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OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Signals and Systems

Lecture 3: Fourier Series(CT)

Farzaneh Abdollahi

Department of Electrical Engineering

Amirkabir University of Technology

Winter 2012

Farzaneh Abdollahi Signal and Systems Lecture 3 1/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

LTI Systems Response to Complex Exponential Signals

Fourier Series for CT Signals

Fourier Series Convergence

Properties of CT Fourier Series

Farzaneh Abdollahi Signal and Systems Lecture 3 2/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Introduction

I Perviously, we have seen that by using convolution, LTI systems can be represented by linear combination of linear impulses. I Now we are going to describe LTI systems based on linear combination of a set of basic signals ( complex exponentials I Based on superposition property of LTI systems, response to any input including linea rcombination of basic signals is the same linea r combination of the individual responses to each of the basic signals I

At rst, CT/DT periodic signals are described by

F ourierSeries

I Then

F ouriertransfo rm

is intro ducedto rep resentap eriodicsignals Farzaneh Abdollahi Signal and Systems Lecture 3 3/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

LTI Systems Response to Complex Exponential Signals I For analyzing LTI system, the signals can be represented as a linear combination of basic signals. I

The basic signals should

1. b eable to b eused to construct a wide range of useful class of signals 2. have simple structure in L TIsystem resp onse. I Complex exponential signals are good candidate for basic signal since I its LTI system response is the same complex exponential with dierent amplitude

Ifor CT:est!H(s)est(H(s): a function ofs)

Ifor DT:zn!H(z)zn(H(z): a function ofz)Farzaneh Abdollahi Signal and Systems Lecture 3 4/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

I Eigenfunction:a signal for which the system output is a modulus of input. I

The constant value which can be complex is called

eigenvalue I

Consider input signal:x(t) =est:

y(t) =R1

1h()es(t)d=estZ1

1 h()es()d |{z}

H(s)=H(s)est

I )x(t) =estis an eigenfunction I

Consider input signal:x[n] =zn:

y[n] =P1 k=1h[k]znk=zn1X k=1h[k]zk |{z}

H(z)=H(z)zn

I )x[n] =znis an eigenfunctionFarzaneh Abdollahi Signal and Systems Lecture 3 5/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

I x(t) =P1 k=1akeskt!y(t) =P1 k=1akH(sk)eskt I x[n] =P1 k=1akznk!y[n] =P1 k=1akH(zk)zn I

For now assume thatsandzare purely imaginary:

I s=j! est=ej!t I z=ej! zn=ei!nFarzaneh Abdollahi Signal and Systems Lecture 3 6/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Fourier Series for CT Signals

I

Fourier series can represent CT periodic signals

I Remember the denition of periodic signals:x(t) =x(t+T) with fundamental periodTand fundamental frequency!0=2T Ix(t) =ej!tis a periodic signal with fundamental freq.!0 IRemember the harmonically related complex exponentials: k(t) =ejk!0t=ejk(2T )t;k= 0;1;2;::: I Each harmonic has fundamental freq. which is a multiply of!0 I A linear combination of harmonically related complex exponentials: x(t) =1X k=1a kejk!0t=1X k=1a kejk2=Tt(1) I it is periodic with periodT

Ik=1 have fundamental freq.!0(rst harmonic)

Ik=Nhave fundamental freq.N!0(Nth harmonic)Farzaneh Abdollahi Signal and Systems Lecture 3 7/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

I

Equation

(1) is F ourierSeries rep resentation: x(t) =1X k=1a kejk!0t=1X k=1a kejk2=Tt I

Now ifx(t) is real:

I x(t) =x(t) =P1 k=1akejk!0t I k=k x(t) =P1 k=1akejk!0t I ak=ak

IIfakis real:ak=ak.

I x(t) =a0+P1 k=1akejk!0t+akejk!0t= a 0+P1 k=1akejk!0t+akejk!0t=a0+ 2P1 k=1Refakejk!0tg I

Describeakby polar representation:ak=Akejk:

I

Second denition of Fourier Series:x(t) =a0+ 2P1

k=1Akcos(k!0t+k) I

Describeakby Cartesian representation:ak=Bk+jCk:

I

Third denition of Fourier Series:

x(t) =a0+ 2P1 k=1Bkcos(k!0t)Cksin(k!0t)Farzaneh Abdollahi Signal and Systems Lecture 3 8/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

How to nd coecientakI

Rewrite Fourier series:x(t) =P1

k=1akejk!0t I

Multiply both sides byejn!0t:x(t)ejn!0t=P1

k=1akejk!0tejn!0 I

TakeRT

0:RT

0x(t)ejn!0tdt=RT

0P 1 k=1akej(kn)!0tdt I

Tis fundamental period ofx(t):

