CHAPTER4

THECONTINUOUS-TIMEFOURIERTRANSFORM第4章連續時間傅立葉變換1Thecontinuous-timeFouriertransform(連續時間傅立葉變換);Relationshipbetween

FourierseriesandFouriertransform(傅立葉級數與傅立葉變換之間的關系);Propertiesofthecontinuous-timeFouriertransform(傅立葉變換的性質);Frequencyresponseandfrequencyanalysis(系統的頻率響應及系統的頻域分析)；Maincontent：2

Inpractice,aratherlargeclassofsignalsareaperiodicsignals.Howtheaperiodicsignalsshouldbedecomposed,whatisthespectrumofaperiodicsignals,howshouldwegettheLTIsystemresponsetoanaperiodicsignalaretheissuestobesolvedinthischapter.3

Inthetimedomain,whenaperiodicsignalwithaninfiniteperiod,thentheperiodicsignalbecomesanaperiodicsignal;ontheotherhand,ifanyaperiodicsignalisperiodicallyextended,wewillformaperiodicsignal.

4.0Introduction4IntheFourierseriesrepresentationofaperiodicsignal,astheperiodincreasesthefundamentalfrequencydecreasesandtheharmonicallyrelatedcomponentsbecomecloserinfrequency.Astheperiodbecomesinfinite,thefrequencycomponentsformacontinuumandtheFourierseriessumbecomesanintegral.5(非周期信號的表示—連續時間傅立葉變換)4.1RepresentationofAperiodicSignals:TheContinuous-TimeFourierTransform4.1.1DevelopmentoftheFouriertransformRepresentationofanAperiodicsignal(1)example:Overoneperiodofthecontinuous-timeperiodicsquarewave:隨T增大而減小并最終趨于0，原分析方法失效，但譜線長度〔振幅〕雖同為無窮小，但它們的大小并不相同，相對值仍有差異，引入新的變量—頻譜密度函數。6不變時7WecanregardTakassamplesofanenvelopefunction,specifically---continuousvariable---theenvelopeofTak---equallyspacedsamplesofthisenvelope8AsTincreaseswithT1fixed,thefundamentalfrequencydecreases.wehavetwoobservations:A.Theharmonicsarepackedmoreandmoreclosertoeachother;B.However,theshape(theenvelope)ofthespectrumremainsthesame.Whatdoweknowfromtheabovegraph?T=4T1T=8T1T=16T19AsTincreaseswithT1fixed,thefundamentalfrequencydecreases,then:

A.thefundamentalfrequencydecreases,theFourierseriescoefficientsmultipliedbyTbecomesmoreandmorecloselyspacedsamplesoftheenvelope,sothatthesetofFourierseriescoefficientsapproachestheenvelopefunctionasT→∞

B.However,theshape(theenvelope)ofthespectrumremainsthesame.10(2)GofromperiodictoaperiodicConstructaaperiodicsignalx(t)fromtheperiodicsignal,letForperiodicsignal,wehaveitsFourierseries:Foraperiodicsignalx(t):WhenHowtorepresenttheaperiodicsignalx(t)?11As,UseX(jω)todenotethisintegral,thenwehave:spectrumofx(t)Fouriertransformofx(t)周期信號的頻譜就是與它相對應的非周期信號頻譜的樣本CTFT12SinceNote:AlthoughX(jω)isoftenabbreviatedas“spectrum〞,itisdifferentfromak,whichisthespectrumofperiodicsignals.X(jω)isinfactspectrum-densityfunction(頻譜密度函數:具有頻譜隨頻率分布的物理含義)13Thus,weobtainInverseFouriertransformInverseCTFT14Fouriertransformpair:orImportant:

Theaperiodicsignalscanstillberepresentedasalinearcombinationofcomplexexponentials.Thecomplexcomponentwithfrequenceωhave“amplitude〞:15Anusefulrelationship:whereX(jω)istheFouriertransformofx(t),akistheFouriercoefficientsof.x(t)isoneperiodoftheperiodicsignalor周期信號的頻譜是對應的非周期信號頻譜的樣本；而非周期信號的頻譜是對應的周期信號頻譜的包絡。16Example:periodicsquarewave:Soaperiodicsignalx(t):17

ThederivationoftheFouriertransformsuggeststhattheconvergenceofFourierseriesshouldapplytotheconvergenceofFouriertransform.

