信号与系统课件第4章 连续时间傅里叶变换

CHAPTER4

THECONTINUOUS-TIMEFOURIERTRANSFORM

4.0INTRODUCTION

Representcontinuous-timeaperiodicsignalsaslinearcombinationsofcomplexexponentials.FouriertransformandinverseFouriertransform.

（傅立叶变换）（傅立叶逆变换）UseFouriermethodstoanalyzeandunderstandsignalsandLTIsystems.

频域分析：从本章开始由时域转入变换域分析，首先讨论傅里叶变换。傅里叶变换是在傅里叶级数正交函数展开的基础上发展产生的，这方面的问题也称为傅里叶分析（频域分析）。将信号进行正交分解，即分解为三角函数或复指数函数的组合。频域分析将时间变量变换成频率变量，揭示了信号内在的频率特性以及信号时间特性与其频率特性之间的密切关系，从而导出了信号的频谱、带宽以及滤波、调制和频分复用等重要概念。主要内容：本章从傅里叶级数正交函数展开问题开始讨论，引出傅里叶变换，建立信号频谱的概念。通过典型信号频谱以及傅里叶变换性质的研究，初步掌握傅里叶分析方法的应用。对于周期信号而言，在进行频谱分析时，可以利用傅里叶级数，也可以利用傅里叶变换，傅里叶级数相当于傅里叶变换的一种特殊表达形式。本章最后研究抽样信号的傅里叶变换，引入抽样定理。4.1REPRESENTATIONOFAPERIODICSIGNALS:THECONTINUOUS-TIMEFOURIERTRANSFORM4.1.1DevelopmentoftheFouriertransformrepresentationofthecontinuoustimeFouriertransform(1)Example(FromFourierseriestoFouriertransform)(2)FouriertransformrepresentationofAperiodicsignalForperiodicsignal:Foraperiodicsignalx(t):T

WhenT,SoFouriertransform:RelationbetweenFourierseriesandFouriertransform:orSincespectrumofx(t)AlthoughX(jω)iscommonlyreferredtoas“spectrum”,itisdifferentfromak,whichisthespectrumofperiodicsignals.

Fouriertransformofx(t)X(jω)isactuallyspectrum-densityfunction(频谱密度函数)FouriertransformpairAnusefulrelationship:whereX(jω)istheFouriertransformofx(t),akistheFouriercoefficientsof.x(t)isoneperiodoftheperiodicsignal形象地说，周期信号与频谱之间存在着一一对应的关系，即例如：1T时域：连续、周期1/40频域：离散、非周期而非周期信号与频谱之间已经不存在这种一一对应的关系了，但存在如下另一种一一对应的关系：1时域：连续、非周期01/4频域：连续、非周期傅里叶变换的物理意义1.周期信号指数型傅里叶级数的物理意义：2.非周期信号傅里叶积分的物理意义：周期信号可以分解为无限多频率为、复振幅为的指数分量的离散和.分解为无限多个频率为，复振幅为的指数分量的连续和。（积分）周期、非周期信号两者所不同的是周期信号：频谱是离散的，且各频率分量的复振幅为有限值；非周期信号：频谱是连续的，且各频率分量的复振幅为无限小量。所以，对非周期信号来说，仅仅去研究那无限小量是没有意义的，其频谱不能直接引用复振幅的概念。由即把理解成各频率分量沿频率轴的分布，具有密度的量纲和概念，故称为频率密度函数。简称频谱密度，或在不发生混淆时简称频谱。（注意与周期信号的频谱概念上不一样）可知，量纲是单位频带的复振幅。与周期信号的傅里叶级数类似，一般为复函数。为称为幅频特性；称为相频特性。总称频率特性幅频特性为频率的偶函数；相频特性为频率的奇函数。且均为频率的连续函数。当信号为实函数时:ConvergenceofFourierTransforms

Dirichletconditions:1.

