2024/3/61biomedicalSignalprocessing

生物醫(yī)學(xué)信號處理

Chapter1Introduction

Goalsofthecourse?Tounderstand–whatbiomedicalsignalsare–whatproblemsandneedsarerelatedtotheiracquisitionandprocessing–whatkindofmethodsareavailableandgetanideaofhowtheyareappliedandtowhichkindofproblems?Togettoknowbasicdigitalsignalprocessingandanalysistechniquescommonlyappliedtobiomedicalsignalsandtoknowtowhichkindofproblemseachmethodissuitedfor(andforwhichnot)biomedicalSignalProcessingSignal:anyphysicalquantitythatvariesasafunctionofanindependentvariable?independentvariableisusuallytimebutmaybespace,distance,...Biomedicalsignal:asignalbeingobtainedfromabiologicsystem/originatingfromaphysiologicprocess(humanoranimal(-medical->patients))Processingofbiomedicalsignalsalltreatment(ofbiomedicalsignals)whichoccursbetweentheirorigininaphysiologicalprocessandtheirinterpretationbytheirobserver(e.g.clinician)ProcessingofbiomedicalsignalsProcessingofbiomedicalsignalsProcessingofbiomedicalsignalsisapplicationofsignalprocessingmethodsonbiomedicalsignals→Allpossibleprocessingalgorithmsmaybeused→Biomedicalsignalprocessingrequiresunderstandingtheneeds(e.g.biomedicalprocessesandclinicalrequirements)andselectingandapplyingsuitablemethodstomeettheseneedsRationalesforbiomedicalsignalprocessing1.Acquisitionandprocessingtoextractaprioridesiredinformation2.Interpretingthenatureofaphysiologicalprocess,basedeitherona)observationofasignal(explorativenature),orb)observationofhowtheprocessaltersthecharacteristicsofasignal(monitoringachangeofapredefinedcharacteristic)(Some)goalsforbiomedicalsignalprocessing?Quantificationandcompensationfortheeffectsofmeasuringdevicesandnoiseonsignal?Identificationandseparationofdesiredandunwantedcomponentsofasignal?Uncoveringthenatureofphenomenaresponsibleforgeneratingthesignalonthebasisoftheanalysisofthesignalcharacteristics–Relatedtomodelling/inversemodellingbutoftenmorepragmaticExample:heartratemetersSensorSignalprocessingUserExample:ISTVivago?WristCareHealthmonitoringNeedforprocessingtodrawanyconclusionsBeat-to-beatheartrateSystolicanddiastolicbloodpressureSignalprocessingmethodsNoisereductionPreprocessingSignalvalidationFeatureextractionDatacompressionSegmentationPatternrecognitionTrenddetectionEventdetectionDecisionsupportDecisionmakingFiltering(linear,nonlinear,adaptive,optimal)StatisticalsignalprocessingFrequencydomainanalysisTime-frequencyanalysisFuzzylogicArtificialneuralnetworksExpertsystems,rule-basedsystemsGeneticandevolutionarymethodsSignalprocessingmethodsSignalmodellingWaveletsandfilterbanksPCA,ICA,SVDClusteringHigher-orderstatisticsChaosandnonlineardynamicsComplexityandfractals∴Chooserightmethodforrightproblem!BiomedicalsignalclassificationOnthebasisof–signalcharacteristics:technicalpointofview–signalsource:fromwhereandhowthesignalisoriginatedandmeasured–biomedicalapplication:neurophysiology,cardiology,monitoring,diagnosis,…Classificationmaybehelpfulintheselectionofprocessingmethods...DefinitionsDeterministic:maybeaccuratelydescribedmathematically,Usuallypredictable(notincaseofchaos!)Periodic:s(t)=s(t+nT)Almostperiodic:patternsrepeatwithsomeunregularityTransient:signalcharacteristicschangewithtimeDefinitionsStochastic:definedbytheirstatisticalproperties(distribution)Stationary:statisticalpropertiesofthesignaldonotchangeovertimeErgodic:statisticalpropertiesmaybecomputedalongtimedistributions(Whitenoise:acf=0exceptforτ=0whereacf=1;flatspectrum)DefinitionsAllreal(bio)signalsmaybeconsideredstochastic–almostdeterministicsignals(e.g.ECG):waveshapesthat(almost)repeatthemselves→characterization(often)bydetectionofcertainmeasuresorwaves–“truly”stochastic(e.g.EEG)→characterizationbystatisticalpropertiesClassificationbysource?biomedicalsignalsdifferfromothersignalsonlyintermsoftheapplication-signalsthatareusedinthebiomedicalfield?Bioelectricsignals:generatedbynervescellsandmusclecells.Singlecellmeasurements(microelectrodesmeasureactionpotential)and‘gross’measurements(surfaceelectrodesmeasureactionofmanycellsinthevicinity)Classificationbysource?Biomagneticsignals:brain,heart,lungsproduceextremelyweakmagneticfields,thiscontainsadditionalinformationtothatobtainedfrombioelectricsignals.CanbemeasuredusingSQUIDs.?Bioimpedancesignals:tissueimpedancerevealsinfoabouttissuecomposition,bloodvolumeanddistributionandmore.UsuallytwoelectrodestoinjectcurrentandtwotomeasurevoltagedropClassificationbysource?Bioacousticsignals:manyphenomenacreateacousticnoise.Forexample,flowofbloodthroughtheheart,itsvalves,orvesselsandflowofairthroughupperandlowerairwaysandlungs,butalsodigestivetract,jointsandcontractionofmuscles.Recordusingmicrophones.?Biomechanicalsignals:motionanddisplacementsignals,pressure,tensionandflowsignals.Avarietyofmeasurements(notalwayssimple,ofteninvasivemeasurementsareneeded).Classificationbysource?Biochemicalsignals:chemicalmeasurementsfromlivingtissueorsamplesanalyzedinalaboratory.Forexamples,ionconcentrationsorpartialpressures(pO2orpCO2)inblood.(lowfrequencysignals,oftenactuallyDCsignals)?Bioopticalsignals:bloodoxygenationbymeasuringtransmittedandbackscatteredlightfromatissue,estimationofheartoutputbydyedilution.Fiberoptictechnology.Biomedicalapplicationdomains?Informationgathering–measurementofphenomenatounderstandthesystem?Diagnosis–detectionofmalfunction,pathology,orabnormality?Monitoring–toobtaincontinuousorperiodicinformationaboutthesystemBiomedicalapplicationdomains?Therapyandcontrol–modifythebehaviourofthesystemandensuretheresult?Evaluation–objectiveanalysis:proofofperformance,qualitycontrol,effectoftreatmentProblemsinbiomedicalsignalprocessingAccessibility–Patientsafety,preferencefornoninvasiveness–Indirectmeasurements(variablesofinterestarenotaccessible)Variance–Inter-individual,intra-individualProblemsinbiomedicalsignalprocessingInter-relationshipsandinteractionsamongphysiologicalsystem–SubsystemofinterestmaynotbeisolatedAcquisitioninterference–InstrumentationandproceduresmodifythesystemoritsstateArtefactsandinterference–Interferencefromotherphysiologicalsystems(e.g.muscleartifactsinEEGrecordings)–Low-levelsignals(e.g.microvoltsinEEG)requireverysensitiveamplifiers;theyareeasilysensitivetointerference,too!