lect4-Kmap Logic simplification

lect4-Kmap Logic simplification - CSE140: Components and...

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1 1 Sources: TSR, Katz, Boriello & Vahid CSE140: Components and Design Techniques for Digital Systems Logic simplification Tajana Simunic Rosing 2 Sources: TSR, Katz, Boriello & Vahid Overview • What we covered thus far: – Number representations – Switches –MOS t r a n s i s t o r s – Logic gates – Boolean algebra – Minterm and maxterm representation of combinational logic • What is next: – Combinatorial logic: • Minimization • Implementation
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2 3 Sources: TSR, Katz, Boriello & Vahid canonical sum-of-products minimized sum-of-products canonical product-of-sums minimized product-of-sums F1 F2 F3 B A C F4 Alternative two-level implementations of F = AB + C 4 Sources: TSR, Katz, Boriello & Vahid 1-cube X 01 Boolean cubes • Key tool for simplification is uniting theorem: A (B’ + B) = A • B. Cubes: visual technique for applying the uniting theorem • n input variables = n-dimensional "cube" 2-cube X Y 11 00 01 10 3-cube X Y Z 000 111 101 4-cube W X Y Z 0000 1111 1000 0111
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3 5 Sources: TSR, Katz, Boriello & Vahid ABF 001 010 101 110 ON-set = solid nodes OFF-set = empty nodes DC-set = × 'd nodes two faces of size 0 (nodes) combine into a face of size 1(line) A varies within face, B does not this face represents the literal B' Mapping truth tables onto Boolean cubes • Uniting theorem combines two “faces" of a cube into a larger “face" • Example: A B 11 00 01 10 F 6 Sources: TSR, Katz, Boriello & Vahid ABC i n C o u t 000 0 0 0 011 1 100 0 1 1 111 1 Three variable example • Binary full-adder carry-out logic A B C 000 111 101
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4 7 Sources: TSR, Katz, Boriello & Vahid F(A,B,C) = Σ m(4,5,6,7) on-set forms a square - a cube of dimension 2 This subcube represents the literal A Higher dimensional cubes • Sub-cubes of higher dimension than 2 A B C 000 111 101 100 001 010 011 110 In a 3-cube (three variables): – a 0-cube, i.e., a single node, yields a term in 3 literals – a 1-cube, i.e., a line of two nodes, yields a term in 2 literals – a 2-cube, i.e., a plane of four nodes, yields a term in 1 literal – a 3-cube, i.e., a cube of eight nodes, yields a constant term "1" In general, – an m-subcube within an n-cube (m < n) yields a term with n – m literals 8
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This note was uploaded on 02/14/2008 for the course CSE 140 taught by Professor Rosing during the Fall '06 term at UCSD.

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lect4-Kmap Logic simplification - CSE140: Components and...

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