x(t)isabsolutelyintegrable;thatis

2.x(t)haveafinitenumberofmaximaandminimawithinanyfiniteinterval.3.x(t)haveafinitenumberofdiscontinuitieswithinanyfiniteinterval.Furthermore,eachofthesediscontinuitiesmustbefinite.Ifimpulsefunctionsarepermittedinthetransform,somesignalswhicharenotabsolutelyintegrableoveraninfiniteinterval,canalsobeconsideredtohaveFouriertransforms.ThiswillbeconvenientinthediscussionofFouriermethods.Considerthesignal.Example4.1|X(jω)|1/α-α

αω-α-π/4-π/2π/2π/4

αargX(jω)ωmagnitudespectrumofx(t)phasespectrumofx(t)Example4.2Considerthesignalx(t)TheFouriertransformofthesignalisx(t)1t2/αX(jω)1/α-ααωExample4.3LetusdeterminetheFouriertransformoftheunitimpulse

Thatis,theunitimpulsehasaFouriertransformconsistingofequalcontributionatallfrequencies.Thisspectrumisreferredtoaswhite-spectrum.(becausethewhitecolorhasthesamespectrum).Example4.4Considertherectangularpulsesignal--π/T1

π/T12T1X(jω)ωExample4.5Considerthesignalx(t)whoseFouriertransformisUsingthesynthesisequation,wecandetermine1-W

WX(jω)ωW/π-π/W

π/Wx(t)t4.2THEFOURIERTRANSFORMFORPERIODICSIGNALSToobtainthegeneralresultforperiodicsignals,letusfirstconsidertheFouriertransformofthecomplexexponentialFromtheanalysisequation,However,thisintegraldoesnotconverge.ConsidertheFouriertransformpairδ(t)and1.ThisequationsaysthattheFouriertransformofunitdcis.ThisequationshowsthattheFouriertransformofthecomplexexponentialsignalisanimpulselocatedatwithitsarea2π.Foranarbitraryperiodicsignalx(t),Firstrepresentingx(t)withtheFourierseriesascalculatingtheFouriertransformonbothsidesofthisequation,

TheFouriertransformofaperiodicsignalwithFourierseriescoefficients{ak}canbeinterpretedasatrainofimpulsesoccurringattheharmonicallyrelatedfrequenciesandforwhichtheareaoftheimpulseatthekthharmonicfrequencykω0is2πtimesthekthFourierseriescoefficientak.Example4.6Consideragaintheperiodicsquarewave.

ItsFourierseriescoefficientsarethenitsFouriertransformis

–ω0

ω0X(jω)

π

22ω

FouriertransformofasymmetricsquarewaveforT=4T1.Example4.7ConsiderandTheFourierseriescoefficientsforx1(t)areTheFourierseriescoefficientsforx2(t)are

-ω0

0

ω0

X1(jω)-π/jπ/jω

-ω00ω0

X2(jω)ππωExample4.8ConsidertheimpulsetrainTheFourierseriescoefficientsforthissignalareThus,itsFouriertransformist1……-2T-T0T2Tx(t)X(jω)ω2π/T……-4π/T-2π/T02π/T

4π/T4.3PROPERTIESOFTHECONTINUOUS-TIMEFOURIERTRANSFORM傅里叶变换具有惟一性。傅氏变换的性质揭示了信号的时域特性和频域特性之间的确定的内在联系。讨论傅里叶变换的性质，目的在于：了解特性的内在联系；用性质求F(ω)；了解在通信系统领域中的应用。4.3.1LinearityIfandthen例：4.3.2TimeShiftingIfthenConsequence:asignalwhichisshiftedintimedoesnothavethemagnitude

ofitsFouriertransformaltered.Theeffectofatimeshiftonasignalistointroduceintoitstransformaphaseshift,namely,–ωt0.Example4.9Determinethe

Fouriertransformofshowedinthefollowingfigure.A02T1tSinceandThenfromthetimeshiftingproperty,wehaveπ

0

ω

…ω

π

0

Phasespectrumforbothx1(t)andx(t)频移性（调制定理）(频移因子)注意：不是乘以说明：应用：通信中调制与解调，频分复用。通信技术中，未经调制的信号经电缆传输后，可能因衰减太大，在接收端得到的接收信号很难分清究竟是信号还是噪声。距离较远时必须先进行调制，即把频谱搬移，然后传输，到达目的地后再解调（反调制）。例：已知矩形调幅信号

