Signal and System chapter 6.ppt

CHAPTER6THELAPLACETRANSFORM6 0INTRODUCTION WithLaplacetransform weexpandtheapplicationinwhichFourieranalysiscanbeused TheLaplacetransformprovidesuswitharepresentationforsignalsaslinearcombinationsofcomplexexponentialsoftheformwiths j TheLaplacetransform 拉普拉斯变换 isageneralizationofthecontinuous timeFouriertransform 6 1THELAPLACETRANSFORM Lets j andusingX s todenotethisintegral weobtain ForsomesignalswhichhavenotFouriertransforms ifwepreprocessthembymultiplyingwitharealexponentialsignal thentheymayhaveFouriertransforms TheLaplacetransformisanextensionoftheFouriertransform theFouriertransformisaspecialcaseoftheLaplacetransformwhen 0 Example6 1Considerthesignal Forconvergence werequirethatRe s 0 orRe s Thus Example6 2Considerthesignal Forconvergence werequirethatRe s 0 orRe s Thus ROCforExample6 1 ROCforExample6 2 Example6 3Considerthesignal UsingEuler srelation wecanwrite Thus Consequently SomeusefulLTpairs MarkingthelocationsoftherootsofN s andD s inthes plane s平面 andindicatingtheROCprovidesaconvenientpictorialwayofdescribingtheLaplacetransform Generally theLaplacetransformisrational i e itisaratioofpolynomialsinthecomplexvariables TherootsofN s arereferredtoasthezeros 零点 ofX s andtherootsofD s arereferredtoasthepoles 极点 ofX s TherepresentationofX s throughitspolesandzerosinthes planeisreferredtoasthepole zeroplot 极零图 ofX s Exceptforascalefactor acompletespecificationofarationalLaplacetransformconsistsofthepole zeroplotofthetransform togetherwithitsROC Abouttheinfinity 无穷远点 ingeneral iftheorderofthedenominatorexceedstheorderofthenumeratorbyk X s willhavekzerosatinfinity Similarly iftheorderofthenumeratorexceedstheorderofthedenominatorbyk X s willhavekpolesatinfinity Example6 4Considerthesignal Pole zeroplotandROC 6 2THEREGIONOFCONVERGENCEFORLAPLACETRANSFORM Property1 TheROCofX s consistsofstripesparalleltothej axisinthes plane Property2 ForrationalLaplacetransforms theROCdoesnotcontainanypoles Property3 Ifx t isoffinitedurationandisabsolutelyintegrable thentheROCistheentires plane Example6 5Let Thepoleats isremovable Property4 Ifx t isrightsided andifthelineRe s 0isintheROC thenallvaluesofsforwhichRe s 0willalsobeintheROC andtheROCofaright sidedsignalisaright halfplane Property5 Ifx t isleftsided andifthelineRe s 0isintheROC thenallvaluesofsforwhichRe s 0willalsobeintheROC andtheROCofaleft sidedsignalisaleft halfplane Property6 Ifx t istwosided andifthelineRe s 0isintheROC thentheROCwillconsistofastripinthes planethatincludesthelineRe s 0 Property7 IftheLaplacetransformX s ofx t isrational thenitsROCisboundedbypolesorextendstoinfinity Inaddition nopolesofX s arecontainedintheROC Property8 IftheLaplacetransformX s ofx t isrational thenifx t isrightsided theROCistheregioninthes planetotherightoftherightmostpole Ifx t isleftsided theROCistheregioninthes planetotheleftoftheleftmostpole Example6 6Let 6 3THEINVERSELAPLACETRANSFORM Multiplyingbothsidesby weobtain Changingthevariableofthisintegrationfrom tosandusingthefactthat isconstant sothatds jd Thus thebasicinverseLaplacetransformequationis TheinverseLaplacetransformequationstatesthatx t canberepresentedasaweightedintegralofcomplexexponentials TheformalevaluationoftheintegralforageneralX s requirestheuseofcontourintegration 围线积分 inthecomplexplane Fortheclassofrationaltransforms theinverseLaplacetransformcanbedeterminedbyusingthetechniqueofpartial fractionexpansion Example6 7Let Performingthepartial fractionexpansion weobtain Example6 8Let Computethex t withcontourintegrationmethod X s hastwofirst orderpoles andasecond orderpole FromtheResidueTheorem 6 4GEOMETRICEVALUATIONOFTHEFOURIERTRANSFORMFROMTHEPOLE