信號與系統(tǒng)課件

11.Signals

Signals:physicalphenomenaorphysicalquantities,whichchangewithtimeorspace.Functionsofoneormoreindependentvariables.example:x(t)

1)DefinitionandMathematicalRepresentationofSignals(信號的定義及其數(shù)學(xué)表示)AsimpleRCcircuit:2Aspeechsignal“Shouldwechase”這句話的聲壓隨時間變化的波形3Apicture一幅黑白照片可用亮度隨二維空間變化的函數(shù)來表示42)ClassificationofSignals(信號的分類)(1)

DeterminateandRandomSignalsAdeterminatesignal——x(t)能夠用確定的時間函數(shù)表示。Arandomsignal——cannotfindafunctiontorepresentit不能用確定時間函數(shù)表示——干擾信號、噪聲信號

5(2)

Continuous-timeandDiscrete-timeSignalscontinuous-timesignals’independentvariableiscontinuous:x(t)

對一切時間t(除有限個不連續(xù)點(diǎn)外)都有確定的函數(shù)值，這類信號就稱為連續(xù)時間信號，簡稱連續(xù)信號。discrete-timesignalsaredefinedonlyatdiscretetimes(onlyforintegervaluesoftheindependentvariable):x[n]

僅在不連續(xù)的瞬間(僅在自變量的整數(shù)值上)有確定函數(shù)值

6RepresentingSignalsGraphically

0x(t)

tGraphicalrepresentationsof(a)continuous-timeand(b)discrete-timesignals(a)-2x[-1]x[0]x[4]-4-3-1012345x[n]

n(b)7(3)PeriodicandAperiodicSignals在較長時間內(nèi)(嚴(yán)格地說，無始無終)每隔一定時間T(或整數(shù)N)按相同規(guī)律重復(fù)變化的信號叫周期信號。

Foracontinuous-timesignalx(t)

x(t)=x(t+mT),(m=0,+1,-1,+2,-2,……)

forallvaluesoft.

Foradiscrete-timesignalx[n]

x[n]=x[n+mN],(m=0,+1,-1,+2,-2,……)

forallvaluesofn.Inthiscase,wesaythatx(t)(x[n])isperiodicwithFundamentalPeriodT(N).

8Examplesofperiodicsignals:sin,cosetc.withtheirfundamentalperiodN0=39Example1.1Determinethefundamentalperiodofthesignalx(t)=2cos(10πt+1)-sin(4πt-1).Fromtrigonometry,weknowthatthefundamentalperiodofcos(10πt+1)isT1=1/5,andsin(4πt-1)isT2=1/2.Whataboutthefundamentalperiodofx(t)?TheanswerisifthereisarationalT,anditisthelowestcommonmultipleofT1andT2,thenwesaythatx(t)isperiodicwithfundamentalperiodT,orelse,x(t)isaperiodic.Forthex(t)inthisexample,thelowestcommonmultipleof0.2and0.5isunit1,anditisrational,sothatthefundamentalperiodofx(t)is1.10

(4)EnergyandPowerSignals

TheinstantaneouspowerisThetotalenergyisTheaveragepoweris11SignalEnergyandPower

A.Continuous-TimeSignalInstantaneousPower:Energyovert1

t

t2:TotalEnergy:AveragePower:12B.Discrete-TimeSignalInstantaneousPower:Energyovern1

n

n2:TotalEnergy:AveragePower:13Withthesedefinitions,wecanidentifythreeimportantclassesofsignals:

A.

FiniteEnergySignal:(P

0)Example：14B.

FinitePowerSignal:(E)Example：C.SignalswithneitherfinitetotalenergynorfiniteaveragepowerExample：15

3)TransformationsoftheIndependentVariableofSignals(信號的自變量的變換)(1)TimeShiftRightshift:x(t-t0)

x[n-n0](Delay)Leftshift:x(t+t0)

x[n+n0](Advance)當(dāng)信號經(jīng)不同路徑傳輸時，所用時間不同，從而產(chǎn)生時移。如電視圖像出現(xiàn)的重影是由于信號傳輸?shù)臅r移造成。16TimeShift(Example)SignalTransformation17SignalTransformation(2)TimeReversalx(-t)

orx[-n]:Reflectionofx(t)orx[n]18SignalTransformation(3)TimeScalingx(at)orx[an]

