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1 COMP541 Combinational Logic Montek Singh Jan 16, 2007

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2Today Basics of digital logic (review) Basic functions Basic functions Boolean algebra Boolean algebra Gates to implement Boolean functions Gates to implement Boolean functions Identities and Simplification (review?)

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3 Binary Logic Binary variables Can be 0 or 1 (T or F, low or high) Can be 0 or 1 (T or F, low or high) Variables named with single letters in examples Variables named with single letters in examples Really use words when designing circuits Really use words when designing circuits Basic Functions AND AND OR OR NOT NOT

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4AND Symbol is dot C = A · B C = A · B Or no symbol C = AB C = AB Truth table -> C is 1 only if Both A and B are 1 Both A and B are 1

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5OR Symbol is + Not addition Not addition C = A + B C = A + B Truth table -> C is 1 if either 1 Or both! Or both!

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6NOT Unary Symbol is bar C = Ā C = Ā Truth table -> Inversion

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7Gates Circuit diagrams are traditional to document circuits Remember that 0 and 1 are represented by voltages

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8 AND Gate Timing Diagrams

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9 OR Gate

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10Inverter

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11 More Inputs Work same way What’s output?

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12 Representation: Schematic Schematic = circuit diagram

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13 Representation: Boolean Algebra For now equations with operators AND, OR, and NOT Can evaluate terms, then final OR Alternate representations next

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14 Representation: Truth Table 2 n rows where n = # of variables

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15Functions Can get same truth table with different functions Usually want ‘simplest’ Fewest gates, or using only particular types of gates Fewest gates, or using only particular types of gates More on this later More on this later

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16Identities Use identities to manipulate functions I used distributive law … … to transform from … to transform from to

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17 Table of Identities

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18Duals Left and right columns are duals Replace AND and OR, 0s and 1s

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19 Single Variable Identities

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20Commutativity Operation is independent of order of variables

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21Associativity Independent of order in which we group So can also be written as and

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22Distributivity 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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23 DeMorgan’s Theorem Used a lot NOR invert, then AND NAND invert, then OR

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24 Truth Tables for DeMorgan’s

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25 Algebraic Manipulation Consider function

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26 Simplify Function Apply

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27 Fewer Gates

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28 Consensus Theorem The third term is redundant Can just drop Can just drop Proof in book, but in 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 must be 1! Then one of the first two terms must be 1!

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29 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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30 Mechanically Go From Truth Table to Function

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31 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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32 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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33 Definition: Minterm Product term in which all variables appear once (complemented or not)

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

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35Maxterms Sum term in which all variables appear once (complemented or not)

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

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37 Sum of Minterms Like the introductory slide OR all of the minterms of truth table row with a 1

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38 Complement of F Not surprisingly, just sum of the other minterms In this case m 1 + m 3 + m 4 + m 6

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

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

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41Recap 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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42 Next Time Lab Prep Demo lab software Demo lab software Talk about FPGA internals Talk about FPGA internals Overview of components on board Overview of components on board Downloading and testing Downloading and testing Karnaugh maps: mechanical synthesis approach (quick)

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