解：因为频谱图：振幅调制一般用乘法器来实现：此外，幅度调制还是频分多路复用的基础。振幅调制又称幅度调制，除此之外，还有频率调制，相位调制等。4.3.3ConjugationandConjugateSymmetrythen(1)共轭性：Ifthen(2)共轭对称性Ifx(t)为实函数|X(jω)|=|X(–jω)|arg

X(jω)=–arg

X(–jω)

even

functionofω

Ifx(t)isreal,then

oddfunctionofωIfx(t)isbothrealandeven,thenX(jω)willalsoberealandeven.Ifx(t)isrealandodd,thenX(jω)ispurelyimaginaryandodd.Re{X(jω)}=Re{X(–jω)}Im{X(jω)}=–Im{X(–jω)}

then(3)IfExample4.10ConsideragaintheFouriertransformevaluationof

andFromthesymmetrypropertiesoftheFouriertransform,wehave4.3.4DifferentiationandIntegrationIfthenConsequence:increasethehigh-frequenciescomponentsinasignal.Consequence:decreasethehigh-frequenciescomponentsinasignal.实际解题时，运用积分性质是反向运行的：即求的频谱遇到困难时，发现其导函数的频谱易求。Example4.11

DeterminetheFouriertransformoftheunitstepx(t)=u(t).时域积分公式说明：时域积分时，有可能把原来的“能量信号”积分成“功率信号”。Example4.12

CalculatetheFouriertransformX(jω)forthesignalx(t)displayedinthefollowingfigure:x(t)t-111

-1t-111t

-11(-1)(-1)+Consider:()()()=¢=¢=¢tgtgtg321If,and,thenProof:(1)(2)Since(1)=(2),andtaking(3)into(1)(3)1.当是能量信号时：2.当是有始功率信号时：3.当是一般的功率信号时：小结一下：(1)求三角函数的频谱密度函数．解:-2-101231t(1)(3)(3)(1)t0求：的频谱。4.3.5TimeandFrequencyScalingIfthenSpecially,whena=-1,wehaveAlinearscalingintimebyafactorofacorrespondstoalinearscalinginfrequencybyafactorof1/a,andviceversa.Especially,意义：(1)

0<a<1时域扩展，频带压缩。(2)a>1时域压缩，频域扩展a倍。例：(1)

0<a<1时域扩展，频带压缩。脉冲持续时间增加a倍，变化慢了，信号在频域的频带压缩a倍。高频分量减少，幅度上升a倍。持续时间短，变化快。信号在频域高频分量增加，频带展宽，各分量的幅度下降a倍。此例说明：信号的持续时间与信号占有频带成反比，有时为加速信号的传递，要将信号持续时间压缩，则要以展开频带为代价。（2）a>1时域压缩，频域扩展a倍。方法一：先标度变换，再时延方法二：先时延再标度变换相同4.3.6Duality

IfthenThispropertyshowsthatforanyFouriertransformpairthereisadualpairwiththetimeandfrequencyvariablesinterchanged.Example4.13

LetusconsiderusingdualitytofindtheFouriertransformG(jω)ofthesignalFromdualityproperty,wehaveIntegrationinfrequency-domain:Differentiationinfrequency-domain:写成实用的形式：重复求导得UsethedifferentiationandintegrationpropertyinfrequencydomaintodeterminetheFouriertransformofthefollowingsignals:Example4.144.3.7Parseval’sRelationParseval’srelationsaysthatthistotalenergymaybedeterminedeitherbycomputingenergyperunittime()andintegratingoveralltimeorbycomputingtheenergyperunitfrequency()andintegratingoverallfrequencies.

energy-densityspectrum

（能量密度谱）Example4.15

ForeachoftheFouriertransformsshowninthefollowingfigure,evaluatethefollowingtime-domainexpressions:-101

x(jω)ω(a)

