aps审核数字信号处理Digital Signal Processing.docx
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aps审核数字信号处理Digital Signal Processing.docx
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aps审核数字信号处理DigitalSignalProcessing
数字信号处理
DigitalSignalProcessing
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
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