信号与系统课件

Ch1Introduction(绪论)本章重点:

Whatisasignal?什么是信号?Whatisasystem?什么是系统?Classificationsofsignals信号的分类

Basicoperationsofsignals信号的基本运算

Basicsignals基本信号

Propertiesofsystems系统的特性Ch1.1WhatisaSignal(信号)?ASpeechsignal–It’samplitudevarieswithtime,dependingonthespokenwordandwhospeaksit.It’saone-dimensionalsignal.x(t)WhatisaSignal?Electrocardiogram(ECG)Signal–Representtheelectricalactivityoftheheart.It’saone-dimensionalsignal.x(t)WhatisaSignal?I(x,y)

Animage

isafunctionoftwospatialcoordinates.It’satwo-dimensionalsignal.WhatisaSignal(信号)?

DefinitionsASignalisformallydefinedasafunctionofoneormorevariablesthatconveysinformationonthenatureofaphysicalphenomenon.信号是一个或多个变量的函数，携带着某个物理现象的信息。Ch1.2WhatisaSystem(系统)?ASystemisformallydefinedasanentitythatmanipulatesoneormoresignalstoaccomplishafunction,therebyyieldingnewsignals.

DefinitionsCh1.3OverviewofSpecificSystems

Elementsofacommunicationsystem.

Thetransmitterchangesthemessagesignalintoaformsuitablefortransmissionoverthechannel.Thereceiverprocessesthechanneloutput(i.e.,thereceivedsignal)toproduceanestimateofthemessagesignal.Examples:CommunicationSystems(通信系统)(a)SnapshotofPathfinderexploringthesurfaceofMars.(b)The70-meter(230-foot)diameterantennalocatedatCanberra,Australia.Thesurfaceofthe70-meterreflectormustremainaccuratewithinafractionofthesignal’swavelength.(CourtesyofJetPropulsionLaboratory.)Examples:ControlSystems(控制系统)Blockdiagramofafeedbackcontrolsystem.Thecontrollerdrivestheplant,whosedisturbedoutputdrivesthesensor(s).Theresultingfeedbacksignalissubtractedfromthereferenceinputtoproduceanerrorsignale(t),which,inturn,drivesthecontroller.Thefeedbackloopistherebyclosed.Examples:BiomedicalSignalProcessing

(生物信号处理)Thetracesshownin(a),(b),and(c)arethreeexamplesofEEGsignalsrecordedfromthehippocampusofarat.Neurobiologicalstudiessuggestthatthehippocampusplaysakeyroleincertainaspectsoflearningandmemory.

(a)Inthisdiagram,thebasilarmembraneinthecochleaisdepictedasifitwereuncoiledandstretchedoutflat;the“base”and“apex”refertothecochlea,buttheremarks“stiffregion”and“flexibleregion”refertothebasilarmembrane.Examples:AuditorySystems(听觉系统)(b)Thisdiagramillustratesthetravelingwavesalongthebasilarmembrane,showingtheirenvelopesinducedbyincomingsoundatthreedifferentfrequencies.学习方法理论与实际相结合，物理概念、数学概念和工程概念并重。掌握信号与系统分析的基本思想和方法；注意问题的提出、分析问题和解决问题的方法。讲、练、做相结合；加强计算机实践环节，用MATLAB进行信号与系统的分析。本课程教学与学习的安排1.课堂教学：讲解重点内容和学生学习中遇到的疑难问题。

2.作业：书面作业（理论）+MATLAB上机作业（实践）。3.期中和期末考试：闭卷形式。主要考察学生对本门课的基本理论基本原理及重点内容的掌握程度。4.课程成绩的组成：

由书面作业、MATLAB作业、期中考试和期末考试4部分组成。主要参考书[1]SimonH.,BarryV.V.SignalsandSystems.JohnWiley&Sons,Inc.1999[2]EdwardW.K.,BonnieS.H.FundamentalsofSignalsandSystemsUsingMATLAB.Prentice-HallInternational,Inc.1997[3]A.V.Oppenheim.SignalsandSystems

或中译本（第2版）.西安交通大学出版社.[4]郑君里，应启珩等.信号与系统.第2版.