(a>0)Stretchifa<1Compressedifa>1Example1.2Givenasignalx(-t/3+2),showninFig.(a),drawthegraphofx(t).如錄像帶慢放時，信號被展寬；快放時，信號被壓縮；倒放時，則信號被反褶。190

12t(a)x(-t/3+2)1

-2/3-1/301x(t+2)x(t/3+2)

1

-2-1004/35/321x(t)201x(-t/3)-6-5-401x(t/3)045604/35/321x(t)0

12t(a)x(-t/3+2)1214)

EvenandOddSignals(偶信號與奇信號)Evensignal:x(-t)=x(t)orx[-n]=x[n]Oddsignal:x(-t)=-x(t)orx[-n]=-x[n]Even-OddDecomposition:or:x(t)=Ev{x(t)}+Od{x(t)}

evenpartofx(t)

oddpartofx(t)22Examples:235)SeveralBasicSignals(幾種基本信號)

(1)Continuous-timeComplexExponentialandSinusoidalSignalsA.RealExponentialSignals

x(t)=Ceat(C,aarerealvalue)a>０a<０24B.PeriodicComplexExponentialandSinusoidalSignals

(a)x(t)=ej

0t

(b)x(t)=Acos(

0t+

)

All

x(t)satisfyforx(t)=x(t+T),andT=2/

0,sox(t)isperiodic.Euler’sRelation(歐拉關(guān)系):

ej

0t=

cos0t+jsin0t

andcos0t=(ej

0t+

e-j

0t)/2

sin0t=(ej

0t-

e-j

0t)/2j

Wealsohave25C.GeneralComplexExponentialSignals

x(t)=Ceat,inwhichC=|C|ej

,a=r+j

0,so

Forr=0,therealandimaginarypartsofx(t)aresinusoidal;Forr>0(r<0),theycorrespondtosinusoidalsignalsmultipliedbyagrowing(decaying)exponetial.26Thedashedcurveistheenvelope(包絡(luò))fortheoscillatorcurve.

(a)Growingsinusoidalsignal,，r>0;(b)decayingsinusoid,，r<0.27(2)Discrete-timeComplexExponentialandSinusoidalSignals(sequences)A.RealExponentialSignalsRealExponentialSignal

x[n]=C

n

(a)

>1(b)0<<1(c)-1<<0(d)<-128B.SinusoidalSignalsComplexexponential:x[n]=ej

0n

=cos

0n+

jsin

0n

Sinusoidalsignal:

x[n]=cos(

0n+

)

29C.GeneralComplexExponentialSignalsComplexExponentialSignal:

x[n]=C

n

inwhichC=|C|ej

,

=|

|ej

0(polarform，極坐標(biāo)),then

x[n]=|C||

|ncos(

0n+

)+j|C||

|nsin(

0n+

)30

(a)Growingsinusoidalsequence(b)Decayingsinusoidalsequence31(3)PeriodicityPropertiesofDiscrete-timeComplexExponentialsSampling:

Discrete-timesignalshavethreemajordifferencesfromitscontinuous-timepartner.Aretheythesame?No!

Forej

0t

，ithastwoproperties:Thelargerthemagnitudeof

0，thehigheristherateofoscillationinthesignal;ej

0tisperiodicforanyvalueof

0．32A.Fordiscrete-timecomplexexponentialsignals,weneedtoconsiderafrequencyintervalof2.(在考慮離散時間復(fù)指數(shù)時，僅需要在某一個2間隔內(nèi)選擇即可）

Thus,ej0nandej(0+

2)narethesamesignals.ej0n不具備隨

0在數(shù)值上的增加而不斷增加其振蕩速率的特性！當(dāng)

0從０開始增加，其振蕩速率愈來愈快，直到

0=

,達(dá)到最大，若繼續(xù)增加

0

，其振蕩速率就下降，直到

0=2

時,又得到與

0=0時同樣的效果(常數(shù)序列).33Lowestoscillationrate:Highestoscillationrate:34Continuous-time:ej

