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1 COMP541 Combinational Logic - II Montek Singh Jan 18, 2012

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2Today Basics of Boolean Algebra (review) Identities and Simplification Identities and Simplification Basics of Logic Implementation Minterms and maxterms Minterms and maxterms Going from truth table to logic implementation Going from truth table to logic implementation

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3Identities Use identities to manipulate functions You can use distributive law … … to transform from … to transform from to

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4 Table of Identities

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5Duals Left and right columns are duals Replace AND and OR, 0s and 1s

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6 Single Variable Identities

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7Commutativity Operation is independent of order of variables

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8Associativity Independent of order in which we group So can also be written as and

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9Distributivity Can substitute arbitrarily large algebraic expressions for the variables Distribute an operation over the entire expression Distribute an operation over the entire expression

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10 DeMorgan’s Theorem Used a lot NOR invert, then AND NAND invert, then OR

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11 Truth Tables for DeMorgan’s

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12 Algebraic Manipulation Consider function

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13 Simplify Function Apply

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14 Fewer Gates

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15 Consensus Theorem The third term is redundant Can just drop Can just drop Proof summary: For third term to be true, Y & Z both must be 1 For third term to be true, Y & Z both must be 1 Then one of the first two terms is already 1! Then one of the first two terms is already 1!

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16 Complement of a Function Definition: 1s & 0s swapped in truth table Mechanical way to derive algebraic form Take the dual Take the dual Recall: Interchange AND and OR, and 1s & 0s Complement each literal Complement each literal

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17 Mechanically Go From Truth Table to Function

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18 From Truth Table to Func Consider a truth table Can implement F by taking OR of all terms that are 1

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19 Standard Forms Not necessarily simplest F But it’s a mechanical way to go from truth table to function Definitions: Product terms – AND ĀBZ Product terms – AND ĀBZ Sum terms – OR X + Ā Sum terms – OR X + Ā This is logical product and sum, not arithmetic This is logical product and sum, not arithmetic

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20 Definition: Minterm Product term in which all variables appear once (complemented or not)

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21 Number of Minterms For n variables, there will be 2 n minterms Like binary numbers from 0 to 2 n -1 Often numbered same way (with decimal conversion)

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22Maxterms Sum term in which all variables appear once (complemented or not)

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23 Minterm related to Maxterm Minterm and maxterm with same subscripts are complements Example

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24 Sum of Minterms Like Slide 18 OR all of the minterms of truth table row with a 1 “ON-set minterms” “ON-set minterms”

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25 Sum of Products Simplifying sum-of-minterms can yield a sum of products Difference is each term need not be a minterm i.e., terms do not need to have all variables i.e., terms do not need to have all variables A bunch of ANDs and one OR

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26 Two-Level Implementation Sum of products has 2 levels of gates

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27 More Levels of Gates? What’s best? Hard to answer Hard to answer More gate delays (more on this later) More gate delays (more on this later) But maybe we only have 2-input gates But maybe we only have 2-input gates So multi-input ANDs and ORs have to be decomposed

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28 Complement of a Function Definition: 1s & 0s swapped in truth table Mechanical way to derive algebraic form Take the dual Take the dual Recall: Interchange AND and OR, and 1s & 0s Complement each literal Complement each literal

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29 Complement of F Not surprisingly, just sum of the other minterms “OFF-set minterms” “OFF-set minterms” In this case m 1 + m 3 + m 4 + m 6

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30 Product of Maxterms Recall that maxterm is true except for its own case So M1 is only false for 001

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31 Product of Maxterms Can express F as AND of all rows that should evaluate to 0 or

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32 Product of Sums Result: another standard form ORs followed by AND Terms do not have to be maxterms Terms do not have to be maxterms

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33Recap Working (so far) with AND, OR, and NOT Algebraic identities Algebraic simplification Minterms and maxterms Can now synthesize function (and gates) from truth table

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34 Next Time Lab on Fri, 1/20: Demo lab software Demo lab software Do 1 st lab assignment Do 1 st lab assignment download software: see website for link couple of simple Verilog programming problems Next Week: More on combinational logic

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