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1、.RESEARCH PAPER.SCIENCE CHINAInformation SciencesJune 2012 Vol.55 No.6:12701279doi:10.1007/s11432-011-4320-xc?Science China Press and Springer-Verlag Berlin Heidelberg A novel fractional wavelet transform and itsapplicationsSHI Jun,ZHANG NaiTong&LIU XiaoPingCommunication Research Center,Harbin Insti
2、tute of Technology,Harbin 150001,ChinaReceived March 30,2010;accepted March 29,2011;published online September 9,2011AbstractThe wavelet transform(WT)and the fractional Fourier transform(FRFT)are powerful tools formany applications in the field of signal processing.However,the signal analysis capabi
3、lity of the former islimited in the time-frequency plane.Although the latter has overcome such limitation and can provide signalrepresentations in the fractional domain,it fails in obtaining local structures of the signal.In this paper,a novelfractional wavelet transform(FRWT)is proposed in order to
4、 rectify the limitations of the WT and the FRFT.The proposed transform not only inherits the advantages of multiresolution analysis of the WT,but also has thecapability of signal representations in the fractional domain which is similar to the FRFT.Compared with theexisting FRWT,the novel FRWT can o
5、ffer signal representations in the time-fractional-frequency plane.Besides,it has explicit physical interpretation,low computational complexity and usefulness for practical applications.The validity of the theoretical derivations is demonstrated via simulations.Keywordstime-frequency analysis,wavele
6、t transform,multiresolution analysis,fractional Fourier transform,time-fractional-frquency analysisCitationShi J,Zhang N T,Liu X P.A novel fractional wavelet transform and its applications.Sci China InfSci,2011,54:12701279,doi:10.1007/s11432-011-4320-x1IntroductionThe wavelet transform(WT)has been s
7、hown to be an appropriate tool for time-frequency analysis.It hasbeen applied in many fields of signal processing,including speech,image,communications,radar 1,etc.However,under the extension of research objects and scope,the WT has been discovered to have short-comings.Since each wavelet component
8、is actually a differently scaled bandpass filter in the frequencydomain,the signal analysis capability of the WT is limited in the time-frequency plane and,therefore,theWT is inefficient for processing signals whose energy is not well concentrated in the frequency domain.For example,chirp-like signa
9、ls 2,which are ubiquitous in nature and man-made systems,are this kindof signals 3.So a series of novel signal processing tools have been proposed to analyze such signals,such as:the fractional wavelet transform(FRWT)47,the fractional Fourier transform(FRFT)8,theshort-time FRFT 9,the Radon-Wigner tr
10、ansform 10 and so on.However,the Radon-Wigner transformhas cross-term problem because it is the quadratic time-frequency representation.Although the FRFThas a number of unique properties,it cannot obtain information about local properties of the signal.Inaddition,the drawback of the short-time FRFT
11、is that its time-and fractional-domain resolutions can notCorresponding author(email:)Shi J,et al.Sci China Inf SciJune 2012 Vol.55 No.61271simultaneously be arbitrarily high.As a generalization of the WT,the FRWT combines the advantagesof the WT and the FRFT,i.e.,it is a linear transformation witho
12、ut cross-term interference and is capableof providing multiresolution analysis and representing signals in the fractional domain.Thus,the FRWTmay be potentially useful in the signal processing community and will attract more and more attention.In 1997,Mendlovic et al.4 first introduced the FRWT as a
13、 way to deal with optical signals.Theidea behind this transform is deriving the fractional spectrum of the signal by using the FRFT andperforming the WT of the fractional spectrum.Since the FRFT tells us the fractional frequencies thatlasts for the total duration of the signal rather than for a part
