分数域信号与信息处理及其应用 (7).pdf

IEEE SIGNAL PROCESSING LETTERS,VOL.4,NO.1,JANUARY 199715Product and Convolution Theoremsfor the Fractional Fourier TransformLu s B.Almeida,Associate Member,IEEEAbstractThe fractional Fourier transform(FRFT)is a gener-alization of the classical Fourier transform(FT).It has recentlyfound applications in several areas,including signal processingand optics.Many properties of this transform are already known,but an extension of the FTs convolution theorem is still missing.The purpose of this paper is to introduce extensions of thistheorem,dealing with the FRFT of a product and of a convolutionof two functions.I.INTRODUCTIONTHE fractional Fourier transform(FRFT)was introducedin the 1920s 1,2 but remained largely unknown,andwas reinvented several times 311.Recently,it has foundapplications in various fields including signal processing andoptics.Many properties of the FRFT are currently known.However,extensions of the convolution theorem for the frac-tional Fourier transform are still unknown.It is the purpose ofthis paper to introduce two such extensions,dealing with thetransforms of a product and of a convolution of two functions.The paper is organized as follows.In Section II we make avery brief introduction to the FRFT to situate the problem andto introduce the notation.In Section III we derive expressionsfor the transform of a product of two functions.In SectionIV we derive expressions for the transform of a convolution.In Section V we extend these results to a more general form.Section VI concludes.II.THEFRACTIONALFOURIERTRANSFORMThe fractional Fourier transform is defined inas(1),shown at the bottom of the next page,whereis a parameter;is the imaginary unit.The FRFT can be interpreted as a rotationof the timefrequency plane by an angle79,and hassome notable properties,among which are the following:The FRFT withcoincides with the Fouriertransform(FT)1.The FRFT withis an identity.Two successive FRFTs with anglesandare equiva-lent to a single FRFT with an angle The inverse of an FRFT with an angleis the FRFTwith angleManuscript received November 29,1995.The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof.D.L.Jones.The author is with the Instituto Superior T ecnico and INESC,Lisboa,Portugal(e-mail:luis.almeidainesc.pt).Publisher Item Identifier S 1070-9908(97)01273-X.1Some authors use as parameter?instead of?so that the FT isobtained for?From the definition above,we see that the FRFT exists,fornot multiple ofwhenever the Fourier transform ofexists.Since the complex exponential in thisexpression has constant magnitude,the FRFT can also bedefined in most domains in which the FT can be defined,notably inin the Wiener algebra(the set of Fouriertransforms of functions inand in the set of tempereddistributions.In this paper,we denote by the square root symbolthesquare root that has an argument inand we definethe Fourier transform asIn what follows,we shall only consider functions in(i.e.functions inwith Fourier transforms also infor simplicity.The results that we obtain could beextended to wider domains,but the treatment of such caseswould need a longer paper.The derivations that follow are notvalid when the transform angles take certain values that aremultiples ofThese cases are explicitly noted in the end ofthe derivations only,for simplicity.The correct results for suchcases can be easily obtained by the reader,if desired,sincethey correspond to known situations;in those cases the FRFTis simply a Fourier transform,a time reversal,a combinationof both or an identity.III.THETRANSFORM OF APRODUCTLet us consider two functions,and makeThe functionis inand thus its FRFT is given by(1),shown at the botom of the next page.Let us computeexpressingin terms of its FRFT,as follows:10709908/97$10.00 1997 IEEE16IEEE SIGNAL PROCESSING LETTERS,VOL.4,NO.1,JANUARY 1997or finally,(2)whereis the FT ofThe latter equation gives the resultwe want:the FRFT of the