ZEROPLOT AgeneralrationalLaplacetransformhastheform Complexplanerepresentationofthevectorss1 a ands1 arepresentingthecomplexnumberss1 aands1 arespectively Let stakeanexampletoshowhowtoevaluatetheFouriertransformfromthepole zeroplot Given Geometrically fromFigure wecanwrite X j isthereciprocaloftheproductofthelengthsofthetwopolevectors 极点矢量 argX j isthenegativeofthesumoftheanglesofthetwovectors zerovectors 零点矢量 6 5PROPERTIESOFTHELAPLACETRANSFORM 6 5 1Linearity 6 5 2TimeShifting then Note ROCisatleasttheintersectionofR1andR2 whichcouldbeempty alsocanbelargerthantheintersection If then 6 5 3Shiftinginthes Domain If then 6 5 4TimeScaling If then Consequence ifx t isrealandifX s hasapoleorzeroats s0 thenX s alsohasapoleorzeroatthecomplexconjugatepoints s0 6 5 5Conjugation Whenx t isreal Consequence 6 5 6ConvolutionProperty then 6 5 7DifferentiationintheTimeDomain If then 6 5 8Differentiationinthes Domain 6 5 9IntegrationintheTimeDomain 6 5 10TheInitial andFinal ValueTheorems 初值和终值定理 Initial valuetheorem Final valuetheorem Example6 9Considerthesignal Weknow Andfromthetimeshiftingproperty Sothat Example6 10DeterminetheLaplacetransformof Since Fromthedifferentiationinthes domainproperty Infact byrepeatedapplicationofthisproperty weobtain Example6 11Usetheinitial valuetheoremtodeterminetheinitial valueof 6 6ANALYSISANDCHARACTERIZATIONOFLTISYSTEMSUSINGTHELAPLACETRANSFORM TheLaplacetransformsoftheinputandtheoutputofanLTIsystemarerelatedthroughmultiplicationbytheLaplacetransformoftheimpulseresponseofthesystem Y s H s X s TheROCassociatedwiththesystemfunctionforacausalsystemisaright halfplane AnROCtotherightoftherightmostpoledoesnotguaranteethatasystemiscausal Forasystemwitharationalsystemfunction causalityofthesystemisequivalenttotheROCbeingtheright halfplanetotherightoftherightmostpole Example6 12Considerasystemwithimpulseresponse Sinceh t 0fort 0 thissystemiscausal Thesystemfunction ItisrationalandtheROCistotherightoftherightmostpole consistentwithourstatement Example6 13Considerthesystemfunction Forthissystem theROCistotherightoftherightmostpole Sincethesystemfunctionisirrational AcausalsystemwithrationalsystemfunctionH s isstableifandonlyifallofthepolesofH s lieintheleft halfofthes plane i e allofthepoleshavenegativerealparts AnLTIsystemisstableifandonlyiftheROCofitssystemfunctionH s includesthej axis i e Re s 0 ROCofh t sLTcontainsj axis ForanLTIsystemwhichisdescribedbyalinearconstant coefficientdifferentialequationoftheform Itssystemfunction transferfunction is Thus thesystemfunctionforasystemspecifiedbyadifferentialequationisalwaysrational Example6 14GiventhefollowinginformationaboutanLTIsystem 1 Thesystemiscausal 2 Thesystemfunctionisrationalandhasonlytwopoles ats 2ands 4 3 Ifx t 1 theny t 0 4 Thevalueoftheimpulseresponseatis4 Determinethesystemfunctionofthesystem Fromfact2 wewrite Fromfact3 p s musthavearootats 0andthusisoftheformp s sq s Fromfact4and1 Thehighestpowersinsinboththedenominatorandthenumeratorareidentical thatis q s mustbeaconstant Weletq s k It seasytofindthatk 4 Sothat 6 7SYSTEMFUNCTIONALGEBRAANDBLOCKDIAGRAMREPRESENTATIONS TheuseoftheLaplacetransformallowsustoreplacetime domainoperationssuchasdifferentiation convolution timeshifting andsoon withalgebraicoperations InthissectionwetakealookatanotherimportantuseofsystemfunctionalgebrainanalyzinginterconnectionsofLTIsystemsandsynthesizingsystemsasinterconnectionsofelementarysystembuildingblocks Example6 15ConsiderthecausalLTIsystemwithsystemfunction Thissystemcanalsobedescribedbythedifferentialequation 1 sisthesystemfunctionofasystemwithimpulseresponseu t i e itisthesystemfunctionofanintegrator