0t

,

T=2/

0Discrete-time:ej

0n

,N=?Calculateperiod:Bydefinition:ej

0n

=

ej

0(n+N)

thusej

0N

=1

or

0N=

2mSoN=

2m/

0

withintergerNConditionofperiodicity:2/0

isrational.B.Periodicityofej

0n

若2/0為一有理數(shù)，ej

0n就是周期的，否則就不是周期的．35C.FinitenumberofdistinctharmonicsForaperiodicsignalwithfundamentalperiodofN,ThereareonlyNdistinctperiodicexponentialsfordiscrete-timesignals.IntheContinuous-timecase,alloftheharmonicallyrelatedcomplexexponentialsaredistinct.36UnitSample(Impulse):UnitStepFunction:(4)TheDiscrete-TimeUnitImpulseandUnitStepSequences37Relationshipbetweenunitsampleandunitstepsequencetheunitsampleisthefirstdifferenceoftheunitstepsequencetheunitstepsequenceistherunningsumoftheunitsampleorhere38writeanydiscrete-timesignalintermsofdelayedunitsampleasSamplingPropertyofUnitSample39

UnitStepFunction:UnitImpulseFunction:(5)TheContinuous-TimeUnitStepandUnitImpulseFunctions40RelationBetweenUnitImpulseandUnitStep41Illustrationsofδ(t)

0ΔuΔ(t)t1

Continuousapproximationtotheunitstep,uΔ(t)

0Δ

1/Δ

δΔ(t)

DerivativeofuΔ(t)δ(t)isthelimitofδΔ(t)asΔ→0.42propertiesof

δ(t)Samplingpropertyδ(t)

isaevenfunction:

Scalingproperty:δ(-t)=δ(t)

Why:43Example1.3Determineandsketchthefirstderivativeofthesignaldepictedinthefollowingfigure.-1x(t)t01241-1-21012444(1)Definition:

Interconnection(互聯(lián))ofComponent,device,subsystem….(Broadestsense廣義)Aprocessinwhichsignalscanbetransformed.(Narrowsense狹義)Continuous-timesystem:bothinputsignalsandoutputsignalsarecontinuous-timesignalsDiscrete-timesystem:transformdiscrete-timeinputsintodiscrete-timeoutputs2.systems1)Continuous-TimeandDiscrete-TimeSystems45(2)RepresentationofSystemPictorialRepresentationContinuous-timesystem

x(t)y(t)Discrete-timesystem

x[n]Y[n]Relationbythenotation

46(3)SimpleExampleofsystemsExample1.8:RCCircuitinFigure1.1:Vc(t)

Vs(t)RCCircuit(System)

vs(t)vc(t)---LinearConstantCoefficientDifferentialEquation(微分方程)47BalanceinBank(System)

x[n]y[n]Example1.10:Balance(余額)inabankaccountfrommonthtomonth:balance(第n個月末的余額)---y[n]netdeposit(第n個月的凈存款)---x[n]interest(利息)---1%soy[n]=y[n-1]+1%

y[n-1]+x[n]ory[n]-1.01y[n-1]=x[n]482)InterconnectionsofSystem(1)Series(cascade)interconnection(串聯(lián)或級聯(lián))(2)Parallelinterconnection(并聯(lián))49Series-Parallelinterconnection(3)Feedbackinterconnection(反饋聯(lián)結(jié))50ExampleofFeedbackinterconnection513)

Basicpropertiesofsystems(Classifications)

(1)SystemswithandwithoutMemoryMemorylesssystem(無記憶系統(tǒng)):Itsoutputisdependentonlyontheinputatthesametime.Features:Nocapacitor,noconductor,nodelayer.Examplesofmemorylesssystem:

y(t)=C

x(t)ory[n]=C

x[n]Examplesofmemorysystem:ory[n]-0.5y[n-1]=2x[n]52(2)InvertibilityandInverseSystemsDefinition:

Asystemissaidtobeinvertible

ifdistinctinputsleadtodistinctoutputs.

