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Chapter7 DiscretetimesignalDescription

*comparisonwithcontinuousdescription

7.1 discretetimesignals

Comefrom:

<1> measurablediscretequantities; example:population

<2> sampledcontinuousquantities; example:electricalquantities

 

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7.2 ∗

a) Unitimpulse(KroneckDelta∗)sequences

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Note:notasingularfunction!

b) Unitstepsequence

101

0

Relationwithunitimpulse∗

1

…

…

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c) Unitrampsequence

00 0

d) Unitalternating

u 1 00 0

e) Unitexponential

f Unitsinusoid

cos

g Complexexponential

cos sin

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7.3 DiscretePeriodicSignals

<1> Forallk

<2> PeriodNisthesmallestnumberforwhichsignalrepeats!

Nowlookat

Then 1⇒ 2

2 2

Thenormalizedfrequencymustberationalfortheperiodicityofdiscretesignals.

Distinctvalueof Ω doesnotalwaysproduceperiodicsignals!

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DISCRETE‐TIMEPERIODICSIGNALS

Adiscrete‐time signal k , k 0, 1, 2, … is said to be periodicwith period P,whereP is a positive integer, if

7.1

forallintegers kin ∞,∞ .If(7.2)holds,then

2 ⋯

foranykandeverypositiveintegerm.Thusif k isperiodicwithperiodP,itisperiodicwithperiod2P,3P,…The

smallestsuchPiscalledthe fundamentalperiod.Unlessstatedotherwise,theperiodwillrefertothefundamental

period.Thefundamentalfrequencyisdefinedas2π/P.

Beforeproceeding,wediscusssomedifferencesbetweensinusoidal functionsandsinusoidalsequences. In the

continuous‐timecase, sin isperiodicforeveryω.Inthediscrete‐timecase,however, sin maynotbeperiodic

foreveryω.Theconditionfor sin tobeperiodicisthatthereexistsapositivePsuchthat

sin sin sin

forallk.thisholdsifandonlyif

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2 2

7.2

forsomeintegerm.Thus sin isperiodicifandonlyif / isarationalnumber.

Inotherwords, sin isperiodicifandonlyifthereexistsanintegermsuchthat

2

7.3

is apositive integer.The smallest suchP is the fundamentalperiodof sinωk.Forexample, sin 2k isnotperiodic

because isnot a rational number. In this case, thereexistsno integerm in P such thatP is integer.The

sequence sin 0.01 is periodic because . is a rational number. Its period is P.

200 200 by choosing 1. The sequence sin 3πk is periodic with period P 2 by choosing

3.

Consider sin 3.2πk.Itcanbesimplifiedas

3.2 2 1.2 2 1.2 2 1.2 1.2

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wherewehaveused the fact that sin2πk 0andcos2πk 1 forevery integerk.This implies thatwhenweare

given sinωk,wecanalwaysreduceωtotherange 0,2π bysubtractingoradding2πoritsmultiple.Thusinthe

discrete‐timecase,wehave

6.2 0.2 2.4 1.6

forallintegerk.Inthecontinuous‐timecase, sin 6.2πt and sin 0.2πt aretwodifferentfunctions.

Inthecontinuous‐timecase,thefundamentalfrequencyof sinωt and cosωt isωinradianspersecond.Inview

of (7.4), the fundamental frequency of sinωk and cosωk may not be equal to ω. To better see the relationship

between the fundamental frequency and ω, we plot in Fig. 7.1 cosωk and cos 1.9πk. The sequence cos πk has

period P 2 andfundamentalfrequency .Inordertofindtheperiodof cos 1.9 ,wecompute

21.9

21.9

The smallest integer m to make P an integer is 19. Thus the period of cos 1.9πk is P 2 •.

20, and the

fundamentalfrequencyis 2π/20 0.1π.Weseethatthisfundamentalfrequencyissmallerthantheoneof cos πk,

thus cos πk changesmorerapidlythan cos 1.9πk asshowninFig7.1.Thusinthediscrete‐timecase, cosω k may

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nothaveahigherfundamentalfrequencythanthatof cosω k evenif ω ω .Thisphenomenondoesnotexistin

thecontinuous‐timecase. Inconclusion, the fundamental frequencyof cosωk isnotnecessarilyequal to ω as in

the continuous‐time case. To compute its fundamental frequency, wemust use (7.3) to compute its fundamental

periodP.thenthefundamentalfrequencyisequalto 2π/P.

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7.4 Basicoperations

 

a) Time‐reversal

b) Time‐scaling

Note:

ahastobeinteger!

Time‐reversal 1

c) Time‐delay

nhastobeinteger!

d) Combinations

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7.5symmetry

  

Even    if   

 

Odd    if   

 

A signal does not have to be even or odd! 

 

An arbitrary signal   

 

2

2

 

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7.6 Representationbyorthonormalfunctions

—Signals are often represented in terms of basis functions 

—Basis functions are often chosen to be orthonormal 

—Basis vectors 

*Choose  , ,   to be basis vectors. 

*Project rector F on basis vector, projection coefficients or weights are  , ,  

① 

orthogonality    • , 0,  

normality          • 1 

orthonormality  • 1, 0,  

 

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Projection 

• ② 

0 

Then 

i           F can be represented by 

ii       are complete

iii     F is in the span of   

① and ② are similar to Fourier Series Expansion. 

 

 

 

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Basis functions   

 

 

Condition: 

— ⇔ | | , | |  

Orthonormal functions: 

∗ ‖ ‖ 10  

Look at 

∗ ∗ ∗  

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WalshFunction

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Examples 

—cos2

∈ 0, 1  

— ∈ 0, 1  

—  

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11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

—

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Basisfunctioncanbechosenas