1176407204 - Chapter 2 Combinational Logic Circuits Meilin...

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Chapter 2 Combinational Logic Circuits Meilin Liu Department of Computer Science Wright State University Homepage: http://www.wright.edu Email: meilin.liu@wright.edu
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CEG 260, Meilin Liu 2 Boolean Expression X=0, Y=0, Z=1, X’Y’Z=1, F=1 X=1, Y=0, Z=0, XY’Z’=1, F=1 X=1, Y=0, Z=1, XY’Z=1, F=1 X=1, Y=1, Z=0, XYZ’=1, F=1 X=1,Y=1,Z=1, XYZ=1, F=1 Combine all these input situations, we get: F=X’Y’Z + XY’Z’+XY’Z+XYZ’+XYZ All these five terms are called product terms. F is represented by sum of products. Truth Table 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 X Y Z Z Y X F + =
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CEG 260, Meilin Liu 3 Boolean Expression X=0, Y=0, Z=0, (X+Y+Z)=0, F=0 X=0, Y=1, Z=0, (X+Y’+Z)=0, F=0 X=0, Y=1, Z=1, (X+Y’+Z’)=0, F=0 For all the other input combinations, F=1 Combine all these input situations, we get: F=(X+Y+Z)(X+Y’+Z)(X+Y’+Z’) All the three terms are called sum terms. F is represented by product of sums. Truth Table 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 X Y Z Z Y X F + =
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CEG 260, Meilin Liu 4 Boolean Expression Boolean equations, truth tables and logic diagrams describe the same function! Truth tables are unique; expressions and logic diagrams are not. This gives flexibility in implementing functions. F=X’Y’Z + XY’Z’+XY’Z+XYZ’+XYZ F=(X+Y+Z)(X+Y’+Z)(X+Y’+Z’) F(X,Y,X) = X + Y’Z These three functions implement the same truth table, but they have different boolean expressions, and the corresponding logic circuits are different and have different complexity. By manipulating a Boolean expression according to Boolean algebraic rules, it is often possible to obtain a simpler expression for the same function.
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CEG 260, Meilin Liu 5 Canonical Forms It is useful to specify Boolean functions in a form that: Allows comparison for equality. Has a correspondence to the truth tables Canonical Forms in common usage: Sum of Minterms (SOM) Product of Maxterms (POM)
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CEG 260, Meilin Liu 6 Minterms Minterms are AND terms with every variable present in either true form or complemented form. It represents exactly one combination of the binary variables in a truth table and it has the value of 1 for that combination and 0 for all others. For each binary combination, there is one corresponding minterm. A symbol m_j corresponds to one minterm, whose corresponding binary comination
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This note was uploaded on 05/03/2010 for the course CEG 260 taught by Professor Staff during the Spring '08 term at Wright State.

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1176407204 - Chapter 2 Combinational Logic Circuits Meilin...

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