4.1.2.ConvergenceofFourierTransforms18ThisshowsthatiftheenergyisfinitethentheFouriertransformmustexist.2.

Dirichlet

conditionsTwosetsofconditions：1.Ifthenexist.(1)x(t)isabsolutelyintegrable.

(2)x(t)haveafinitenumberofmaximaandminimawithinanyfiniteinterval.(3)x(t)haveafinitenumberofdiscontinuitywithinanyfiniteinterval.Furthermore,eachofthesediscontinuitiesmustbefinite.19

Note2:

Thesetwosetsofconditionsarenotequivalent.Forexample:isasquareintegrable,butnotabsolutelyintegrable.

Note1:thetwosetsofconditionsaresufficienttoguaranteethatasignalhasaFouriertransform.Ifimpulsefunctionsarepermittedinthetransform,somesignalssuchasperiodicsignals,whichareneitherabsolutelyintegrablenorsquareintegrableoveraninfiniteinterval,canbeconsideredtohaveFouriertransforms.204.1.3.ExamplesofContinuous-TimeFT：1.010212.

Conclusion:TheFouriertransformof

real-evensignalisreal-evenfunction.

Inthiscase10223.001

Thatis,theunitimpulsehasaFouriertransformconsistingofequalcontributionatallfrequencies.Therefore,theunitimpulseresponsecanfullydescribethecharacteristicsofaLTIsystem,andhasanimportantsignificanceinthesignalandsystemanalysis.物理意義：在時域中變化異常劇烈的沖激函數包含幅度相等的所有頻率分量。因此，這種頻譜常稱為“均勻譜“或〞白色譜“。(white-spectrum)23

Clearly,theofreplacedbyandthenmultipliedby.Thenwecangetthecorrespondingperiodicsignalspectrum

4.rectangularpulsesignal:24101000TheimpactofspectrumondifferentpulsewidthTherelationshipbetweenthetimeandfrequencydomainsisinverse.2526(稱為理想低通濾波器)5.Ideallow-passfilter10027ForallωInthiscaseTherelationshipbetweenthetimeandfrequencydomainsisinverse.28

Atthesametimewecanseethatthereisainverserelationshipbetweenthetimeandfrequencydomains.AsWincreases,becomesbroader,whilethemainpeakofx(t)att=0becomeshigherandthewidthofthefirstlobeofthissignalbecomesnarrower,andviceversa.

Ontheexample,if,willbecomeanimpulseand29Dualitycanbeillustratedasfollows

:101000Therelationshipbetweenthesincfunctionandarectangularpulseisduality

.30If,then∵∴6.317.324.1.4.信號的帶寬(BandwidthofSignals)(complementary)由信號的頻譜可以看出：信號的主要能量總是集中于低頻分量。另一方面，傳輸信號的系統都具有自己的頻率特性。因而，工程中在傳輸信號時，沒有必要一定要把信號的所有頻率分量都有效傳輸，而只要保證將占據信號能量主要局部的頻率分量有效傳輸即可。為此，需要對信號定義帶寬。通常有如下定義帶寬的方法:33

下降到最大值的時對應的頻率范圍,此時帶內信號分量占有信號總能量的1/2。1.0例如342.對包絡是形狀的頻譜，通常定義主瓣寬度(即頻譜第一個零點內的范圍)為信號帶寬。以矩形脈沖為例，按帶寬的定義，可以得出，脈寬乘以帶寬等于常數C(脈寬帶寬積)。這清楚地反映了頻域和時域的相反關系。

