CompArch-Lec02-Combinational-Logic

CompArch-Lec02-Combinational-Logic - COSC3330 Computer...

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Lecture 2. Combinational Logic COSC3330 Computer Architecture Instructor: Weidong Shi (Larry), PhD Computer Science Department University of Houston
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Zoom-in a System Component
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Introduction A logic circuit is composed of square4 Inputs square4 Outputs square4 Functional specification Relationship between inputs and outputs square4 Timing specification Delay from inputs to outputs 5 inputs outputs functional spec timing spec
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Circuits Nodes square4 Inputs: A , B , C square4 Outputs: Y , Z square4 Internal: n1 Circuit elements square4 E1, E2, E3 6 A E1 E2 E3 B C n1 Y Z
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Types of Logic Circuits Combinational Logic square4 Outputs are determined by current values of inputs square4 Thus, it is memoryless Sequential Logic square4 Outputs are determined by previous and current values of inputs square4 Thus, it has memory 7 inputs outputs functional spec timing spec
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Rules of Combinational Composition A circuit is combinational if square4 Every node of the circuit is either designated as an input to the circuit or connects to exactly one output terminal of a circuit element square4 The circuit contains no cyclic paths Every path through the circuit visits each circuit node at most once square4 Every circuit element is itself combinational Select combinational logic? 8
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Boolean Equations The functional specification of a combination logic is usually expressed as a truth table or a Boolean equation square4 Truth table is in a tabular form square4 Boolean equation is in a algebraic form Example: S = F( A , B , C in ) C out = F( A , B , C in ) 9 Truth Table Boolean equation
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Terminology The complement of a variable A is A square4 A variable or its complement is called l iteral AND of one or more literals is called a product or implicant square4 Example: AB, ABC, B OR of one or more literals is called a sum square4 Example: A + B Order of operations square4 NOT has the highest precedence, followed by AND, then OR Example: Y = A + BC 10
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Minterms 11 A minterm is a product (AND) of literals involving all of the inputs to the function Each row in a truth table has a minterm that is true for that row (and only that row )
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Sum-of-Products (SOP) Form The function is formed by ORing the minterms for which the output is true square4 Thus, a sum (OR) of products (AND terms) All Boolean equations can be written in SOP form 12 A B Y 0 0 0 1 1 0 1 1 0 1 0 1 minterm A B A B A B A B Y = F(A, B) = AB + AB
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Boolean Algebra We learned how to write a boolean equation given a truth table square4 But, that expression does not necessarily lead to the simplest set of logic gates One way to simplify boolean equations is to use boolean algebra square4 Set of theorems square4 It is like regular algebra, but in some cases simpler because variables can have only two values (1 or 0) square4 Theorems obey the principles of duality : ANDs and ORs interchanged , 0’s and 1’s interchanged 13
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Boolean Theorems of One Variable The prime (’) symbol denotes the dual of a statement 14
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