部分傅里叶变换在信号处理中的研究发展中英翻译.docx

部分傅里叶变换在信号处理中的研究发展中英翻译.docx

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部分傅里叶变换在信号处理中的研究发展中英翻译

毕业设计（论文）外文资料翻译

系别：

电子信息系

专业：

通信工程

班级：

B090310

姓名：

孙春甫

学号：

B09031015

外文出处：

知网

附件：

1.原文；2.译文

2013年05月

ResearchProgressoftheFractionalFourierTransform

inSignalProcessing

ABSTRACT

ThefractionalFouriertransformisageneralizationoftheclassicalFouriertransform,whichisintroducedfromthemathematicaspectbyNamiasatfirstandhasmanyapplicationsinopticsquickly.Whereasitspotentialappearstohaveremainedlargelyunknowntothesignalprocessingcommunityuntil1990s.ThefractionalFouriertransformcanbeviewedasthechirp-basisexpansiondirectlyfromitsdefinition,butessentiallyitcanbeinterpretedasarotationinthetime-frequencyplane,i.e.theunifiedtime-frequencytransform.Withtheorderfrom0increasingto1,thefractionalFouriertransformcanshowthecharacteristicsofthesignalchangingfromthetimedomaintothefrequencydomain.Inthisresearchpaper,thefractionalFouriertransformhasbeencomprehensivelyandsystematicallytreatedfromthesignalprocessingpointofview.OuraimistoprovideacoursefromthedefinitiontotheapplicationsofthefractionalFouriertransform,especiallyasareferenceandanintroductionforresearchersandinterestedreaders.

Whilesolvingaheatconductionproblemin1807,aFrenchscientistJeanBaptisteJosephFourier,suggestedtheusageoftheFouriertheorem.Thereafter,theFouriertransform（FT）hasbeenappliedwidelyinmanyscientificdisciplines,andhasplayedimportantroleinalmostallthescienceandtechnologydomains.However,withtheextensionofresearchobjectsandscope,theFThasbeendiscoveredtohaveshortcomings.SincetheFTisakindofholistictransform,i.e.,throughwhichthewholespectrumisobtained,itcannotobtainthelocaltime-frequencycharacterthatisessentialandpivotalforprocessingnonstationarysignals.Soaseriesofnovelsignalanalysistheorieshavebeenputforwardtoprocessnonstationarysignals,suchas:

thefractionalFouriertransform,theshort-timeFouriertransform,Wigner-Villedistribution,Gabortransform,wavelettransform,cyclicstatistics,AM/FMsignalanalysisandsoon.HereintothefractionalFouriertransform（FRFT）,asageneralizationoftheclassicalFT,hascaughtmoreandmoreattentionforitsinherentpeculiarities.Inthelastdecade,researchintotheFRFTtheoryandapplicationwasfruitful,resultinginanupsurgeinthestudyoftheFRFT.

In1980,NamiasintroducedtheFRFTasawaytosolvecertainclassesofordinaryandpartialdifferentialequationsarisinginquantummechanicsfromclassicalquadraticHamiltonians.HisresultswerelaterrefinedbyMcBrideandKerr.TheydevelopedanoperationalcalculustodefinetheFRFTwhichwasthebasefortheopticalversionoftheFRFT.In1993,MendlovicandOzaktasofferedtheopticalrealizationoftheFRFTtoprocesstheopticalsignal,whichwaseasytoberealizedwithsomeopticalinstruments.SotheFRFThasmanyapplicationsinoptics.AlthoughtheFRFTmaybepotentiallyuseful,itappearstohaveremainedlargelyunknowntothesignalprocessingcommunityforthelackofphysicalilluminationandfastdigitalcomputationalgorithmuntiltheinterpretationasarotationinthetime-frequencyplaneandtheefficientdigitalcomputationalgorithmoftheFRFTemergedin1993and1996respectively.Thereafter,manyrelevantresearchpapershavebeenpublished.ThestudyoftheFRFTdidnotstarttoolateathome,butstillstayedattheimmaturestageinviewofthenumberandcontentoftherelevantpapers.Inearly1996,somereviewpapersabouttheFRFTappearedathome,yetthepotentialoftheFRFTwasjustexploredthen.Whatismore,noreviewpaperoftheFRFTfromtheaspectofsignalprocessinghasbeenpublishedoverseassofar.SothispapertriestosummarizetheresearchprogressoftheFRFTinsignalprocessing,andexpatiatethetheoreticsystemoftheFRFTinthefoundation,application-foundationandapplicationfieldstoprovidethereferencetorelevantresearchers.

Theorganizationofthispaperisasfollows:

wefirstprovidethedefinitionoftheFRFTanditsmeaning.ThepropertiesandtherelationbetweentheFRFTandtheconventionaltime-frequencydistributionaredepictedinsection2,aswellastheuncertaintyprincipleintheFRFTdomain.WeconsidertheFRFTdomaintobeinterpretedastheunifiedtime-frequencytransformdomain.Insection3,wesystematicallysummarizesomesignalanalysistoolsbasedontheFRFT.WesummarizetheapplicationsoftheFRFTinsignalprocessinginsection4.Finally,thispaperisconcludedinsection5.