Z T 0 x(t)ejn!0tdt=1X k=1a kZ T 0 ej(kn)!0tdt(2) I

Use Euler's formula:RT

0ej(kn)!0tdt=RT

0cos((kn)!0t)dt+jRT

0sin((kn)!0t)dt

I cos((kn)!0t) andsin((kn)!0t) are periodic with fundamental period: TjknjFarzaneh Abdollahi Signal and Systems Lecture 3 9/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

I )ej(kn)!0t=T k=n 0k6=n I

Substituting the above equation to

(2) :RT

0x(t)ejn!0tdt=anT

I

Analysis Equation of Fourier Series:an=1T

R T

0x(t)ejn!0tdt

I

Synthesis Equation of Fourier Series:x(t) =P1

k=1akejk!0t I dc or constant component ofx(t):a0=1T R Tx(t)dtFarzaneh Abdollahi Signal and Systems Lecture 3 10/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Example: Fourier Series of periodic square wave

I x(t) =1jtj0T1 jtj T=2 I dc gain:a0=1T R T2 T2 x(t)dt=2T1T I k6= 0:ak=1T R T2 T2 x(t)ejk!0tdt=1T R T1

T1ejk!0tdt=

1jk!0Tejk!0tjT1T1=sink!0T1kFarzaneh Abdollahi Signal and Systems Lecture 3 11/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Example Cont'd

I a)T= 4T1, b)T= 8T1Farzaneh Abdollahi Signal and Systems Lecture 3 12/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Fourier Series Convergence

I

How can one guarantee thatP1

k=1akejk!0trepresent a periodic signal x(t)? I Let us use an engineering tool named "approximation error": I Assume that a periodic signal can be expressed by linear combination of limited complex exponential terms:xN(t) =PN k=Nakejk!0t I

Approximation erroreNis dened:eN=x(t)xN(t)

I

The energy of the error in one period:EN=R

TjeN(t)j2dt

I It can be shown that to achieve minEN, one should dene (show it!): a k=1T R

Tx(t)ejk!0tdt

I

N" EN#

I N! 1 limEN!0Farzaneh Abdollahi Signal and Systems Lecture 3 13/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Fourier Series Convergence

I

Ifak! 1the approximation will diverge

I Even for boundedakthe approximation may not be applicable for all periodic signals. I Convergence Conditions of Fourier Series Approximation 1.

E nergyof signal should b ea nite in a p eriod:R

Tjx(t)j2dt<1

I

This condition only guaranteesEN!0

IIt does not guarantee thatx(t) equals to its Fourier series at each momentt 2.

Dric hletConditions

I Over any period,x(t) must be absolutely integrable:R

Tjx(t)jdt<1

- with square integrability condition, it guaranteesjakj<1 IIn a single period,x(t) should have nite number of max and min IIn any nite interval of time, there are only a nite number of discontinuities. Each discontinuity should be nite. Farzaneh Abdollahi Signal and Systems Lecture 3 14/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Drichlet Conditions

Farzaneh Abdollahi Signal and Systems Lecture 3 15/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Gibbs Phenomenon

I It happens for FS of piecewise continuous periodic signals with bounded discontinuities. I FS produces ripples in the vicinity of discontinuity. I

By Increasing N

I the ripples concentrates around discontinuities.

ITotal energy of the ripples decreases

IBUT the amplitude of the largest ripple does not changeFarzaneh Abdollahi Signal and Systems Lecture 3 16/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Gibb's Phenomenon

Farzaneh Abdollahi Signal and Systems Lecture 3 17/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Properties of CT Fourier Series

I Considerx(t) andy(t): periodic signals with same fundamental periodT I x(t),ak;y(t),bk I

Linearity:Ax(t) +By(t),Aak+Bbk

I

Time Shifting:x(tt0),akejk!0t0

I In time shifting magnitude ofFscoecient remains the same but its angel is changed I

Time Reversal:x(t),ak

Ievenx(t),evenak:ak=ak

Ioddx(t),oddak:ak=ak

I

Time Scalingx(t),akbut fundamental periodT

Farzaneh Abdollahi Signal and Systems Lecture 3 18/19

OutlineL TISystems Resp onseto Complex Exp onentialSignals Fourier Series for CT SignalsProp ertiesof CT F ourierSeries

Properties of CT Fourier Series

I

Multiplication

:x(t)y(t),hk=P1 l=1albkl(DT convolution between coecients) I

Conjugation and Conjugate Symmetry

I

Realx(t),ak=ak(conjugate symmetric)

IReal and evenx(t),ak=ak(real and evenak)

IReal and oddx(t),ak=ak(purely imaginary and oddak),a0= 0

IEven part ofx(t),Refakg(show it!)

IOdd part ofx(t),jImfakg(show it!)

I

Periodic convolution

:x(t)y(t) (for one period) =RT=2

T=2x()y(t)d,Takbk

I

Parseval's Relation

:R

Tjx(t)j2dt=P1

k=1jakj2 I

Total average power

sum of average p owerin all ha rmoniccomp onents

IEnergy in time domain equals to energy in frequency domainFarzaneh Abdollahi Signal and Systems Lecture 3 19/19

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