product ofandis obtainedby multiplying the FRFT ofby a chirp,convolving with the(scaled)FT ofand multiplying again by a chirp and by ascale factor.Other useful forms of this result can be obtained throughchanges of the integration variable.We first make the changeresulting in(3)We then make the further changeresulting in(4)Equations(2)(4)are valid ifis not a multiple ofOfcourse,the roles ofandcan be interchanged.In SectionV we will see how to extend these results to a more general,albeit more complex,form.IV.THETRANSFORM OF ACONVOLUTIONLet us again take two functionsandboth inTheir convolutionis inthuswhereandare inWe know thatis theFRFT ofwith angleWe can therefore use(2)to obtain(5)which is our first expression for the transform of a convolution.The FRFT of a convolution can therefore be obtained bytaking the FRFT of one of the signals,multiplying by a chirp,convolving with a scaled version of the other signal,andmultiplying again by a chirp and by a scale factor.This expression can take some other forms,which may befound useful.The first is obtained by making the change ofvariableresulting in(6)For the second one,we make the further change of variableobtaining(7)Equations(5)(7)are valid ifis not a multiple ofV.MOREGENERALFORMSTo obtain a more general expression for the FRFT of aproduct,we make againassuming once more thatand we choosetwo anglesandsuch thatThenThe integral on the right hand side of this equation istimes the Fourier transform,with argumentof the product ofandThereforeHoweverifis not a multiple ofifif(1)ALMEIDA:FRACTIONAL FOURIER TRANSFORM17andcan be put in a similar form.Thereforeor finally(8)This result is valid ifandare not multiples ofTwospecial cases are notable.The caseresults inwhich is expression(2)with the change of variableThe caseresults in(4)withandinterchanged.The expressions of the transform of a convolution can alsobe generalized in the same way.Take againwithand follow the same path as in SectionIV,but using(8)instead of(2).The result is(9)The anglesandmust now be related byThis result is valid ifandare not multiples ofLike(8),(9)also has two notable special cases.Withwe obtainwhich is(5)with the change of variableWithwe obtain(7)withandinterchanged.VI.CONCLUSIONWe have introduced expressions for the FRFTs of a productand of a convolution of two functions.These expressionsare extensions of the convolution theorem of the FT to thefractional domain.ACKNOWLEDGMENTWe wish to acknowledge useful discussions with A.F.Santos regarding some validity aspects of the derivations.REFERENCES1 H.Weyl,“Quantenmechanik und gruppentheorie,”Ztsch.f.Physik,vol.46,pp.147,1927.2 N.Wiener,“Hermitian polynomials and Fourier analysis,”J.Math.Phys.MIT,vol.8,pp.7073,1929.3 E.U.Condon,“Immersion of the Fourier transfom in a continuousgroup of functional transformations,”in Proc.Nat.Acad.Sci.USA,vol.23,pp.158164,1937.4 A.L.Patterson,“FunctionspacesbetweencrystalspaceandFouriertransform space,”Z.Krist.,vol.112,pp.2232,1959.5 V.Bargmann,“On a Hilbert space of analytic functions and an asso-ciated integral transform,part I,”Commun.Pure Appl.Math.,vol.14,pp.187214,1961.6 V.Namias,“The fractional Fourier transform and its application toquantum mechanics,”J.Inst.Math.Appl.,vol.25,pp.241265,1980.7 L.B.Almeida,“An introduction to the fractional Fourier transform,”inProc.1993 IEEE Int.Conf.Acoust.,Speech,Signal Processing,vol.III,Apr.1993,pp.257260.8,“The fractional Fourier transform and time-frequency repre-sentations,”IEEE Trans.Signal Processing,vol.42,pp.30843091,1994.9 A.W.Lohmann,“Image rotation,Wigner rotation,and the fractionalFourier transform,”J.Opt.Soc.Amer.,vol.10,pp.21812186,1993.10 D.Mendlovic and H.M.Ozatkas,“Fractional Fourier transforms andtheir optical implementation,”J.Opt.Soc.Amer.,vol.A 10,pp.18751881,25222531,1993.11 O.Seger,“Model building and restoration with applications in confocalmicroscopy,”Ph.D.dissertation,Link oping Univ.,Sweden,1993.