14、icular time,the fractional spectrum of thesignal cannot be ascertained when those fractional frequencies exist.Therefore,the FRWT 4 fails inobtaining information about local properties of the signal and,thus,it has been applied only for imageentropy 5,6 so far.Shortly after,in 1998,Huang and Suter 7
15、 proposed the concept of the fractionalwave packet transform(FRWPT).Unfortunately,the FRWPT did not receive much attention for thelack of physical interpretation and high computational complexity.In some works,the WT based on thefractional B-splines is also called as the FRWT 11.Since this transform
16、 is actually a scaled bandpassfilter in the frequency domain and cannot represent signals in the fractional domain,it still belongs tothe traditional WT.The aim of this paper is to introduce a novel fractional wavelet transform(NFRWT)to circumvent the limitations of the WT and the FRFT.The proposed
17、transform has explicit physicalinterpretation,i.e.,each fractional wavelet component is actually a differently scaled bandpass filter inthe fractional domain,and it has much lower computational complexity than those introduced in 4,7.The rest of the paper is organized as follows.In the next section,
18、definitions of the WT and the FRFTare briefly introduced and the fractional convolution theorem of the FRFT is given.In Section 3,the novelfractional wavelet transform(NFRWT)is proposed.Moreover,basic properties,the inverse formula,andthe admissibility condition of the NFRWT are also derived.In Sect
19、ion 4,the time-fractional-frequencyanalysis of the NFRWT is discussed.Some applications of the NFRWT are examined and simulationsare given in Section 5.Conclusions appear at the end of the paper.2Preliminaries2.1Wavelet transformThe classical convolution of two time domain functions f(t)and g(t)is d
20、efined asf(t)g(t)=?+f()g(t )d=?f(),g(t )?,(1)where andin the subscript denote the classical convolution operator and the complex conjugate,respectively,and?,?indicates the inner product.Furthermore,the classical convolution theorem of theFourier transform(FT)is given byf(t)g(t)F2F()G(),(2)where F de
21、notes the FT operator.F()and G()stands for the FT of f(t)and g(t),respectively.The wavelet transform(WT)1 of a signal x(t)L2(R)can be defined as a classical convolution,i.e.,Wx(a,b)=x(t)?a12(t/a)?=?x(),a,b()?,(3)where the kernel a,b(t)is a continuous affine transformation of the mother wavelet(t),i.
22、e.,a,b(t)=1a?t ba?,(4)where a R+and b R are respectively scaling and translation parameters.It follows from the classicalconvolution theorem and the inverse FT that the WT of the signal x(t)can be also expressed asWx(a,b)=?+2aX()(a)ejbd,(5)1272Shi J,et al.Sci China Inf SciJune 2012 Vol.55 No.6where
23、X()and()denote the FT of x(t)and(t),respectively.Since(0)=?+(t)dt=0,the WT is actually a bandpass filter in the frequency domian.Thus,signal analysis associated with it islimited to the time-frequency plane.As for signals whose energy is not well concentrated in the frequencydomain,results obtained
24、using the WT will be not optimal.2.2Fractional Fourier transform and fractional convolution theoremThe fractional Fourier transform(FRFT)of a signal x(t)L2(R)is defined as 8X(u)=Fx(u)=?+x(t)K(u,t)dt(6)where Fdenotes the FRFT operator and the kernel K(u,t)is given byK(u,t)=?1jcot2e(j/2)(u2+t2)cotjutc
25、sc,?=k,(t u),=2k,(t+u),=2k+1,(7)where denotes the order of the FRFT,=/2 is the rotation angle.The u axis is regarded as thefractional domain,and the variable u is the fractional frequency.The inverse FRFT is the FRFT withorder.Whenever=1,the FRFT reduces to the FT.The FRFT is different from the FT b
26、ecause of the unique properties,and it has been widely appliedin signal processing community in recent years 3,8,1216.However,the FRFT tells us the fractionalfrequencies that exist across the whole duration of the signal but not the fractional frequencies whichexist only at a particular time.This me
27、ans that the FRFT is a global transformation so that it fails inobtaining any local information of the signal,which are essential and pivotal for processing non-stationarysignals 10.The fractional convolution of the FRFT for functions x(t)and h(t)can be defined as 12x(t)h(t)=e(j/2)t2cot?x(t)e(j/2)t2
