Chapter-10-The-z-Trans.ppt

Chapter10Thez Transform 罗欣UESTC 2012 2013 1 Thez Transform Anothertransform anyend Motivation analogoustoLTinCTSo thedefinitionandpropertiesofthez TransformcloselyparallelthoseoftheLaplaceTransformPayattentiontotherelationanddifferencebetweenLTandzT Contents Thez Transform bilateralandunilateral andtheinversez transformTheRegionofConvergenceforthezTPropertiesofthez TransformAnalysisDTLTIsystemusingzT Review 10 1Thez Transform assumingitconverges Definition The Bilateral z Transform Whenz ej unitmagnitude H ej correspondstotheFTofh n Whenz rej complex H z correspondstothezTofh t RelationshipbetweenDTFTandzT SimilartoFTandLT Unitcircle r 1 intheROC DTFTX ej exists x n r n ROC z rej atwhich dependsonlyonr z justliketheROCins planeonlydependsonRe s RelationshipbetweenLTandZT If andsampledsignal suppose Wecanget 1 2 1 2 Then z transformasDTversionofLaplacetransformwithz esT s plane z planerelationship j axisins plane s j z esT 1 aunitcircleinz planeLHPins plane Re s 0 z esT 1 outsidethe z 1circle Specialcase Re s z Averticallineins plane Re s constant esT constant acircleinz plane z esT Example10 1 Specially TheROCofsignalsinexample10 1 right sidedsignal unitcircle ROCoutside z 1unstable include z 1stable ROCoutside z 1unstable include z 1stable Example10 2 TheROCofExample10 2 aleft sidedsignal 0 a 1 z plane ROCinclude z 1stable inside z 1unstable Example TheROCistheentirez plane Example10 3 Example10 4 Rationalz Transforms x n linearcombinationofDTexponentials X z isrational polynomialsinz characterized exceptforaprefactor byitspolesandzeros sometimes itisconvenientforX z tobeexpressedintermsofpolynomialsinz 1 n 0 x n 0 Property1 TheROCofX z consistsofaringinthez planecenteredabouttheorigin Property2 theROCdoesnotcontainanypoles 10 2TheRegionofConvergenceforz Transform equivalenttoaverticalstripinthes plane sameasinLT z plane Property3 Ifx n isoffiniteduration thentheROCistheentirez plane exceptpossiblyz 0and orz Example z 1 ROC z 0 ROC z CTcounterpart Property4 Ifx n isrightsided andifthecircle z r0isintheROC thenallfinitevaluesofzforwhich z r0willalsointheROC z plane TheROCofaright sidedsignal Prove r0 convergesslowerthan Property5 Ifx n isleftsided andifthecircle z r0isintheROC thenallvaluesofzforwhich0 z r0willalsointheROC TheROCofaleft sidedsignal Property6 Ifx n istwosided andifthecircle z r0isintheROC thentheROCwillconsistofaringinthez planethatincludesthecircle z r0 Normally r1 z r2 r1 r2 z plane Quiz WhattypesofsignalsdothefollowingROCcorrespondto right sided left sided two sided Example10 6 or ROC zeros Then poles z 0 N 1 storder z a Note iscancelled z plane Example10 7 Clearly ROCdoesnotexistifb 1 Noz transformforb n Property7 IftheztransformX z ofx n isrational thenitsROCisboundedbypolesorextendstoinfinity Inaddition nopolesofX z arecontainedintheROC Property8 Ifthez transformX z ofx n isrational ifx n isrightsided thentheROCistheregioninthez planeoutsidetheoutmostpolei e outsidethecircleofradiusequaltothelargestmagnitudeofthepolesofX z Furthermore ifx n iscausal thentheROCalsoincludesz Property9 Ifthez transformX z ofx n isrational ifx n isleftsided thentheROCistheregioninthez planeinsidetheinnermostnonzeropolei e insidethecircleofradiusequaltothesmallestmagnitudeofthepolesofX z otherthananyatz 0andextendinginwardtoandpossiblyincludingz 0 inparticular ifx n isanticausal