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1 Purpose of Combinational Logic Design • Give you a solid theoretical foundation for the analysis and synthesis of combinational logic circuits • Give you a foundation that will be doubly important later when we move on to sequential circuits

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2 Combinational Logic Defined • Outputs as a function of inputs • Output behavior depends only on the current inputs • memoryless z 1 z 2 z m M x 1 x 2 x n M ( ) () 11 1 2 22 1 2 12 ,, , ,,, , n n mm n zf x x x x x x x x x = = = L L M M L
3 Sequential Logic Defined • Output behavior depends on the sequence of inputs • Memory • Combination Lock

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4 Combinational Logic Design • Combinational Logic & Simplification – Refer to Chapter 2&3 • Combinational Logic Technologies – Refer to Chapter 4 & 5
5 Combinational Logic & Simplification • Laws & Theorems • Two-level Logic – Canonical Forms – Karnaugh Maps • Two-level Simplification • Multi-level Logic

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6 Definition of Boolean Algebra • Differing in a major way from ordinary algebra – Boolean constants and variables are allowed to have only two possible values • expressing the relationship between a logic circuit’s inputs and outputs using basic operators – Operations: NOT, AND, OR – Complement is always applied first, followed by AND, followed by OR – Binary Operators: + (logical addition), (logical multiplication), or (complement)
7 Boolean Function • Uniquely maps some number of inputs over the set {0,1} into an output set {0,1} • Can be expressed – By Truth Table – In terms of AND, OR and NOT – By Logic Gates

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8 Laws or Axioms (1) • Commutative 交换律 – X+Y=Y+X – XY=YX • Associative 结合律 – (a+b)+c=a+(b+c)=a+b+c – (ab)c=a(bc)=abc 恒等性 – A+0=A –A 1=A
9 Laws or Axioms (2) • Distributive 分配律 – a+(b c)=(a+b)(a+c) – a(b+c)=ab+ac • Complement 互补 – a+a’=1 –a a’=0

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10 Theorems based on Laws • Tools to simplify Boolean Expressions • Duality 对偶 – A dual of an expression is derived from the original expression by • replacing AND operations by OR operations and vice versa • replacing constant logic 0s by logic 1s and vice versa • while leaving the literals unchanged – fundamental theorem – any statement that is true about a expression is also true for its dual – (X+0=X)D=(X 1=X)
11 Useful Laws & Theorems (1) • Operations with 0 & 1 – A+0=A, A+1=1 –A 1=A, A 0=0 (Dual for A+1=1) • Idempotent theorem 重叠定理 – X+X=X; X X=X (Dual for X+X=X) • Involution theorem 对合定理 –(X ’) ’=X • Theorem of Complementarity 互补定理 – X+X’=1; X X’=0

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12 Useful Laws & Theorems (2) • Commutative Law 交换律 – X+Y=Y+X – XY=YX • Associative Law 结合律 – (X+Y)+Z=X+(Y+Z)=X+Y+Z – (XY)Z=X(YZ)=XYZ • Distributive Law 分配律 –X+(Y Z)=(X+Y)(X+Z) – X(Y+Z)=XY+XZ
13 Useful Laws & Theorems (3) • DeMorgan’s Law

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## This note was uploaded on 03/28/2011 for the course EE 30230563 taught by Professor Rongluo during the Spring '09 term at Tsinghua University.

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