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Combinational Logic 1

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**Topics Basics of digital logic Basic functions**

Boolean algebra Gates to implement Boolean functions Identities and Simplification

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**Binary Logic Binary variables Basic Functions**

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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**AND Operator Symbol is dot Or no symbol Truth table ->**

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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**OR Operator Symbol is + Truth table -> Z is 1 if either 1**

Not addition Z = X + Y Truth table -> Z is 1 if either 1 Or both!

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**NOT Operator Unary Symbol is bar (or ’) Truth table -> Inversion**

Z = X’ Truth table -> Inversion

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**Gates Circuit diagrams are traditionally used to document circuits**

Remember that 0 and 1 are represented by voltages

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AND Gate Timing Diagrams

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OR Gate

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Inverter

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More Inputs Work same way What’s output?

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**Digital Circuit Representation: Schematic**

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**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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**Digital Circuit Representation: Truth Table**

2n rows where n # of variables

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**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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**Identities Use identities to manipulate functions**

On previous slide, I used distributive law to transform from to

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Table of Identities

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

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**Single Variable Identities**

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Commutative Order independent

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**Associative Independent of order in which we group**

So can also be written as and

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Distributive Can substitute arbitrarily large algebraic expressions for the variables

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**DeMorgan’s Theorem Used a lot NOR equals invert AND**

NAND equals invert OR

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**Truth Tables for DeMorgan’s**

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**Algebraic Manipulation**

Consider function

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Simplify Function Apply Apply Apply

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Fall 2005 Fewer Gates

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**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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**Complement of a Function**

Definition: 1s & 0s swapped in truth table

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**Truth Table of the Complement of a Function**

X Y Z F = X + Y’Z F’ 1

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**Algebraic Form for Complement**

Mechanical way to derive algebraic form for the complement of a function Take the dual Recall: Interchange AND & OR, and 1s & 0s 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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**Example: Algebraic form for the complement of a function**

F = X + Y’Z To get the complement F’ Take dual of right hand side X . (Y’ + Z) 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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**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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**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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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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**Definition: Minterm (continued)**

Each minterm represents exactly one combination of the binary variables in a truth table.

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**Truth Tables of Minterms**

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**Number of Minterms For n variables, there will be 2n minterms**

Minterms are labeled from minterm 0, to minterm 2n-1 m0 , m1 , m2 , … , m2n-2 , m2n-1 For n = 3, we have m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7

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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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**Definition: Maxterms (continued)**

mmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm,m xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ,mmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmm Maxterm

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**Truth Tables of Maxterms**

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**Minterm related to Maxterm**

Minterms and maxterms with same subscripts are complements Example

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**Standard Form of F: Sum of Minterms**

OR all of the minterms of truth table for which the function value is 1 F = m0 + m2 + m5 + m7

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**Complement of F Not surprisingly, just sum of the other minterms**

In this case F’ = m1 + m3 + m4 + m6

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**Product of Maxterms Recall that maxterm is true except for its own row**

So M1 is only false for 001

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**Product of Maxterms or F = m0 + m2 + m5 + m7 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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**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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