28、cot?h(t)?=e(j/2)t2cot?x()e(j/2)()2cot,h(t )?,(8)where denotes the fractional convolution operator.Let X(u)indicate the FRFT of x(t),andH(ucsc)be the FT(with its argument scaled by csc)of h(t).Then,the fractional convolutiontheorem of the FRFT is given byx(t)h(t)F2X(u)H(ucsc),(9)which demonstrates th
29、at a time-domain fractional convolution is equivalent to a simple fractional-domainmultiplication.Particularly,when=1,(8)reduces to the classical convolution as given by(1).3Novel fractional wavelet transform3.1Definition of novel fractional wavelet transformMotivated by the relationship between the
30、 WT and the classical convolution in(3),one would naturallyexpect that there exists a similar relationship between the FRWT and the fractional convolution.Thus,we define a novel fractional wavelet transform(NFRWT)with an order of a square integrable signalx(t)asWx(a,b)=x(t)?a12(t/a)?=e(j/2)b2cot?x()
31、e(j/2)()2cot,a,b()?=?+x(t),a,b(t)dt,(10)Shi J,et al.Sci China Inf SciJune 2012 Vol.55 No.61273where the kernel,a,b(t)satisfies,a,b(t)=e(j/2)(t2b2)cota,b(t),(11)where a,b(t)is expressed in(4).Note that when=1,the NFRWT coincides with the WT.It follows from(9)and the inverse FRFT that the NFRWT can be
32、 expressed in terms of the FRFTX(u)of the signal x(t)Wx(a,b)=?+2aX(u)(aucsc)K(u,b)du,(12)where(ucsc)denotes the FT(with its argument scaled by csc)of(t).(12)states that each fractionalwavelet component is actually a differently scaled bandpass filter in the fractional domain.This meansthat the NFRWT
33、 can overcome the weakness of the WT whose analysis is limited to the time-frequencyplane and circumvent the above mentioned drawback of the FRFT,as will be elaborated in detail insection 4.Further,the definition of the NFRWT in(10)can be rewritten asWx(a,b)=e(j/2)b2cot?+?x(t)e(j/2)t2cot?a,b(t)dt.(1
34、3)Evidently,the computation of the NFRWT corresponds to the following steps:1)a product by a chirp signal,i.e.,x(t)x(t)=x(t)e(j/2)t2cot.2)a traditional WT,i.e.,x(t)W x(a,b).3)another product by a chirp signal,i.e.,W x(a,b)Wx(a,b)=W x(a,b)e(j/2)b2cot.Since the input signal(e.g.,a digital signal)is pr
35、ocessed by a digital computing machine,it is prudentto define the discrete version of each step.First,multiply the function x(t)with a chirp signal e(j/2)t2cot,and sample the multiplied signal x(t).Next,discretize the scaling and translation parameters a and b.Themost popular approach of discretizin
36、g a and b is using a=2m,b=n2mwhere m and n are integers.Then,perform a discrete WT on the samples of x(t).Finally,after multiplying another chirp signal e(j/2)b2cot,the NFRWT of the signal x(t)is obtained.It follows that the computational complexity of the NFRWTdepends on that of the WT which has O(
37、N)(N is the length of the sequence)implementation time.So,the computational complexity of the NFRWT is O(N).In contrast,the complexity of computation for theFRWT introduced in 4 and 7 is O(N3log2N)and O(N+N log2N),respectively.Thus,the NFRWTis easy to implement for practical applications.Figure 1 sh
38、ows the comparison of the computationalcomplexity of the NFRWT and the FRWT introduced in 4,7.Moreover,the computation of the FRWT4,7 needs the discrete algorithm of the FRFT,and there still exists no satisfactory discrete algorithmfor the FRFT so far 13.The NFRWT can be implemented by the discrete
39、algorithm of the WT anddoes not need the discrete algorithm of the FRFT.3.2Basic properties of novel fractional wavelet transfrom3.2.1Linearity propertyIf x(t)=k1x1(t)+k2x2(t),and x1(t)Wx1(a,b),x2(t)Wx2(a,b),then it is easy to verify thatWx(a,b)=k1Wx1(a,b)+k2Wx2(a,b).(14)3.2.2Scaling propertyIf x(t)
40、Wx(a,b),then?+x(ct),a,b(t)dt=?+x(ct)e(j/2)(t2b2)cot1a?t ba?dt=1c?+x(t?)e(j/2)(t?2(bc)2)cot1ac?t?bcac?dt?=1cWx(ac,bc),(15)where c R+,=arccot(c2cot),and =/(/2).1274Shi J,et al.Sci China Inf SciJune 2012 Vol.55 No.6Figure 1Comparison of the computation complexity of the NFRWT and the existing FRWT.3.2.
