基于小波修剪阈值法的消噪外文翻译全文Word文档格式.docx

基于小波修剪阈值法的消噪外文翻译全文Word文档格式.docx

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）generatesubspace

suchthat...

（1）

Thereexistsawavelet

ittranslatesanddilates

produceabasisofthe‘detail’subspace

togive

.Sowecanget

...，

Andasignalx（n）canbedecomposedby

（2）

（3）

Where

arethediscretedetailcoefficientsofthesignalatlevel

and

aretheapproximationcoefficientsatlevel

andarelow-passfilterandhigh-passfilterrespectivelycorrespondingtosomewaveletbasisandtheyareconnectedby

（4）

WhereNisthelengthofthefilters.

Thealgorithmofthereconstructofthesignalis

（5）

willbegotten,whichistheoriginalsignal

whenrepeatingthereconstructformula（5）.

III.DE-NOISINGBYWAVELETTHRESHOLDING

Themethodofwaveletthresholdde-noisingisbasedontheprincipleofthe

multiresolutionanalysis.Thediscretedetailcoefficientsandthediscreteapproximationcoefficientscanbeobtainedbyamulti-levelwaveletdecompose[2].

Anoisyone-dimensionalsignalmodelcanbeexpressedasfollows:

Amongthem,

isthesignalwithnoise.

istheusefulsignal,

isthenoisesignal.Hereweconsider

asaGaussianwhitenoiseoflevel1.Itisusuallyahigh-frequencysignal.Butinengineeringpractice

isusuallyalow-frequencysignal,orsomestablesignal.Therefore,wecanusethefollowingmethodofde-noising:

Firstusewaveletdecompositiontothesignal（seeinFigure1）,obtainedbythewaveletdecompositionlayerisapartofthesignalofseriesoflarge-scaleapproximationandthedetailssection.InFigure1CA3iscalledtheapproximationsignalorthesmoothingsignal,itiscorrespondedtothelowfrequencysignal.CD1,CD2,CD3arecalledthedetailsignal.Theyarecorrespondedtothesignalfrequencycomponents,thenoisepartisusuallyincludedintheCDI,CD2,CD3.Therefore,wecanprocessthewaveletcoefficientswiththresholdformandthenreconstructthesignaltoremovethenoise.Thepurposeofremovingnoisefrom

istosuppressthenoisepartofthesignaltorecoverthetruesignal

from

.

Figure1Threewaveletdecomposition

Waveletdecompositiontransformssignalfromtime-domaintotime-scaledomain,anditcandescribethelocalfeaturewellinbothtimedomainandfrequencydomain.Becausetheamplitudeofthediscretedetailcoefficientsofthenoisedecreaseswiththelevelincreasing,wecanselectathreshold,modifyandprocessallofthediscretedetailcoefficientsatallscalebythresholdmethodsoastoremovenoise.Thede-noisingprocedureproceedsinthreesteps:

（1）Decomposition

（2）Thresholddetailcoefficients

（3）Reconstruction.

Generally,themostpopularthresholdingmethodsarehardandsoftthresholding.Wecanexpectthatthetechniqueofsoftthresholdingwouldintroducemoreerrororbiasthanhardthresholdingdoes.Butontheotherhand,softthresholdingismoreefficientinde-noising.Exampleswillbeillustratedlater.Toachieveacompromisebetweenthetwomethods,thetrimmedthresholdingmethodisproposedinthis

paper.

Grossmann[3]provedthatthevariancesandamplitudesofthedetailsofthewhitenoiseatthevariouslevelsdecreaseregularlyasthelevelincrease.Ontheotherhand,theamplitudeandvariancesofwavelettransformoftheavailablesignalarenotrelatedtothechangeofscale.Accordingtothepropertiesofwavelettransformofthenoiseandtheavailablesignal,wecanweaken,andevenremovenoise.Inthewaveletthresholdimgde-noising,weshouldfirstselectathresholdandprocessthecomponentsofwavelettransformofthenoisysignalinordertoimprovesignal-to-noiseratio（SNR）.

