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Microprocessors vs. Custom Digital Circuits
Designers that work with digital phenomena often buy an off-the-shelf microprocessor and program it. Microprocessors are readily available, inexpensive, easy to program, and easy to reprogram Why would anyone ever need to design new digital circuits?
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Microprocessors vs. Custom Digital Circuits
Designers that work with digital phenomena often buy an off-the-shelf microprocessor and program it. Microprocessors are readily available, inexpensive, easy to program, and easy to reprogram Why would anyone ever need to design new digital circuits? Microprocessors are sometimes: Too slow; Too big; Consume too much power; Too costly
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Combinatorial Logic Circuits
A digital circuit whose output depends solely on the present combination of input values is called a combinatorial circuit Logic gates – building blocks of logic circuits AND OR NOT Boolean Algebra Boolean algebra is a branch of mathematics that uses variables whose values can only be 1 or 0 (“true” or “false”, respectively) and whose operators, like AND, OR, NOT, operate on such variables and return 1 or 0. We can build circuits by doing math
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Example: Seatbelt warning light
Logic Gates Truth Tables Example: Seatbelt warning light Design a system for an automobile that illuminates a warning light whenever the driver’s seatbelt is not fastened, and the key is in the ignition Boolean equation: w = NOT(s) AND k
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Notation and Terminology
Operators: NOT(a) is typically written as a’ a OR b is typically written as a + b a AND b is typically written as a * b (or a b) w = NOT(s) AND k = s’k Precedence rule: Expression in parentheses AND, NOT OR w = (a + b) * (c’) + d
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Properties of Boolean Algebra
Commutative a + b = b + a a * b = b * a Distributive a * (b + c) = a * b + a * c a + (b * c) = (a + b) * (a + c) Associative (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Identity 0 + a = a + 0 = a 1 * a = a * 1 = a Complement a + a’ = 1 a * a’ = 0
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Additional Properties
Null elements a + 1 = 1 a * 0 = 0 Idempotent Law a + a = a a * a = a Involution Law (a’)’ = a De Morgan’s Law (a + b)’ = a’ b’ (a b)’ = a’ + b’ Example: Simplification of an automatic sliding door system f = h c’ + h’ p c’
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Additional Properties
Null elements a + 1 = 1 a * 0 = 0 Idempotent Law a + a = a a * a = a Involution Law (a’)’ = a De Morgan’s Law (a + b)’ = a’ b’ (a b)’ = a’ + b’ Example: Simplification of an automatic sliding door system f = h c’ + h’ p c’ f = c’ (h + p)
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Boolean Functions Converting a truth table to an equation
Boolean function is a mapping of each possible combination of input values to either 0 or 1. Boolean function can be represented as an equation, a circuit, and as a truth table. Converting a truth table to an equation F = a b + a’ F = a’ b’ + a’ b + a b For any function, there may be many equivalent equations, and many equivalent circuits, but there is only one truth table!
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