Download presentation

Presentation is loading. Please wait.

Published byKamron Goard Modified about 1 year ago

1
Logical Systems Synthesis

2
Logical Synthesis

3
Basic Gate Synthesis f = a. b f = a + b f = a’

4
Basic Gate Synthesis f = (a. b)’ f = (a + b)’ f = a

5
F = A + B. C Correct Incorrect Operator Precedence in Synthesis

6
F = (A + B). C Incorrect Correct Operator Precedence in Synthesis

7
An example of Step by Step Synthesis Single Output Circuits

8
F = A + (B’ + C)(D’ + BE’) Reducing it to a more abstract form P = BE’ M = B’ + C F = A + M(D’ + P) N = D’ + P F = A + MN Z = MN F = A + Z

9

10
F = A + MN

11
F = A + Z F = A + MN F = A + M(D’ + P)

12
F = A + (B’ + C)(D’ + BE’)

13

14
An example of Multi-Output Synthesis

15
f = ab + a’c g = ab + a’c’ These are the common terms in both logic expressions, therefore, they can be used to as a common gate when synthesizing the complete digital system. Sharing Gates

16
Idea of sharing of Gates Circuits for f and g have a common product term (Implicant) namely ab. We are going to make it a common factor i.e. we are going to use the corresponding gate to drive both outputs f & g. f = a’c + ab g = a’c’ + ab

17
Symbols

18
From Digital Design, 5th Edition by M. Morris Mano and Michael Ciletti

19
Boolean Algebra Manipulation of Switching Functions Boolean/Logic/SwitchingExpression/Function

20
Boolean Functions Variables and constants take on only one of the two values : 0 or 1 There are three operators – ORwritten as a + b – ANDwritten as a. b – NOTwritten as a ’

21
() Operator Precedence ’. + Precedence Sequence

22
Examples s = (((x. y’) + (x’. y))’. z) + (((x. y’) + (x’. y)). z’) c = (x. y’ + x’. y). z + x. y f = ( a.b + a.c + b.c’)’. (a + b + c) + a.b.c f = (((a.b) + (a.c) + (b.c’))’. (a + b + c)) + (a.b.c) s = ((x. y’ + x’. y)’. z) + ((x. y’ + x’. y). z’) c = (((x. y’) + (x’. y)). z) + (x. y)

23
Metatheorem Duality Any Theorem or Identity in Switching Algebra remains true if 0 & 1 are swapped and. & + are swapped throughout the expression A Theorem about Theorems Note: Duality does not imply logical equivalence. It only states that one fact leads to another fact.

24
Axioms / Postulates 0. 0 = = = 1. 0 = 0 x = 0 implies x’ = = = = = 1 x = 1 implies x’ = 0

25
Theorems (Single Variable) x. 0 = 0 x. 1 = x x. x = x x. x’ = 0 (x’)’ = x x + 1 = 1 x + 0 = x x + x = x x + x’ = 1 (x’)’ = x Null Elements Identities Idempotency Complements Involution

26
Theorems (2- & 3- Variable) x. y = y. x x. (y. z) = (x. y). z x. (y + z) = x. y + x. z x + (x. y) = x (x. y) + (x. y’) = x x + y = y + x x + (y + z) = (x + y) + z x + (y. z) = (x + y). (x + z) x. (x + y) = x (x + y). (x + y’) = x Commutativity Associativity Distributivity Covering Combining

27
Theorems (2- & 3- Variable) (x. y)’ = x’ + y’ (x. y) + (x’. z) + (y. z) = (x. y) + (x’. z) (x + y)’ = x’. y’ (x + y). (x’ + z). (y + z) = (x + y). (x’ + z) DeMorgan’s Theorem Consensus Theorem

28
Theorems (n-Variable) x. x x = x (x 1. x x n )’ = x 1 ’ + x 2 ’ x n ’ x + x x = x (x 1 + x x n )’ = x 1 ’. x 2 ’ x n ’ Generalized Idempotency DeMorgan’s Theorem

29
Theorems (n-Variable) [F(x 1, x 2, -----, x n,., + )]’ = F(x 1 ’, x 2 ’, -----, x n ’, +,. ) Generalized DeMorgan’s Theorem