(b)UseParseval’srelation,wemayevaluateEinthefrequencydomain:Ea=5/8Eb=1

ToevaluateDinthefrequencydomain,firstusethedifferentiationproperty:Since

Da=0

Db=-1-0.50.51

x(jω)ω4.4THECONVOLUTIONPROPERTYConsidertheconvolutionintegral:TheFouriertransformofy(t)is:

Interchangingtheorderofintegrationandnotingthatx(τ)doesnotdependont,wehave

TheFouriertransformmaps

theconvolutionoftwosignalsintotheproductoftheirFouriertransforms.H(jω),thefrequencyresponse,istheFouriertransformoftheimpulseresponse.Itcapturesthechange

incomplexamplitudeoftheFouriertransformoftheinputateachfrequencyω.ThefrequencyresponseH(jω)alsocancharacterizeanLTIsystem,justasitsinversetransform,theunitimpulseresponseh(t).x(t)H1(jω)y(t)H2(jω)x(t)H1(jω)H2(jω)y(t)x(t)H2(jω)y(t)H1(jω)ThreeequivalentLTIsystems.

Here,eachLTIsystemisrepresentedbythefrequencyresponse.ThefrequencyresponsecannotbedefinedforeveryLTIsystem.Sinceessentiallyallphysicalorpractical

signalssatisfythelasttwoconditionsinDirichletconditions,theconditionofabsolutelyintegrablebecomesthedeterminingfactorwhichcanguaranteetheexistenceoftheFouriertransformH(jω)ofh(t).Thatis,only

astableLTIsystemhasafrequencyresponseH(jω).Example4.16

Consideranintegrator―thatis,anLTIsystemspecifiedbytheequationSince

Theimpulseresponseforthissystemistheunitstepu(t).thefrequencyresponseofthesystemisUsingtheconvolutionproperty,wehavetotheinputsignalExample4.17ConsidertheresponseofanLTIsystemwithimpulseresponsetheFouriertransformsofx(t)andh(t)areTherefore,ExpandingY(jω)inapartial-fractionexpansion（部分分式展开）

whereAandBareconstantstobedetermined.

Whenb≠a

Therefore,Theinversetransformforeachofthetermscanberecognizedbyinspection,thenwehave

Whenb=a

RecognizingthisasWecanusethedifferentiationinthefrequency-domainproperty.Thus,Consequently,Example4.18Determinetheresponseofanideallow-passfiltertoaninputsignalx(t)thathastheformofasincfunction.Thatis,Theimpulseresponseoftheideallow-passfilterisofasimilarform:Therefore,whereω0=min(ωi

,ωc).Finally,theinverseFouriertransformofY(jω)isgivenbyThatis,dependingonwhichofωi

andωcissmaller,theoutputisequaltoeitherx(t)orh(t).4.5THEMULTIPLICATIONPROPERTYamplitudemodulationproperty

（幅度调制定理）Example4.19Lets(t)beasignalwhosespectrumS(jω)isdepictedinFigure(a).Also,considerthesignal

(a)Then-ω1

ω1S(jω)Aω-ω0

ω0

P(jω)

π

πω(b)thespectrumR(jω)ofr(t)=s(t)p(t)isobtainedbyanapplicationofthemultiplicationproperty:Thespectrumofr(t)consistsofthesumoftwoshiftedandscaledversionsofS(jω).-ω0

ω0(-ω0-ω1)(-ω0+ω1)(ω0-ω1)(ω0+ω1)R(jω)=[S(jω)*P(jω)]/2π

A/2ωExample4.20Letusconsiderr(t)asobtainedinExample4.19,andletg(t)=r(t)p(t).