D.L.Donohoproposedaverysimplemethodofwaveletthresholdde-noising.Themethodinthesenseofminimummeansquareerroriseffectiveandhasbettervisualeffect.Thebasicideaofthismethodis:

doafewcontinuouswaveletdecompositiontonoisysignal

iscorrespondedtothescalewaveletcoefficients

.Somespecificlocationshavealargevalue.Thesepointsarecorrespondedtothelocationoftheoriginalsignal

ofoddchangesandimportantinformation.Whilemostothersmallerlocationsofwaveletcoefficients.Forwhitenoise

itiscorrespondedtowaveletcoefficientsx.Theamplitudeofwaveletcoefficientsxindistributionofeachscaleisuniform.Anditdecreaseswiththescaleincreasing.Sothenumber

canbefoundasasuitablethreshold.When

islessthanthethreshold,weconsiderthat

ismainlycausedbynoise.Anditissettozeroandabandoned.When

islargerthanthethreshold,weconsiderthat

ismainlycausedbythesignal.Wecandirectlyretainthispartof

orshrinktozerobyafixedamounttogetwaveletcoefficients

andthenreconstructthenewwaveletcoefficients

togetthede-noisedsignal.Anditformsthehardandsoftthresholding.

A.Softandhardthresholding[4]

LetΔtdenotethegiventhreshold.Thesoftthresholdingisdefinedby

（6）

Forthehardthresholding

（7）

Hardthresholdingcanbedescribedastheusualprocessofsettingtozerotheelementswhoseabsolutevaluesarelowerthanthethreshold.Softthresholdingisanextensionofhardthresholding,firstsettingtozerotheelementswhoseabsolutevaluesarelowerthanthethreshold,andthenshrinkingthenonzerocoefficientstowards0.

B.Wavelettrimmedthresholding

Motivatedbyfindingamoregeneralcasethatincorporatesthesoftandhardthresholding,weproposedthefollowingthresholdingrule

（8）

Δtischosenasanestimateofnoiselevel.Whenα=1,itisequivalenttosoftthresholding;

whenα→

itisequivalenttohardthresholding.Figure1graphicallyshowsitsrelationwithsoftandhardthresholding.Itcanbeclearlyseenthattrimmedthresholdingissomethingbetweenhardandsoftthresholding.Withcarefultuningofparameterαforaparticularsignal,onecanachievebestde-noisingeffectwithinthresholdingframework.

C.Thresholdselectionrule

Accordingtothenoisemodel,therearefourthresholdselectionrules[3].TheThresholdselectionrulebasedonStein’sunbiasedestimatedofriskisusedinthispaper.Letnbethelengthofvectorxinthealgorithmofthefollowingthresholdselectionrule.WegetanestimateofriskforaparticularthresholdvalueΔt.MinimizingtherisksinΔtgivesaselectionofthethresholdvalue.Sorttheabsolutevalueofthevectortobeestimatedfromminimumtomaximum,andthenextracttherootofthesortedvectorandanewvectorNVisobtained.Andthealgorithmofriskattheindexkis

（9）

Thecorrespondingthresholdis

Inthethresholdselecting,weshouldnotignorethedetailcoefficientsineverylevelthatprobablyinfluencetherobustnessofthethresholdestimating.Sowehavetorescaleaselectedthresholdinsomelevel.Inthispaper,thethresholdisdependentonthedetailcoefficientsateverylevel.

Figure2hard,softandtrimmedthresholdingfunction

IV.EXPERIMENTANDRESULT

Adiscretesignalsequenceintimedomain,whichisshowninFigure3,istheoriginalsignal.

Figure3originalsignal

Thissignalgeneratedfromthefunction

.Theoriginalsignalconsistsofslowchangeandfastchangecomponents,soit’sappropriatefortestingthede-noisingperformanceofhard,softandtrimmedthresholding.ThenoisysignalwithSNRof15dB,whichisshowninFigure3,isartificiallycontaminatedbystochasticnoisesgeneratedfromnormalGaussianwhitenoisewithzero-means

Figure4contaminatedsignalwiththeSNRof15dB

Inallexperiments,thesym5waveletischosen,performlevel5decomposition.Figure5showsthede-noisedsignal