Example6 16ConsideracausalLTIsystemwithsystemfunction Herez t istheoutputofthefirstsubsystem y t istheoutputoftheoverallsystem Example6 17Considerasecond orderLTIsystemwithsystemfunction Direct form 直接型 representationforthesysteminExample6 17 Cascade form 串联型 representationforthesysteminExample6 17 y t x t Parallel form 并联型 representationforthesysteminExample6 17 6 8SignalFlowGraphRepresentation 信号流图 Formally asignalflowgraphisanetworkofdirectedbranchesthatconnectatnodes Associatedwitheachnodeisavariableornodevalue Input output nodevalue direction branch j k Twospecialtypesofnodes Sourcenodes 源结点 nodesthathavenoenteringbranches whichareusedtorepresenttheinjectionofexternalinputsorsignalsourcesintoagraph Sinknodes 汇结点 nodesthathaveonlyenteringbranches whichareusedtoextractoutputsfromagraph Loop 环 self loop 自环 forwardpath 前向通路 mixednode 混合结点 Example6 18Consideragainthesysteminexample6 17 Usesignalflowgraphtorepresentit Mason sFormula 梅森规则 Hereisthegraphdeterminant 特征行列式 Inasignalflowgraph thetransfervalue transferfunction betweenanysourcenodeandsinknodeormixednodecanbedeterminedbythefollowingequation isthegainofeachloop isthemultiplicationofthegainsoftwoloopswhichhavenosharednodesandbranches isthegraphdeterminantoftheleftgraphafterremovingthek thforwardparth isthegainofthek thforwardparthbetweenthesourcenodeandthesinknode Example6 19ComputethetransferfunctionsbetweennodesAandB andnodesAandCinthefollowingsignalflowgraph AtoB AfterremovingG1 theleftgraphis Thus AtoC AfterremovingG1 theleftgraphis Thus 6 9THEUNILATERALLAPLACETRANSFORM bilateralLaplacetransform 双边拉普拉斯变换 unilateralLaplacetransform 单边拉普拉斯变换 Thelowerlimitofintegration signifiesthatweincludeintheintervalofintegrationanyimpulsesorhigherordersingularityfunctionsconcentratedatt 0 奇异函数 Thebilateraltransformdependsontheentiresignalfromt tot whereastheunilateraltransformdependsonlyonthesignalfromto Thebilateraltransformandtheunilateraltransformofacausalsignalareidentical TheROCfortheunilateraltransformisalwaysaright halfplane Example6 20Considerthesignal ThebilateraltransformX s forthisexamplecanbeobtainedfromExample6 1andthetime shiftingproperty Bycontrast theunilateraltransformis TheevaluationoftheinverseunilateralLaplacetransformsisalsothesameasforbilateraltransforms withtheconstraintthattheROCforaunilateraltransformmustalwaysbearight halfplane Infact wecouldrecognizeasthebilateraltransformofx t u t Since and Thus Fortheunilateraltransform theROCmustbetheright halfplanetotherightoftherightmostpoleof Inthiscase theROCconsistsofallpointsswithRe s 1 Thus unilateralLaplacetransformprovideuswithinformationaboutsignalsonlyfor Example6 22ConsidertheunilateralLaplacetransform Takinginversetransformsofeachtermresultsin 6 9 1PropertiesoftheUnilateralLaplaceTransform Timescaling Convolution assumingthatx1 t andx2 t areidenticallyzerofort 0 Differentiationinthetimedomain Proofofthispropertyforfirst derivativeofx t Similarly theunilateralLaplacetransformofsecond derivativeofx t canbeobtainedbyrepeatingusingtheproperty 6 9 2SolvingDifferentialEquationsUsingtheUnilateralLaplaceTransform Applyingtheunilateraltransformtobothsidesofthedifferentialequation weobtain orequivalently Thus weobtain TheunilateralLaplacetransformisofconsiderablevalueinanalyzingcausalsystemswhicharespecifiedbylinearconstant coefficientdifferentialequationswithnonzeroinitialconditions i e systemsthatarenotinitiallyatrest 6 9 3RepresentationofCircuitsins domain ApplyunilateralLaplacetransformtoeachequationtoobtain Foracircuit ifweobtaintherepresentationforthebasicelementsinthecircuitinthes domain thenwealsoobtainthecircuitinthes domain Aninductorwithinduct