信號(hào)與系統(tǒng)課件第3章周期信號(hào)的傅里葉級(jí)數(shù)表示

CHAPTER3FOURIERSERIESREPRESENTATIONOF

PERIODICSIGNALS

3.0INTRODUCTION

Representationofcontinuous-timeanddiscrete-timeperiodicsignals—Fourierseries（傅立葉級(jí)數(shù)）.UseFouriermethodstoanalyzeandunderstandsignalsandLTIsystems.

3.1THERESPONSEOFLTISYSTEMSTOCOMPLEXEXPONENTIALS

Importantconcept—signaldecomposition

basicsignals:possesstwoproperties1.

Thesetofbasicsignalscanbeusedtoconstructabroadandusefulclassofsignals.2.ItshouldbeconvenientforustorepresenttheresponseofanLTIsystemtoanysignalconstructedasalinearcombinationofthebasicsignals.complexexponentialsignals

indiscretetime:incontinuoustime:

Eigenfunction（特征函數(shù)）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（指數(shù)型傅立葉級(jí)數(shù)）

Given

periodicx(t)withfundamentalperiodT,itscomplexexponentialFourierseriesis:Thesignalsinthesetareharmonicallyrelatedcomplexexponentials.

:fundamentalcomponents

or

thefirstharmoniccomponents

（基波分量）

（一次諧波分量）theNthharmoniccomponents（N次諧波分量）wherethecoefficients

ak

isgenerallyacomplexfunctionof.

theFourierseriescoefficientsaredeterminedbyequation:synthesisequation:analysisequation:Fourierseriescoefficientsorspectrumofx(t):（傅立葉系數(shù)）（頻譜）phasespectrum:（相位頻譜）magnitudespectrum:

（幅度頻譜）constantcomponentordc

ofx(t):

（直流分量）

小結(jié)：2.當(dāng)為實(shí)函數(shù)時(shí)，，即與為一對(duì)共軛復(fù)數(shù)。有：相角是的奇函數(shù)。傅氏復(fù)系數(shù)的模是的偶函數(shù)；為直流分量，一般仍要單獨(dú)1.計(jì)算；3.2.2TrigonometricFourierSeries

（三角型傅立葉級(jí)數(shù)）orTherelationshipsbetweenakandBk,Ck,Ak,θkare:直流分量余弦分量的幅度正弦分量的幅度關(guān)系曲線稱為幅度頻譜圖關(guān)系曲線稱為相位頻譜圖圖中，每條豎線代表該頻率分量的振幅（相角），稱為譜線。連接各譜線頂點(diǎn)的曲線稱為包絡(luò)線，它反映了各分量振幅（相角）變化的情況。兩者合稱頻譜圖。...一個(gè)周期信號(hào)與它的頻譜（幅度頻譜和相位頻譜）之間存在一一對(duì)應(yīng)的關(guān)系。...試畫出其振幅譜和相位譜例：一個(gè)周期信號(hào)可表示為解：先將含有相同頻率的正弦項(xiàng)與余弦項(xiàng)合并為一個(gè)余弦項(xiàng)，且所有項(xiàng)都表示為帶正振幅的余弦項(xiàng)。521234567

1注意：（1）三角型傅里葉級(jí)數(shù)必須統(tǒng)一用余弦函數(shù)表示；（2）振幅頻譜必然位于橫軸的上方；（3）相位頻譜中的角度的絕對(duì)值不能大于。01234567

