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1、IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.65,NO.18,SEPTEMBER 15,20174797A Sampling Theorem for Fractional WaveletTransform With Error EstimatesAbstractAsageneralizationoftheordinarywavelettransform,the fractional wavelet transform(FRWT)is a very promising toolfor signal analysis and processing.Many
2、 of its fundamental prop-erties are already known;however,little attention has been paidto its sampling theory.In this paper,we first introduce the conceptof multiresolution analysis associated with the FRWT,and thenpropose a sampling theorem for signals in FRWT-based multires-olution subspaces.The
3、necessary and sufficient condition for thesampling theorem is derived.Moreover,sampling errors due totruncation and aliasing are discussed.The validity of the theoreti-cal derivations is demonstrated via simulations.Index TermsFractional Fourier transform,fractional wavelettransform,multiresolution
4、analysis,sampling theorem.I.INTRODUCTIONTHE fractional Fourier transform(FRFT)1 is a general-ized form of the ordinary Fourier transform(FT)with anangle parameter and is identical to the FT when the angle is equal to/2.The FRFT can be interpreted as a projection inthe time-frequency plane onto a lin
5、e that makes an angle of with respect to the time axis 2,as shown in Fig.1.The FRFTof a signal or function f(t)L2(R)is defined as 2F(u)=Ff(t)(u)=?Rf(t)K(u,t)dt(1)whereK(u,t)=Aeju2+t22cot jut csc,?=m(t u),=2m(t+u),=(2m 1)(2)Manuscript received August 15,2016;revised January 31,2017 and April23,2017;a
6、ccepted May 25,2017.Date of publication June 12,2017;date ofcurrent version July 10,2017.The associate editor coordinating the review ofthis manuscript and approving it for publication was Dr.Eleftherios Kofidis.This work was supported in part by the National Natural Science Foundationof China under
7、 Grants 61501144 and 61671179,in part by the FundamentalResearch Funds for the Central Universities under Grant 01111305,and in partby the National Basic Research Program of China under Grant 2013CB329003.(Corresponding author:Xiaoping Liu.)J.Shi,X.Liu,and X.Sha are with the Communication Research C
8、en-ter,Harbin Institute of Technology,Harbin 150001,China(e-mail:;).Q.Zhang is with the Shenzhen Graduate School,Harbin Institute of Tech-nology,Shenzhen 518055,China(e-mail:).N.Zhang is with the Communication Research Center,Harbin Instituteof Technology,Harbin 150001,China,and also with the Shenzh
9、en Gradu-ate School,Harbin Institute of Technology,Shenzhen 518055,China(e-mail:).Color versions of one or more of the figures in this paper are available onlineat http:/ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2017.2715009Fig.1.The time-fractional-frequency plane.isthetransformkern
10、elwithA=?(1 j cot)/2 andm Z.TheinverseFRFTwithrespecttoangleistheFRFTatangle,i.e.,f(t)=FF(u)(t)=?RF(u)K(u,t)du(3)where the superscript denotes the complex conjugate.By ex-tending the frequency concept to the FRFT domain,the ar-gument u is termed the fractional frequency 8.From thisviewpoint,the FRFT
11、 might be taken as the fractional spec-trum 9.It has proven to be a useful tool in areas such asoptics,radar,communications,and signal processing 29,among others.However,the FRFT has one major drawbackdue to using global kernel,i.e.,it only provides such fractionalspectral content without indication
12、 about the time localizationof the fractional spectral components.Therefore,the analysisof non-stationary signals whose fractional spectral character-istics change with time requires joint signal representations inboth time and fractional-frequency domains,rather than just afractional-frequency doma
