19

THELAPLACETRANSFORM

拉普拉斯變換Maincontent：TheLaplaceTransform(雙邊拉普拉斯變換〕TheRegionofConvergenceforLaplaceTransform〔雙邊拉普拉斯變換的收斂域〕Pole-ZeroPlot〔零極點圖〕ThePropertiesoftheLaplaceTransform〔雙邊拉普拉斯變換的性質(zhì)〕SystemFunction(系統(tǒng)函數(shù))TheUnilateralLaplaceTransform(單邊拉普拉斯變換)9.0Introduction(p.654)

WithLaplacetransform,weexpandtheapplicationinwhichFourieranalysiscanbeused.

TheLaplacetransformisageneralizationofthecontinuous-timeFouriertransform.

TheLaplacetransformprovidesuswitharepresen-tationforsignalsaslinearcombinationsofcomplexexponentialsoftheformwith.(9.1)where(9.2)9.1TheLaplaceTransform(拉普拉斯變換)

(p.655)

For

s

imaginary(i.e.,),theintegralineq.(9.2)correspondstotheFouriertransformof.

istheeigenfunctionofcontinuous–timeLTIsystems.(Section3.2)

TheresponseofaLTIsystemwithimpulseresponse toaninputoftheformis

9.1.1DefinitionofTheLaplaceTransform

(拉普拉斯變換的定義)(p.655)(9.3)

When,eq.(9.3)becomes

Itisafunctionoftheindependentvariable.(9.5)

whichcorrespondstotheFouriertransformof.(9.4)

Thecomplexvariablecanbewrittenas.

Thatis,theLaplaceTransformofcanbeinterpretedastheFouriertransformofaftermultiplicationbyarealexponentialsignal.Therealexponentialmaybedecayingorgrowingintime,dependingonwhetherispositiveornegative.7Theindependentvariable

representsnotonlythefrequencybutalsothegrowing(ordecaying)rateoftheoscillation.Theindependentvariablerepresentsthefrequencyoftheoscillationonly.

isabsolutelyintegrableforsomevaluesofbutnotforothers.

isnotabsolutelyintegrableTimedomain?

ComplexFrequencydomainTimedomain?FrequencydomainLaplaceTransformFourierTransformTheintegralconvergesfor

.Example9.1For,theFourierTransformofconvergesandisgivenby

Infact,if,thenX(s)canbeevaluatedattoobtain9For:with,then

No!For:theFourierTransformdoesnotexist,theLaplaceTransformstillexist.?whyExample9.2

ComparingExample1and2,weseethatthealgebraicexpressionfortheLaplaceTransformisidenticalforbothofthesignals,butthesetofvaluesofsforwhichtheexpressionisvalidisverydifferent.Example9.1JustastheFouriertransformdoesnotforallsignals,theLaplacetransformmayconvergeforsomevaluesofRe{s}andnotforothers.Therangeofvaluesofsforwhichtheintegralineq.(9.3)convergesisreferredtoasregionofconvergence(whichweabbreviateasROC)oftheLaplacetransform.Note：IftheROCoftheLaplacetransformconcludesthe

-axisofs-plane,thenNote：ThealgebraicexpressionfortheLaplaceTransformisidenticalforbothofthesignals,butthesetofvaluesofsforwhichtheexpressionisvalidisverydifferent.InspecifyingtheLaplaceTransformofasignal,boththealgebraicexpressionandtheROCarerequired.9.1.2TheROCandPole-ZeroPlotforLaplaceTransforms(拉氏變換的ROC及零極點圖)Example9.3(p.658)

