0,1,1) f (1,0,0) f (1,0,1) f (1,1,0) f (1,1,1) x f (0,0,0) f (0,0,1) f (0,1,1) f (0,1,0) f (1,0,0) f (1,0,1) f (1,1,1) f (1,1,0) 46 yz 11 10

47 Three-Variable K-map Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f (x, y, z)

48 Three-Variable K-map Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f (x, y, z) yz x

49 Three-Variable K-map Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f (x, y, z) yz x

50 Three-Variable K-map Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f (x, y, z) yz x

51 Three-Variable K-map Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f (x, y, z) F = x z + xy x yz 11 10

52 Three-Variable K-map Example 2 What s the minimized f(x,y,z) = (0,1,6,7)? Egemen K. Çetinkaya 52

53 Three-Variable K-map Example 2 What s the minimized f(x,y,z) = (0,1,6,7)? Egemen K. Çetinkaya yz x

54 Three-Variable K-map Example 2 What s the minimized f(x,y,z) = (0,1,6,7)? Egemen K. Çetinkaya yz x

55 Three-Variable K-map Example 2 What s the minimized f(x,y,z) = (0,1,6,7)? Egemen K. Çetinkaya yz x

56 Three-Variable K-map Example 2 What s the minimized f(x,y,z) = (0,1,6,7)? Egemen K. Çetinkaya yz x F = x y + xy 56

57 Karnaugh Maps Simplification Rules Group together adjacent cells containing ones 1s grouping is done for a group of: 2 0, 2 1, 2 2 1s that means no 3, 5, 7 groupings Every one must be in at least one group Each group should be as large as possible Fewest number of groups possible Overlapping of 1s is allowed Zeros are not allowed unless you are doing sum terms Diagonal groupings are not allowed 57

58 Karnaugh Maps Four-Variable K-map Four-variable k-map has 2 4 cells Egemen K. Çetinkaya w x y z f (x, y, z) f (0,0,0,0) f (0,0,0,1) f (0,0,1,0) f (0,0,1,1) f (0,1,0,0) f (0,1,0,1) f (0,1,1,0) f (0,1,1,1) f (0,0,0,0) f (1,0,0,1) f (1,0,1,0) f (1,0,1,1) f (1,1,0,0) f (1,1,0,1) f (1,1,1,0) f (1,1,1,1) 58

59 Karnaugh Maps Four-Variable K-map Four-variable k-map has 2 4 cells w x y z f (x, y, z) f (0,0,0,0) f (0,0,0,1) f (0,0,1,0) yz f (0,0,1,1) f (0,1,0,0) f (0,1,0,1) wx f (0,0,0,0) f (0,0,0,1) f (0,0,1,1) f (0,0,1,0) f (0,1,0,0) f (0,1,0,1) f (0,1,1,1) f (0,1,1,0) f (1,1,0,0) f (1,1,0,1) f (1,1,1,1) f (1,1,1,0) f (1,0,0,0) f (1,0,0,1) f (1,0,1,1) f (1,0,1,0) f (0,1,1,0) f (0,1,1,1) f (0,0,0,0) f (1,0,0,1) f (1,0,1,0) f (1,0,1,1) f (1,1,0,0) f (1,1,0,1) f (1,1,1,0) f (1,1,1,1) 59