–LimitedpossibilitiesforshieldingorotherprotectionNonlinearityandobscurityofthesystemunderstudyArtefactsandinterference–basicallyallbiologicalsystemsexhibitnonlinearitieswhilemostofthemethodsarebasedontheassumptionoflinearity→approximation–exactstructuresandtruefunctionofmanyphysiologicalsystemsareoftennotknownSignalacquisitionShort-termHRVandBPV2024/3/633signalprocessingApplicationsofsignalprocessing:entertainment,communications,spaceexploration,medicine,archaeology(考古學(xué)),etc.Drivenbytheconvergenceofcommunications,computersandsignalprocessing.2024/3/634signalprocessingSignalprocessingisbenefitedfromaclosecouplingbetweentheory,application,andtechnologiesforimplementingsignalprocessingsystems.Signalprocessingisconcernedwiththerepresentation,transformation,andmanipulationofsignalsandtheinformationtheycontain.2024/3/635ContinuousandDigitalSignalProcessingPriorto1960:continuous-timeanalogsignalprocessing.Digitalsignalprocessingiscausedby:theevolutionofdigitalcomputersandmicroprocessorsImportanttheoreticaldevelopmentssuchasthefastFouriertransformalgorithm(FFT)2024/3/636DigitalandDiscrete-timeSignalProcessingIndigitalsignalprocessingSignalsarerepresentedbysequencesoffinite-precisionnumbersProcessingisimplementedusingdigitalcomputationDigitalsignalprocessingisaspecialcaseofdiscrete-timesignalprocessing2024/3/637DigitalandDiscrete-timeSignalProcessingContinuous-timesignalprocessing:timeandsignalarecontinuousDiscrete-timesignalprocessing:timeisdiscrete,signaliscontinuousDigitalsignalprocessing:timeandsignalarediscrete2024/3/638Discrete-timeProcessingDiscrete-timeprocessingofcontinuous-timesignalReal-timeoperationisoftendesirable:outputiscomputedatthesamerateatwhichtheinputissampled2024/3/639ObjectsofSignalProcessingProcessonesignaltoobtainanothersignalSignalinterpretation:Characterizationoftheinputsignal,Example:speechrecognitiondigitalpreprocessing(filtering,parameterestimation,etc)speechsignalpatternrecognitionexertsystemphonemictranscriptionfinalsignalinterpretation2024/3/640ObjectsofSignalProcessingSymbolicmanipulationofsignalprocessingexpression:signalandsystemsarerepresentedandmanipulatedasabstractdataobjects,withoutexplicitlyevaluatingthedatasequence2024/3/641WhydoWeLearnDSPSoftware,suchasMatlab,hasmanytoolsforsignalprocessingItseemsthatitisnotnecessarytoknowthedetailsofthesealgorithms,suchasFFTAgoodunderstandingoftheconceptsofalgorithmsandprinciplesisessentialforintelligentuseofthesignalprocessingsoftwaretools2024/3/642ExtensionMultidimensionalsignalprocessingimageprocessingSpectralAnalysisSignalmodelingAdaptivesignalprocessingSpecializedfilterdesignSpecializedalgorithmforevaluationofFouriertransformSpecializedfilterstructureMultiratesignalprocessingWalettransform2024/3/643HistoricalPerspective17thcenturyTheinventionofcalculusScientistdevelopedmodelsofphysicalphenomenaintermsoffunctionsofcontinuousvariableanddifferentialequationsNumericaltechniqueisusedtosolvetheseequationsNewtonusedfinite-differencemethodswhicharespecialcasesofsomediscrete-timesystems2024/3/644HistoricalPerspective18thcenturyMathematiciansdevelopedmethodsfornumericalintegrationandinterpolationofcontinuousfunctionsGauss(1805)discoveredthefundamentalprincipleoftheFastFourierTransform(FFT)evenbeforethepublication(1822)ofFourier'streatiseonharmonicseriesrepresentationoffunction(proposedin1807)2024/3/645HistoricalPerspectiveEarly1950ssignalprocessingwasdonewithanalogsystem,implementedwithelectronicscircuitsormechanicaldevices.firstusesofdigitalcomputersindigitalsignalprocessingwasinoilprospecting.Simulatesignalprocessingsystemonadigitalcomputerbeforeimplementingitinanaloghardware,ex.vocoder2024/3/646HistoricalPerspectiveWithflexibilitythedigitalcomputerwasusedtoapproximate,orsimulate,ananalogsignalprocessingsystemThedigitalsignalprocessingcouldnotbedoneinrealtimeSpeed,cost,andsizearethreeoftheimportantfactorsoftheuseofanalogcomponents.Somedigitalflexiblealgorithmhadnocounterpartinanalogsignalprocessing,impractical.all-digitalimplementationtempting2024/3/647HistoricalPerspectiveFFTdiscoveredbyCooleyandTukeyin1965anefficientalgorithmforcomputationofFouriertransforms,whichreducethecomputingtimebyordersofmagnitude.FFTmightbeimplementedinspecial-purposedigitalhardwareManyimpracticalsignalprocessingalgorithmsbecametobepractical2024/3/648HistoricalPerspectiveFFTisaninherentlydiscrete-timeconcept.FFTstimulatedareformulationofmanysignalprocessingconceptsandalgorithmsintermsofdiscrete-timemathematics,whichformedanexactsetofrelationshipsinthediscrete-timedomain,sothereemergedafieldofdiscrete-timesignalprocessing.493/6/202449Chapter2Discrete-TimeSignalsandSystems2.0Introduction2.1Discrete-TimeSignals:Sequences2.2Discrete-TimeSystems2.3LinearTime-Invariant(LTI)Systems2.4PropertiesofLTISystems2.5LinearConstant-CoefficientDifferenceEquations503/6/202450Chapter2Discrete-TimeSignalsandSystems2.6Frequency-DomainRepresentationofDiscrete-TimeSignalsandsystems2.7RepresentationofSequencesbyFourierTransforms2.8SymmetryPropertiesoftheFourierTransform2.9FourierTransformTheorems2.10Discrete-TimeRandomSignals2.11Summary513/6/2024512.0IntroductionSignal:somethingconveysinformationSignalsarerepresentedmathematicallyasfunctionsofoneormoreindependentvariables.Continuous-time(analog)signals,discrete-timesignals,digitalsignalsSignal-processingsystemsareclassifiedalongthesamelinesassignals:Continuous-time(analog)systems,discrete-timesystems,digitalsystemsDiscrete-timesignalSamplingacontinuous-timesignalGenerateddirectlybysomediscrete-timeprocess523/6/2024522.1Discrete-TimeSignals:SequencesDiscrete-TimesignalsarerepresentedasInsampling,1/T(reciprocalofT):samplingfrequencyCumbersome,sojustuse533/6/202453Figure2.1Graphicalrepresentationofadiscrete-timesignalAbscissa:continuousline:isdefinedonlyatdiscreteinstants54Figure2.2EXAMPLESamplingtheanalogwaveform553/6/202455SumoftwosequencesProductoftwosequencesMultiplicationofasequencebyanumberαDelay(shift)ofasequenceBasicSequenceOperations563/6/202456BasicsequencesUnitsamplesequence(discrete-timeimpulse,impulse)573/6/202457BasicsequencesarbitrarysequenceAsumofscaled,delayedimpulses583/6/202458BasicsequencesUnitstepsequenceFirstbackwarddifference593/6/202459BasicSequencesExponentialsequencesAandαarereal:x[n]isrealAispositiveand0<α<1,x[n]ispositiveanddecreasewithincreasingn-1<α<0,x[n]alternateinsign,butdecreaseinmagnitudewithincreasingn:x[n]growsinmagnitudeasnincreases603/6/202460EX.2.1CombiningBasicsequencesIfwewantanexponentialsequencesthatiszeroforn<0,thenCumbersomesimpler613/6/202461BasicsequencesSinusoidalsequence623/6/202462ExponentialSequencesComplexExponentialSequencesExponentiallyweightedsinusoidsExponentiallygrowingenvelopeExponentiallydecreasingenvelopeisreferedto633/6/202463Frequencydifferencebetweencontinuous-timeanddiscrete-timecomplexexponentialsorsinusoids:frequencyofthecomplexsinusoidorcomplexexponential:phase643/6/202464PeriodicSequencesAperiodicsequencewithintegerperiodN653/6/202465EX.2.2