Ifsystemisinvertable(可逆),thenaninversesystemexists.Aninversesystemcascaded(級聯(lián))withtheoriginalsystem,yieldsanoutputequaltotheinput.5354(3)Causality(因果性)Definition:AsystemiscausalIftheoutputatanytimedependsonlyonvaluesoftheinputatthepresenttimeandinthepast.Forcausalsystem,ifx(t)=0fort<t0,theremustbey(t)=0fort<t0.(nonanticipative(不可預(yù)測的))Memorylesssystemsarecausal.x(t)y(t)t1t255(4)Stability(穩(wěn)定性)Definition:

Iftheinputtoastablesystemisbounded(i.e.,ifitsmagnitudedoesnotgrowwithoutbound),thentheoutputmustalsobebounded.Finiteinputleadtofiniteoutput:if|x(t)|<M,then|y(t)|<N.Examples:Stablependulum Motionofautomobile56x(t)y(t)x(t-t0)y(t-t0)(5)TimeInvariance(時不變性)Definition:

Asystemistimeinvariantifthebehaviorandcharacteristicsofthesystemarefixedovertime.Timeinvariantsystem:Continuous-time:Ifx(t)y(t),thenx(t-t0)

y(t-t0).Discrete-time:Ifx(n)y(n),thenx(n-n0)

y(n-n0).57(6)Linearity(線性)

Definition:Thesystempossessestheimportantpropertyofsuperposition:A.Additivityproperty:Theresponsetox1(t)+x2(t)isy1(t)+y2(t).B.Scalingorhomogeneityproperty:Theresponsetoax1(t)isay1(t).(whereaisanycomplexconstant,a0.)Continuous-time:Discrete-time:58Lx1(t)x2(t)y1(t)y2(t)ax1(t)x1(t)+x2(t)ax1(t)+bx2(t)ay1(t)y1(t)+y2(t)ay1(t)+by2(t)Representedinblock-diagram:594)LTISystem

(線性時不變系統(tǒng))

LTIx(t)y(t)x(t-t0)ax(t)+bx(t-t0)y(t-t0)ay(t)+by(t-t0)LinearandTime-invariantsystemContinuous-time:Ifx1(t)y1(t),x2(t)y2(t),thenDiscrete-time:Ifx1(n)y1(n),x2(n)y2(n),then60Examples:1(Example1.12).Checkingthecausalityoftwosystems.ThefirstsystemisdefinedbyThesystemisnotcausal.

ThesecondsystemisdefinedbyCausal!

Itisimportanttodistinguishcarefullytheeffectsoftheinputfromthoseofanyotherfunctionsusedinthedefinitionofthesystem.2(

Example1.13).Checkthestabilityofthefollowingtwosystems3(Example1.15).Considerthetimeinvariancepropertyofthediscrete-timesystem61However,4.Considerasystemwhoseinputx[n]andoutputy[n]arerelatedbyThus,thesystemistime-varying.Todeterminewhetherornotthissystemislinear.Letx3[n]bealinearcombinationofx1[n]andx2[n]:Thus,thesystemisnonlinear.62SUMMARY1.Theconceptsofsignalsandsystems;2.Thegraphicalandmathematicalrepresentationsofsignals;3.Theclassificationsofsignals;4.Transformationsoftheindependentvariableofsignals;5.Theperiodicityofsignals;6.Severalbasicsignals;7.Interconnectionofsystems;8.Blockdiagramofsystems;9.Propertiesofsystems.63INTRODUCTION

LTIsystemspossessesthesuperpositionproperty.Representsignalsaslinearcombinationsofdelayedimpulses.Convolutionsumorconvolutionintegral.linearconstant-coefficientdifferenceordifferentialequations.641.Discrete-TimeLTI:ConvolutionSum1)TheRepresentationofDiscrete-timeSignalsinTermsofImpulses(Ch.2.1.1)2)TheDiscrete-timeUnitImpulseResponseofLTISystems(Ch.2.1.2)3)TheDiscrete-timeResponseofLTISystemstoanyInputSignal:ConvolutionSum651)TheRepresentationofDiscrete-timeSignalsinTermsofImpulsesIfx[n]=u[n],thenSiftingPropertyofUnitSample:1.Discrete-TimeLTI:ConvolutionSum662)TheDiscrete-timeUnitImpulseResponseofLTISystemsLTIx[n]=[n]y[n]=h[n]