高等教育出版社，2000.

主要参考书[5]吴湘淇等.信号、系统与信号处理(上).第2版.

电子工业出版社，2001[6]吴湘淇,郝晓莉等.信号、系统与信号处理——软硬件实现.电子工业出版社，2002[7]

陈后金等.信号与系统.清华大学出版社，

2003[8]陈后金等.信号与系统学习指导与习题精解.

清华大学出版社，2004Ch1.4classificationsofsignals

(信号的分类)

1.continuous-timeanddiscrete-timesignals

连续时间信号和离散时间信号2.periodicandnon-periodicsignals

周期信号和非周期信号3.deterministicandrandomsignals

确定信号和随机信号4.Energyandpowersignals

能量信号和功率信号Continuous-timeandDiscrete-timesignals

连续时间信号和离散时间信号(a)Continuous-timesignalx(t).(b)Representationofx(t)

asadiscrete-timesignalx[n].

Continuous-timeandDiscrete-timesignals

连续时间信号和离散时间信号Discrete-timesignal:asignalifitisdefinedonlyatdiscreteinstantsoftime.

离散时间信号：若信号仅在某些离散时刻处有定义,用x[n]表示。Continuous-timesignal:asignalifitisdefinedfor

alltimet.连续时间信号：若信号在所有时间t处都有定义,用x(t)表示。Definitions

Continuous-timeandDiscrete-timesignals

连续时间信号和离散时间信号离散信号可以由连续信号取样(sampling)得来：

x[n]=x(t)|t=nT

=x(nT)

T称为取样间隔

periodicandnon-periodicsignals

周期信号和非周期信号(a)SquarewavewithamplitudeA=1andperiodT=0.2s.

(b)RectangularpulseofamplitudeAanddurationT1.PeriodicandNon-PeriodicSignals

(周期信号与非周期信号)A

Periodicsignalsisafunctionoftimethatsatisfiesthecondition:

x(t+T)=x(t)forallt.Definition

FundamentalPeriod:T0，thesmallestvalueofTthatsatisfiesthedefinition.Period：PeriodicandNon-PeriodicSignals

(周期信号与非周期信号)Definition

FrequencyFundamentalFrequencyAngularFrequencyPeriodicandNon-PeriodicSignals

(周期信号与非周期信号)(a)SquarewavewithamplitudeA=1andperiodT=0.2s.

(b)RectangularpulseofamplitudeAanddurationT1.Problem:Triangularwavealternativebetween–1and+1

PeriodicandNon-PeriodicSignals

(周期信号与非周期信号)Adiscrete

Periodicsignalsisafunctionoftimethatsatisfiesthecondition:

x[n+N]=x[n]forintegern.Definition

FundamentalPeriod:N，thesmallestintegerthatsatisfiesthedefinition.PeriodicandNon-PeriodicSignals

(周期信号与非周期信号)Definition

PeriodFundamentalFrequency(a)Discrete-timesquarewavealternativebetween–1and+1.

(b)Non-periodicdiscrete-timesignalconsistingofthree

nonzerosamples.periodicandnon-periodicsignals

(周期信号和非周期信号)DeterministicandRandomSignals

确定性信号和随机信号(非确定性信号)DefinitionsArandomsignalisasignalaboutwhichthereisuncertaintybeforeitoccurs.随机信号：再出现之前具有不确定性的信号。Adeterministicsignalisasignalaboutwhichthereisnouncertaintywithrespecttoitsvalueatanytime.确定性信号：在任意时刻都有确定的值的信号。DeterministicandRandomSignals