13、in representation.A common approach to obtain a joint signal representation inboth time and fractional-frequency domains is to cut the signalfirst into slices,followed by doing an FRFT analysis on theseslices.The resulting joint signal representation is referred to asthe short-time FRFT(STFRFT)10,11
14、.However,the short-coming of the STFRFT is that its time and fractional-domainresolutions cannot simultaneously be arbitrarily high.This isdue to the uncertainty principle 12,13 of the FRFT,whichstates that a signal cannot be simultaneously concentrated inboth time and fractional-frequency domains.A
15、s a generaliza-tion of the ordinary wavelet transform(WT),the notion of the1053-587X 2017 IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See http:/www.ieee.org/publications standards/publications/rights/index.html for more information.4798IEEE TRANSACTIONS O
16、N SIGNAL PROCESSING,VOL.65,NO.18,SEPTEMBER 15,2017fractional wavelet transform(FRWT)was first introduced byMendlovic and David 14 as a way to deal with optical sig-nals 1519,and the FRWT 14 was defined as a cascade ofthe FRFT and the WT.Unfortunately,this transform can not beregardedasakindofjointti
17、me-fractional-frequencyrepresenta-tionsincetheresnoexplicit2-Ddomain(asillustratedinFig.1)for it,and time information is lost in the transform.Recently,Prasad and Mahato 20 expressed the ordinary WT of a signalin terms of the FRFTs of the signal and mother wavelet,andthey also called the expression
18、the FRWT.Actually,the authors20 found an equivalent expression of the ordinary WT in theFRFTdomain.Later,Shietal.21proposedanewdefinitionofthe FRWT using the concept of fractional convolution7.TheFRWT with an angle of a function f(t)L2(R)is definedas 21Wf(a,b)=?Rf(t),a,b(t)dt,(4)and the transform ke
19、rnel is given by,a,b(t)=1a?t ba?ejt2b22cot(5)where a R+,b R.Note that when =/2,the FRWTreduces to the ordinary WT.It is well-known that the ordinary WT is based on rectan-gular tessellations of the time-frequency plane,as illustratedin Fig.2(a),where we consider the t plane as the time-frequency pla
20、ne.It was shown in 21 that the FRWT tilesthe time-frequency plane in a parallelogram fashion shown inFig.2(b),whichmakesitbeaunifiedtime-frequencytransform.The interpretation of the FRWT may be twofold.Consideringa chirped signal f(t)ejt22cot,the FRWT can be viewed as theordinaryWTofthechirpedsignal
21、,whichcontainsachirpfactorejb22cot.Fast computation of the FRWT using the fast WTalgorithms is thus possible.On the other hand,invoking the Par-seval theorem of the FRFT 3,(4)has an equivalent expressionin the FRFT domain,i.e.,Wf(a,b)=?R2aF(u)(aucsc)K(u,b)du(6)where(ucsc)denotes the FT(with its argu
22、ment scaled bycsc)of(t).Moreover,the so-called admissibility condition22 of the FRWT implies that(0)=0,i.e.,?R(t)dt=0.Consequently,the continuous fractional wavelet basis functionsdefined in(5)must oscillate and behave as bandpass filters inthe FRFT domain.Therefore,if one defines a set of fixed sca
23、les,the FRWT can be interpreted as a non-uniform filterbank in theFRFT domain.Thereisalsoadirectlinktothewell-knownchirplettransform(CT),defined as 23Cf(tc,fc,log,)=?Rf(t)ctc,fc,log,(t)dt(7)withctc,fc,log,(t)=1g?t tc?ej(ttc)2+j2fc(ttc)(8)Fig.2.Tiling of the time-frequency plane:(a)WT and(b)FRWT.wher
24、e g(t)is a window function,tcis the time center,0 isthe effective time spread,fcis the frequency center,and is thechirp rate.In the time-frequency plane,the CT employs atomswith oriented time-frequency supports.Therefore,as noted inFig.2(b),the CT can be regarded as a special case of the FRWT.Now,we
25、 describe how to modify the FRWT to obtain the CT.Actually,(7)can be rewritten asCf(tc,fc,log,)=ejt2c?Rf(t)ejt21g?t tc?ej2(fctc)(ttc)dt(9)which implies that the CT is identical to a FRWT,with cot=2,b=tc,a=,and(ttc)=g(ttc)ej2(fctc)(ttc).Since the CT can be useful in a number of different applications
