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1、IEEE SIGNAL PROCESSING LETTERS,VOL.15,2008365Generalization of the Fractional Hilbert TransformRan Tao,Senior Member,IEEE,Xue-Mei Li,and Yue WangAbstractIn this letter,we generalize the fractional Hilberttransform of a real signal to get an analytic version which containsno negative spectrum while m

2、aintaining the essential informationof the real signal.We also present a secure single-sideband(SSB)modulation system in which the angle of the fractional Fouriertransform and the phase of the fractional Hilbert transform areused as double keys for demodulation.Index TermsFractional Fourier transfor

3、m,fractional Hilberttransform,single-sideband(SSB).I.INTRODUCTIONTHE fractional Fourier transform(FRFT)has many impor-tant applications in the solution of quantum physics,op-ticalsystems,and signal processing27inrecent years.As ageneralization ofthestandardFouriertransform(FT),theFRFToperationindica

4、tesarotationofasignalwithangleinthetime-frequency plane.The FRFT of a signalis defined as2,3(1)where,indicates the rotation angle ofthe transformed signalfor the FRFT;thetransformationkernel,and.Whenis an integer,four special cases of the FRFT are:,and,whereis an integer anddenotes the FT of.Whenvar

5、ies from 0 to,the FRFT develops from its original function to its FT in thetime-frequency plane.The FRFT has many useful propertieswhich can be seen as the generalization of those properties inManuscript received August 14,2007;revised January 13,2008.This workwassupportedinpartbytheNationalScienceF

6、oundationofChinaunderGrants60232010 and 60572094 and in part by the National Science Foundation ofChina for Distinguished Young Scholars under Grant 60625104.The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof.Weifeng Su.R.Tao and Y.Wang are with

7、 the Department of Electronic Engineering,Bei-jing Institute of Technology,Beijing 100081,China(e-mail:).X.-M.Li is with the Department of Electronic Engineering,Beijing Instituteof Technology,Beijing 100081,China,and also with the Department of Elec-tronicandInformationEngineering,BeijingElectronic

8、ScienceandTechnologyInstitute,Beijing 100070,China(e-mail:li-).Digital Object Identifier 10.1109/LSP.2008.919814the Fourier sense 2,3.The inverse transformation of theFRFT is(2)The Hilbert transform based on the FT has widely been ap-plied in many areas,such as optical system,modulation,andedge dete

9、ction 1,811,13,14,etc.In communication,the Hilbert transform is used to construct the analytic signalneeded for SSB modulation from a real signal.One of the mostimportant properties of the analytic signal is that it contains nonegative frequency components of the real signal in the Fourierdomainmain

10、tainingthenecessaryinformationoftherealsignal,which makes it possible to reconstruct the real signal from itsanalytic version.Thus,the bandwidth of the real signal can besaved.The fractional Hilbert transform was proposed in 9 viatwo different definitions.One is a modification of the standardHilbert

11、 transform which can be seen as a phase shifter withparameter,and the other definition is basedon the FRFT.The fractional Hilbert transform can produce theimage edge enhancement or the image compression in differentways when both parameters(the angle of the FRFT and thephase of the fractional Hilber

12、t transform)are varying.Usingthe fractional Hilbert transform which was defined by the firstdefinition 9,Tseng and Pei proposed an analytic presentationand constructed an SSB modulation by taking the parameter ofthe fractional phase as a secret key 11.Zayed generalized theHilbert transform(GHT)in a

13、different way and obtained theanalytic part of a signal by suppressing the negative frequencycomponent of the signal in the fractional Fourier domain 10.In this letter,the fractional Hilbert transform is associated withthe first definition in 9.As we know,theFRFT of a real signal has noconjugate-sym

14、-metry property;therefore,suppressing the negative frequencyportion of the real signal might lead to a situation where the re-constructionoftherealsignalisimpossible.Inthisletter,weareinterested in applying the fractional Hilbert transform to savethe bandwidth of a real signal in the fractional Four

15、ier domain.Firstly,the fractional Hilbert transform and analytic signal ofa real signal are generalized.The analytic signal contains onlypositive components in the fractional Fourier domain with in-dispensable information.Secondly,a secure SSB communica-tion system is proposed in which two parameter

16、s(the angle ofthe FRFT and the phase of the fractional Hilbert transform)areused as secret keys for demodulation.Finally,simulations aregiven.1070-9908/$25.00 2008 IEEE转载http:/366IEEE SIGNAL PROCESSING LETTERS,VOL.15,2008II.MAINRESULTSDefinition 1:The generalized fractional Hilbert transform ofa rea

17、l signalis defined as(3)where,is the phase of the fractional Hilberttransform;the angle of the FRFT,andthe conventionalHilbert transform of,i.e.,and the FT ofis.Definition 2:The analytic version of a real signalasso-ciated with the generalized fractional Hilbert transform can bedefined as(4)Equation

18、s(3)and(4)are equivalent to the multiplication ofthe conventional fractional Hilbert transform and the analyticversion by a chirp signal.This is similar to the new samplingformulae proposed in 14.Here we apply the fractional Hilberttransform to save the bandwidth of a real signal in the fractionalFo

19、urier domain,leading to the following theorem.Theorem 3:A real signaland its generalized fractionalHilbert transform can be used to construct an analytic versionwhich has no negative frequency components in the fractionalFourier domain for anglewithout loss of the essential infor-mation contained in

20、.Remarks:Taking the FRFT for angleat bothsides of(4),we have(5)whereis the FT of.The FRFT ofis(6)Substituting(6)into(5)results in.(7)Equation(7)shows that the FRFT of the analytic signalhasonlypositivefrequencycomponentsinthefractionalFourierdomain for angle.Whenand,contains the indispensable half s

