APS审核数字信号处理DigitalSignalProcessing.docx

APS审核数字信号处理DigitalSignalProcessing.docx

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APS审核数字信号处理DigitalSignalProcessing

Signal

Signalisanyvariablethatcarriesorcontainssomekindofinformationthatcanberecorder，conveyed，displayed，ormanipulated

Example：

speech，sound，music，image，raderandsonarsignal

DSP

DSPisconcernedwith：

Thedigitalrepresentationofsignal

Theanalysis，modificationandextractionofinformationfromthesesignals

ThepurposeofDSP

Removeinterferenceandnoise（filtering）fromsignal

Obtainandanalyzethespectrumofasignal

Transformthesignalintomoresuitableformat

DigitalSignalProcessing（DSP）isconcernedwiththerepresentation,transformationandmanipulationofsignalsonacomputer.DigitalSignalProcessingbeginswithadiscussionoftheanalysisandrepresentationofdiscrete-timesignalsystems,includingdiscrete-timeconvolution,differenceequations,thez-transform,andthediscrete-timeFouriertransform.Emphasisisplacedonthesimilaritiesanddistinctionsbetweendiscrete-time.Thecourseproceedstocoverdigitalnetworkandnon-recursive（finiteimpulseresponse）digitalfilters.DigitalSignalProcessingconcludeswithdigitalfilterdesignandadiscussionofthefastFouriertransformalgorithmforcomputationofthediscreteFouriertransform.

1.1Discrete-TimeSignals：

Time-DomainRepresentation

Thearrowisplacedunderthesampleattimeindexn=0

1.2Sampling:

Adiscrete-timesequence{x[n]}maybegeneratedbyperiodicallysamplingacontinuous-timesignalatuniformintervalsoftime.

ThespacingTbetweentwoconsecutivesamplesiscalledthesamplingperiod.ReciprocalofsamplingintervalT,denotedasFT,iscalledthesamplingfrequency（UNIT:

HZ）

Samplingtheorem（Shannon）:

Xa（t）canberepresenteduniquelybyitssampledversion{x[n]}ifthesamplingfrequency

ischosentobegreaterthan2timesthehighestfrequencycontainedinXa（t）

1.3ClassificationofSequences

1.3.1{x[n]}isarealsequence,ifthen-thsamplex[n]isrealforallvaluesofn.Otherwise,{x[n]}isacomplexsequence

Example：

isarealsequence;

isacomplexsequence

1.3.2Asingle-input,single-output（SISO）discrete-timesystemoperatesonasequence,calledtheinputsequence,accordingsomeprescribedrulesanddevelopsanothersequence,calledtheoutputsequence,withmoredesirableproperties。

1.3.4BasicOperations：

Addition

Multiplication

Time-shifting

Example:

1.3.4

Totalenergyofasequencex[n]isdefinedby

Theaveragepowerofanaperiodicsequenceisdefinedby

1.4Discrete-TimeSystems:

1.4.1Classification

•LinearSystems

•Shift-InvariantSystems

•CausalSystems

•StableSystems

•PassiveandLosslessSystems

1.4.2ImpulseandStepResponses

•Theresponseofadiscrete-timeLTIsystemtoaunitsamplesequence{d[n]}iscalledtheunitsampleresponseor,simply,theimpulseresponse,andisdenotedby{h[n]}

•Theresponseofadiscrete-timeLTIsystemtoaunitstepsequence{m[n]}iscalledtheunitstepresponseor,simply,thestepresponse,andisdenotedby{s[n]}

•

Example-Theimpulseresponseofthesystem

isobtainedbysettingx[n]=d[n]resultingin

Example-Theimpulseresponseofthediscrete-timeaccumulator

isobtainedbysettingx[n]=d[n]resultingin

1.4.3Input-OutputRelationship–Aconsequenceofthelinear,time-invariancepropertyisthatadiscrete-timeLTIsystemiscompletelycharacterizedbyitsimpulseresponse

Example:

Computeitsoutputy[n]fortheinput:

Asthesystemislinear,wecancomputeitsoutputsforeachmemberoftheinputseparatelyandaddtheindividualoutputstodeterminey[n].

Sincethesystemistime-invariant

Becauseofthelinearitypropertyweget

Hence,theresponsey[n]toaninput

willbe

ConvolutionSum

1.4.4FIRandIIR

Iftheimpulseresponseh[n]isoffinitelength,i.e.,

thenitisknownasafiniteimpulseresponse（FIR）discrete-timesystem.

Iftheimpulseresponseisofinfinitelength,thenitisknownasaninfiniteimpulseresponse（IIR）discrete-timesystem

Example-Thediscrete-timeaccumulatordefinedby

isanIIRsystem

•NonrecursiveSystem-Heretheoutputcanbecalculatedsequentially,knowingonlythepresentandpastinputsamples

•RecursiveSystem-Heretheoutputcomputationinvolvespastoutputsamplesinadditiontothepresentandpastinputsamples

1.5Discrete-timeFourierTransform（DTFT）

Thediscrete-timeFouriertransform（DTFT）ofasequencex[n]isgivenby

andarecalledthemagnitudeandphasespectra

1.6DiscreteFourierTransform

•

UsingthenotationtheDFTisusuallyexpressedas:

•TheinversediscreteFouriertransform（IDFT）isgivenby

1.7Z-Transform

1.8FilterDesign