信号与系统课件第3章周期信号的傅里叶级数表示

CHAPTER3FOURIERSERIESREPRESENTATIONOF

PERIODICSIGNALS

3.0INTRODUCTION

Representationofcontinuous-timeanddiscrete-timeperiodicsignals—Fourierseries（傅立叶级数）.UseFouriermethodstoanalyzeandunderstandsignalsandLTIsystems.

3.1THERESPONSEOFLTISYSTEMSTOCOMPLEXEXPONENTIALS

Importantconcept—signaldecomposition

basicsignals:possesstwoproperties1.

Thesetofbasicsignalscanbeusedtoconstructabroadandusefulclassofsignals.2.ItshouldbeconvenientforustorepresenttheresponseofanLTIsystemtoanysignalconstructedasalinearcombinationofthebasicsignals.complexexponentialsignals

indiscretetime:incontinuoustime:

Eigenfunction（特征函数）Definingtwoquantities:H(s)andH(z)

H(s)orH(z)isingeneralafunctionofthecomplexvariablesorz.Eigenfunction(ofthesystem):aninputsignalforwhichthesystemoutputisaconstanttimestheinput.Foraspecificvalueofskorzk,or:eigenvalue.

（特征值）(1)ContinuoustimeLTIsystemh(t)x(t)=esty(t)=H(s)est(systemfunction)(2)DiscretetimeLTIsystemh[n]x[n]=zny[n]=H(z)zn(systemfunction)(3)InputasacombinationofComplexExponentialsContinuoustimeLTIsystem:DiscretetimeLTIsystem:3.2FOURIERSERIESREPRESENTATIONOFCONTINUOUS-TIMEPERIODICSIGNALS3.2.1ComplexExponentialFourierSeries（指数型傅立叶级数）

Given

periodicx(t)withfundamentalperiodT,itscomplexexponentialFourierseriesis:Thesignalsinthesetareharmonicallyrelatedcomplexexponentials.

:fundamentalcomponents

or

thefirstharmoniccomponents

（基波分量）

（一次谐波分量）theNthharmoniccomponents（N次谐波分量）wherethecoefficients

ak

isgenerallyacomplexfunctionof.

theFourierseriescoefficientsaredeterminedbyequation:synthesisequation:analysisequation:Fourierseriescoefficientsorspectrumofx(t):（傅立叶系数）（频谱）phasespectrum:（相位频谱）magnitudespectrum:

（幅度频谱）constantcomponentordc

ofx(t):

（直流分量）

小结：2.当为实函数时，，即与为一对共轭复数。有：相角是的奇函数。傅氏复系数的模是的偶函数；为直流分量，一般仍要单独1.计算；3.2.2TrigonometricFourierSeries

（三角型傅立叶级数）orTherelationshipsbetweenakandBk,Ck,Ak,θkare:直流分量余弦分量的幅度正弦分量的幅度关系曲线称为幅度频谱图关系曲线称为相位频谱图图中，每条竖线代表该频率分量的振幅（相角），称为谱线。连接各谱线顶点的曲线称为包络线，它反映了各分量振幅（相角）变化的情况。两者合称频谱图。...一个周期信号与它的频谱（幅度频谱和相位频谱）之间存在一一对应的关系。...试画出其振幅谱和相位谱例：一个周期信号可表示为解：先将含有相同频率的正弦项与余弦项合并为一个余弦项，且所有项都表示为带正振幅的余弦项。521234567

1注意：（1）三角型傅里叶级数必须统一用余弦函数表示；（2）振幅频谱必然位于横轴的上方；（3）相位频谱中的角度的绝对值不能大于。01234567

Constructionofthesignalx(t)asalinearcombinationoftheharmonicallyrelatedsinusoidalsignals(当)(当)再强调一遍：指数型和三角形两种傅氏级数的n，取值范围是不同的。指数型和三角形两种傅氏级数间的关系单边谱与双边谱的关系：1.振幅谱：直流分量一样，其它情况双边谱振幅是单边谱振幅的一半。2.相位谱两者在n>0时相同。3.双边振幅谱偶对称，相位谱奇对称。称为相位谱。由于指数型傅里叶谱在正负频率处均存在，故它又叫双边谱，三角型傅里叶谱又叫单边谱。Example3.1Considerarealperiodicsignalx(t),withfundamentalfrequency2π,thatisexpressedinthecomplexexponentialFourierseriesaswhereUsethetrigonometricformtoexpressthesignalx(t).Example3.2Considerthesignal

Plotthemagnitudespectrumandphasespectrumofx(t).Thus,theFourierseriescoefficientsforthisexampleare:11/2

-3-2-10123kπ/4karctan(1/2)

-21

-3-1023

Plotsofthemagnitudespectrumandphasespectrumofthesignalx(t)Example3.3Theperiodicsquarewaveisdefinedoveroneperiodas:……

-T-T/2-T1T1T/2Ttx(t)RepresentitinFourierseries.ForT=4T1,thecoefficientsare:ForT=8T1,thecoefficientsare:

