Linear Systems in LabVIEW Linear Systems in LabVIEW

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1、ApplicationNote039LinearSystemsinLabVIEW®J.Kodosky,E.PérezPh.D.IntroductionThisapplicationnotedescribesthedifferenceequationsfordiscrete-time,time-invariantlinearsystemsandhowtouseLabVIEWblockdiagramstoimplementtheseequations.ThisapplicationnotealsodiscussestheZtransform,signalflowgraphs,discret

2、einputandoutputsequences,andanexamplethatmodelsasimple,linear,time-invariantsystem.SignalsandSystemsTheconceptsofsignalsandsystemsariseinawidevarietyoffieldssuchastelecommunications,aerospaceengineering,biomedicalengineering,acoustics,chemistry,geophysics,andimageprocessing.Asignalisavariationof

3、someform,andasystemisanyprocesswhichtransformsasignal(Figure1).Figure1.SignalsandSystemThemathematicalmodelofasignalissimplyafunctionofoneormoreindependentvariables,forexampleh(t)andg(x,y),andthemathematicalmodelofasystemisatransformationT{•}thatoperatesonafunction.Thediscussioninthisapplication

4、noteislimitedtooneindependentvariablebecausethesameprinciplesapplytotwoormoreindependentvariables.Theindependentvariabletisreferredtoastime.Therelationshipbetweensignalsandsystemissummarizedinequation(1)forthecaseofasingleindependentvariabley(t)=T{x(t)}(1)wherex(t)istheinputsignalandy(t)isthesys

5、tem'sresponsetotheinputsignal.340478-01©Copyright1993NationalInstrumentsCorporation.Allrightsreserved.February1993Alinearsystemisasystemthatobeysthesuperpositionprinciple.Thatis,iftheinputtothesystemisalinearcombinationofsignals,thentheoutputisthelinearcombination(superposition)oftheresponsesgen

6、eratedbyeachsignal.Thesuperpositionprincipleismathematicallysummarizedinequations(2)through(4).x(t)=a1x1(t)+a2x2(t)+…+anxn(t)(2)y(t)=a1y1(t)+a2y2(t)+…+anyn(t)(3)yi(t)=T{xi(t)},i=1,2,…,n(4)whereallaiarescalarvalues.Atime-invariantsystemisasystemwhoseshiftedinputsignalcausesashiftedoutputsignal.y(

7、t-t0)=T{x(t-t0)}(5)wheret0isthetimeshift.Naturallyoccurringlinearsystemsdonotexist,andstrictmathematicalmodelingofnonlinearsystemscanbecomplicatedandinmanycasesunrealistic.However,formostpracticalapplications,youcanstudynonl