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Combinational Logic 1

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2 Topics Basics of digital logic Basic functions ♦ Boolean algebra ♦ Gates to implement Boolean functions Identities and Simplification

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

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4 AND Operator Symbol is dot ♦ Z = X · Y Or no symbol ♦ Z = XY Truth table -> Z is 1 only if ♦ Both X and Y are 1

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5 OR Operator Symbol is + ♦ Not addition ♦ Z = X + Y Truth table -> Z is 1 if either 1 ♦ Or both!

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6 NOT Operator Unary Symbol is bar (or ’) ♦ Z = X’ Truth table -> Inversion

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7 Gates Circuit diagrams are traditionally used 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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10 Inverter

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11 More Inputs Work same way What’s output?

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12 Digital Circuit Representation: Schematic

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13 Digital Circuit 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 Digital Circuit Representation: Truth Table 2 n rows where n # of variables

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

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16 Identities Use identities to manipulate functions On previous slide, I used distributive law to transform from to

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17 Table of Identities

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

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19 Single Variable Identities

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20 Commutative Order independent

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

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22 Distributive Can substitute arbitrarily large algebraic expressions for the variables

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23 DeMorgan’s Theorem Used a lot NOR equals invert AND NAND equals invert 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 Proof in book, but in summary ♦ For third term to be true, Y & Z both 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

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30 Truth Table of the Complement of a Function XYZ F = X + Y ’ ZF’F’ 00001 00110 01001 01101 10010 10110 11010 11110

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31 Algebraic Form for Complement Mechanical way to derive algebraic form for the complement of a function 1.Take the dual Recall: Interchange AND & OR, and 1s & 0s 2.Complement each literal (a literal is a variable complemented or not; e.g. x, x’, y, y’ each is a literal)

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32 Example: Algebraic form for the complement of a function F = X + Y’Z To get the complement F’ 1.Take dual of right hand side X. (Y’ + Z) 2.Complement each literal: X’. (Y + Z’) F’ = X’. (Y + Z’)

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Mechanically Go From Truth Table to Function

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34 From Truth Table to Function Consider a truth table Can implement F by taking OR of all terms that correspond to rows for which F is 1 “Standard Form” of the function

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

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36 Definition: Minterm Product term in which all variables appear once (complemented or not) For the variables X, Y and Z example minterms : X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ

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37 Definition: Minterm (continued) Min Term Each minterm represents exactly one combination of the binary variables in a truth table.

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38 Truth Tables of Minterms

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39 Number of Minterms For n variables, there will be 2 n minterms Minterms are labeled from minterm 0, to minterm 2 n -1 ♦ m 0, m 1, m 2, …, m 2 n -2, m 2 n -1 For n = 3, we have ♦ m 0, m 1, m 2, m 3, m 4, m 5, m 6, m 7

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40 Definition: Maxterm Sum term in which all variables appear once (complemented or not) For the variables X, Y and Z the maxterms are: X+Y+Z, X+Y+Z’ …., X’+Y’+Z’

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41 Definition: Maxterms (continued) mmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm,m xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx,mmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmm Maxterm

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42 Truth Tables of Maxterms

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43 Minterm related to Maxterm Minterms and maxterms with same subscripts are complements Example

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44 Standard Form of F: Sum of Minterms OR all of the minterms of truth table for which the function value is 1 F = m 0 + m 2 + m 5 + m 7

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

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

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47 Product of Maxterms F = m 0 + m 2 + m 5 + m 7 Remember: ♦ M1 is only false for 001 ♦ M3 is only false for 011 ♦ M4 is only false for 100 ♦ M6 is only false for 110 Can express F as AND of M1, M3, M4, M6 or

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

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