TheROCoftheLaplaceTransformisthecommonpartoftheROCs.s-planetherootsofthenumeratorpolynominalN(s)—zeros(零點)therootsofthedenominatorpolynominalD(s)—poles(極點)Ifisrational,i.e.,Pole-zeroplotandROCs-planePole-zeroplot：+M+ROCTheROCofX(s)consistsofstripsparalleltothe-axisinthes-plane.(p.662)ForrationalLaplacetransforms,theROCdoesnotcontainanypoles.(p.663)Ifx(t)isoffinitedurationandisabsolutelyintegrable,thentheROCistheentires-plane.Properties:9.2TheRegionofConvergence(ROC)forLaplaceTransforms(拉氏變換的收斂域)

(p.662)17

Theintegralconvergesforanyvalueofsinthes-plane,thatisROCistheentires-plane.pole

？

Example9.6(p.664)(9.41)Ifx(t)isrightsided,andifthelineisintheROC,thenallvaluesofsforwhichwillalsobeintheROC.(p.665)Thenif ,willalsobeabsolutelyintegral.

Sinceisright-sided,,andisintheROC,thenisabsolutelyintegral,i.e.(9.45)Ifx(t)isleftsided,andifthelineisintheROC,thenallvaluesofsforwhichwillalsobeintheROC.(p.666)

Sinceisleft-sided,,andisintheROC,thenisabsolutelyintegral,i.e.Thenif,willalsobeabsolutelyintegral.Example9.7(p.668)()Ifx(t)istwosided,andifthelineisintheROC,thentheROCwillconsistofastripinthes-planethatincludestheline.(p.666)Ifb≤0,thereisnocommonregionofconver-gence,thusx(t)hasnoLaplaceTransform.Figure9.12Ifb>0,theLaplaceTransformofx(t)is(9.51)itsROCisboundedbypolesorextendstoinfinity.Inaddition,nopolesofX(s)arecontainedintheROC.ifx(t)isrightsided,theROCistheregioninthes-planetotherightoftherightmostpole;ifx(t)isleftsided,theROCistheregioninthes-planetotherightoftheleftmostpole.IftheLaplacetransformX(s)ofx(t)isrational,thenProperties(p.669)TherearethreepossibleROCs：

x(t)isrightsided.HowmanypossibleROCsaretherewhentherearethreedifferentpoles?x(t)isleftsided.x(t)istwosided.Example9.8(p.669)24Example25Example26Example27Example28ExampleTheintegraldoesnotconverge,i.e.,theLaplaceTransformdoesnotexist.9.3.1DefinitionFromwhenisintheROC,then:9.3TheInverseLaplaceTransform

(拉普拉斯反變換)(p.670)Thisequationstatesthatx(t)canberepre-sentedasaweightedintegralofcomplexexponentials.——ThebasicinverseLaplaceTransformequation〔拉氏反變換根本關(guān)系式〕(9.56)Withfixed,from,thenwhenvaryingfromto,

s:

9.3.2Solving:Residue-basedmethod〔留數(shù)法〕Partial-fractionexpansion-basedmethod(局部分式展開法)

FromtheROCofX(s),theROCofeachoftheindividualtermscanbeinferred;ByuseofTable9.2,theinverseLaplaceTrans-formofeachofthesetermscanbedetermined.poles：HowmanysignalshaveaLaplacetransformthatmaybeexpressedasbelowinitsregionofconvergence?RightsidedsignalLeftsidedsignalTwosidedsignalExample9.9(p.671)(9.58)Example9.11(p.673)(9.69)349.4GeometricEvaluationoftheFourierTransformfromthePole-ZeroPlot

(由零極點圖對傅里葉變換幾何求值)(p.674)AgeneralrationalLaplacetransformhastheform:where

arezerosandpolesofX(s),respectively.anditcanbefactoredintotheform:(9.70)35Figure9.15Complexplanerepresentationofthevectorss1,a,ands1–arepresentingthecomplexnumberss1,aands1–arespectively.0s-plane36Let’stakeanexampletoshowhowtoevaluatetheFouriertransformfromthepole-zeroplot:Given