-202kk-404k-808

PlotsoftheFourierSeriescoefficientsfortheperiodicsquarewavewithT1fixedandforseveralvaluesofT:(a)T=4T1;(b)T=8T1;(c)T=16T1.

spectrumof

periodicsquarewave当频谱的谱线无限密集，频谱振幅无限趋小，这时，周期信号已经向非周期信号转化。1.随着重复周期T的增大,则信号谱线间隔相应地渐趋密集；最大的频谱幅度(形象化称为主峰高度)渐趋减小。可以想象其频谱的形状没有改变。讨论：减小，谱线间隔不变；主峰高度减小；第2.一个零交点增加，也就是信号的频带宽度加大。可见，信号的频带宽度与脉宽成反比。2谐波性每条谱线只能出现在基波频率的整数倍的频率上，频谱中不可能存在任何频率为基波频率非整数倍的分量；3收敛性各次谐波的振幅，总的趋势是随着谐波次数的增高而逐渐减小的。在时域中是连续的周期函数，它的频谱在频域中是离散的非周期函数。周期信号频谱的特点：1离散性频谱由不连续的谱线组成，每一条线代表一个正弦分量，这样的频谱称为不连续频谱或离散频谱。据此可画出单边谱023单边相位谱0123321单边幅度谱23-3-201231.51-1-3023单边谱012332123-3-2双边谱01231.51-1-33.3CONVERGENCEOFTHEFOURIERSERIES

（傅立叶级数的存在性）Anapproximationofx(t)isThereexisterrorbetweentheoriginalsignalx(t)andtheapproximationxN(t)andwiththeNincreases,theerrordecreases.

TheDirichletconditions（狄里赫利条件）

areasfollows:Condition1:Overanyperiod,x(t)mustbeabsolutelyintegrable

Condition2:Inanyfiniteintervaloftime,x(t)isofboundedvariation;thatis,therearenomorethanafinitenumberofmaximaandminimaduringanysingleperiodofthesignal.Condition3:Inanyfiniteintervaloftime,thereareonlyafinitenumberofdiscontinuities.Furthermore,eachofthesediscontinuitiesisfinite.狄里赫利条件：（实际遇到的信号都满足）1.一个周期内只有有限个不连续点；2.一个周期内只有有限个极大值、极小值；3.一个周期内绝对可积，即Foraperiodicsignalthathasnodiscontinuities,theFourierseriesrepresentationconvergesandequalstheoriginalsignalateveryvalueoft.Foraperiodicsignalwithafinitenumberofdiscontinuitiesineachperiod,theFourierseriesrepresentationequalsthesignaleverywhereexceptattheisolatedpointsofdiscontinuity,atwhichtheseriesconvergestotheaveragevalueofthesignaloneithersideofthediscontinuity.Gibbsphenomenon（吉布斯现象）.ConvergenceoftheFourierseriesrepresentationofasquarewave:anillustrationoftheGibbsphenomenon.

Anycontinuity:xN(t1)x(t1)Vicinityofdiscontinuity:ripplespeakamplitudedoesnotseemtodecreaseDiscontinuity:overshoot9%Gibbs’sconclusion:3.4PROPERTIESOFCONTINUOUS-TIMEFOURIERSERIESWegenerallyuseashorthandnotationtoindicatetherelationshipbetweenaperiodicsignalanditsFourierseriescoefficients,thatis3.4.1LinearityIf

then

3.4.2TimeShiftingIf

then

Whenaperiodicsignalisshiftedintime,themagnitudespectrumremainsunaltered.

3.4.3TimeReversalIf

then

Timereversalappliedtoacontinuous-timesignalresultsinatimereversalofthecorrespondingsequenceofFourierseriescoefficients.Ifx(t)iseven:Ifx(t)isodd:

3.4.4TimeScalingIf

then

TheFouriercoefficientsforeachofthosecomponentsremainthesame.However,theharmoniccomponentschangewiththechangeinthefundamentalfrequency.3.4.5MultiplicationIf

then

hkistheconvolutionsumofthesequencerepresentingtheFouriercoefficientsofx(t)andthesequencerepresentingtheFouriercoefficientsofy(t).3.4.6ConjugationandConjugateSymmetry

（共轭对称）If

then

ifx(t)real,a0

isrealifx(t)isrealandeven,thensoareitsFourierseriescoefficients.ifx(t)isrealandodd,thenitsFourierseriescoefficientsarepurelyimaginaryandodd.

3.4.7Parseval’sRelationforContinuous-TimePeriodicSignals

（帕色伐尔定理）Parseval’srelationstatesthatthetotalaveragepowerinaperiodicsignalequalsthesumoftheaveragepowersinallofitsharmoniccomponents.Example3.4DeterminetheFourierseriesrepresentationofg(t)whichisshowninthefollowingfigure:

-2-112-1/2g(t)t1/2

g(t)=x(t-1)–1/2,wherex(t)istheperiodicsquarewaveinExample3.3,andT=4andT1=1.

timeshiftingproperty:theFouriercoefficientsofx(t-1)

is,whereakistheFouriercoefficientsofx(t).

