Engin112-F07-L06-boolean

Engin112-F07-L06-boolean - Engin112 Lectures 6,7 Binary...

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1 Maciej Ciesielski Department of Electrical and Computer Engineering 09/17/2007 Engin112 – Lectures 6,7 Binary Logic and Boolean Algebra Engin112 – 09/17/2007 2 Recap from last lecture ± Codes y Assignment of bit sequence to represent discrete elements y BCD code and addition y Weighted code ± Binary storage and registers y Register transfer operations ± Today’s lecture y Binary logic y Boolean algebra
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2 Engin112 – 09/17/2007 3 Register Transfer Operations ± Example: addition Designing digital logic is focus of rest of course Engin112 – 09/17/2007 4 Binary Logic ± Binary logic uses two possible values y ‘1’ and ‘0’ y ‘yes’ and ‘no’ y ‘true’ and ‘false’ y Can be represented by variables: A, B, C, x, y, z, … ± Logic functions modify input values y What is the minimum number of inputs? » 1 input -> pretty uninteresting, inverter or wire only » 2 inputs -> basic logic functions ± What are possible logic functions? y NOT, AND, OR y Others can be derived from those (NAND, XOR, etc.)
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3 Engin112 – 09/17/2007 5 NOT Function ± Complement operation y Single input y Inverts value of input y Symbolized by prime x’ or overbar ± Truth table y Input combinations on the left y Output of function on the right ± Graphic symbol y NOT gate y Little circle indicates inversion 0 1 1 0 x' x x input output Engin112 – 09/17/2007 6 AND Function ± Operation to check if two conditions are met y Two inputs y Output = 1 if and only if both inputs are 1 y Symbolized by dot or absence of operator x·y or xy ± Truth table y Needs to consider 2 2 = 4 input combinations ± Graphic symbol y AND gate 1 1 1 0 0 1 0 1 0 0 0 0 z y x inputs output
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4 Engin112 – 09/17/2007 7 OR Function ± Operation to check if at least one condition is met y Two inputs y Output = 1 if any one or both inputs are 1 y Symbolized by “plus” sign: x + y ± Truth table ± Graphic symbol y OR gate 1 1 1 1 0 1 1 1 0 0 0 0 z y x Engin112 – 09/17/2007 8 Comparison of Functions ± Timing diagrams (waveforms) y Horizontal axis is time ( t ) y Vertical axis shows signals, each with two different voltage levels t
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5 Engin112 – 09/17/2007 9 Multiple Inputs ± Two inputs might not be enough ± 3-input AND gate: ± 4-input OR gate: ± What are the rules for aggregating functions? y Boolean algebra Engin112 – 09/17/2007 10 Algebras ± What is an algebra? y
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This note was uploaded on 04/16/2008 for the course ENGIN 112 taught by Professor Ciesielski during the Spring '08 term at UMass (Amherst).

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Engin112-F07-L06-boolean - Engin112 Lectures 6,7 Binary...

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