ExamplesofPeriodicSequencesSupposeitisperiodicsequencewithperiodN663/6/202466SupposeitisperiodicsequencewithperiodNEX.2.2

ExamplesofPeriodicSequences673/6/202467EX.2.2

Non-PeriodicSequencesSupposeitisperiodicsequencewithperiodN683/6/202468HighandLowFrequenciesinDiscrete-timesignal(a)w0=0or2(b)w0=/8or15/8(c)w0=/4or7/4(d)w0=693/6/2024692.2Discrete-TimeSystemDiscrete-TimeSystemisatrasformationoroperatorthatmapsinputsequencex[n]intoauniquey[n]y[n]=T{x[n]},x[n],y[n]:discrete-timesignalT{?}x[n]y[n]Discrete-TimeSystem703/6/202470EX.2.3

TheIdealDelaySystemIfisapositiveinteger:thedelayofthesystem.Shifttheinputsequencetotherightbysamplestoformtheoutput.Ifisanegativeinteger:thesystemwillshifttheinputtotheleftbysamples,correspondingtoatimeadvance.713/6/202471x[m]mnn-5dummyindexmEX.2.4MovingAverageforn=7,M1=0,M2=5723/6/202472PropertiesofDiscrete-timesystems

2.2.1Memoryless(memory)systemMemorylesssystems:theoutputy[n]ateveryvalueofndependsonlyontheinputx[n]atthesamevalueofn733/6/202473PropertiesofDiscrete-timesystems

2.2.2LinearSystemsIfT{?}T{?}T{?}T{?}T{?}additivitypropertyhomogeneityorscaling同(齊)次性propertyprincipleofsuperpositionandonlyIf:743/6/202474ExampleofLinearSystemEx.2.6Accumulatorsystemforarbitrarywhen753/6/202475Example2.7NonlinearSystemsMethod:findonecounterexample

counterexample

For

counterexample

For763/6/202476PropertiesofDiscrete-timesystems

2.2.3Time-InvariantSystemsShift-InvariantSystemsT{?}T{?}773/6/202477ExampleofTime-InvariantSystemEx.2.8Accumulatorsystem783/6/202478ExampleofTime-varyingSystemEx.2.9ThecompressorsystemT{?}0T{?}000T{?}793/6/202479PropertiesofDiscrete-timesystems

2.2.4CausalityAsystemiscausalif,foreverychoiceof,theoutputsequencevalueattheindexdependsonlyontheinputsequencevaluefor803/6/202480Ex.2.10ExampleforCausalSystemForwarddifferencesystemisnotCausalBackwarddifferencesystemisCausal813/6/202481PropertiesofDiscrete-timesystems

2.2.5StabilityBounded-InputBounded-Output(BIBO)Stability:everyboundedinputsequenceproducesaboundedoutputsequence.ifthen823/6/202482Ex.2.11TestforStabilityorInstabilityifthenisstable833/6/202483AccumulatorsystemEx.2.11TestforStabilityorInstabilityAccumulatorsystemisnotstable843/6/2024842.3LinearTime-Invariant(LTI)SystemsImpulseresponseT{?}T{?}853/6/202485LTISystems:ConvolutionRepresentationofgeneralsequenceasalinearcombinationofdelayedimpulseprincipleofsuperpositionAnIllustrationExample（interpretation1）

863/6/202486873/6/202487ComputationoftheConvolutionreflectingh[k]abouttheorigiontoobtainh[-k]Shiftingtheoriginofthereflectedsequencetok=n（interpretation2）883/6/202488Ex.2.1289ConvolutioncanberealizedbyReflectingh[k]abouttheorigintoobtainh[-k].Shiftingtheoriginofthereflectedsequencestok=n.Computingtheweightedmovingaverageofx[k]byusingtheweightsgivenbyh[n-k].903/6/202490Ex.2.13AnalyticalEvaluationoftheConvolutionForsystemwithimpulseresponseh(k)0inputFindtheoutputatindexn913/6/202491h(k)00h(n-k)x(k)h(-k)0923/6/202492h(-k)0h(k)0933/6/202493h(-k)0h(k)0943/6/202494953/6/2024952.4PropertiesofLTISystemsConvolutioniscommutative(可交換的)h[n]x[n]y[n]x[n]h[n]y[n]Convolutionisdistributedoveraddition963/6/202496Cascadeconnectionofsystemsx