21、pectrum ofbesides a com-plex factorin the positive frac-tional Fourier domain.Since the chirp signalissymmetric about,andis conjugate-symmetric,the real signalcan be reserved by.Thus,we canprocessarealsignaltosuppressthespectrumbyhalfinthefrac-tional Fourier domain.The generalized fractional Hilbert

22、 transformand an-alytic signalhave two parameters:and.When,they correspond to the ones depending on only the phase.When,they reduce to the forms associated with thefractional Fourier sense,and when,both of thembecome the conventional definitions.Even though they providetwo degrees of freedom(and)whi

23、ch are the same as thosein9,thereremainsomedifferences.Firstly,we givethedefini-tion ofin the time domain,whereas the definition is inthe transform domain in 9.Secondly,our work is to apply thegeneralizedfractionalHilberttransformandtheanalyticversionofa realsignal forSSB communicationapplicationinw

24、hichthetwo parameters can be seen as double keys.In 9,the fractionalHilbert transform was used to produce different improvementsin the image processing.Torecoverfromtheanalyticversionin(4),wesetto obtain(8)It can be seen that the real part ofis.Thus,to recover,we apply the following expression:(9)Ba

25、sed on(4)and(9),a generalized SSB communication systemisillustratedinFig.1.isthecarrierfrequencyinthefractionalFourier domain,the angle of the FRFT,andthe phase of thefractional Hilbert transform.Fig.1(b)istheblockdiagramforthemodulationprocess.Theoutput signalcan be written as(10)where.According to

26、 the con-volution structure in the fractional Fourier domain 4,we canpresent the FRFT ofas(11)whereandistheconvolutionoperator.The FRFT ofcan be written as(12)Substituting(12)into(11)yields(13)Equation(13)describes the relationship between the spectrumofand the spectrum ofin the fractional Fourierdo

27、main.中国科技论文在线http:/TAO et al.:GENERALIZATION OF THE FRACTIONAL HILBERT TRANSFORM367Fig.1.(a)Generalized SSB communication system.(b)Block diagram of the modulator.(c)Block diagram of the demodulator.In the demodulator shown in Fig.1(c),is obtained basedon(9).Whenthereceivedsignaliscorruptedbyanaddit

28、ivenoise,the output signal will be,wherecanbe given as(14)Equation(14)reveals that noise gain is mainly determined bythe factor,which is similar to that in 11.It is better tochoose the phase parameterin the interval;thismakesto lie within the range 1,2.Thus,the effect ofnoise is very small.The SSB c

29、ommunication system in Fig.1 is a generalizationof the conventional one.It should be noted that the parameters(and)must be known to the demodulator in advance,or elseit will be difficult to recoverfrom the received signal.If wereplaceandbyand,respectively,in(9),combiningwith(3)and(4),the recovered r

30、eal signal can be written as(15)Settingyields(16)Due to,replacingbyin(16),we can presentthe recovered real signal as follows:(17)Fig.2.Recovered signal?.(a)Without a chirp signal?.(b)Witha chirp signal?.It can be seen thatonly whenand;otherwise,it will be difficult to recover.Thus,they canbe used as

31、 double keys in the SSB communication system toenhance the security of the system in some situation.III.SIMULATIONSAssuming a real signal is rectangular and has uniform ampli-tude,theangleofitsFRFTisandthephaseofthefrac-tional Hilbert transform is given as.We use two differentways to transmitin the

32、SSB communication system.Oneis to apply the real signal and the fractional Hilbert transformwithout a chirp signalfor constructing an analyticversion which suppresses the negative content of the FRFT ofdirectly.The other utilizes the analytic version of the realsignalbasedon(4)withthechirpsignal.Fig

33、.2showstherecoveredsignalsobtainedfromthetwodifferentcases,ignoring the effect of noise in the demodulator.In Fig.2(a),is distorted greatly due to the fact that theanalytic version does not contain exactly the same informationasafter suppressing the spectrum components by half inthe fractional Fouri

34、er domain,while in Fig.2(b),is recon-structed perfectly based on the work in this letter.Another example is about measuring the performance ofthe different parameters(angleand phase)for recoveringthe real signal(the same as in the above example)inthe demodulator.Equation(13)indicates that the error

35、ofrecoveringis zero only whenand.Wecalculate the mean square root of amplitude estimation errors中国科技论文在线http:/368IEEE SIGNAL PROCESSING LETTERS,VOL.15,2008Fig.3.Mean square root of amplitude estimation errors.by,whereandare thesample-sequences of the recovered signaland the desiredreal signal;N is s

36、ampling points;varies fromto,andfromto.Fig.3 shows the mean square rootof the amplitude errors with the different values of angleandphase.In Fig.3,it is clear that bothandaffect the errors.1)When the phase of the fractional Hilbert transform is,the errors are mainly dependent on the param-eter.There

37、is a narrowrange,in which theerrors drop sharply to zero.2)When the angle of the FRFT is,the errorsaremainlydependentontheparameter.Atthesametime,largeerrorsappearonbothsidesofthecurveanddropslackto the sink.3)Whenand,the errors are large.The error iszero only whenand.Thus,the parametersandcan be us

38、ed as secret keys in thesecurity SSB communication system.IV.CONCLUSIONIn this letter,the definitions of fractional Hilbert transformand analytic version of a real signal based on the FRFT are gen-eralized.The analytic version of a real signal occupying onlythe positive frequency components in the f

39、ractional Fourier do-main contains all essential information which is indispensableto reconstruct the real signal.Based on Theorem 3 in this letter,a secure SSB modulation system in which the two parameterscan be seen as double keys is illustrated.Simulations show thevalidity of this letter.REFERENC

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