确定性信号和随机信号x[n]0Nn……DeterministicandRandomSignals

确定性信号和随机信号tttX(t)EnergyandPowersignals

(能量信号和功率信号)Definitions:

EnergyAveragePowerEnergyandPowersignals

(能量信号和功率信号)Definitions:

EnergySignalPowerSignalEnergyandPowersignals

能量信号和功率信号Problem:Determinethetotalenergyofthediscrete-timesignal.EnergyandPowersignals

(能量信号和功率信号)Problem:Determinetheaveragepowerofthesquarewave.

amplitudeA=1andperiodT=4s,T0=1s.)(txtT0T-0TT-10EnergyandPowersignals

(能量信号和功率信号)Energysignal?Powersignal?Ch1.5basicoperationsonsignals

信号的基本运算

1.基于从变量（信号本身或信号之间）的运算幅度变化；相加和相乘；连续信号的微积分，离散信号的差分与累加

2.基于自变量的运算连续信号的翻转、展缩和平移离散信号的翻转、展缩和平移operationsperformedondependentvariables

(基于从变量的运算)

1.

Amplitudescaling(幅度比例变化)

x(t)

cx(t)x[n]

cx[n](c为常数)

波形不变，幅度成比例放大或缩小。

Example：x(t)=sin(2π×10t);y(t)=5x(t)=5sin(2π×10t);

operationsperformedondependentvariables

(基于从变量的运算)2.

Addition(信号相加)

y(t)=x1(t)+x2(t) y(n)=x1[n]+x2[n]3.

Multiplication(信号相乘)

y(t)=x1(t)

x2(t) y(n)=x1[n]

x2[n]operationsperformedondependentvariables

(基于从变量的运算)4.Differentiation(连续信号的微分)

5.

integration（连续信号的积分）operationsperformedondependentvariables

(基于从变量的运算)Example：电感两端的电压与其电流为微分关系：

Example：电容两端的电压与其电流为积分关系：operationsperformedonindependentvariables

(基于信号自变量的运算)(a)continuous-timesignalx(t)

(b)versionofx(t)compressedbyafactorof2,

(c)versionofx(t)expandedbyafactorof2.1.Timescaling（尺度展缩）:y(t)=x(at)a>0

若0<a<1，则x(at)是x(t)扩展1/a倍。若a>1，

则x(at)是x(t)压缩a倍。

Timescaling（尺度展缩）

(a)discrete-timesignalx[n]

(b)versionofx[n]compressedbyafactorof2,withsomevaluesoftheoriginalx[n]lostasaresultofthecompression.y[n]=x[kn]operationsperformedonindependentvariables

(基于信号自变量的运算)(a)continuous-timesignalx(t)

(b)reflectedversionofx(t)abouttheorigin.

2.reflection（翻转）:

y(t)=x(-t)

x(-t)表示将x(t)以纵轴为中心作180

翻转。Reflection（翻转）Problem:Findthereflectedversionof

x[n]andy[n]2.reflection（

翻转）:

x[n]

x[-n]

x[n]以纵轴为中心作180

翻转operationsperformedonindependentvariables

(基于信号自变量的运算)(a)continuous-timesignalintheformofarectangularpulseofamplitude1.0andduration1.0,symmetricabouttheorigin;

(b)time-shiftedversionofx(t)by2timeshifts.

3.Timeshifting(时移):

y(t)=x(t-t0)

x(t-t0)表示信号

x(t)右移t0单位；

x(t+t0)表示信号x(t)左移t0单位。Timeshifting(时移)Problem:Findthetime-shiftedsignal

y[n]=x[n+3]

y[n]=x[n

k]，k>0

x[n+k]，左移k单位；

x[n-k]，右移k单位。operationsperformedonindependentvariables

(基于信号自变量的运算)

总结公式：operationsperformedonindependentvariables

(基于信号自变量的运算)Example:

已知x(t)的波形如图所示，试画出x(2t)、x(t/3)、x(t+6)、x(-t)、x(6-2t)的波形。operationsperformedonindependentvariables(基于信号自变量的运算)Example:已知x(n)的波形如图所示，求x[n+2]、x[-n]

、x[-n-2]、x[-n/3]、x[2n]

的波形。Ch1.6BasicSignals

基本信号ExponentialSignals指数信号SinusoidalSignals正弦信号ExponentialDampedSinusoidalSignals按指数衰减的正弦信号StepSignals阶跃信号ImpulseSignals冲激信号DerivativesofTheImpulse冲激信号的导数RampFunction斜坡函数ExponentialSignals

指数信号

(a)Decayingexponentialformofcontinuous-timesignal.

(b)Growingexponentialformofcontinuous-timesignal.实指数信号：ExamplesofExponentialSignals

指数信号Lossycapacitor,withthelossrepresentedbyshuntresistanceR.

ExponentialSignals(指数信号)

(a)Decayingexponentialformofdiscrete-timesignal.

(b)Growingexponentialformofdiscrete-timesignal.

实指数信号：Sinusoidalsignal(正弦信号)(a)SinusoidalsignalAcos(ωt+Φ)withphaseΦ=+

/6radians.

(b)SinusoidalsignalAsin(ωt+Φ)withphaseΦ=+

/6radians.Sinusoidalsignal：

x(t)=Acos(ωt+Φ)ExamplesofSinusoidalsignal

正弦信号ParallelLCcircuit,assumingthattheinductorLandcapacitorCarebothideal.

Sinusoidalsignal

正弦信号Discrete-timesinusoidalsignal.RelationbetweenSinusoidalandComplexExponentialSignals

正弦信号和复指数信号的关系Complexplane,showingeightpointsuniformlydistributedontheunitcircle.Exponentiallydampedsinusoidalsignal

按指数衰减的正弦信号ExponentiallydampedsinusoidalsignalAe-atsin(ωt),withA=60andα=6.x(t)=Ae-atsin(ωt)，α>0Stepfunction

阶跃函数Continuous-timeversionoftheunit-stepfunctionofunitamplitude.Definitions:Stepfunction(阶跃函数)(a)Rectangularpulsex(t)ofamplitudeAanddurationof1s,symmetricabouttheorigin.(b)Representationofx(t)asthedifferenceoftwostepfunctionsofamplitudeA,withonestepfunctionshiftedtotheleftby½andtheothershiftedtotherightby½;thetwoshiftedsignalsaredenotedbyx1(t)andx2(t),respectively.Notethatx(t)=x1(t)–x2(t).ExamplesofStepfunction

阶跃函数(a)SeriesRCcircuitwithaswitchthatisclosedattimet=0,therebyenergizingthevoltagesource.(b)Equivalentcircuit,usingastepfunctiontoreplacetheactionoftheswitch.

(t)=0,t

0Definitions:UnitImpulse

单位冲激信号(a)Evolutionofarectangularpulseofunitareaintoanimpulseofunitstrength(i.e.,unitimpulse).(b)Graphicalsymbolforunitimpulse.

(c)RepresentationofanimpulseofstrengthathatresultsfromallowingthedurationΔofarectangularpulseofareaatoapproachzero.ExamplesofUnitImpulse

冲激信号(a)Seriescircuitconsistingofacapacitor,adcvoltagesource,andaswitch;theswitchisclosedattimet=0.