[n]h1[n]h2[n]y

[n]x

[n]h2[n]h1[n]y

[n]x

[n]h1[n]]h2[n]y

[n]973/6/202497Parallelconnectionofsystems983/6/202498StabilityofLTISystemsLTIsystemisstableiftheimpulseresponseisabsolutelysummable.CausalityofLTIsystemsHW:proof,Problem2.62993/6/202499ImpulseresponseofLTIsystemsImpulseresponseofIdealDelaysystemsImpulseresponseofAccumulator1003/6/2024100ImpulseresponseofMovingAveragesystems101ImpulseresponseofForwardDifferenceImpulseresponseofBackwardDifference102Finite-durationimpulseresponse(FIR)systemsTheimpulseresponseofthesystemhasonlyafinitenumberofnonzerosamples.TheFIRsystemsalwaysarestable.suchas:103Infinite-durationimpulseresponse(IIR)Theimpulseresponseofthesystemisinfiniteinduration.StableIIR

System:104Equivalentsystems105Inversesystem1062.5LinearConstant-CoefficientDifferenceEquationsAnimportantsubclassoflineartime-invariantsystemsconsistofthosesystemforwhichtheinputx[n]andoutputy[n]satisfyanNth-orderlinearconstant-coefficientdifferenceequation.107Ex.2.14DifferenceEquationRepresentationoftheAccumulator108Blockdiagramofarecursivedifferenceequationrepresentinganaccumulator109Ex.2.15DifferenceEquationRepresentationoftheMoving-AverageSystemwithrepresentation1anotherrepresentation1110111DifferenceEquationRepresentationoftheSystemAnunlimitednumberofdistinctdifferenceequationscanbeusedtorepresentagivenlineartime-invariantinput-outputrelation.112SolvingthedifferenceequationWithoutadditionalconstraintsorinformation,alinearconstant-coefficientdifferenceequationfordiscrete-timesystemsdoesnotprovideauniquespecificationoftheoutputforagiveninput.113SolvingthedifferenceequationOutput:Particularsolution:oneoutputsequenceforthegiveninput

