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7 - ECSE-2610 Computer Components Operations(COCO Fall 2006...

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ECSE-2610 Computer Components & Operations (COCO) Fall 2006 Part 6a : Synthesis of Combinational Circuits
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Manipulating Circuits Given an algebraic representation of a logic circuit, we manipulate the algebraic representation to Convert it to standard forms. Redesign with more efficient gates like NAND or NOR. Simplify or even minimize the expression.
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What is Minimization? Minimization is the process of reducing a complex logic expression or equation into its simplest form (with the fewest number of terms) by removing redundancies and terms having no effect on the output. Similar to simplifying and reducing complex expressions in Algebra by combining and collecting like terms. As in algebraic reduction, rules must be followed that guarantee the value of the expression is not changed by the simplification.
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Rationale for Logic Minimization Reduce the complexity of the gate level implementation Reduce the complexity of the gate level implementation Reduce the number of literals (gate inputs). Reduce the number of gates. Reduce the number of levels of gates. Fewer inputs implies faster gates in some technologies. Fan-ins (number of gate inputs) are limited in some technologies. Fewer levels of gates implies reduced signal propagation delays. Minimum delay configuration often requires more gates.
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Minimization Example Apply the laws and theorems to simplify Boolean equations Example: Full adder's carry out function C out = A' B C in + A B' C in + A B C in ' + A B C in Simplify by minimizing the number terms
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Recall the Combining Theorem I A B F A B G 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 X Y' + X Y = X(Y'+Y) = X F = A B' + A B = A G = A' B' + A B' = B' Is there a systematic way of recognizing this pattern, so we can use this theorem to simplify?
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