1 convergence of fourier series can we get fourier series representation for all periodic signals....

1

Convergence of Fourier Series

• Can we get Fourier Series representation for all periodic signals.

• I.e. are the coefficients from eqn 3.39 finite or in other words the integrals do not diverge.

• If the coefficients are finite and substituted in the synthesis eqn, will the resulting infinite series converge or not to the original signal x(t).

2


N

Nk

tjkkeat 0)(x

-:form theof series finite aby is,that -:lsexponentiacomplex relatedly harmonical of

number finite a ofn combinatiolinear aby x(t)signal periodicgiven theingApproximat

N

3


t.coefficien seriesFourier our is which .)(1a

-:ist coefficien thefor choice particular theso, be tofor this shown that becan It

error. in theenergy theminimise tois objective The

dt. |(t)e |E

period. oneover error in theenergy theuse togoing are We

-x(t)(t)x-x(t)(t)e isthat

error.ion approximat thedenote (t)eLet

Tk

2N

T

NN

N

0

0

dtetxT

ea

tjk

N

Nk

tjkk

4

Convergence of Fourier SeriesIf x(t) has a Fourier Series representation, the best approximation is obtained by truncating the Fourierseries to the desired term

.|x(t)|

-:period single aover energy finite havemust signal theseriesFourier by the blerepresenta be tosignal periodic aFor

T

2

5

Dirichlet Conditions

Series.Fourier by the blerepresenta becannot signal theso violated,being conditions 3 theseof examples theshows 199 page 3.8 Figure

finite. is itiesdiscontinu theseofeach e,Furthermor ities.discontinu ofnumber finite aonly are there time,of interval finiteany In 3)

interval. in the minima and maxima ofnumber infinitean havenot should

function The signal. theof period singleany during minima and maxima ofnumber finite a than more no are there, isthat ; variationbounded is x(t) time,of interval finiteany In )2

.dt|x(t)| i.e.

,integrable absolutely bemust x(t)period,any Over 1)

T

6

Gibbs Phenomenon• The partial sum in the vicinity of the

discontinuity exhibits ripples.• The peak amplitudes of these ripples does not

seem to decrease with increasing N.• As N increases, the ripples in the partial sums

become compressed towards the discontinuity, but for any finite value of N, the peak amplitude of the ripples remains constant. Fig 3.9 pg 201 illustrates this phenomenon.

7

Properties of Continuous-time Fourier Series.

kkk

k

k

k

BbAa cThen ,c z(t)

By(t).Ax(t) z(t) T. period with periodic be also willsignals two theofn combinatiolinear Any

ion.represenat SeriesFourier thedenotes where,b y(t)

a t) x( T. period same with thesignals periodic twodenotes y(t) and Let x(t)

-:Linearity (1)

8


kT

2jk-

kjk-

0

2jk-

jk-)(

0

0k

k,

k

0

aeae )t-x(ty(t)Then

,e

)(T1e)(

T1

-:have weT,duration of intervalanover range also will and ,t-t Letting

.)t-x(tT1b

b y(t) ,a t) x(

).t-x(t y(t) preserved. is signal

theof T period the x(t),signal periodic a toapplied isshift timeaWhen -:Shifting Time (2)

000

000

00000

0

tt

k

tT

jk

kt

T

jkt

T

tjk

T

tjk

aea

dexdex

dte

9


.a ) x(-t ,a x(t)if is,That

.b

ax(-t)y(t)

-:have we-m,k ngSubstituti

a x(-t)

b y(t) ,a t) x( ).x(-t y(t)

preserved. is signal theof T period the x(t),signal periodic a toapplied is reversal timeaWhen

-:Reversal Time (3)

k-

k

k

-m

2

m-

-k

2

k

k,

k

k

tT

jm

tT

jk

a

e

e

10


-k

)(k

0

0

0at) x(

same. remain the tscoefficienFourier The . positivefor frequency lfundamenta and

,

T period with periodic is t)then x(

,2

frequency lfundamenta and period thebeing T with periodic is x(t)Ifsignal. theof period thechanges scaling timeofoperation The