60 Karnaugh Maps Four-Variable K-map The grid is toroidally connected [ref: 60

61 Four-Variable K-map Typical One-Variable Elimination 1 Simplified functions? yz yz wx wx

62 Four-Variable K-map Typical One-Variable Elimination 1 Simplified functions? yz yz wx wx F = wxz F = xy z 62

63 Four-Variable K-map Typical One-Variable Elimination 2 Simplified functions? yz yz wx wx

64 Four-Variable K-map Typical One-Variable Elimination 2 Simplified functions? yz yz wx wx F = w xz F = x yz 64

65 Four-Variable K-map Typical Two-Variable Elimination 1 Simplified functions? yz yz wx wx

66 Four-Variable K-map Typical Two-Variable Elimination 1 Simplified functions? yz yz wx wx F = wy F = y z 66

67 Four-Variable K-map Typical Two-Variable Elimination 2 Simplified functions? yz yz wx wx

68 Four-Variable K-map Typical Two-Variable Elimination 2 Simplified functions? yz yz wx wx F = xz F = x z 68

69 Four-Variable K-map Typical Three-Variable Elimination 1 Simplified functions? yz yz wx wx

70 Four-Variable K-map Typical Three-Variable Elimination 1 Simplified functions? yz yz wx wx F = w F = y 70

71 Four-Variable K-map Typical Three-Variable Elimination 2 Simplified functions? yz yz wx wx

72 Four-Variable K-map Typical Three-Variable Elimination 2 Simplified functions? yz yz wx wx F = x F = z 72

73 Karnaugh Maps K-maps for Sum Terms Can we simplify Boolean functions using 0s? 73

74 Simplified functions? Karnaugh Maps K-maps for Sum Terms yz yz wx wx

75 Simplified functions? Karnaugh Maps K-maps for Sum Terms yz yz wx wx F = x F = x + z 75

76 Karnaugh Maps Cell Designations 76

77 Karnaugh Maps K-maps Example 1 Simplified functions for F(w,x,y,z)= (0,2,4,6,12,14)? 77

78 Karnaugh Maps K-maps Example 1 Simplified functions for F(w,x,y,z)= (0,2,4,6,12,14)? yz yz ???? wx wx 01 11???????? ???? 78

79 Karnaugh Maps K-maps Example 1 Simplified functions for F(w,x,y,z)= (0,2,4,6,12,14)? yz yz wx wx

80 Karnaugh Maps K-maps Example 1 Simplified functions for F(w,x,y,z)= (0,2,4,6,12,14)? yz yz wx wx F = w z + xz 80

81 Karnaugh Maps Constructing n-bit Gray Code Construction can be done via reflect and prefix Egemen K. Çetinkaya [ref: 81

82 Karnaugh Maps Five-Variable K-map Five-variable k-map has 2 5 cells 82

83 Karnaugh Maps Six-Variable K-map Six-variable k-map has 2 6 cells 83

84 Karnaugh Maps Many Variable K-maps can be useful in minimizing few variables Many variable functions can be challenging 84

85 Karnaugh Maps Don t Care Conditions Three-variable incomplete Boolean function example x y z f x y z fˈ

86 Karnaugh Maps Don t Care Conditions Three-variable incomplete Boolean function example x y z f don t care conditions x y z fˈ

87 Karnaugh Maps Don t Care Conditions Describing incomplete Boolean function example x y z f minterm canonical formula f (x,y,z) = (0,4) + dc(2,5) maxterm canonical formula f (x,y,z) = (1,3,6,7) + dc(2,5) 87

88 Karnaugh Maps Don t Care Conditions Some input states never occur output states are irrelevant Input states may occur but output don t need to be specified In k-map, dc can be used for minterms or maxterms To simplify with dc conditions: assign a dc condition a 0 or 1, then simplify In a k-map, dc conditions are shown as: -, d, x 88

89 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f

90 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f yz x

91 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f yz x

92 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f you can assign a dc a 0 or a x yz 11 10

93 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f you can assign a dc a 0 or a x yz 11 10

94 Don t Care Conditions Example 1 What s the minimized function? Egemen K. Çetinkaya x y z f F = y x yz 11 10

95 Minimization Strategies Other Methods Other than K-maps, methods exist for minimizing CAD-based tools (Computer Automated Design) heuristic: good solution, close to optimal but not necessarily optimal exact algorithm: yields optimal solution Cubical representations Tabular method (Quine-McCluskey) 95

96 Karnaugh Maps Example 96

97 Karnaugh Maps Example 97

98 Karnaugh Maps Example 98

99 Karnaugh Maps Summary Karnaugh maps used to simplify truth tables Group together adjacent cells containing ones 1s grouping is done for a group of: 2 0, 2 1, 2 2 1s Every one must be in at least one group Each group should be as large as possible Fewest number of groups possible Overlapping of 1s is allowed Zeros are not allowed Diagonal groupings are not allowed For DC conditions; assign a 0 or a 1 and then simplify 99

100 References and Further Reading [V2011] Frank Vahid, Digital Design with RTL Design, VHDL, and Verilog, 2nd edition, Wiley, [BV2009] Stephen Brown and Zvonko Vranesic, Fundamentals of Digital Logic with VHDL Design, 3rd edition, McGraw-Hill, [MKM2016] M. Morris Mano, Charles R. Kime, Tom Martin, Logic and Computer Design Fundamentals, 5th edition, Pearson, [W2006] John F. Wakerly, Digital Design Principles and Practices, 4th edition, Prentice Hall, [G2003] Donald D. Givone, Digital Principles and Design, McGraw-Hill,

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