-ω0

ω0R(jω)

A/2ω

A/4A/4-2ω0-ω1

ω12ω0G(jω)A/2ωExample4.21DeterminetheFouriertransformofthesignal

Recognizex(t)astheproductoftwosincfunctions:ApplyingthemultiplicationpropertyoftheFouriertransform,weobtainNotingthattheFouriertransformofeachsincfunctionisarectangularpulse,thusX(jω)canbeobtainedbyconvolvingthesepulses:

½-3/2-1/21/23/2X(jω)ω10t0t解：(1)利用对称性求解t1(2)利用调制定理求解(3)利用频域卷积定理4.6SYSTEMSCHARACTERIZEDBYLINEARCONSTANT-COEFFICIENTDIFFERENTIALEQUATIONS前面讨论了信号分析的内容。傅里叶变换有两大作用：一.信号的频谱分析（时域、频域全面了解了一个信号）二.信号作用于线性系统时，频域求解其零状态响应；直观了解输入、输出信号频谱和系统的频率特性。讨论信号作用于线性系统时在频域中求解零状态响应的方法，又称频域分析法。频域分析法的理论基础是时域卷积定理。ConsiderthequestionofdeterminingthefrequencyresponseofanLTIsystemwhichisdescribedbyalinearconstant-coefficientdifferentialequationoftheform(assumingthatthesystemisstable):Fromthelinearproperty,thisbecomes一.系统函数的意义fromthedifferentiationproperty,orequivalently,Fromtheconvolutionproperty(Y(jω)=H(jω)X(jω)),H(jω)

isaratioofpolynomialsin(jω).coefficientsofthenumeratorpolynomial=coefficientsappearingontherightsideofthedifferentialequation.coefficientsofthedenominatorpolynomial=coefficientsappearingontheleftsideofthedifferentialequation.系统函数可以从微分方程直接求得。它是响应傅氏变换与激励傅氏变换的比。它同样表征了系统自身的特性，与输入波形无关。其分子、分母均为j

的多项式之比。傅氏分析法的步骤：1.求取的傅氏变换；2.确定系统函数；3.计算；4.取的反变换，得。可见，意义重大，下面重点讨论它。1.由当时，由，得2.设激励假设

为参变量（一个确定的实数）由时域卷积分析法，得:这说明，虚指数信号作用于系统时，其零状态响应仍为同频率的虚指数信号，不同的仅仅是零状态响应是激励加了个权。所以，系统函数也可定义为在激励下的响应由此可以更深一步理解傅氏变换的物理意义：实质上就是把信号分解为无穷多个虚指信号分量的和。则在范围内:的分量为则响应的分量为把无穷多个响应分量叠加起来，得即频域分析法（傅里叶分析法），是把信号分解为无穷多个无时限虚指数信号之和，即单元信号是，先求取各个单元信号作用于系统的响应，再叠加。时域分析法，是把信号分解为无穷多个冲激信号之和，即单元信号是；然后求取各单元信号作用于系统的响应，在进行叠加；另外，卷积分析法是直接在时域中求解系统的零状态响应，而傅里叶分析法是在频域中求解系统的零状态响应的傅里叶变换后，在反变换到时域中去。这两种分析方法通过F变换的时域卷积定理联系起来。1.从微分方程直接求解;(方程两边取傅氏变换)2.从系统的冲激响应3.设激励为求其响应；4.由电路模型求得。例：已知微分方程求：系统函数。二.的求法(2)若由第二章已经求得冲激响应为对冲激响应求傅氏正变换，得当然，很多情况下是反向运作，用来求的。解：(1)对方程两边求傅氏变换，可得响应与激励之比，为(3)设这时的响应为,代入原微分方程，得三.

系统的频率特性总称系统的频率特性即：幅频特性是偶函数；相频特性是奇函数。可见，系统

的频率特性与实信号的频谱密度函数的特性相类似。但也有不相同的地方：系统带宽(不同于信号带宽)一般定义为等于最大值的处的频率为(称为半功率角频率，或截止频率或3分贝频率)作为系统带宽的根据。如系统为低通滤波器时系统带宽为等等。由公式可以清楚的看到：响应的频谱取决于激励的频谱和系统的频率特性。Example4.22ConsiderastableLTIsystemthatischaracterizedbythedifferentialequation

Determineitsimpulseresponse.ThefrequencyresponseisTodeterminethecorrespondingimpulseresponse,weusethemethodofpartial-fractionexpansion:

Thus,theimpulseresponseisExample4.23DeterminetheoutputofthesysteminExample4.20,andsupposethattheinputis