UnitImpulseResponseh[n]:responseoftheLTIsystemtotheunitsampleδ[n].δ[n]→h[n]Whydoweneedit?673)TheDiscrete-timeResponseofLTISystemstoanyInputSignal:ConvolutionSumLTIx[n]y[n]=?Solution:Question:[n]h[n][n-k]h[n-k]x[k][n-k]x[k]h[n-k]Theresponsey[n]tox[n]istheweightedlinearcombinationofdelayedunitsampleresponses.68ConvolutionSumSoRepresentingtheconvolutionoperationsymbolicallyas:y[n]=x[n]*h[n]---ConvolutionSum

Thatis,theunitimpulseresponse--h[n]

canfullycharacterizeanLTIsystem.SummaryoncalculatingconvolutionsumTimeInversal: h[k]h[-k]TimeShift: h[-k]h[n-k]Multiplication: x[k]h[n-k]Summing:69Example2.1

ConsideraLTIsystemwithunitsampleresponseh[n]andinputx[n],asillustratedinFigure(a).Calculatetheconvolutionsum(convolution)ofthesetwosequencesgraphically.

nx[n]

012nh[n]-202(a)

122kx[k]

012kh[-k]-202

(b)22170kx[k]

0122kh[-k]-20221n=0kh[-1-k]-3-20121n=-1kh[1-k]-1012321n=171Example2.2

Consideraninputx[n]andaunitsampleresponseh[n]givenbyDetermineandplottheoutputUsingthegeometricalsumformulatoevaluatelastequation,wehave

72n……21y[n]732.Continuous-TimeLTI:ConvolutionIntegral1)TheRepresentationofContinuous-timeSignalsinTermsofImpulses(Ch.2.2.1)2)TheContinuous-timeUnitImpulseResponseofLTISystems(Ch.2.2.2)3)TheContinuous-timeResponseofLTISystemstoanyInputSignal:ConvolutionIntegral741)TheRepresentationofContinuous-timeSignalsinTermsofImpulsesDiscrete-time:Continuous-time:Why?t┉┉-Δ0Δ2ΔkΔ

┉x(t)Staircaseapproximationtoacontinuous-timesignalx(t)2.Continuous-TimeLTI:ConvolutionIntegral75Therefore:Whatisthis?DefineWehavetheexpression:as,thesummingapproachesanintegralandistheunitimpulsefunction

762)TheContinuous-timeUnitImpulseResponseofLTISystemsLTIx(t)=(t)y(t)=h(t)UnitImpulseResponseh(t)

:

theresponseoftheLTIsystemtotheinput.

3)TheContinuous-timeResponseofLTISystemstoanyInputSignal:ConvolutionIntegralLTIx(t)y(t)=?77Givetheastheresponseofacontinuous-timeLTIsystemtotheinput,thentheresponseofthesystemtopulseis

Thus,theresponse

toisAs,inaddition,thesummingbecomesanintegral.Therefore,---ConvolutionIntegral

78Representconvolutionintegraloftwosignalsx(t)andh(t)symbolicallyas:ConvolutionIntegralAcontinuous-timeLTIsystemiscompletelycharacterizedbyitsunitimpulseresponseh(t).ComputationofConvolutionIntegral:

TimeInversal:h()h(-)TimeShift:h(-)h(t-)Multiplication:x()h(t-)Integrating:79Example2.3

Considertheconvolutionofthefollowingtwosignals,whicharedepictedin(a):

2x(t)1h(t)

012t

0123t-1(a)

x(τ)h(-τ)-20123τ

t=0

x(τ)h(t-τ)-20123τt0<t<1Whent<1:x(τ)h(t-τ)=0So

80

x(τ)h(t-τ)-201t23τ1<t<2

x(τ)h(t-τ)-2012t3τ2<t<3

x(τ)h(t-τ)-20123t4τ3<t<481

x(τ)h(t-τ)-201234t5τ4<t<5

y(t)

