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Ship encountering the superposition of 3 waves. Fourier Series Representation of Periodic Signals. Signals can be represented as 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. - PowerPoint PPT Presentation

### Transcript of Ship encountering the superposition of 3 waves.

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Ship encountering the superposition of 3 waves.

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Fourier Series Representation of Periodic Signals

•Signals can be represented as 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.

4

Fourier Analysis

• Both properties are provided for by the complex exponential signals in the continuous and discrete time

• I.e. Signal of the form:-

numbers.complex are z & s where

time.discrete in the z

time.continuousin n

ste

5

The principle of superposition for linear systems.

LTI Systemx(t)

x[n]

y(t)

y[n]

case. time-discrete for the

...)(a)(ay(t) :Then

linear is system and

)( )(

......)(a)(a x(t)if

2211

k

2211

Similarly

tt

tt

tt

k

6

Criteria for choosing a set of basic signals in terms of which to decompose the input

compute easy to are ' toresponses -

.' of

ncombinatiolinear a as dconstructe

becan signals of class broad a-

: that so ][or )( Choose

...)(a)(ay(t) :Then

linear is system and

)( )(

......)(a)(a x(t)if

k

k

2211

k

2211

s

s

nt

tt

tt

tt

kk

k

7

Choice for basic signals that led to the convolution integral and convolution sum for

LTI Systems

Sumn Convolutio

][][

][][ :T-D

Integraln Convolutio

)()(

)()( :T-C

k

k

knhn

knn

ktht

ktt

k

k

8

Complex exponentials as a set of basic signals

.transforms-z complex z

s transformLaplace complex s

][ 1|z| :T-D

)( s :T-C

:AnalysisFourier

complex z ][

complex s )(

k

k

kk

kk

kk

k

nj

tjk

nk

tsk

k

k

k

en

etj

zn

et

9

Eigenfunction (est , zn) & Eigenvalues (H(s) , H(z))

LTI Systemx(t)

x[n]

y(t)

y[n]

amplitude.in change aonly with

lexponentiacomplex same theisinput lexponentia

complex a tosystem LTIan of response the

fact that thefrom stems systems LTI ofstudy

in the lsexponentiacomplex of importance The

H(z)zy[n] z x[n]: timediscrete

)()(e x(t): timenn

st

stesHtycontinuous

10

Showing complex exponential as eigenfunction of system

.)(constantcomplex a is H(s)

whereH(s)ey(t) form theof is e toreponse the

converge, RHSon integral that thessuming

.)(

.)(

)()()(

einput x(t) with integraln convolutio theFrom

stst

-

)(

st

deh

A

dehe

deh

dtxhty

s

sst

ts

11

Showing complex exponential as eigenfunction of system

k

kn

kn

n

zkh

A

zkhz

zkh

knxkhny

z

][constantcomplex a is H(z)

whereH(z)zy[n] form theof is z toreponse the

converges, RHSon summation that thessuming

][

][

][][][

,input x[n] with sumn convolutio theFrom

-:case Time-D for theSimilarly

nn

12

Example x(t) to be linear combination of 3 complex exponentials

tsts

tsts

tsts

tststs

esHe

esHe

esHe

eee

33

22

11

321

)(aa

)(aa

)(aa

-:is separatelyeach toresponse the

property,ion eigenfunct theFrom

aaax(t)

333

222

111

321

13

Example x(t) is linear combination of 3 complex exponentials

kk

kk

kk

kk

332211

321

.)(ay[n] be loutput wil then the

,ainput x[n] if time-Dfor similarly

)(ay(t) be loutput wil then the

,ainput x(t) if Generally,

)(a)(a)(ay(t)

-:responses theof sum theis sum the toresponse the

property,ion superposit theFrom

aaax(t)

321

321

nkk

nk

tsk

ts

tststs

tststs

zzH

z

esH

e

esHesHesH

eee

k

k

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Fourier Series Representation of Continuous-time Periodic Signals.

A signal is periodic if x(t)=x(t+T), for all t.T is the fundamental period.

,e x(t)lexponentiacomplex and

)cos( x(t)sinusoidal signals,

periodic basic 2 before studied have We

0j

0

0

t

t

T

15

.components harmonics second theis 2k component,

harmonicfirst theis 1k T. period with periodic also is and series

Fourier theis, x(t)lexponentia

complex relatedly harmonical ofn combinatiolinear a Thus,

T). offraction a is period lfundamenta the

2,kfor ( T with periodic iseach and of multiple a is

thatfrequency lfundamenta a has signals theseofEach

2,...1,0,k ,)(

ls,exponentiacomplex relatedly harmonical ofset the

is signal lexponentiacomplex basic with thisAssociated

-k

2

-k

,0

2

k

00

0

tT

jk

ktjk

k

tT

jktjk

eaea

eet

16

above. x(t)signal peroidic aconstruct toup added becan

waveformscosine thesehow sillustrate OWN 188 pg 3.4 Figure

.6cos3

24cos2cos

2

11)(

-:below form cosine get the to

),e(e2

1cos ip,relationshEuleruse

.3

1,

2

1,

4

1,1 where

,)(

3.2 Example

j-j

3322110

3

3

2

ttttx

aaaaaaa

eatxk

tjkk

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Trigonometric forms of Fourier Series.