thentheROCalsoincludesz 0 Example10 8 HowmaypossibleROCsinthisFigure TherearethreepossibleROCsshowninFigurebefore ROC1 ROC2 ROC3 leftsided rightsided twosided Homework 10 210 310 610 7 From 10 3TheInverseZ Transform So Let Integrationaroundacounterclockwiseclosedcircularcontourcenteredattheoriginandwithradiusr Thecalculationforinversez Transform 1 Integrationofcomplexfunctionbyequation 2 ComputebyPFE PartialFractionExpansion 3 power seriesexpansion PartialFractionExpansionofrationalX z Normally TheROCisoutsidez aitheinverseZTis TheROCisinsidez aitheinverseZTis Example10 9 Then wehaveknown So Example10 10thesameX z inexample10 9 ROC So Because When ROC Example10 12 Fromthepower seriesdefinition or Example10 13 Consider ROC1 longdivision ROC2 longdivision Homework 10 910 1010 24 10 4GeometricEvaluationofTheFTFromthePole zeroPlot lowpass highpass bandpass bandstop all pass Examplefirstordersystems Examplesecondordersystems 0 r 0 95 r 0 75 10 5 1Linearity R1 R2 R1 R2 If Then Note R1 R2maybelargerthanR1orR2 10 5ThePropertiesofZ Transform 10 5 2TimeShifting R If R exceptthepossibleadditionordeletionoftheoriginorinfinity Then Example Then From 10 5 3Scalinginz Domain R If Then z0 R Specially R R if 10 5 4TimeReversal R If 1 R Then Example Then and From 10 5 5TimeExpansion R If R Then 10 5 6Conjugate R If R Then Note Ifx n isreal X z X z ThusifX z hasapole orzero atz z0 itmusthaveapole orzero atthecomplexconjugatepointz z0 10 5 7TheConvolutionProperty R1 R2 If Then Note ifR1 R2 isnotexisted R1 R2 Example R If Then R z 1 10 5 8Differentiationinthez Domain R If R Then Example Then From 10 5 9TheInitial andFinal ValueTheorems Ifx n 0forn 0 ItszTX z Then ROC z r1 TheROCof z 1 X z contains z 1 10 6SomeZTPairs Page776Table10 2 Example Wecanget notexist Homework 10 1110 1610 1710 31 ConsideraLTIsystem 10 7AnalysisandCharacterizationofLTIsystemsUsingZT 10 7 1Causality 1 Acausalsystem z r1 ROC 2 Forrational ROC includinginfinity exteriorofacircle a exteriorofacircleoutsidetheoutmostpole r1 z r1 b TheorderofthenumeratorN z cannotbegreaterthantheorderoftheD z denominator Acausalsystem Example10 20 notcausalsystem Example10 21 causalsystem ROC 10 7 2Stability 1 Astablesystem z 1 theunitcircle 2 Acausalstablesystemwithrational Allpolesliesinsidetheunitcircleofz plane ROC Includes Example10 22 24Readbyyourself 10 7 3LTISystemsCharacterizedbyLinearConstant CoefficientDifferenceEquations Z transform rational Usually apracticalsystemiscausalandstable Example10 25 ROC1 Wecanget ROC2 10 7 4ExamplesRelatingSystemBehaviortotheSystemFunction Example10 26 SupposethatwearegiventhefollowinginformationaboutanLTIsystem 1 2 DeterminethesystemfunctionH z Solution And So For ROC Then Example10 27astable causalsystem H z rational containsapolez 1 2andazerosomewhereontheunitcircle Otherzerosandpolesareunknown Whethercanwedefinitelysaythatitistrueorfalseeachoffollowingstatements a convergence b forsomewhere T T c h n hasfiniteduration d h n isreal F Insufficientinformation T e istheimpulseresponseofastablesystem 10 8 1SystemFunctionsforInterconnectionsofLTISystems Parallel Series cascade 10 8SystemFunctionAlgebraandBlockDiagramRepresentations Feed back Basicelements adder unitdelay multiplicationbyacoefficient 10 8 2BlockDiagramRepresentationforcausalLTISystemsDescribedbyDifferenceEquationsandRationalSystemFunctions Example10 28 Let Th