0135τ-2

x(τ)

h(t-τ)-2012345tτ

t≥5Whent≥5:x(τ)h(t-τ)=0So

823.PropertiesofLTISystems1)TheCommutativeProperty(Ch.2.3.1)2)TheDistributiveProperty(Ch.2.3.2)3)TheAssociativeProperty(Ch.2.3.3)4)LTISystemwithandwithoutMemory(Ch.2.3.4)5)InvertibilityofLTIsystem(Ch.2.3.5)6)CausalityforLTIsystem(Ch.2.3.6)7)StabilityforLTIsystem(Ch.2.3.7)8)TheUnitStepResponseofLTIsystem(Ch.2.3.8)83h(t)orh[n]completelycharacterizesanLTIsystemWhatpropertyshouldh(t)orh[n]havefortheLTIsystemtobestable,causal,memorylessandinvertible?3.PropertiesofLTISystems841)TheCommutativePropertyDiscretetime:

x[n]*h[n]=h[n]*x[n]Continuoustime:x(t)*h(t)=h(t)*x(t)h(t)x(t)y(t)=x(t)*h(t)x(t)h(t)y(t)=h(t)*x(t)

2)TheDistributivePropertyDiscretetime:x[n]*{h1[n]+h2[n]}=x[n]*h1[n]+x[n]*h2[n]Continuoustime:

x(t)*{h1(t)+h2(t)}=x(t)*h1(t)+x(t)*h2(t)3.PropertiesofLTISystems85h1(t)+h2(t)x(t)y(t)=x(t)*{h1(t)+h2(t)}h1(t)x(t)y(t)=x(t)*h1(t)+x(t)*h2(t)h2(t)

3)TheAssociativePropertyDiscretetime:

x[n]*{h1[n]*h2[n]}={x[n]*h1[n]}*h2[n]Continuoustime:

x(t)*{h1(t)*h2(t)}={x(t)*h1(t)}*h2(t)86h1(t)*h2(t)x(t)y(t)=x(t)*{h1(t)*h2(t)}h1(t)x(t)y(t)=x(t)*h1(t)*h2(t)h2(t)4)LTISystemwithandwithoutMemoryMemorylesssystem:Discretetime:y[n]=kx[n],h[n]=?Continuoustime:y(t)=kx(t),h(t)=?k

(t)

x(t)y(t)=kx(t)=x(t)*k(t)k

[n]

x[n]y[n]=kx[n]=x[n]*k[n]875)InvertibilityofLTIsystemOriginalsystem:h(t)Reversesystem:h1(t)(t)

x(t)x(t)*(t)=x(t)So,fortheinvertiblesystem:

h(t)*h1(t)=(t)orh[n]*h1[n]=[n]h(t)

x(t)x(t)h1(t)

6)CausalityforLTIsystemDiscretetimesystemsatisfy:

h[n]=0forn<0Continuoustimesystemsatisfy:

h(t)=0fort<0Why?887)StabilityforLTIsystemDefinitionofstability:Everyboundedinputproducesaboundedoutput.If|x[n]|<B,thesufficientandnecessaryconditionfor|y[n]|<AisDiscretetimesystem:Continuoustimesystem:If|x(t)|<B,theconditionfor|y(t)|<Ais898)TheUnitStepResponseofLTIsystemTheunitstepresponse,s[n]ors(t),istheoutputofanLTIsystemwheninputx[n]=u[n]orx(t)=u(t).A.Thestepresponseofadiscrete-timeLTIsystemistherunningsumofitssampleresponse:B.Theimpulseresponseofadiscrete-timeLTIsystemisthefirstdifferenceofitsstepresponse:h[n]/h(t)

[n]/(t)h[n]/h(t)u[n]/u(t)s[n]=u[n]*h[n]/s(t)=u(t)*h(t)90C.Theunitstepresponseofacontinuous-timeLTIsystemistherunningintegralofitsimpulseresponse:D.Theunitimpulseresponseofacontinuous-timeLTIsystemisthefirstderivativeoftheunitstepresponse:E.Propertiesofconvolutionintegral:Derivativeproperty:Integralproperty:Combiningthetwoproperties,wehave914.CausalLTISystemsdescribedbyDifferentialandDifferenceEquations1)Continuous-timesystem:DifferentialEquation(Ch.2.4.1)2)Discrete-timesystem:DifferenceEquation(Ch.2.4.2)3)BlockDiagramRepresentations(Ch.2.4.3)92:inputsignal;

:outputsignal.