Constructionofthesignalx(t)asalinearcombinationoftheharmonicallyrelatedsinusoidalsignals(當(dāng))(當(dāng))再強(qiáng)調(diào)一遍：指數(shù)型和三角形兩種傅氏級(jí)數(shù)的n，取值范圍是不同的。指數(shù)型和三角形兩種傅氏級(jí)數(shù)間的關(guān)系單邊譜與雙邊譜的關(guān)系：1.振幅譜：直流分量一樣，其它情況雙邊譜振幅是單邊譜振幅的一半。2.相位譜兩者在n>0時(shí)相同。3.雙邊振幅譜偶對(duì)稱，相位譜奇對(duì)稱。稱為相位譜。由于指數(shù)型傅里葉譜在正負(fù)頻率處均存在，故它又叫雙邊譜，三角型傅里葉譜又叫單邊譜。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當(dāng)頻譜的譜線無限密集，頻譜振幅無限趨小，這時(shí)，周期信號(hào)已經(jīng)向非周期信號(hào)轉(zhuǎn)化。1.隨著重復(fù)周期T的增大,則信號(hào)譜線間隔相應(yīng)地漸趨密集；最大的頻譜幅度(形象化稱為主峰高度)漸趨減小?？梢韵胂笃漕l譜的形狀沒有改變。討論：減小，譜線間隔不變；主峰高度減小；第2.一個(gè)零交點(diǎn)增加，也就是信號(hào)的頻帶寬度加大。可見，信號(hào)的頻帶寬度與脈寬成反比。2諧波性每條譜線只能出現(xiàn)在基波頻率的整數(shù)倍的頻率上，頻譜中不可能存在任何頻率為基波頻率非整數(shù)倍的分量；3收斂性各次諧波的振幅，總的趨勢(shì)是隨著諧波次數(shù)的增高而逐漸減小的。在時(shí)域中是連續(xù)的周期函數(shù)，它的頻譜在頻域中是離散的非周期函數(shù)。周期信號(hào)頻譜的特點(diǎn)：1離散性頻譜由不連續(xù)的譜線組成，每一條線代表一個(gè)正弦分量，這樣的頻譜稱為不連續(xù)頻譜或離散頻譜。據(jù)此可畫出單邊譜023單邊相位譜0123321單邊幅度譜23-3-201231.51-1-3023單邊譜012332123-3-2雙邊譜01231.51-1-33.3CONVERGENCEOFTHEFOURIERSERIES

（傅立葉級(jí)數(shù)的存在性）Anapproximationofx(t)isThereexisterrorbetweentheoriginalsignalx(t)andtheapproximationxN(t)andwiththeNincreases,theerrordecreases.

TheDirichletconditions（狄里赫利條件）

areasfollows:Condition1:Overanyperiod,x(t)mustbeabsolutelyintegrable

Condition2:Inanyfiniteintervaloftime,x(t)isofboundedvariation;thatis,therearenomorethanafinitenumberofmaximaandminimaduringanysingleperiodofthesignal.Condition3:Inanyfiniteintervaloftime,thereareonlyafinitenumberofdiscontinuities.Furthermore,eachofthesediscontinuitiesisfinite.狄里赫利條件：（實(shí)際遇到的信號(hào)都滿足）1.一個(gè)周期內(nèi)只有有限個(gè)不連續(xù)點(diǎn)；2.一個(gè)周期內(nèi)只有有限個(gè)極大值、極小值；3.一個(gè)周期內(nèi)絕對(duì)可積，即Foraperiodicsignalthathasnodiscontinuities,theFourierseriesrepresentationconvergesandequalstheoriginalsignalateveryvalueoft.Foraperiodicsignalwithafinitenumberofdiscontinuitiesineachperiod,theFourierseriesrepresentationequalsthesignaleverywhereexceptattheisolatedpointsofdiscontinuity,atwhichtheseriesconvergestotheaveragevalueofthesignaloneithersideofthediscontinuity.Gibbsphenomenon（吉布斯現(xiàn)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

（共軛對(duì)稱）If

then

ifx(t)real,a0

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

3.4.7Parseval’sRelationforContinuous-TimePeriodicSignals

（帕色伐爾定理）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（沖激串）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（系統(tǒng)函數(shù)）

IfRe{s}=0,s=jω.

If|z|=1,.frequencyresponse（頻率響應(yīng)）

Incontinuoustime,letx(t)beaperiodicsignalwithFourierseriesrepresentationgivenbyThen,theoutputis

Thatis,theeffectoftheLTIsystemistomodifyindividuallyeachoftheFouriercoefficientsoftheinputthroughmultiplicationbythevalueofthefrequencyresponseatthecorrespondingfrequency.Indiscretetime,letx[n]beape