26、ranging from sparsely representing audio 24,modeling au-ditory processing 25,characterizing visual evoked potentials26,sparse detection 27,to instantaneous frequency estima-tion 2830,the aforementioned link implies that the FRWTis a very promising tool for signal analysis and processing.Many propert
27、ies of the FRWT are already known,however,there is currently no existing work on the sampling theory of theFRWT.Asisknowntoall,digitalsignalprocessingapplicationsareoftenconcerned withtheabilitytostoreandprocessdiscretesets of numbers,which are related to continuous-time signalsSHI et al.:SAMPLING T
28、HEOREM FOR FRACTIONAL WAVELET TRANSFORM WITH ERROR ESTIMATES4799through an acquisition process.One major goal,which is at theheart of digital signal processing,is the ability to reconstructcontinuous-time functions,by properly processing their avail-able samples.Therefore,a fundamental problem of th
29、e FRWTis how to represent a continuous signal in terms of a discretesequence.Our objective in this paper is to establish a samplingtheory associated with the FRWT.The remainder of this paper is organized as follows.InSection II,after starting notation and some facts of the FRFT,weintroduce theconcep
30、t ofthemultiresolutionanalysis(MRA)associatedwiththeFRWT.InSectionIII,asamplingtheoreminmultiresolution subspaces for the FRWT is proposed.The nec-essary and sufficient condition for which the proposed samplingtheorem holds is derived.Some properties of the sampling the-orem are also discussed.Sampl
31、ing errors due to truncation andaliasing are estimated in Section IV.Some applications of thederived results are presented in Section V.Concluding remarksare drawn in Section VI.II.PRELIMINARIESA.Notation and DefinitionsWe use the following notation throughout:R,Z,Z+,L10,2,L2(R),L20,2,L0,2,?2(Z),and
32、?(Z)de-note the set of real numbers,the set of integers,the set ofpositive integers,the space of absolutely integrable functionson 0,2,the space of all square-integrable functions on R,the space of all square-integrable functions on 0,2,the spaceof all essentially bounded functions on 0,2,the space
33、ofall square-summable sequences on Z,and the space of allbounded sequences on Z,respectively.Continuous signals aredenoted with parentheses,e.g.,f(t),t R,and discrete sig-nals with brackets,e.g.,cn,n Z.We denote the innerproduct between two continuous functions f(t)and g(t)inL2(R)by?f,g?L2=?Rf(t)g(t
34、)dt.Similarly,the inner prod-uct between two sequences cn and dn in?2(Z)is givenby?c,d?2=?nZcndn.Correspondingly,we denote thenorm in L2(R)by?f?2L2=?f,f?L2,and the norm in?2(Z)by?c?2?2=?c,c?2.Let H be a Hilbert space and n(t)nZbe a completeset of functions in H.The set is a Riesz basis for H if ando
35、nly if there exist constants 0 A B +such that for allcn?2(Z),A?c?2?2?n Zcnn(t)?2L2 B?c?2?2(10)where the equality holds if and only if the basis is orthonormal,i.e.,when A=B=131.For a measurable function f(t)on R,let?f?=esssup|f(t)|and?f?0=essinf|f(t)|(11)denote the essential supremum and infimum of|
36、f(t)|,respec-tively.ThecharacteristicfunctionofameasurablesubsetE Ris given byE(x)=?1,x E0,otherwise.(12)The support set of a function F(x),denoted by suppF(x),isthe closure of the set x:F(x)?=0,where F(x)does notvanish.The discrete-time FRFT(DTFRFT)of a sequence qn?2(Z)is defined as 33?Q(u)=?Fqn(u)
37、?n ZqnK(u,n)(13)where?FdenotestheDTFRFToperator,andK(,)isdefinedin(2).If =/2,the DTFRFT reduces to the discrete-time FT(DTFT)3.Conversely,the inverse DTFRFT is written asqn=?I?Q(u)K(u,n)du,I?0,2 sin.(14)Combining(13)and(2),we have?Q(u)eju22cot=?n Zcnejn22cot ejun csc(15)where cn=Aqn.Plugging(2)into(
38、14)leads tocn=12 sin?I?Q(u)eju22cot ejn22cot+jun csc du.(16)Note that if?Q(u)is in L2(I),its product by the chirp signaleju22cot is also in L2(I).Eqs.(15)and(16)imply that func-tions belonging to L2(I)in the FRFT domain can be expandedinto series defined in(15).B.Multiresolution Analysis Associated