Ci(t)VsR+–1)Continuous-timesystem:DifferentialEquationLinearconstant-coefficientdifferentialequationLinearconstant-coefficientdifferential(difference)equation

providesanimplicit

relationshipbetweentheinputandoutputratherthananexplicitexpressionforthesystemoutputasafunctionoftheinput.4.CausalLTISystemsdescribedbyDifferentialandDifferenceEquations93Howtofindthesystemoutputgivenaninputsignal?naturalresponse

Forcedresponse

Wemustspecifyoneormoreauxiliaryconditionstosolveadifferential(difference)equation.Initialrest:foracausalLTIsystem,ifx(t)=0fort<t0,theny(t)mustalsoequal0fort<t0.Itisimportanttoemphasizethattheconditionofinitialrestdoesnotspecifyazeroinitialconditionatafixedpointintime,butratheradjuststhispointintimesothattheresponseiszerountiltheinputbecomesnonzero.Thus,ifx(t)=0fort≤t0foracausalLTIsystem,theny(t)=0fort≤t0,andwewouldusetheinitialconditiony(t0)=0tosolvefortheoutputfort>t0.94AgeneralNth-orderlinearconstant-coefficientdifferentialequation:orandinitialcondition:

y(t0),y’(t0),……,y(N-1)(t0)(Nvalues)ForacausalLTIsystem:952)Discrete-timesystem:DifferenceEquationAgeneralNth-orderlinearconstant-coefficientdifferenceequation:orandinitialcondition:

y[0],y[-1],……,y[-(N-1)](Nvalues)Underinitialrest,thesystemdescribedbylinearconstant-coefficientdifferential(difference)equationiscausalandLTI.96Generalsolutionstosuchdifferenceequations:laterinChapter5or10.

Secondresolution:(recursivemethod)Firstresolution:N

auxiliaryconditions:973)BlockDiagramRepresentations(1)Dicrete-timesystemBasicelements:A.AnadderB.MultiplicationbyacoefficientC.AnunitdelayFirst-orderdifferenceequation:

additiondelaymultiplication

98Example:y[n]+ay[n-1]=bx[n]

(2)Continuous-timesystemFirst-orderdifferentialequation:differentiationThreebasicelementsinblockdiagram:adder,multiplierandintegrator.99Example:y’(t)+ay(t)=bx(t)

Suchblockdiagramscanalsobedevelopedforhigherordersystems.1005.SingularityFunctions1)Theunitimpulseasanidealizedshortpulse(1)(2)Important:forsmallΔ,theybothbehavesthesamefromanLTIsystem,seeFigure2.34.1012)Definingtheunitimpulsethroughconvolution---Operationaldefinition（運(yùn)算定義）Or，equivalently,Theprimaryimportanceoftheunitimpulseisnotwhatitisateachvalueoft,butratherwhatitdoesunderconvolution.1023)Differentiator,unitdoublet

（單位沖激偶）

103SUMMARY1.Arepresentationofanarbitrarydiscrete-timesignalasweightedsumsofshiftedunitsamples;2.Convolutionsumrepresentationfortheresponseofadiscrete-timeLTIsystems;3.Arepresentationofanarbitrarycontinuous-timesignalaweightedintegralsofshiftedunitimpulses;4.Convolutionintegralrepresentationforcontinuous-timeLTIsystems;1045.RelatingLTIsystemproperties,includingcausality,stability,tocorrespondingpropertiesoftheunitimpulse(sample)response;6.Someofthepropertiesofsystemsdescribedbylinearconstant-coefficientdifferential(difference)equations;7.Understandingoftheconditionofinitialrest.SUMMARY105INTRODUCTION

Representationofcontinuous-timeanddiscrete-timeperiodicsignals—Fourierseries.UseFouriermethodstoanalyzeandunderstandsignalsandLTIsystems.1061.TheResponseofLTISystemstoComplexExponentials

1)Importantconcept—signaldecomposition(1)basicsignals:possesstwopropertiesA.

Thesetofbasicsignalscanbeusedtoconstructabroadandusefulclassofsignals.

B.ItshouldbeconvenientforustorepresenttheresponseofanLTIsystemtoanysignalconstructedasalinearcombinationofthebasicsignals.(2)complexexponentialsignals

indiscretetime:

incontinuoustime:1072)TheResponseofanLTISys