30
f = ( a. b )’ = a’ + b’ f = ( a + b )’ = a’. b’ Implications of DeMorgan’s Theorem

31
f = { [a. ( b’ + c ) ]’ }’ = { a’ + ( b’ + c )’ }’ = { a’ + b. c’ }’ f’ = a’ + ( b. c’ ) Implications of DeMorgan’s Theorem

32
Function Manipulation Using Axioms and Theorems of Boolean Algebra

33
Proof of Consensus Theorem x.y + x’.z + y.z = x.y + x’.z x.y + x’.z + y.z = x.y + x’.z + y.z.1 = x.y + x’.z + y.z.(x + x’) = x.y + x’.z + x.y.z + x’.y.z = x.y + x.y.z + x’.z + x’.y.z = x.y.1 + x.y.z + x’.z.1 + x’.y.z = x.y.(1 + z) + x’.z.(1 + y) = x.y.1 + x’.z.1 = x.y + x’.z Theorem Used a = a.1 1 = a + a’ a.(b + c) = a.b + a.c a = a.1 a.b + a.c = a.(b + c) 1 + a = 1 a.1 = a

34
Find DeMorgan’s Equivalent of the Following

35
F = ( ( B. E’ ) + D’ ). ( B’ + C ) + A = { [ ( ( ( B. E’ ) + D’ ). ( B’ + C ) ) + A ]’ }’ = { ( ( ( B. E’ ) + D’ ). ( B’ + C) )’. A’ }’ = { ( ( B. E’ ) + D’ )’ + ( B’ + C )’. A’ }’ = { ( B. E’ )’. D ) + ( B. C’ ). A’ }’ = { ( B’ + E ). D ) + ( B. C’ ). A’ }’ = { ( ( B’ + E ). D ) ) + ( ( B. C’ ). A’ ) }’ F = { ( B’ + E ). D ) + ( B. C’ ). A’ }’ F’ = ( B’ + E ). D ) + ( B. C’ ). A’ DeMorgan’s Equivalent

36
F = ( ( B. E’ ) + D’ ). ( B’ + C ) + A DeMorgan’s Equivalent

37
F’ = ( B’ + E ). D ) + ( B. C’ ). A’ DeMorgan’s Equivalent

38
A circuit developed using NAND gates only can be converted into a circuit employing only NOR gates by directly replacing the NAND gates with the NOR gates and inverting all the inputs and outputs of the circuit NAND NOR Inter-Conversion The Converse of this operation is also true Using DeMorgan’s Theorem

39
F = ((((B.E’)’. D)’. (B.C’)’)’. A’)’

40
F’ = ((((B’ + E)’ + D’)’ + (B’ + C)’)’ + A)’

41
Proof by Example F = ((((B.E’)’. D)’. (B.C’)’)’. A’)’ = ((((B’ + E). D)’. (B’ + C))’. A’)’ = ((((B’ + E)’ + D’). (B’ + C))’. A’)’ = ((((B’ + E)’ + D’)’ + (B’ + C)’). A’)’ = ((((B’ + E)’ + D’)’ + (B’ + C)’)’ + A) F’ = ((((B’ + E)’ + D’)’ + (B’ + C)’)’ + A)’ Using DeMorgan’s Theorem

42
F = A + (B’ + C). (D’ + B.E’) F = A + (B’ + C)(D’ + BE’) Can also be written in the following notation

43
Forms of Logic Expressions

44
Generalized Forms of Logic Expressions SOP Sum of Products POS Product of Sums

45
Terminology Literal Product Term Sum Term Product of Sums Sum of Products

46
Definitions Literal – A variable in either it’s complemented or uncomplemented form Sum Term – One or more Literals connected by an OR operator Product Term – One or more Literals connected by an AND operator

47
Definitions Product of Sums – One or more Sum Terms connected by an AND operator Sum of Products – One or more Product Terms connected by an OR operator

48
SOP Expression

49

50
POS Expression

51

52
More SOP Expressions ABC + A’BC’ AB + A’BC’ + C’D’ + D A’B + CD’ + EF + GK + HL’ More POS Expressions (A + B’)(A’ + C’ + D)F’ (A + C)(B + D’)(B’+C)(A + D’ + E) (A + B’ + C)(A + C)