-:Scaling Time (4)

tjke

T

11


-:tionMultiplica (5)

domain.frequency in then convolutio toequivalent isdomain timein thetion Multiplica

.h)()(

b y(t),a x(t)

k

k,

k

llklbatytx

12


-:symmetry conjugate andn Conjugatio (6)

*k-

*

**

k

a

-:symmetry conjugate be willtscoefficien seriesFourier ).(x x(t)real, is x(t)

)(

,a x(t)

k

k

a

ThetIf

atxthen

13


-:Relation sParseval' (7)

k

kTadttx

T.|||)(|1 22

The total average power in a periodic signalequals the sum of the average powers in all of its harmonic components.

14

The Discrete-time Fourier Series •In the previous lectures you have learned about:-

–Fourier Series Pair Equations for the continuous-time periodic signals.–And also their properties.

•The derivation is through using:-–Signals as represented by linear combinations of basic signals with the following 2 properties.–The set of basic signals can be used to construct a broad and useful class of signals.–The response of an LTI system is a combination of the responses to these basic signals at the input.

15

Parallel Between The Continous-time & The Discrete-time.

LTI

x(t) y(t)x[n] y[n]

Sai ......aaayThen

.......aaa x-:asInput Decompose

33221

332211

Fi

16

Parallel Between The Continous-time & The Discrete-time.

)(

)(

)(

-:Time-Continuouscompute. easy to s' toResponse -

d.constructe becan Signals of Class Broad -: thatso ][or )( Choose

k

k

ktj

jtjtj

tj

k

He

dethee

et

nt

k

kkk

k

17

Discrete-time

ValueEigen Function xEigen )(

][

][

knj

r

rjnjnj

njk

He

erhee

en

k

kkk

k

18

Linear Combinations of Harmonically Related Complex Exponentials,

1).-(N.2,........1,0,k ,][

-:bygiven is N period withsignals lexponentiacomplex time-discrete all ofSet

N. with periodic is e lexponentiacomplex theE.g.

frequency. lfundamenta theis 2N,integer positivesmallest theis period lfundementa The

N].x[n x[n]ifN period with periodic is x[n]signal time-discreteA

)/2(k

)/2(

0

0

nNjknjk

nNj

een

N

19

Linear Combinations of Harmonically Related Complex Exponentials.

ts.coefficien seriesFourier theare s'alues.integer va Nany of

nscombinatioany on can takesummation thewhere

aa][ax[n]

-:bygiven is SeriesFourier time-Discretefor Notation 1.-.N0,1,2,....k where

aa][ax[n]

-:follows as ,][

lsexponentiacomplex relatedly harmonical of nscombinatiolinear aby drepresente becan x[n]sequence periodic general moreA

k

)/2(

)(k

)(kk

(N)kk

)/2(kkk

kk

)/2(k

0

0

0

nNjk

Nk

njk

Nk

nNjk

k

njk

k

nNjknjk

een

een

een

20

Discrete-time Fourier Series

njk

N

njk

Nkk

njk

kk

n

nNkjnjk

enxN

ea

eanx

eeN

0

0

0

00

][1a

unknowns N with equations N

x[n]

1-0,1,2,...Nk ,][

integeran isn since ,1e where

frequency.in periodic is

,2frequency lFundamenta

N period with in time periodic x[n]

k

N2jN

)(

0

21

Discrete-time Fourier series pair

.][1][1a

,aax[n]

. frequencyfor symbol same back the Using

)/2(k

)/2(kk

00

0

0

Nn

nNjk

Nn

njk

nNjk

Nk

njk

Nk

enxN

enxN

ee

22

Continuous-time & Discrete-time Fourier Series

njk

Nn

njk

Nkk

T

tjkk

k

tjkk

enxN

Analysis

ea

dtetxT

a

eatx

0

0

0

0

0

][1a

:

x[n]

-:Synthesis-:FS time-Discrete

.)(1

,)(

-:FS time-Continuous

k

0

23

True for Discrete-time case

kin periodic akin periodic e

nin periodic e

nin periodic ][

k

jk

jk

0

0

n

n

nx

Not true for C-T

True also for C-T

24


• Continuous time:– x(t) square integrable– or Dirichlet conditions.