-1/2is

thedcoffseting(t).(supposingthattheconstantcomponentinx(t)isa0,thentheconstantcomponenting(t)isa0–1/2.)linearproperty

:Example3.5Considerthetriangularwavesignalx(t)

whichisshowninthefollowingfigure.x(t)t-221Thederivativeofx(t)isthesignalg(t)inlastexamplewejustconsidered.Denotingthecoefficientsofg(t)bydkandthoseofx(t)byek,thenwehave:(differentiationproperty:)Thus,

Fork=0,e0

canbedeterminedbyfindingtheareaunderoneperiodofx(t)anddividingbythelengthoftheperiod:Example3.6（冲激串）DeterminetheFourierseriesrepresentationoftheimpulsetrain,whichisperiodicwithperiodTandisexpressedas:1……

-TTtx(t)1……

-T-T/2-T1T1T/2Ttg(t)-11……

-T/2-T1T1T/2Ttg’(t)g’(t)=x(t+T1)–x(t–T1).Example3.7

Givingthefollowingfactsaboutasignalx(t):

1.x(t)isarealsignal;

2.x(t)isperiodicwithperiodT=4,andithasFourierseriescoefficients;3.for;4.ThesignalwithFouriercoefficientsisodd;5..Determinethesignal.or3.5FOURIERSERIESREPRESENTATIONOFDISCRETE-TIMEPERIODICSIGNALS3.5.1LinearCombinationofHarmonicallyRelatedComplexExponentialsGiven

periodicx[n]withfundamentalperiodN,itsFourierserieshastheform:Since

finite

series

Thismeansthatdiscrete-timecomplexexponentialswhichdifferinfrequencybyamultipleof2πareidentical.

Consequently,thereareonlyNdistinctsignalsintheset

ThesummationneedonlyincludetermsoverarangeofNsuccessivevaluesofk.Weusetoindicatethis.Then,3.5.2DeterminationoftheFourierSeriesRepresentationofaPeriodicSignalMultiplyingbothsidesofthediscrete-timeFourierseriesequationbyandsummingoverNterms,weobtainInterchangingtheorderofsummationontheright-handside,wehave

theFourierseriescoefficientsaredeterminedbyequation:synthesisequation:analysisequation:periodic

Discreteness↔PeriodicityExample3.8Considerthesignalx[n]=sin3(2π/5)n,drawthegraphofcoefficients.ThissignalisperiodicwithperiodN=5.

-1/2j1/2j-7-238

-8-32712

……kFouriercoefficientsforx[n]=sin3(2π/5)n.k1/2-9-8-7-5-4-301234567891011

-6-2-112

……

Magnitudeofthecoefficients.(magnitudespectrum)-π/2π/2-9-7-5-4-2013456891012

-8-6-3-12711

……kPhaseofthecoefficients.(phasespectrum)Example3.9Considerthediscrete-timeperiodicsquarewave:n1–N–N10N1N

……Therearenoconvergenceissueswiththediscrete-timeFourierseriesingeneral,becauseanydiscrete-timeperiodicsequencex[n]iscompletelyspecifiedbyafinite

numberNofparameters.

FourierseriescoefficientsfortheperiodicsquarewaveofExample3.9;plotsofNakfor2N1+1=5and(a)N=10;(b)N=20;(c)N=40

3.6PROPERTIESOFDISCRETE-TIMEFOURIERSERIES3.6.1MultiplicationIfandthenperiodicconvolution

3.6.2FirstDifferenceIfthen3.6.3Parseval’sRelationforDiscrete-TimePeriodicSignals

istheaveragepowerinthekthharmoniccomponentofx[n].Parseval’srelationstatesthattheaveragepowerinaperiodicsignalequalsthesumoftheaveragepowersinallofitsharmoniccomponents.Differentfromthecontinuoustimecase,indiscretetime,thereareonlyNdistinctharmoniccomponents.Example3.10FindtheFourierseriescoefficientsofthesequencex[n]showninthefigure:…

-505x[n]

21…n…

-505x2[n]1…n…

-505x1[n]

1…nRepresentingx[n]asasumofthesquarewavex1[n]andthedcsequencex2[n]Example3.11

Givingthefollowingfactsaboutasequencex[n]:1.

x[n]isperiodicwithperiodN=6.2.3.4.x[n]hastheminimumpowerperperiodamongthesetofsignalssatisfyingtheprecedingthreeconditions.Determinethesequencex[n].3.7FOURIERSERIESANDLTISYSTEMSsystemfunction（系统函数）

IfRe{s}=0,s=jω.

If|z|=1,.frequencyresponse（频率响应）

Incontinuoustime,letx(t)beaperiodicsignalwithFourierseriesrepresentationgivenbyThen,theoutputis

Thatis,theeffectoftheLTIsystemistomodifyindividuallyeachoftheFouriercoefficientsoftheinputthroughmultiplicationbythevalueofthefrequencyresponseatthecorrespondingfrequency.Indiscretetime,letx[n]beape