數(shù)字信號(hào)處理 實(shí)驗(yàn)一 系統(tǒng)響應(yīng)與系統(tǒng)穩(wěn)定性

ExperimentOnetheresponseandstabilityofsystems

1.Aims

(1)Knowinghowtocomputetheresponseofasystemtoaninput.

(2)Havingalevelofsolidunderstandingofcharacterizationintime-domainofsystems.

(3)Toobserve,verifyandanalyzethestabilityofsystems.

2.Principleandmethod:

Discrete-TimeLTIsystemscanberepresentedasalinearconstdifferenceequationortheimpulseresponseintimedomainandasatransformfunctioninfrequency-domain.Theresponseofthesystemtoanarbitraryinputcanbecomputedknowingthelinearconstdifferenceequationortheimpulseresponse.

Inthisexperiment,wecomputetheresponseofthesystemtoanarbitraryinputbytwoways.

1)OutputcomputationusingMATLAB,knowingthelinearconstdifferenceequation.

2)Outputcomputationusinglinearconvolution,knowingtheimpulseresponseofthesystems.

SignalprocessingtoolboxinMATLABprovidesusaconvenientandefficientfunctiontoreachourobjectives.

Characterizationintime-domainofsystemsreferstothelinear,shift-invariant,causalandstableofasystem.Wefocusonthestabilityofsystems,includingthesteadyandtransientresponse.

Wedefinedadiscrete-timesystemtobestableifandonlyifforeveryboundedinput,theoutputisalsobounded,or,itsimpulseresponsesequenceisabsolutelysummable.Thestabilityofadiscrete-timesystemdependsonthecoefficientsofitsdifferenceequation.

Itisimpossibletoexaminethesystemwhethertheoutputisboundornotwhenallofwhoseinputisbound,orexaminetheresponseofthesystemsatisfytheabsolutelysummablewhenexaminingwhetherthesystemisstableornotinpractice.Oneofthefeasiblewayistoaddtheunitstepsequencetotheinputend,sowecandecidethesystemisstableifitsoutputtendstoaconstant(includingzero).Thesteadystateoutputistheoutputasntendstoinfinity.Ifitisstable,theoutputattheinitialperiodoftimeisdefinedtotransienteffect,withnincreasing,themagnitudetendstobestable,toreachthestableoutput.

Payattentionthattheinitialstateofthesystemsaresupposedtobezerointhefollowingexperiment.

3.Thecontentsandsteps

(1)Givenadifferenceequationoflow-passfilterfollowing:

theinputsignal:

①Solvetheresponseofthesystemrespectivelywhenand,anddrawuptheirwaveform.

②Solvetheunitimpulseresponseofthesystem,anddrawuptheirwaveform.

(2)Giventheunitimpulseresponseofthesystemfollowing,

Solvetheoutputresponseofthesystemh1(n)andh2(n)bylinearconvolutionrespectivelywhentheinput,anddrawupthewaveform.

(3)Giventhedifferenceequationofaresonatorasfollowing

Let,thefrequencyoftheresonatoris0.4rad.

①Examinewhetherthesystemisstableornotbyexperiment.Whentheinputisu(n),drawupthewaveformofthesystemoutput.

②Giventheinputsignalis

Solvetheoutputresponseofthesystem,anddrawupitswaveform.

4.Questionforreview

(1)Iftheinputsareinfinitelongsequence,buttheunitimpulseresponseofsystemsarefinitelongsequence,cantheresponseofthesystemsbesolvedbylinearconvolution?How?

(2)Ifthesignalpassingalow-passfilter,filtratingthecomponentsofthehighfrequency,whatwillthechangeoftheTime-Domainsignal.Pleaseanalyzeandillustratebytheresultsoftheformerexperiment.

5.Therequirementofthereport

(1)SketchthemethodofsolvingtheresponseofsystemsinTime-Domain;

(2)Sketchthemethodofjustifyingthestabilityofsystemsbyexperiment,analyzethestableoutputwaveformoftheabove