Fractional Fourier Transform and Application

8/2/2019 Fractional Fourier Transform and Application

1/17

The Fractional Fourier Transform and Applications

Author(s): David H. Bailey and Paul N. SwarztrauberSource: SIAM Review, Vol. 33, No. 3 (Sep., 1991), pp. 389-404Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031440 .

Accessed: 14/04/2011 20:07

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at .http://www.jstor.org/action/showPublisher?publisherCode=siam. .

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend

access to SIAM Review.

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=siamhttp://www.jstor.org/stable/2031440?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=siamhttp://www.jstor.org/action/showPublisher?publisherCode=siamhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2031440?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=siam


2/17

SIAM REVIEW ?) 1991 Society for Industrial and Applied MathematicsVol. 33, No. 3, pp. 389-404, September 1991 002

THE FRACTIONAL FOURIER TRANSFORM AND APPLICATIONS*DAVID H. BAILEYt AND PAUL N. SWARZTRAUBERtAbstract. This paper describes the "fractionalFourier transform,"which admits computationby an algorithm hat has complexityproportional o the fast Fourier transform lgorithm. Whereasthe discrete Fourier transform DFT) is based on integral roots of unity e-27i/n, the fractionalFourier transform s based on fractionalroots of unity -27ia, where a is arbitrary.The fractionalFouriertransform nd the corresponding astalgorithm re useful for uch applications as computingDFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequenceswithnoninteger eriodicities,performing igh-resolution rigonometricnterpolation, etecting inesin noisy mages, and detecting ignals with inearlydrifting requencies. n many cases, the resultingalgorithms re fasterby arbitrarily arge factors han conventional echniques.

Key words. discrete Fourier transform, astFourier transformAMS(MOS) subject classifications. 65T20, 65T10, 65Y201. Introduction. The conventional astFourier ransformFFT) algorithmswidely sed tocomputediscrete ourier ransformsDFTs) and discrete onvolutions,and to perform rigonometricnterpolation.However, n some applicationsof theFFT, either he input is only partiallynonzero,or only part of the DFT result srequired,rboth. Nonetheless, heFFT algorithms ordinarily sed unless he desiredresults an be moreefficientlyomputeddirectly rom he definition f theDFT. Wepresenthere a technique hat permitsmanyof these applicationsto be computed

moreefficiently.his same technique an also be appliedin other ituations hat donotadmit efficientolutionusing tandardFFTs.The centralconcepthere is a generalization f the DFT that is termedthefractional ourier ransformFRFT). It is defined n them-long omplexsequencex = (xj, 0 < j < m) asm-1(1) Gk(X, aJ) = 3 e-e27rijkej=O

The parameter a will not be restricted o rationalnumbers nd in factmaybeany complexnumber.Although his transforms defined or ll integers , we willusually compute he firstm nonnegative alues, .e., for < k < m. Straightforwardevaluation f hesem valuesusing 1) requires m2floating ointoperations, ssumingtheexponential actors ave beenprecomputed.Note that the ordinaryDFT and its inverse re special cases of the fractionalFourier ransform:m-1

(2) Fk(X) = 3 e-e27rijk/mj=O

* Received by the editors May 10, 1990; accepted forpublication (in revisedform) October 30,1990.t Numerical AerodynamicSimulation (NAS) SystemsDivision at NASA Ames Research Center,Moffett ield, California94035.t National Center forAtmosphericResearch, Boulder,Colorado 80307, which s sponsored by theNational Science Foundation. This work was completedwhile the author was visitingthe ResearchInstitute for Advanced Computer Science (RIACS) at NASA Ames, and was fundedby the NASSystems Division via Cooperative AgreementNCC 2-387 between NASA and the Universities paceResearch Association389


3/17

390 DAVID H. BAILEY AND PAULN. SWARTZTRAUBER(3) = Gk(x, 1/m), 0 < k


4/17

FRACTIONAL FOURIER TRANSFORM 391m-1(12) = e 7rik%y2 3 jZk-jj=O

where hem-long equencesy and z are defined y(13) yj = Xje-ii2e7rij 0C(14) zj = e

Since the summation 12) is in the form f a discrete onvolution,t suggestsevaluation using the well-known FT-based procedure. However, he usual DFTmethod evaluatescircular onvolutions,wherein k-j = Zk-j+m. This condition snot satisfied ere,but instead k-j = Zj-k when k j < 0. Fortunately,t is possibleto convert his ummation nto formhat s a circular onvolution. uppose we wishto computethe the FRFT for0 < k < m. Firstselect an integer > m - 1, andextendthe sequencesy and z to length p as follows:(15) Yj = 0, m < j < 2p,(16) z; = 0, m< j < 2p-m,(17) zi=eri(j-2p)2 2p-m


