Ship encountering the superposition of 3 waves.

transcript

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.

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

The principle of superposition for linear systems.

LTI Systemx(t)

case. time-discrete for the

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

linear is system and

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

Similarly

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

compute easy to are ' toresponses -

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

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

LTI Systems

Sumn Convolutio

][][ :T-D

Integraln Convolutio

)()( :T-C

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 )(

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

LTI Systemx(t)

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

stesHtycontinuous

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

dtxhty

Showing complex exponential as eigenfunction of system

knxkhny

][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

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

tststs

-:is separatelyeach toresponse the

property,ion eigenfunct theFrom

aaax(t)

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

332211

.)(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)

tststs

esHesHesH

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

frequency.radian lfundamenta theis ,2

.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

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

waveformscosine thesehow sillustrate OWN 188 pg 3.4 Figure

.6cos3

24cos2cos

-:below form cosine get the to

),e(e2

1cos ip,relationshEuleruse

1,1 where

3.2 Example

3322110

aaaaaaa

Trigonometric forms of Fourier Series.

][)( -:3.25Eqn From

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.........)(

eaeaatx

eqneatx

Trigonometric forms of Fourier Series.

].sincos[2)(

,a lettingby formother the

, )cos(A2ax(t)

}.2)( ,a Expressing

}.2][)(

tkCtkBatx

eRe{AatxeA

eRe{aaeaeaatx

tjktjkk

Solving for Fourier Series Coefficients(Analysis)

TTT tnkj

T tnkjk

T tjntjkk

tjntjk

tdtnkjtdtnkdte

dteadtetx

dteeadtetx

eeaetx

0 00 00

)sin()cos(However

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

have we,2

T to0 fromequation of sidesboth gIntegratin

-:by eqn Synthesis SeriesFourier theof sideboth gMultiplyin

Solving for Fourier Series Coefficients(Analysis)

jnx(t)e

-:nk taking3.34Equ From

nk T,T0 dt

n)j(ke

-:have weOverall

TT0 0dtjT

0 1dtT0 dt

n)j(ke

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

Summarizing for Fourier Series Pair Representation

kk eaeatx

x(t)eT

jkx(t)e

Fourier Series Synthesis:- eqn 3.38

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

.2||0,2

-:are tscoefficien seriesFourier

11(1)(

11x(t)

-:)e(e2

1sin and

1cosfor iprelationshEuler use

2cos(cos2sin1)(

3.4 Example

2)4/(2)4/(

4/24/2

000000

kforaeaea

eeeeej

eeeeee

tjjtjjtjtj

tjtjtjtjtjtj

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

Magnitude of Fourier Coefficients.

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

Phase of Fourier Coefficients.

kaangle

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.

Example 3.5 continued

0k i.e. first, period, aover

valueaverageor ermconstant t dc, get the usLet

x(t)eT

jkx(t)e

Example 3.5 continued

TjkTjk

)sin()sin(2]

0.k have we

-:harmonicsorder higher andorder first lfundamentaFor

x(t)eT

jkx(t)e

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.

Example 3.5 Continued

1a 3.42eqn from and

zeros. all are sa' thesk'even For

0,k ,)2/sin(

a -: 3.44eqn From

2 ,T4TFor

553311

aaaaaa

Plot of the Coefficients with T1 Fixed and T varied.

a) T=4T1

b) T=8T1

c) T=16T1

Ship encountering the superposition of 3 waves.

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