毕业设计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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毕业设计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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毕业设计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’ss
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