Lecture 3

Lecture 3 - 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 Creating Boolean Functions We stated in the last lecture that digital circuits solve ALL problems by performing Boolean transformations on binary numbers. ny computer function can be created by combinations of Boolean Any computer function can be created by combinations of Boolean variables . Boolean functions created in this way are called combinational logic, and they are time-independent (we will discuss sequential, or time-dependent, logic in a week or so). When a digital system is being designed, an engineer usually starts with a specification (“spec”), which describes how the system performs (“when the system inputs are this, the output is that”). We go from “spec” truth table, from which we can derive a Boolean expression, from which a circuit can be made. We will look at the entire process later: Let’s now examine the © N. B. Dodge 09/09 1 Lecture #3: Boolean Algebra and Combinational Digital Logic p Boolean function circuit design process.
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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 Circuit Derived from Boolean Expression Assume we have some Boolean functions that represent 1: computer circuits that we want to design. How do we go from the Boolean function to the mputer circuit? 2: computer circuit? Consider these functions: a + b + c = f a + b = f 3: (a · b) + c = f (finally, a more complicated one!): (a + b) · (c + d) = f 4: © N. B. Dodge 09/09 2 Lecture #3: Boolean Algebra and Combinational Digital Logic
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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 Exercise 1 Now let’s try a few digital circuit designs from the Boolean expressions: · b · c = f a b c f (a + b) · c = f a + b · c · d = f © N. B. Dodge 09/09 3 Lecture #3: Boolean Algebra and Combinational Digital Logic a + (b · c) + d = f
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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 Complex Boolean Functions: “Combinational Logic Any Boolean function f maps a total of n inputs (x 1 , x 2 , x 3 , . ..... x n ) into one output (0,1) . That is, a Boolean function f of n variables roduces ngle output for each unique combination f puts produces a single output for each unique combination of inputs. Remember: There are only two Boolean variables: 0 or 1 . A Boolean function in which the output f depends only on the input variables is called combinational logic . No matter how complex, a Boolean function can always be defined by sing a uth table as we have done previously using a truth table , as we have done previously. Principle: Any combinational logic function may be represented by two levels of logic : one level of AND gates and another level of OR ates (input version ay also be required, but this is not © N. B. Dodge 09/09 4 Lecture #3: Boolean Algebra and Combinational Digital Logic gates (input inversion
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Lecture 3 - Th The University of Texas at Dallas Erik...

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