02_Boolean&TwoLevel

02_Boolean&TwoLevel - Representations of Boolean...

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20 Representations of Boolean Functions Readings: 2.5,2.5.2-2.10.3 Boolean Function: F = X + YZ Truth Table: XYZF 000 001 010 011 100 101 110 111 Circuit Diagram: 21 Why Boolean Algebra/Logic Minimization? Logic Minimization: reduce complexity of the gate level implementation reduce number of literals (gate inputs) reduce number of gates reduce 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 number of gates (or gate packages) influences manufacturing costs
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22 Basic Boolean Identities: X + 0 = X * 1 = X + 1 = X * 0 = X + X = X * X = X + X = X * X = X = 23 Basic Laws Commutative Law: X + Y = Y + X XY = YX Associative Law: X+(Y+Z) = (X+Y)+Z X(YZ)=(XY)Z Distributive Law: X(Y+Z) = XY + XZ X+YZ = (X+Y)(X+Z)
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24 Boolean Manipulations Boolean Function: F = XYZ + XY + XYZ Truth Table: XYZF 000 001 010 011 100 101 110 111 Reduce Function: 25 Advanced Laws (Absorbtion) X+XY = XY + XY = X+XY = X(X+Y) = (X+Y)(X+Y) = X(X+Y) =
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26 Boolean Manipulations (cont.) Boolean Function: F = XYZ + XZ Truth Table: XYZF 000 001 010 011 100 101 110 111 Reduce Function: 27 Boolean Manipulations (cont.) Boolean Function: F = (X+Y+XY)(XY+XZ+YZ) Truth Table: Reduce Function:
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28 DeMorgan’s Law (X + Y) = X * Y (X * Y) = X + Y Example: Z = A B C + A B C + A B C + A B C Z = (A + B + C) * (A + B + C) * (A + B + C) * (A + B + C) DeMorgan's Law can be used to convert AND/OR expressions to OR/AND expressions X 0 0 1 1 Y 0 1 0 1 X 1 1 0 0 Y 1 0 1 0 X + Y X•Y X 0 0 1 1 Y 0 1 0 1 X 1 1 0 0 Y 1 0 1 0 X + Y X•Y 29 DeMorgan’s Law example If F = (XY+Z)(Y+XZ)(XY+Z), F =
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30 Boolean Equations to Circuit Diagrams
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This document was uploaded on 02/11/2012.

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02_Boolean&TwoLevel - Representations of Boolean...

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