Inthiscase,thepartial-fractionexpansiontakestheformwhereA11,A12,andA2areconstantstobedetermined.Sothat例：某系统的微分方程为试求全响应。该系统的齐次微分方程为零输入响应的通解为频域分析法小结：1.只能求零状态响应；(由傅氏变换定义，无法表示初始条件)2.反变换有时不太容易;(如激励是)

故一般情况下求到就为止了。3.从频域的观点来看激励与响应的差异概念十分清楚；4.可以用代数方程代替微分方程卷积求解。部分分式展开法注意点：需展开的分式必须是真分式。如得最后得4.7Frequency-SelectiveFilters

Filtering:

aprocessinwhichtherelativeamplitudesofthefrequencycomponentsinasignalarechangedorsomefrequencycomponentsareeliminatedentirely.Frequency-selectivefilters（频选滤波器）

:systemsthataredesignedtopasssomefrequencies

essentiallyundistortedandsignificantlyattenuateoreliminateothers.Typesoffrequency-selectivefilters

low-passfilter（低通滤波器）

high-passfilter

（高通滤波器）

band-passfilter（带通滤波器）

band-stopfilter

（带阻滤波器）

Ideally,thefrequencyresponsesofalow-passfilter,ahigh-passfilter,aband-passfilterandaband-stopfilterareillustratedinthefollowingfigures,respectively:Passband（通带）

H(jω)

1–ωc

0ωcωStopbandStopband（阻带）cutofffrequency（截止频率）

H(jω)

1–ωc

0ωcω

H(jω)

1–ωc2–ωc1

0ωc1

ωc2ω

H(jω)

1–ωc2–ωc1

0ωc1

ωc2ωuppercutofffrequency（上截止频率）

lowercutofffrequency（下截止频率）

Theimpulseresponseoftheideallow-passfilteris:Thus,theimpulseresponseoftheidealhigh-passfilteris:

Thefrequencyresponseoftheidealhigh-passfiltercanberepresentedintermsofthefrequencyresponseofthelow-passfilteras:h(t)forbothlow-passandhighpassfiltersarenotcausal,sotheyareideal.4.7.1ASimpleRCLow-passFiltervs(t)+

vr(t)–

R+C

vc(t)

–+–Input:sourcevoltagevs(t);Output:capacitorvoltagevc(t)

–1/RC01/RC|H(jω)|1ωnonideallow-passfilter

t

s(t)

11-1/eτ=RC

τ=RC

h(t)1/τ1/τe4.7.2ASimpleRCHigh-passFilterInput:sourcevoltagevs(t);Output:resistorvoltagevr(t)–1/RC01/RC|H(jω)|

1ω

t

τ=RC

s(t)11/enonidealhigh-passfilter

4.8TransmissionwithoutDistortion（无失真传输）Lineardistortion:withoutnewfrequencycomponentsproduced.Typesoflineardistortion:magnitudedistortionandphasedistortion.Input:x(t);Output:y(t)Ify(t)=K

x(t-t0),wesayx(t)istransmittedwithoutdistortion.无失真传输系统应满足的两个条件：k信号的无失真传输条件SupposeThen

Forbeingtransmittedwithoutphasedistortion,theremustbeThereforeSupposeThen

Forbeingtransmittedwithoutphasedistortion,theremustbeTherefore4.9Samplingxp(t)x(t)p(t)Χ

x(t)0

t

T1

p(t)0

t

T0

xp(t)

x(0)

x(T)

t

Impulse-trainsampling

4.9.1Impulse-trainsampling（冲激抽样）

-ωM

ωMωX(jω)1

P(jω)2π/T

-2ωs-ωs

0ωs2ωs

ω…Xp(jω)1/T

-ωM0ωMωsω(ωs–ωM)…Xp(jω)1/T0ωs

ω

(ωs–ωM)…Effectinthefrequencydomainofsampling

SamplingTheorem（抽样定理）:Letx(t)beaband-limitedsignalwithX(jω)=0for|ω|>ωM.Thenx(t)isuniquelydeterminedbyitssamplesx(nT),n=0,±1,±2,…,if