39、With the FRWTDefinition 1:AMRAVkk Z1associatedwiththeFRWTis a family of subspaces of L2(R),which satisfies 22i)Vk Vk+1,?k ZVk=L2(R),and?k ZVk=0;ii)f(t)Vkif and only iff(2t)ej(2t)2t22cot Vk+1,(17)orf(21t)ej(21t)2t22cot Vk1;(18)iii)There exists a function(t)L2(R)(called the frac-tional scaling functio
40、n)with(t)ejt22cot V0suchthat?0,n,(t)=(t n)ejt2n22cot?n Zforms aRiesz basis of V0.1The subspace Vkis spanned by 22?k,n,(t)?2k2(2kt n)ejt2?n2k?22cot?k Z.4800IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.65,NO.18,SEPTEMBER 15,2017For any(t)L2(R),let(ucsc)denote the FT(withits argument scaled by csc)of the
41、 fractional scaling function(t),and takeG,(u)=?k Z|(ucsc+2k)|2?12.(19)Then,G,(u)=G,(u+2 sin)L1(I).It is shown in22 that0?G,(u)?0?G,(u)?1/2.For anyf(t)V0,it follows thatf(t)=?m Zcm(t m)ejt2m22cot(35)in L2(R),where the scalar sequence cm?2(Z).The seriesdefined in(35)point-wise converges to a continuou
42、s function inV0due to the fact that?m Zcm(t m)ejt2m22cot?2?m Z|cm|2?m Z|(t m)|2.(36)Therefore,without loss of generality,we can take any f(t)V0as a continuous function.SHI et al.:SAMPLING THEOREM FOR FRACTIONAL WAVELET TRANSFORM WITH ERROR ESTIMATES4801Theorem 1:Let(t)L2(R)be the fractional scalingf
43、unction of a MRA Vkk Zassociated with the FRWT suchthat the sampling sequence n at integers of(t)belongsto?2(Z).Then,there exists a function s(t)L2(R)withs(t)ejt22cot V0and J Z+?0 such thatf(t)=?n Zf?n2J s?2Jt n?ejt2(n2J)22cot(37)holds for all f(t)V0in the L2(R)sense if and only if12?(ucsc)EJ(u)L2(I
44、)(38)holds.The interpolation function s(t)in(37)satisfiesS(ucsc)=(u csc)2?(u csc)for a.e.u EJ,where S(ucsc)and(ucsc)denote the FTs(with their argument scaled bycsc)of(t)and s(t),respectively.Proof:Sufficiency:First assume that(38)holds.Thus,?(ucsc)?=0 holds for a.e.u EJ.Because of(15),thereexists a
45、sequence cn?2(Z)such that12?(ucsc)EJ(u)=?n Zcnejn22cot jun csc(39)holds in the L2(I)sense.Next,since?(ucsc)is a periodicfunction with period 2 sin,we have?R?(ucsc)2?(ucsc)EJ(u)?2du=?k Z?I?(ucsc+2k)2?(ucsc)?2EJ(u)du=?IG2,(u)|2?(ucsc)|2EJ(u)du(40)from which along with(20),it follows that?R?(ucsc)2?(uc
46、sc)EJ(u)?2du?G,(u)?2?I1|2?(ucsc)|2EJ(u)du(41)which implies that(u csc)2?(u csc)EJ(u)belongs to L2(R).Thus,we can deriveS(ucsc)=Fs(t)(ucsc)?(ucsc)2?(ucsc)EJ(u),(42)where F denotes the FT operator.Further,it follows that(ucsc)EJ(u)=2S(ucsc)?(ucsc).(43)Then,substituting(39)into(42)gives rise toS(ucsc)=
47、(ucsc)?n Zcnejn22cot jun csc.(44)Combining(44)and the relationship between the FT and theFRFT 32 yieldsF!s(t)ejt22cot(u)=2Aeju22cot Fs(t)(ucsc)=2(ucsc)?n ZcnK(u,n)=2?C(u)(ucsc)(45)where?C(u)denotes the DTFRFT of cn.Now taking theinverse FRFTs of both sides of(45),we haves(t)ejt22cot=?n Zcn(t n)ejt2n
48、22cot(46)which implies that s(t)ejt22cot belongs to V0due to thefact that(t n)ejt2n22cot n Zis a Riesz basis of V0.Moreover,combining(43)and(31)results inJ(u)(ucsc)EJ(u)=J(u)2S(ucsc)?(ucsc),(47)i.e.,J(u)(ucsc)=J(u)2S(ucsc)?(ucsc).(48)Then,using(48),(29),and(33),we obtain?2Jucsc?=2?J?2Jucsc?S(ucsc)(4
49、9)so that(ucsc)=2?J(ucsc)S?ucsc2J?.(50)Using Poissons summation formula of the FT 3,we have?J(ucsc)=?k Z(ucsc+2J+1k)=2J?m Z?m2J ejm2Ju csc.(51)Then,taking the inverse FT of both sides of(50)and using(51)yield(t)=?m Z?m2J s?2Jt m?.(52)For any function f(t)V0,there exists a sequence ak?2(Z)such thatf(t)=?k Zak(t k)ejt2k22cot.(53)Inserting(52)into(53)yieldsf(t)=?k Zak?m Z?m2J s?2J(t k)m?ejt2k22cot.(54)4802IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.65,NO.18,SEPTEMBER 15,2017Then,by setting n=2Jk+m in(54),we obtainf(t)=?n Zs(2Jt n)?k Zak?n2J k ejt2k22cot.(55)Next,due to(53),we letfn=?k Zakn kejn2k