-2-1ReImω

s-planeGeometrically,

fromFigure,wecanwrite|X(jω)|isthereciprocal〔倒數(shù)〕oftheproductofthelengthsofthetwopolevectors(極點矢量);argX(jω)isthenegativeofthesumoftheanglesofthetwovectors.zerovectors(零點矢量)37IfandNote:

ROCisatleasttheintersectionofR1andR2,whichcouldbeempty,alsocanbelargerthantheintersection.9.5PropertiesofTheLaplaceTransform(拉普拉斯變換性質(zhì))(p.682)9.5.1LinearityoftheLaplaceTransform(p.683)then(9.82)389.5.2TimeShifting(時移性質(zhì))(p.684)9.5.3Shiftinginthes-Domain(s域平移)(p.685)IfthenIfthen(9.87)(9.88)39Example:(complementary)

ConsiderthesignalWeknowAndfromthetimeshiftingproperty,SothatHere,thepoleats=0isremovable.40IfConsequence:ifx(t)isrealandifX(s)hasapoleorzeroats=s0,thenX(s)alsohasapoleorzeroatthecomplexconjugatepoints=s0*.9.5.5Conjugation(共軛)(p.687)Whenx(t)isreal:

Consequence:9.5.4TimeScaling(時域尺度變換)(p.685)then(9.90)(9.93)41Ifand9.5.7DifferentiationintheTimeDomain(時域微分)(p.687)Ifthen(9.95)then(9.98)9.5.6ConvolutionProperty(卷積性質(zhì))(p.687)42(9.100)9.5.8Differentiationinthes-Domain(s域微分)(p.688)(9.106)9.5.9IntegrationintheTimeDomain(時域積分)(p.685)43SinceFromthedifferentiationinthes-domainproperty,Infact,byrepeatedapplicationofthisproperty,weobtainExample9.14

DeterminetheLaplacetransformof(9.101)(9.102)(9.104)44Conditions:

x(t)=0fort<0andthatx(t)containsnoimpulsesorhigherordersingularitiesattheorigin.Initial-valuetheorem:(9.110)Final-valuetheorem:(9.111)9.5.10TheInitial-andFinal-ValueTheorems(初值和終值定理)(p.690)45IntheTimeDomain:Example9.16(p.691)

Usetheinitial-valuetheoremtodeterminetheinitial-valueof(9.24)46Weknow,inthetimedomain,theinputandtheoutputofanLTIsystemarerelatedthroughConvolutionbytheimpulseresponseofthesystem.Thus

y(t)=h(t)*x(t)suppose9.7

AnalysisandCharacterizationofLTISystemsUsingTheLaplaceTransform(用拉普拉斯變換分析和表征LTI系統(tǒng))(p.693)

9.7.0Systemfunction47FromConvolutionProperty

Y(s)=H(s)X(s)

For,H(s)isthefrequencyresponse(頻率響應(yīng))oftheLTIsystem.y(t)=h(t)*x(t)systemfunction/transferfunction(系統(tǒng)函數(shù)/傳輸函數(shù))48TheROCassociatedwiththesystemfunctionforacausalsystemisaright-halfplane.(p.693)AnROCtotherightoftherightmostpoledoesnotguaranteethatasystemiscausal(asillustratedinExample9.19).ForacausalLTIsystem,theimpulseresponseiszerofort<0andthusisrightsided.9.7.1Causality(因果性)(p.693)49Forasystemwitharationalsystemfunction,causalityofthesystemisequivalenttotheROCbeingtheright-halfplanetotherightoftherightmostpole.(p.694)Asystemisanticausalifitsimpulseresponseh(t)=0,fort>0.TheROCassociatedwiththesystemfunctionforaanticausalsystemisaleft-halfplane.50Example:

ConsiderasystemwithimpulseresponseSinceh(t)=0fort<0,thissystemiscausal.Thesystemfunction:ItisrationalandtheROCistotherightoftherightmostpole,consistentwithourstatement.51Forthissystem,theROCistotherightoftherightmostpole.Sincethesystemfunctionisirrational(無理的).(9.115)Theimpulseresponseassociatedwiththesystemwhichisnonzerofor–1<t<0.Hence,thesystemisnotcausal.(9.117)Example9.19(p.694)