53
nd = ((((b.e’)’. d)’. (b.c’)’)’. a’)’ nr = ((((b’ + e)’ + d’)’ + (b’ + c)’)’ + a)’ s = (((x. y’) + (x’. y))’. z) + (((x. y’) + (x’. y)). z’) f = (((a.b) + (a.c) + (b.c’))’. (a + b + c)) + (a.b.c) c = (((x. y’) + (x’. y)). z) + (x. y) Not an SOP or POS Expression

54
Canonical Forms of Logic Expressions C-SOP Sum of Standard Product Terms C-POS Product of Standard Sum Terms aka Canonical Sum aka Canonical Product

55
More Terminology Minterm Maxterm Canonical Product Canonical Sum

56
Definitions Mintermaka Standard Product Term – A Product Term composed of all the variables of the given problem Maxtermaka Standard Sum Term – A Sum Term composed of all the variables of the given problem

57
Definitions Canonical Sum – One or more Minterms connected by an OR operator Canonical Product – One or more Maxterms connected by an AND operator

58
C-SOP Expression

59

60
C-POS Expression

61

62
Writing Algebraic Expressions From Truth Tables

63
OR = Decomposing into Minterms Implementing 1’s

64
a.b’ + a’.b = f OR = = m2 + m1 = f

65
Minterms Notation

66

67
Expressions involving Minterms

68

69

70

71

72

73
AND = Decomposing into Maxterms Implementing 0’s

74
AND = ( a + b’ ) ( a’ + b ) = f AND = M1. M2 = f

75
Maxterms Notation

76

77
Expressions involving Maxterms

78

79

80

81

82

83
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = ? Using Maxterms f = ?

84
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = m2 + m5 + m6 Using Maxterms f = M0. M1. M3. M4. M7

85
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = m2 + m5 + m6 f’ = ? Using Maxterms f = M0. M1. M3. M4. M7 f’ = ?

86
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = m2 + m5 + m6 f’ = m0 + m1 + m3 + m4 + m7 Using Maxterms f = M0. M1. M3. M4. M7 f’ = M2. M5. M6

87
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = m2 + m5 + m6 f’ = m0 + m1 + m3 + m4 + m7 Using Maxterms f = M0. M1. M3. M4. M7 f’ = M2. M5. M6 f = ∑ (2, 5, 6) = ∏ (0, 1, 3, 4, 7) f’ = ∏ (2, 5, 6) = ∑ (0, 1, 3, 4, 7) Alternative Notation

88
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = ? f’ = ? Using Maxterms f = ? f’ = ?

89
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table Using Minterms f = m1 + m5 + m6 + m8 + m9 + m10 + m14 + m15 f’ = m0 + m2 + m3 + m4 + m7 + m11 + m12 + m13 Using Maxterms f = M0. M2. M3. M4. M7. M11. M12. M13 f’ = M1. M5. M6. M8. M9. M10. M14. M15

90
Write Expressions using Minterm/Maxterm Terminology for the following Truth Table f = ∑ (1, 5, 6, 8, 9, 10, 14, 15) f = ∏ (0, 2, 3, 4, 7, 11, 12, 13) f’ = ∑ (0, 2, 3, 4, 7, 11, 12, 13) f’ = ∏ (1, 5, 6, 8, 9, 10, 14, 15)

91
Can you intuitively establish a relationship within the following set of results? f = m2 + m5 + m6 f’ = m0 + m1 + m3 + m4 + m7 f = M0. M1. M3. M4. M7 f’ = M2. M5. M6 f = M0. M2. M3. M4. M7. M11. M12. M13 f’ = M1. M5. M6. M8. M9. M10. M14. M15 f = m1 + m5 + m6 + m8 + m9 + m10 + m14 + m15 f’ = m0 + m2 + m3 + m4 + m7 + m11 + m12 + m13