(b)Equivalentcircuit,replacingtheactionoftheswitchwithastepfunctionu(t).ExamplesofUnitImpulse

冲激信号①筛选特性PropertiesofUnitImpulse

冲激信号的性质②取样特性证明：利用筛选特性PropertiesofUnitImpulse

冲激信号③TimeScaling(展缩特性)推论：冲激信号是偶函数。根据d(t)泛函定义证明取a=-1,可得d(t)=d(-t)PropertiesofUnitImpulse

冲激信号的性质TheTime-scalingPropertyofUnitImpulse

冲激信号的时间展缩Stepsinvolvedinprovingthetime-scalingpropertyoftheunitimpulse.(a)RectangularpulsexΔ(t)ofamplitude1/ΔanddurationΔ,symmetricabouttheorigin.(b)PulsexΔ(t)compressedbyfactora.(c)Amplitudescalingofthecompressedpulse,restoringittounitarea.④冲激信号与阶跃信号的关系PropertiesofUnitImpulse

冲激信号的性质Problems

solution

Definitions:DerivativesofTheImpulsefunction

冲激函数的导数（冲激偶）Properties:(取样特性)(筛选特性)(展缩特性)DerivativesofTheImpulsefunction

冲激函数的导数（冲激偶）Rampfunction(斜坡函数)Definitions:RelationBetweenUnitFunctionandRampfunction

阶跃函数与斜坡函数的关系(1)(1)(2)(2)解：ProblemsRelationsofImpulseFunction,UnitFunctionandRampfunction

冲激函数、阶跃函数与斜坡函数的关系Ch1.7SystemsViewedasInterconnectionsofOperations

BlockdiagramrepresentationofoperatorHfor(a)continuoustimeand(b)discretetime.Discrete-time-shiftoperatorSk,operatingonthediscrete-timesignalx[n]toproducex[n–k].Ex:Moving-AverageSystems

滑动平均系统：y(n)=[x(n)+x(n-1)+x(n-2)]/3

Twodifferent(butequivalent)implementationsofthemoving-averagesystem:(a)cascadeformofimplementationand(b)parallelformofimplementation.Ex:Moving-AverageSystems

滑动平均系统：y(n)=[x(n)+x(n-1)+x(n-2)]/3Ch1.8PropertiesofSystems

1.Stability(稳定性）

2.Causality（因果性）

3.Invertibility（可逆性）

4.Time-Invariance(时不变性)

5.linearity(线性)

1.Stability(稳定性）稳定系统：BoundedInput-BoundedOutput（有界输入产生

有界输出，BIBO稳定）

不稳定系统：系统的输入有界而输出无界。Stability(稳定性）

Ex:Moving-AverageSystems(滑动平均系统).ShowthattheSystemisBIBOstable:y(n)=[x(n)+x(n-1)+x(n-2)]/3.Solution：∣y(n)∣=∣x(n)+x(n-1)+x(n-2)∣/3

≤（Mx+Mx+Mx）/3=Mx∣y(n)∣有界，系统稳定。Solution：∣x(n)∣≤Mx<∞∣y(n)∣=∣rn∣•∣x(n)∣,由于r>1，∣rn∣→∞，∣y(n)∣无界，系统不稳定Ex:UnstableSystem.y(n)=rnx(n)

,r>1DramaticphotographsshowingthecollapseoftheTacomaNarrowssuspensionbridgeonNovember7,1940.(a)Photographshowingthetwistingmotionofthebridge’scenterspanjustbeforefailure.

(b)Afewminutesafterthefirstpieceofconcretefell,thissecondphotographshowsa600-ftsectionofthebridgebreakingoutofthesuspensionspanandturningupsidedownasitcrashedinPugetSound,Washington.Notethecarinthetopright-handcornerofthephotograph.AnUnstableSystem2.Causality（因果性）

因果：输出不领先于输入，即现时刻的输出仅取决于当前时刻的输入和（或）过去时刻的输入

Ex:Moving-AverageSystems(滑动平均系统)

y(n)=[x(n)+x(n-1)+x(n-2)]/3Isthissystemcausal?Causality（因果性）

SeriesRCcircuitdrivenfromanidealvoltagesourcev1(t),producingoutputvoltagev2(t).Ex:ConsidertheRCcircuit.Isthissystemcausalornon-causal?3.Invertibility(可逆性)Thenotionofsysteminvertibility.ThesecondoperatorHinvistheinverseofthefirstoperator