# Lecture 4 - Th The University of Texas at Dallas Erik...

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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Exclusive OR/Exclusive NOR (XOR/XNOR) XOR and XNOR are useful logic functions. Both have two or more inputs . The truth table for two inputs is shown at right. XOR/XNOR Truth Table a b a XOR b a XNOR b 00 0 1 aXORb = 1 if and only if (iff) a b . a XNOR b = 1 if and only if (iff) a = b . Both may also have many inputs. For >2 01 1 0 10 1 0 inputs, the XOR output is 1 for an odd number of 1 inputs; XNOR has a 1 output for an even number of 1 inputs . ymbols are shown below and to the right 1 1 0 1 XNOR = (ab+ab) Symbols are shown below and to the right. a b a b XOR Like NAND and NOR, XOR and XNOR are not asic Boolean functions, © N. B. Dodge 09/09 1 Lecture #4: More Complex Combinational Logic Circuits a b a b XNOR XOR = ab+ab basic Boolean functions, but can be made from AND , OR and NOT.

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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas More Boolean Expressions and Circuits Let us look at a few more Boolean expressions and the circuits that result. Consider the following SOP expression gp and truth table: f = abc+abc+abc: a b c f 0 0 0 0 a 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 b c f 1 0 1 1 1 1 0 1 1 1 1 1 he circuit is shown at right. © N. B. Dodge 09/09 2 Lecture #4: More Complex Combinational Logic Circuits The circuit is shown at right. Can this circuit be simplified?
Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Simplifying the Circuit The circuit is simplifiable. Thus: f = abc+abc+abc = abc+abc+abc+abc , = (abc+abc)+(abc+abc) = ab(c+c)+ac(b+b) . a () ()( ) We got the blue result using the identity x+x = x (thus abc = abc + abc). Grouping the expressions (turquoise), we then factor b c f riginal Circuit terms, remembering that xy + xz = x(y + z) to get the red expression. Note that all the expressions are equivalent . ow in the red expression we know that Original Circuit b Now, in the red expression, we know that the items in parentheses = 1 (x+x = 1). Thus f = ab + ac . he original and simplified circuits are implified Circuit f a c © N. B. Dodge 09/09 3 Lecture #4: More Complex Combinational Logic Circuits The original and simplified circuits are shown at right. Simplified Circuit

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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Deriving the Expression from the Circuit Sometimes we may develop an experimental circuit to perform a function, and then want to simplify it. i lif th i it il d To simplify the circuit easily, we need the Boolean expression for that circuit . We can then simplify the logic design using that expression. he circuit at right is an example: The Boolean expression is: Top gate: Output = ab.
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Lecture 4 - Th The University of Texas at Dallas Erik...

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