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Logic Functions and their Representation

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Slide 2 Combinational Networks x1x1 x2x2 xnxn f

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Logic Functions and their Representation Slide 3 Logic Operations Truth tables xy AND x y OR x y NOT x NAND x y NOR x y EXOR x y 00000110 01010101 10011101 11111000

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Logic Functions and their Representation Slide 4 SOP and POS Definition: A variable x i has two literals x i and x i. A logical product where each variable is represented by at most one literal is a product or a product term or a term. A term can be a single literal. The number of literals in a product term is the degree. A logical sum of product terms forms a sum-of-products expression (SOP). A logical sum where each variable is represented by at most one literal is a sum term. A sum term can be a single literal. A logical product of sum terms forms a product-of-sums expression (POS).

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Logic Functions and their Representation Slide 5 Minterm A minterm is a logical product of n literals where each variable occurs as exactly one literal A canonical SOP is a logical sum of minterms, where all minterms are different. Also called canonical disjunctive form or minterm expansion

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Logic Functions and their Representation Slide 6 Maxterm A maxterm is a logical sum of n literals where each variable occurs as exactly one literal A canonical Pos is a logical product of maxterms, where all maxterms are different. Also called canonical conjunctive form or maxterm expansion Show an example

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Logic Functions and their Representation Slide 7 Shannon Expansion Theorem: An arbitrary logic function f(x 1,x 2,…,x n ) is expanded as follows: f(x 1,x 2,…,x n ) = x 1 f(0,x 2,…,x n ) x 1 f(1,x 2,…,x n ) (Proof) When x 1 = 0, = 1 f(0,x 2,…,x n ) 0 f(1,x 2,…,x n ) = f(0,x 2,…,x n ) When x 1 = 1, similar

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Logic Functions and their Representation Slide 8 Expansions into Minterms Example: Expand f(x 1,x 2,x 3 ) = x 1 (x 2 x 3 ) Example: minterm expansion of an arbitrary function Relation to the truth table Maxterm expansion (duality)

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Logic Functions and their Representation Slide 9 Reed-Muller Expansions EXOR properties (x y) z = x (y z) x(y z) = xy xz x y = y x x x = 0 x 1 = x

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Logic Functions and their Representation Slide 10 Reed-Muller Expansions Lemma xy = 0 x y = x y (Proof) ( ) Let xy = 0 x y = xy x y = ( xy xy) (x y xy) = x y ( ) Let xy ≠ 0 x = y = 1. Thus x y = 0, x y = 1 Therefore, x y ≠ x y

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Logic Functions and their Representation Slide 11 An arbitrary 2-varibale function is represented by a canonical SOP f(x 1,x 2 ) = f(0,0) x 1 x 2 f(0,1) x 1 x 2 f(1,0)x 1 x 2 f(1,1) x 1 x 2 Since the product terms have no common minterms, the can be replaced with f(x 1,x 2 ) = f(0,0) x 1 x 2 f(0,1) x 1 x 2 f(1,0)x 1 x 2 f(1,1) x 1 x 2 Next, replace x 1 = x 1 1, and x 2 = x 2 1 Show results!

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Logic Functions and their Representation Slide 12 PPRM An arbitrary n-variable function is uniquely represented as f(x 1,x 2,…,x n ) = a 0 a 1 x 1 a 2 x 2 … a n x n a 12 x 1 x 2 a 13 x 1 x 3 … a n-1,n x n-1 x n … a 12…n x 1 x 2 …x n

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