41、3Inner product theoremTheorem 1.Let Wx(a,b)and Wg(a,b)denote the th NFRWT of x(t),g(t)L2(R),and let()denote the FT of(t).If()satisfiesC=?+|()|2d .(16)Then,?+Wx(a,b)?Wg(a,b)?daa2db=2C?x(),g()?.(17)Proof.It follows from(12)thatWx(a,b)=?+2aX(u)(aucsc)K(u,b)du,(18)Wg(a,b)=?+2aG(u?)(au?csc)K(u?,b)du?.(19
42、)Next,inserting(18)and(19)into(17)results in?+Wx(a,b)?Wg(a,b)?daa2db=2?+|(aucsc)|2ada?+X(u)G(u)du.(20)Then,by utilizing the inner product theorem of the FRFT 8 and(20),(17)can be established.Thiscompletes the proof of the Theorem 1.3.2.4Parsevals relationTheorem 2.For any function x(t)L2(R),let Wx(a
43、,b)denote the NFRWT of x(t).Then,?+|x(t)|2dt=12C?+|Wx(a,b)|2daa2db,(21)where one can define an energy density called a fractional scalogram,denoted by|Wx(a,b)|2.The proofof Theorem 2 can be easily deduced by setting g(t)=x(t)in Theorem 1 and is omitted.Theorem 2 states that the th fractional scalogr
44、am shows how the energy of the signal is distributedin the th time-scale plane.This is a generalization of the fact that the scalogram of the WT measuresthe energy of the signal in the ordinary time-scale plane.Shi J,et al.Sci China Inf SciJune 2012 Vol.55 No.612753.3Inversion formula and admissibil
45、ity conditionTheorem 3.Let()denote the FT of(t).If x(t),(t)L2(R),and()satisfies(16),then theinversion formula of the NFRWT is given byx(t)=12C?+Wx(a,b),a,b(t)daa2db.(22)The proof of Theorem 3 can be easily derived by setting g(t)=(t)in Theorem 1 and is omitted here.Both Theorem 3 and Theorem 2 hold
46、for C which is called the admissibility condition for theNFRWT.It follows from the condition that(0)=0,and therefore,the NFRWT is intrinsically a bankof multiscale bandpass filters in the fractional domain.4Time-fractional-frequency analysis4.1Constant-Q propertyAccording to(10)and(12),if the kernel
47、 of the NFRWT,a,b(t)is supported in the time domain,thenthe inner product of x(t)and,a,b(t)can ensure that Wx(a,b)is supported in the time domain too.Therefore,Wx(a,b)contains information about x(t)near b.Similarly,if(ucsc)is bandpass,i.e.,()satisfies the admissibility condition in(16),then,the mult
48、iplication of X(u)and(aucsc)canprovide fractional-domain local properties of x(t).This implies that the NFRWT is capable of providingthe time-and fractional-domain information simultaneously,hence giving a time-fractional-frequencyrepresentation of the signal.To be specific,both(t)and its FT()must h
49、ave sufficiently fast decayso that they can be used as window functions.Suppose that(t)and()are functions with finitecenters Eand Eand finite radii and.Then,the center and radii of the time-domain windowfunction,a,b(t)of the NFRWT are respectively given byE,a,b(t)=?+t|,a,b(t)|2dt?+|,a,b(t)|2dt=?+t|a
50、,b(t)|2dt?+|a,b(t)|2dt=E a,b(t)=b+aE,(23),a,b(t)=?+(t b aE)2|,a,b(t)|2dt?+|,a,b(t)|2dt?12=?+(t b aE)2|a,b(t)|2dt?+|a,b(t)|2dt?12=a,b(t)=a,(24)where E and denote the expectation and deviation operator,respectively.Similarly,the centerand radii of the frequency-domain window function(a)of the WT can b