Considerthesystemfunction52ThestabilityofanLTIsystemisequivalenttoitsimpulseresponsebeingabsolutelyintegrable,inwhichcasetheFouriertransformoftheimpulseresponseconverges.sinceAnLTIsystemisstableifandonlyiftheROCofitssystemfunctionH(s)includesthe-axis[i.e.,Re{s}=0].9.7.2Stability(穩(wěn)定性)(p.695)53AcausalsystemwithrationalsystemfunctionH(s)isstableifandonlyifallofthepolesofH(s)lieintheleft-halfofthes-plane―i.e.,allofthepoleshavenegativerealparts.stableh(t)isabsolutelyintegrableh(t)hasFTROCofh(t)’sLTcontains-axis54

H(s)havetwopoles:s1=-1,s2=2Ifthesystemisknowntobecausal,theROCwillbe,thus,thesystemisunstable.Ifthesystemisknowntobestable,theROCis,thus ,IftheROCofH(s)is,thenanticausalandunstableExample9.20

ConsideranLTIsystemwithsystemfunctionthesystemisnotcausal.55ForanLTIsystemwhichisdescribedbyalinearconstant-coefficientdifferentialequationoftheformThus,thesystemfunctionforasystemspecifiedbyadifferentialequationisalwaysrational.9.7.3LTISystemCharacterizedbyLinearConstant-CoefficientDifferentialEquations(p.698)(由線性常系數(shù)微分方程表征的LTI系統(tǒng)〕(9.131)systemfunction(transferfunction):(9.133)56Thesystemiscausal.Thesystemfunctionisrationalandhasonlytwopoles,ats=–2ands=4.Ifx(t)=1,theny(t)=0.Thevalueoftheimpulseresponseatis4.Determinethesystemfunctionofthesystem.Fromfact2,wewriteExample9.26

GiventhefollowinginformationaboutanLTIsystem:57Fromfact3,p(s)musthavearootats=0andthusisoftheform p(s)=sq(s)Fromfact4and1,Thehighestpowersinsinboththedenominatorandthenumeratorareidentical,thatis,q(s)

mustbeaconstant.Weletq(s)=k.

It’seasytofindthatk=4.Sothat58TheuseoftheLaplacetransformallowsustoreplacetime-domainoperationssuchasdifferentiation,convolution,timeshifting,andsoon,withalgebraicoperations.InthissectionwetakealookatanotherimportantuseofsystemfunctionalgebrainanalyzinginterconnectionsofLTIsystemsandsynthesizingsystemsasinterconnectionsofelementarysystembuildingblocks.9.8SystemFunctionAlgebraandBlockDiagramRepresentations

(系統(tǒng)函數(shù)的代數(shù)屬性與方框圖表示)(p.706)59h1(t)H1(s)x(t)y(t)=x(t)*h1(t)+x(t)*h2(t)h2(t)H2(s)y(s)=x(s)H1(s)+x(s)H2(s)x(s)Theparallelinterconnectionoftwosystems:9.8.1SystemFunctionsforInterconnectionsofLTISystems(LTI系統(tǒng)互聯(lián)的系統(tǒng)函數(shù))(p.707)h(t)=h1(t)+h2(t)(9.155)H(s)=Y(s)/X(s)=H1(s)+H2(s)(9.156)Figure9.30(a)ParallelinterconnectionoftwoLTIsystems60h(t)=h1(t)*h2(t)(9.157)H(s)=Y(s)/X(s)=H1(s)H2(s)(9.158)Theseriesinterconnectionoftwosystems:h1(t)H1(s)x(t)h2(t)H2(s)x(s)y(t)Y(s)Figure9.30(b)SeriescombinationoftwoLT