][][ N,p ,][

x[n]

time-Discrete

^

termsp

^0

0

nxnxeanx

ea

njkk

njk

Nkk

25

Fourier Series Representation of Discrete-time Periodic Signals

• Fourier series of continuous-time periodic signals are infinite series.

• Fourier series of discrete-time periodic signals are of finite series in nature.

• So for the fourier series of discrete-time periodic signals, mathematical issues of convergence do not arise.

26

.2/1a and 2/1a thusNh repeat wit

,21

21

21

21 x[n]ls,exponentiacomplex twoof sum a as signal the

expanding and iprelationsheigen theUsing.sin x(t)time-continuous toanalogousexactly isresult and N period lfundamenta with periodic is x[n],/2when

i.e. Ninteger an is/2 when caseFor integers. of ratio aor integer an is /2 ifonly periodic is ][ ,sin x[n]signal heConsider t

3.10 Example

1-N1N

11

)/2()/2(

0

0

0

00

jjtsCoefficienj

aandj

aequationsynthesisFrom

ej

ej

tN

nxn

nNjnNj

27

.)21()

21(

)21

23()

21

23(1x[n]

-: termsCollecting

].[21

][23][

211 x[n]

ls,exponentiacomplex as signal theexpanding and iprelationsheigen theUsing

N. period lfundamenta with periodic is x[n],/2when

)2/2cos(cos3sin1 x[n]signal heConsider t3.11 Example

)/2(22/)/2(22/

)/2()/2(

)2//4()2//4(

)/2()/2()/2()/2(

0

000

nNjjnNjj

nNjnNj

NnjNnj

nNjnNjnNjnNj

eeee

ej

ej

ee

eeeej

N

nnn

28

ts.coefficienFourier theseof phase magnitude/or imaginary real/ showsbook in the 15.321 ,

21),

21

23(a

),21

23(a and ,1a thusNh repeat wit

k. of seother valufor 0

,21

,21

)21

23(

)21

23(

,1:

3.11 Example

222211N

11-N0N

2

2

1

1

0

Figure

jaajaaj

a

jaatsCoefficien

awith

ja

ja

ja

ja

aequationsynthesisFrom

NN

k

29

0-N N

0 N-N

Re(ak)

Im (ak)

13/2

1/2

k

k

Example 3.11

30

0-N N

0 N-N

|(ak)|

phase(ak)

1

/2

k

k

2/10

1/2

2

Example 3.11

31

0-N N

|

n

Example 3.12

1

N1-N1

11

11

2

0

)/2()/2(

2

0

))(/2(

1

.1

.1

.Nnm

N

m

mNjkNNjkk

N

m

NmNjkk

eeN

a

eN

a

Let

1

1

)/2(k

1a

-:equation analysis FromNn

Nn

nNjkeN

32

0-N N

|

n

Example 3.12

1

N1-N1

2N,....N,0,k ,12

,...2,N0,k ,)/sin(

]/)2/1(2sin[1

1

1

NNa

and

NNk

NNkN

a

k

k

1

1

)/2(k

1a

-:equation analysis FromNn

Nn

nNjkeN

33

| Example 3.12

20,....,100,k ,2/110

12*2

,...20,100,k ,)10/sin(

]10/)2/12(2sin[101

k

k

a

andk

ka

04 7

-1 12 3

5 6 8 9 10

Case N=10, 2N1+1=5.1/2 1/2

34

| Example 3.12

0,....4,200,k ,4/120

12*2

,...40,200,k ,)20/sin(

]20/)2/12(2sin[201

k

k

a

andk

ka

0 47

-1 1 2 356 8

20

Case N=20, 2N1+1=5.1/4 1/4

-2 9101112

19

35

| Example 3.12

0,....8,400,k ,8/140

12*2

,...80,400,k ,)40/sin(

]40/)2/12(2sin[401

k

k

a

andk

ka

0 4 7-1 123 56 8 40

Case N=40, 2N1+1=5.1/8 1/8

-29

1 convergence of fourier series can we get fourier series representation for all periodic signals....

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