1DefinitionoftheFRFT

TheFRFTisdefinedas：

（1）

where

（2）

where

indicatestherotationangleofthetransformedsignalforFRFT,pisthetransformorderoftheFRFT,andtheFRFToperatorisdesignatedby

.ItisobviousthattheFRFTisperiodicwithperiod4.Ifandonlyif

thentheFRFTisjustthesameastheFT.Let

and

.Theneq.

（1）isequivalentto

（3）

eq.（3）showsthatthecomputationoftheFRFTcorrespondstothefollowingthreesteps:

a.aproductbyachirp,

;

b.aFT（withitsargumentscaledby

）,

with

c.anotherproductbyachirp,

ItturnsupthattheFRFTof

existsinthesameconditionsinwhichitsFTexists;inotherwords,if

exits,

exitstoo.UsingthecomputationstepsaboveobtainedtheunifiedsamplingtheoremfortheFRFT.Basedonchirp-periodicityErsegheetal.[11]generalizedthecharacteroftheFT（continuous-time,periodiccontinuous-time,discrete-time,periodicdiscrete-time）tofourcorrespondingversionsoftheFRFT,anddeducedtheunifiedsamplingtheoremfortheFRFT.

TheFRFTcanbeconsideredasadecompositionofthesignal,fortheinverseFRFTisdefinedas

（4）

where

isexpressedbyaclassoforthonormalbasisfunction

withweightfactors

.Thebasisfunctionsarecomplexexponentialswithlinearfrequencymodulation（LFM）.Fordifferentvaluesofu,theyonlydifferbyatimeshiftandbyaphasefactorthatdependson

:

（5）

2PropertiesofthefractionalFouriertransform

2.1Basicproperties

TheFRFTisageneralizationoftheFT,somostofthepropertiesoftheFThavetheircorrespondinggeneralizationversionsoftheFRFT.ThebasicpropertiesoftheFRFTarelistedintheappendix.Animportantproperty,convolutiontheoremoftheFRFT,hasnotbeenlistedintheappendix,foritisnotobtainedsimply.Interestedreadersmayrefertorefs.AnotherimportantpropertywillbeintroducedthattheFRFTcanbeinterpretedasarotationinthetime-frequencyplanewithangleα.ThepropertyestablishesthedirectrelationshipbetweentheFRFTandthetime-frequencydistribution,andfoundsthetheorythattheFRFTdomaincanbeinterpretedasauniformtime-frequencydomain,whichofferstheFRFTtheadvantagetobeusedinsignalprocessing.WiththeWignerdistributionastheexample,let

denotetheoperatortorotatea2-Dfunctionclockwise:

（6）

Thentherelationshipisasfollows:

（7）

where

expresstheWignerdistributionof

respectively.Suchrelationsstillremainavailablefortheambiguityfunction,themodifiedshort-timeFouriertransformandthespectrogram.Lohmanngeneralizedeq.（7）,andobtainedtherelationshipbetweentheFRFTandRadon-Wignertransform:

（8）

where

istheoperatoroftheRadonTransform,expressingtheintegralprojectionofa2-Dfunctionwithangle

toaxist.eq.（8）canalsobeunderstoodasmarginalintegralafterarotationofthereferenceframewithangle

namely:

（9）

SincetheFRFThassuchrelationshipwithconventionaltime-frequencydistributions,wewanttoknowwhetheramoregeneralexpressionexists.Let

（10）

where

isthetransformkernel,

and

aretheWignerdistributionandtheCohenclassoftime-frequencydistributionof

respectively.Onlyifthetransformkernel

isrotationallysymmetricaroundtheorigin,then

thetime-frequencydistributionoftheFRFTof

isarotatedversionofthetime-frequencydistributionof

.Thus,theFRFTcorrespondstorotationofarelativelylargeclassoftime-frequencyrepresentations.

FromtherelationshipbetweentheFRFTandthetime-frequencydistributionsmentionedabove,weseethattheFRFToffersanintegrativedescriptionofthesignalfromthetimedomaintothefrequencydomain.TheFRFTcanprovidemorespacefortime-frequencyanalysisofsignals.

2.2Uncertaintyprinciple

SincetheFRFTdomainisaunifiedtime-frequencytransformdomain,whatisthegeneralizationoftheconventionaluncertainprincipleintheFRFTdomain?

UsingtheconventionaluncertainprincipleandthethreedecompositionstepsoftheFRFTmentionedinsection1,wecanobtaintheuncertainprinciplebetweenthetwoFRFTdomainswithdifferenttransformorders.

3Fractionaloperatorandtransform

BecausetheFRFTisaunitedtime-frequencyanalysistool,andcanbeinterpretedasarotationinthetime-frequencyplane,wecandefinesomeusefulfractionaloperatorsandtransformsbasedontheFRFT.

3.1Fractionaloperators

Convolutionandcorrelationarethetwokindsofsignalprocessingoperatorsincommonuse.Thefractionalconvolutionandfractionalcorrelationoperatoraredefinedinthetimedomainandtransformdomainrespectivelyadaptedtosignaldetectionandparameterestimation;adaptedtofilterdesign,beamformingandpatternrecognition.

Inthetime-frequencyanalysistheory,theunitaryoperatorandhermitianoperatoraretwoimportantoperators.Unitarityisoneofthefactorsneededtoconsiderindesigningatransformoperator.Anddifferenttransformdomainsusuallycanberelatedbysomehermitianoperators.Thus,itattractsthepeople’sstron