92
Relationship between Minterms and Maxterms f = m0 + m3 f = x’ y’ + x y = ( x’ x + y’ x ) ( x’ y + y’ y ) = ( x y’ ) ( x’ y ) f’ = [ (x y’) (x’ y) ]’ = x’ y + x y’ = m1 + m2 f’ = m1 + m2 f = m0 + m3 f = x’ y’ + x y f’ = ( x’ y’ + x y )’ = ( x’. y’ )’ ( x. y )’ = ( x + y ) ( x’ + y’ ) f’ = ( x + y ) ( x’ + y’ ) = M0. M3 f’ = M0. M3 f = m0 + m3 f’ = m1 + m2 = x’ y + x y’ f = ( x’ y + x y’ )’ = ( x’ y )’ ( x y’ )’ = ( x + y’ ) ( x’ + y) = M1. M2 f = M1. M2

93
To implement a function we can use the logic 1’s in the output to obtain a Minterms’ SOP or use the logic 0’s to obtain a Maxterms’ POS Where, T = {Set of all indices} = {0 to 2 n - 1} R = {Set of indices of required to implement C-SOP of F(x1, x2, …., xn)} S = T – R F(x1, x2, …., xn) = ∑m R = ∏M S [F(x1, x2, …., xn)]’ = ∑m S = ∏M R Note : Observe the Duality inherent in this Generalization

94
Use Boolean Algebra to establish the relationship between the following set of Expressions f = m2 + m5 + m6 f’ = m0 + m1 + m3 + m4 + m7 f = M0. M1. M3. M4. M7 f’ = M2. M5. M6 f = M0. M2. M3. M4. M7. M11. M12. M13 f’ = M1. M5. M6. M8. M9. M10. M14. M15 f = m1 + m5 + m6 + m8 + m9 + m10 + m14 + m15 f’ = m0 + m2 + m3 + m4 + m7 + m11 + m12 + m13

95
Converting SOP to POS f = x y + x’ z(SOP Form) Using Distributive Law f = x y + x’ z f = ( x y + x’ ) (x y + z) f = ( x + x’ ) ( y + x’ ) ( x + z ) ( y + z ) f = ( y + x’ ) ( x + z ) ( y + z ) f = ( y + x’ ) ( x + z ) ( y + z )(POS Form)

96
Converting POS to SOP f = ( x + y ) ( x’ + z )(POS Form) Using Distributive Law f = ( x + y ) ( x’ + z ) f = x ( x’ + z ) + y ( x’ + z ) f = x x’ + x z + y x’ + y z f = x z + y x’ + y z f = x z + y x’ + y z(SOP Form) Note : To obtain the expression f = x y + x’ z as in the previous slide apply Consensus Theorem

97
Converting SOP to C-SOP F = A + B’C(SOP Form) Expanding both terms separately using {x = x. 1} & {1 = (x + x’)} A = A (B + B’) = AB + AB’ = AB (C + C’) + AB’ (C + C’) A = ABC + ABC’ + AB’C + AB’C’ B’C = B’C (A + A’) = AB’C + A’B’C Combining the expressions F = ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C = ABC + ABC’ + AB’C + AB’C + AB’C’ + A’B’C Using {x + x = x} F = ABC + ABC’ + AB’C + AB’C’ + A’B’C F = ABC + ABC’ + AB’C + AB’C’ + A’B’C (C-SOP Form)

98
Converting POS to C-POS f = ( x’ + y ) ( x + z ) ( y + z ) (POS Form) Expanding the three terms separately using {x = x + 0} & {0 = x x’} ( x’ + y ) = x’ + y + z z’ = ( x’ + y + z ) ( x’ + y + z’ ) ( x + z ) = x + z + y y’ = ( x + z + y ) ( x + z + y’ ) = ( x + y + z ) ( x + y’ + z ) ( y + z ) = y + z + x x’ = ( y + z + x ) ( y + z + x’ ) = ( x + y + z ) ( x’ + y + z ) Combining the expressions f = ( x’ + y + z ) ( x’ + y + z’ ) ( x + y + z ) ( x + y’ + z ) ( x + y + z ) ( x’ + y + z ) f = ( x’ + y + z ) ( x’ + y + z ) ( x + y + z ) ( x + y + z ) ( x’ + y + z’ ) ( x + y’ + z ) Using {x. x = x} f = ( x’ + y + z’ ) ( x + y + z ) ( x’ + y + z’ ) ( x + y’ + z ) f = ( x’ + y + z’ ) ( x + y + z ) ( x’ + y + z’ ) ( x + y’ + z )(C-POS Form)

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google