Homogenoussolution:solutionforthehomogenousequation():whereistherootsof114SolvingthedifferenceequationrecursivelyIftheinputandasetofauxiliaryvaluearespecified.y(n)canbewritteninarecurrenceformula:115Example2.16RecursiveComputationofDifferenceEquation116Example2.16RecursiveComputationofDifferenceEquation117ExampleforRecursiveComputationofDifferenceEquationThesystemisnoncausal.Thesystemisnotlinear.Thesystemisnottimeinvariant.118DifferenceEquationRepresentationoftheSystemIfasystemischaracterizedbyalinearconstant-coefficientdifferenceequationandisfurtherspecifiedtobelinear,timeinvariant,andcausal,thesolutionisunique.Inthiscase,theauxiliaryconditionsarestatedasinitial-restconditions(初始松弛條件).Theauxiliaryinformationisthatiftheinputiszerofor,thentheoutput,isconstrainedtobezerofor119SummaryThesystemforwhichtheinputandoutputsatisfyalinearconstant-coefficientdifferenceequation:Theoutputforagiveninputisnotuniquelyspecified.Auxiliaryconditionsarerequired.120SummaryIftheauxiliaryconditionsareintheformofNsequentialvaluesoftheoutput,latervaluecanbeobtainedbyrearrangingthedifferenceequationasarecursiverelationrunningforwardinn,121Summaryandpriorvaluescanbeobtainedbyrearrangingthedifferenceequationasarecursiverelationrunningbackwardinn.122SummaryLinearity,timeinvariance,andcausalityofthesystemwilldependontheauxiliaryconditions.Ifanadditionalconditionisthatthesystemisinitiallyatrest,thenthesystemwillbelinear,timeinvariant,andcausal.123Example2.16withinitial-restconditionsIftheinputis,againwithinitial-restconditions,thentherecursivesolutioniscarriedoutusingtheinitialcondition124DiscussionIftheinputis,withinitial-restconditions,Notethatfor,initialrestimpliesthatItdoesmeanthatif.Initialrestdoesnotalwaysmeans1252.6Frequency-DomainRepresentationofDiscrete-TimeSignalsandsystems2.6.1EigenfunctionandEigenvalueforLTIiscalledastheeigenfunctionofthesystem,andtheassociatedeigenvalueisIf126EigenfunctionandEigenvalueComplexexponentialsistheeigenfunctionfordiscrete-timesystems.ForLTIsystems:frequencyresponseeigenvalueeigenfunction127Frequencyresponseiscalledasfrequencyresponseofthesystem.Magnitude,phaseRealpart,imaginepart128Example2.17FrequencyresponseoftheidealDelayFromdefination(2.109):129Example2.17FrequencyresponseoftheidealDelay130Linearcombinationofcomplexexponential131Example2.18SinusoidalresponseofLTIsystems132SinusoidalresponseoftheidealDelay133PeriodicFrequencyResponseThefrequencyresponseofdiscrete-timeLTIsystemsisalwaysaperiodicfunctionofthefrequencyvariablewithperiod134PeriodicFrequencyResponseThe“l(fā)owfrequencies”arefrequenciesclosetozeroThe“highfrequencies”arefrequenciesclosetoMoregenerally,modifythefrequencywith,risinteger.Weneedonlyspecifyover135Example2.19

Ideal

Frequency-SelectiveFiltersFrequencyResponseofIdealLow-passFilter136FrequencyResponseofIdealHigh-passFilter

137FrequencyResponseofIdealBand-stopFilter138FrequencyResponseofIdealBand-passFilter139Example2.20FrequencyResponseoftheMoving-AverageSystem140141FrequencyResponseoftheMoving-AverageSystemM1

=0andM2=4相位也取決于符號，不僅與指數(shù)相關(guān)1422.6.2SuddenlyappliedComplexExponentialInputsInpractice,wemaynotapplythecomplexexponentialinputs

ejwntoasystem,butthemorepractical-appearinginputsoftheform

x[n]=ejwnu[n]i.e.,x[n]suddenlyappliedatanarbitrarytime,whichforconveniencewechoosen=0.For

causalLTIsystem:1432.6.2SuddenlyappliedComplexExponentialInputsForn≥0ForcausalLTIsystem1442.6.2SuddenlyappliedComplexExponentialInputsSteady-stateResponseTransientresponse1452.6.2SuddenlyAppliedComplexExponentialInputs(continue)Forinfinite-durationimpulseresponse(IIR)Forstablesystem,transientresponsemustbecomeincreasinglysmallerasn

,Illustrationofarealpartofsuddenlyappliedcomplexexponential

Input

with

IIR146Ifh[n]=0exceptfor0n

M

(FIR),

thenthetransientresponseyt[n]=0forn+1>M.Forn

M,onlythesteady-stateresponseexists2.6.2SuddenlyAppliedComplexExponentialInputs(continue)

Illustrationofarealpartofsuddenlyappliedcomplexexponential

Input

with

FIR1472.7RepresentationofSequencesbyFourierTransforms(Discrete-Time)FourierTransform,DTFT，analyzingIfisabsolutelysummable,i.e.thenexists.(Stability)InverseFourierTransform,synthesis148FourierTransformrectangularformpolarform149PrincipalValue（主值）isnotuniquebecauseanymaybeaddedtowithoutaffectingtheresultofthecomplexexponentiation.Principlevalue:isrestrictedtotherangeofvaluesbetween.Itisdenotedas:phasefunctionisreferredasacontinuousfunctionoffor150ImpulseresponseandFrequencyresponseThefrequencyresponseofaLTIsystemist