][)( -:3.25Eqn From

.or ,

3.25, eqn. with comparing and (t)

k,-by k replacing ,(t)

x(t).(t) x x(t),realFor ,(t)

-:have we3.25, eqn. of sideboth ofn conjugatio Taking

. SeriesFourier the

of Form lExponentiaComplex theis

Equation) (Synthesis 25.3.........)(

1k0

**k-

*

*

***

00

0

0

0

0

tjkk

tjkk

kkk

k

tjkk

k

tjkk

k

tjkk

k

tjkk

eaeaatx

aaaa

eax

eax

eax

eqneatx

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Trigonometric forms of Fourier Series.

].sincos[2)(

,a lettingby formother the

, )cos(A2ax(t)

}.2)( ,a Expressing

}.2][)(

01

00

k

1k0k0

1

)(0k

10

1k

*0

0

000

tkCtkBatx

jCB

tk

eRe{AatxeA

eRe{aaeaeaatx

kk

k

kk

k

k

tkjk

jk

k

tjkk

tjktjkk

kk

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Solving for Fourier Series Coefficients(Analysis)

TTT tnkj

k

T tnkjk

tjn

k

T tjntjkk

tjn

tjntjk

kk

tjn

tjntjk

kk

tjn

tjn

tdtnkjtdtnkdte

eeaetx

e

0 00 00

)(

0

)(T

0

0

T

0

T

0

T

0

0

)sin()cos(However

3.34quation .........e ][)(

][)(

)(

have we,2

T to0 fromequation of sidesboth gIntegratin

.)(

-:by eqn Synthesis SeriesFourier theof sideboth gMultiplyin

0

00

000

000

000

0

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Solving for Fourier Series Coefficients(Analysis)

T0 dt

t0

jnx(t)e

T

1n

a

.

-:nk taking3.34Equ From

nk 0.

nk T,T0 dt

t0

n)j(ke

-:have weOverall

TT0 0dtjT

0 1dtT0 dt

t0

n)j(ke

0t0

n)sin(k 1andt 0

n)cos(k n,kFor

zeros. are termcosine and sinefor

periods lfundamenta of numbersinteger eover whol nsintegratio The

).|n-k|

T( period lfundamenta with

sinusoids periodic aret n)sin(k andt n)cos(k n,kFor 00

21

Summarizing for Fourier Series Pair Representation

tT

jk

kk

tjk

kk eaeatx

2

0)(

TT

dtt

T

2jk

x(t)eT

1dt

t0

jkx(t)e

T

1k

a

Fourier Series Synthesis:- eqn 3.38

Fourier Series Analysis or F.S. Coefficients 0r Spectral Coefficents:- eqn 3.39

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.2||0,2

1,

2

1

),2

11(),

2

11(,1

-:are tscoefficien seriesFourier

)2

1)

2

1)

2

11()

2

11(1)(

][2

1][][

2j

11x(t)

-:)e(e2

1sin and

)e(e2

1cosfor iprelationshEuler use

)4

2cos(cos2sin1)(

3.4 Example

)4/(2

)4/(2

110

2)4/(2)4/(

4/24/2

j-j

j-j

000

0000

000000

kforaeaea

ja

jaa

eeeeej

ej

tx

eeeeee

j

ttttx

kjj

tjjtjjtjtj

tjtjtjtjtjtj

23

0

0.2

0.4

0.6

0.8

1

1.2

k= -3 -2 -1 0 1 2 3 4

Magnitude of Fourier Coefficients.

|| ka

24

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k= -3 -2 -1 0 1 2 3 4

Phase of Fourier Coefficients.

kaangle

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Example 3.5

T/2T1-T/2 -T1

This periodic signal x(t) repeats every T seconds.x(t)=1, for |t|<T1 , and x(t)=0, for T1 <|t|< T/2

Fundamental period= T,Fundamental frequency Choosing the period of integration to be between-T/2 and +T/2. Use eqn 3.39 to get at Fourier Series Coefficients.

26

Example 3.5 continued

T

Tdt

T

T

TT

12

T

10

a

0k i.e. first, period, aover

valueaverageor ermconstant t dc, get the usLet

dtt

T

2jk

x(t)eT

1dt

t0

jkx(t)e

T

1k

a

1

1

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Example 3.5 continued

k

Tk

Tk

Tk

j

ee

Tk

eTjk

TjkTjk

TT

tjkT

T

TT

)sin()sin(2]

2[

2k

a

,|1

dtt

0jk

eT

1k

a

0.k have we

-:harmonicsorder higher andorder first lfundamentaFor

dtt

T

2jk

x(t)eT

1dt

t0

jkx(t)e

T

1k

a

10

0

10

0

0

1010

1

1

01

1

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Example 3.5 Continued

• In this example the coefficients are real values.

• Generally the coefficients are complex.

• In this case we can represent a single plot of magnitude of coefficient against k.

• Generally we will have the magnitude plot and the phase plot of the coefficients.

29

Example 3.5 Continued

.2

1a 3.42eqn from and

zeros. all are sa' thesk'even For

,5

1,

3

1,

1

0,k ,)2/sin(

a -: 3.44eqn From

,2

2 ,T4TFor

0

553311

k

1101

aaaaaa

k

k

TT

T

30

Plot of the Coefficients with T1 Fixed and T varied.

2

1x

a) T=4T1

b) T=8T1

c) T=16T1