10035(周期信號的傅立葉變換)

havedeveloped:FourierseriesforperiodicsignalsandFouriertransformforaperiodicsignals.WecanalsodevelopFouriertransformrepresentationforperiodicsignalstoallowustoconsiderbothperiodicandaperiodicsignalswithinaunifiedcontext.However,periodicsignaldoesnotsatisfyDirichletconditions,andthereforeitisimpossibletodirectlyobtaintheFouriertransformfromitsdefinition.4.2TheFourierTransformationofPeriodicSignals36considerIf

thenThisequationshowsthattheFouriertransformofthecomplexexponentialsignalisanimpulselocatedatwithitsarea2π.37Foranarbitraryperiodicsignalx(t),firstrepresentingx(t)withtheFourierseriesasSoTheFouriertransformofperiodicsignal38TheFouriertransformofaperiodicsignalwithFourierseriescoefficients{ak}is:atrainofimpulsesoccurringattheharmonicallyrelatedfrequencies;theareaoftheimpulseatthekthharmonicfrequencykω0is2πtimesthekthFourierseriescoefficientak.Foraperiodicsignalx(t)withtheperiodT=239

求取方法：２．利用定義求周期信號的頻譜１．利用傅立葉變換求周期信號的頻譜其中408：

9：4110：均勻沖激串010Therelationshipbetweenthetimeandfrequencydomainsisinverse.4211.Periodicrectangularpulse01

–ω0

ω0X(jω)

π

22ω43(連續時間傅立葉變換的性質)Providingasignificantamountofinsightinto:1)thetransform;2)therelationshipbetweenthetime-domainandfrequency-domaindescriptionsofasignal;3)therelationshipbetweentheCFSandCTFTrepresentationofaperiodicsignal.ReducingthecomplexityoftheevaluationofFouriertransformorinversetransform.4.3PropertiesoftheContinuous-TimeFourierTransform4412:1.Linearity(線性):thenIf452.TimeShifting(時移):thenIfConsequence:

asignalwhichisshiftedintimedoesnothavethemagnitude

ofitsFouriertransformaltered,butintroduceintoaphaseshift.Refertop302example4.94613E047frequencyshifting(頻移〕ifthenExample:48頻移性質493.ConjugationandConjugateSymmetry(共軛對稱性)thenIfSo50

Ifisarealsignal,thenthen

Fromrealpartisanevenfunctionimaginarypartisanoddfunction

FromThemagnitudeisanevenfunctionThephaseisanoddfunctionForarealsignal:51

IfThesignalisbothrealandevenConsequently：TheFouriertransformofrealevensignalisanrealevenfunction.

If∵∴ThesignalisbothrealandoddConsequently：TheFouriertransformofrealoddsignalisanimaginaryandodd.52

Ifthen:Forarealsignal:5314:Thespectrumof101/20-1/21/20canbebrokenintoevenandoddpart54554.DifferentiationandIntegration(時域微分與積分)Thisisparticularlyimportantpropertybecauseitreplacestheoperationofdifferentiationinthetimedomainwiththatofmultiplicationbyjωinthefrequencydomain.(bydifferentiatingbothsidesof)Example:determinetheFToftheunitstepFrompropertyofintegration:Weobtain:thenIf56E57585.時域和頻域的尺度變換:ScalingWhen,weobtainThescalingpropertyisanotherexampleoftheinverserelationshipbetweenthetimeandfrequencydomain.thenIfCompression(expansion)inthetimedomaincorrespondingtoextension(compression)inthefrequencydomainTherelationshipbetweenthetimeandfrequencydomainsisinverse.5917:606.對偶性:DualityIfthenProof1：Proof2:

6162FromWeobtainFrequencyShiftingFromusingdualityusingTimeShiftingusingdualityByduality:somepropertiesinthetimedomaincorrespondstotheonesinthefrequencydomain(1)、timeshiftingfrequencyshiftingdualityTimeshiftingTimereversalduality63FromWeobtainSo2、DifferentiationinFrequencyProof164FromWeobtaindifferentiationintimedualityDifferentiationinFrequencyProof2:usingthedualitypropertydualityTimedomaindualityTimereversal65設求16:663、integrationinfrequencyintegrationintimedualityfromWeobtainintegrationinfrequencydualityTimedomaindualityTimereversal6718:6819:697.Parseval’sRelation:Ifthen

Parseval’srelationsaysthatthistotalenergymaybedeterminedeitherbycomputingenergyperunittime()andintegratingoveralltimeorbycomputingtheenergyperunitfrequency()andintegratingoverallfrequencies.energy-densityspectrum〔能量譜密度〕704.4TheConvolutionProperty(卷積性質)一.TheConvolutionProperty：thenIf71InfactWeobtaincanbedecomposedintoalinearcombinationofcomplexexponentialsignals,andforeachtheresponseoftheLTIsystemwillbetheweightedcomplexexponentialmultipliedbycorrespondingeigenvalue.Thiseigenvalueis:Soshows：Proof172theconvolutionintegral:TheFouriertransformofy(t)is:

Proof2731.TheFouriertransform

mapstheconvolutionoftwosignals

into

theproductoftheirFouriertransforms.2.ThefrequencyresponseH(jω),istheFouriertransformoftheimpulseresponse,

alsocancharacterizeanLTIsystem,justasitsinversetransform,theunitimpulseresponseh(t).3.ThefrequencyresponsecannotbedefinedforeveryLTIsystem.Conclusion：74isoneofthreeDirichletconditions.Assumingthath(t)satisfiestheothertwoconditions,thenweseethatastableLTIsystemhasafrequencyresponseH(jω).InordertoexamineunstableLTIsystem,wewilldeveloptheLaplacetransform.IfanLTIsystemisstable,thenitsimpulseresponseintegrable;thatis75二.FrequencyResponseofLTIsystem(LTI系統的頻域分析法):

ToanalyzetheLTIsystemsinfrequencydomain

,wehavethefollowingsteps:

1.Determinethe

Fouriertransform

2.find

3.4.76totheinputsignalConsidertheresponseofanLTIsystemwithimpulseresponseTheFouriertransformsofx(t)andh(t)areTherefore,

ExpandingY(jω)inapartial-fractionexpansion

whereAandBareconstantstobedetermined.

Whenb≠a

20:77Therefore,78

Whenb=a

Consequently,Wecanusethedifferentiationinthefrequency-domainproperty.Thus,79Determinetheresponseofanideallow-passfiltertoaninputsignalx(t)thathastheformofasincfunction.Thatis,Theimpulseresponseoftheideallow-passfilterisofasimilarform:21:80Therefore,Finally,theinverseFouriertransformofY(jω)isgivenbyThatis,dependingonwhichofωiandωcissmaller,theoutputisequaltoeitherx(t)orh(t).814.5TheMultiplicationProperty(相乘性質)Ifthen例1:FrequencyShifting82ThispropertycanbederivedUsingdualitypropertytogetherwiththeconvolutionproperty.s1s2s3s483Multiplicationofonesignalbyanothercanbethoughtofasusingonesignaltoscaleor

modulate

theamplitudeoftheother,andconsequently,themultiplicationoftwosignalsisoftenreferredtoas

amplitudemodulation.Themodulationproperty〔調制定理〕:s(t)p(t)r(t)modulationproperty〔調制定理〕84例2.Sinusoidalamplitudemodulation(正弦幅度調制)108501/2

正弦幅度調制等效于在頻域將調制信號的頻譜搬移到載頻位置。86例3.

Synchronousdemodulation(同步解調):1/21/41/487此時，用一個頻率特性為的系統即可從恢復出。20只要即可。具有此頻率特性的LTI系統稱為理想低通濾波器。例4.

Frequency-SelectFilteringwithVariableCenterFrequency(中心頻率可變的帶通濾波器)：88A1理想低通的頻率響應891等效帶通濾波