5/17

392 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBERWe will assumethat 5q og2 floating oint operations re required or q-pointFFT. This is the costofthe commonly sedradix-2 omplexFFT. Ifq is not a powerof 2, the cost is somewhathigher, ependingon the factors fq. If m is a powerof 2, the obviouschoiceforp is p = m. Then the total computational ost oftheabove algorithms 20mlog2m + 44m floating oint operations, ssumingthat theexponential actorsn (19) and theFFT of the z sequencehave been precomputed.Note thatthis cost is approximatelyour imesthe cost ofthe usual m-point FT.The efficiencyf theabove algorithm ependson theefficiencyf theunderlyingcircular onvolution lgorithm sed to evaluate (18). A variety f fastconvolutionalgorithms xist in the literature see [2] and [11]), and one of these maybe moreattractive han usingFFTs, depending n the computerhardware nd the applica-tion. Furthermore,FTs overfields ther hanthecomplexnumbers see [11])maybe profitablymployed n some instances. However, or implicityn assessingthe

computational ostsofalgorithms, e willassumethat ordinary omplexFFTs areemployed. We will also assume when assessingcosts that the sizes of DFTs andFRFTs arepowers f2, although healgorithmsobe discussed re validfor nysize.Hereafternlythedominant ermofthesecomputational osts willbe listed, incelower-ordererms re generally vershadowed y implementationetails,especiallyfor argesizes.3. Computing longer and shorter segments of results. Formula 19) givesthe firstm valuesof thefractional ourier ransform.Multiple egments f m valuescan be computedeven more efficiently.or example,supposewe wish to computethe first resultsof an FRFT, wherer = qm. These can be efficientlyomputedby repeating ormulas19)-(22) for = 0, m, 2m,* *, q - 1)m. In thiscase it isreasonable o assumethatfor ach value of s, theDFT oftheappropriate sequencehas beenprecomputed. ignificantly,heDFT of hey sequenceneedsto be computedonlyonce. Thus eachsegment fm results an be computedbymultiplyingheDFTofthey sequencebythe precomputed FT ofthe appropriate sequence,performingan inverse p-point FT on the result, nd multiplyingy exponentials s in (19).The dominant ostof computing results n thismanner s lOr og2m floating ointoperations whenr is largecompared o m).The discussion n the previousparagraphassumedthat we are computing hefirst values of the FRFT. However, nyset ofr results hatis theunionofm-longcontiguous egments an be computed n thismanner, nd the computational ostis the same. It is not evennecessary o assume that the startingndicess of thesegments remultiples fm.Segments f theFRFT shorter hanm in length an be efficientlyomputedbyusinga decimation cheme imilar o thatused for omputing short egment f aDFT. Sincethe fast lgorithmor short egment fa DFT is not widelyknown, twill be presented irst.Suppose thatwe wish to computethe firstm values of the DFT of the n-longsequencex, wheren = qm. We can write

m-1 q-1(23) Fk(X) = E xj+lqe-27i(j+lq)k/n1=0 j=0q-1 m-1(24) E e-27rijk/n Xj+lqe-27rilk/mj=0 1=0


6/17

FRACTIONAL FOURIER TRANSFORM 393q-1(25) Ze2Xiik/TFk(yi), 0 < k < r,j=O

whereyj denotesthe m-long equence Xj+lq, 0 < 1< mi). This meansthat thefirstm valuesofthen-pointDFT ofx can be computedbywritingheinput as a q x mmatrix n columnmajor order,performing -pointFFTs on each row,multiplyingthe resultsbycertain xponentials, nd thensumming ach column. The dominantcomputational ostof thisprocedures 5n og2m floating oint operations,which sless than thatof omputing full-sized-point FT. Unfortunately,his avings snotsignificantn the mportantpecialcasem = n/2,a casewe have already ncountered:recallthatonlythefirst alfof theDFT resultsn (19) are required.Formula 25) maybe easilygeneralizedo computemvaluesoftheDFT startingat any indexs. We thenobtainq-1(26) Fk?8(x) = Ze-2-ii(k+s)/n Fk(Zj), 0 < k < m,j=0

where j (xj+1qe-2-il1/m, 0 < 1< m).The schemedescribedby (25) can be seen as an abbreviatedform fan FFTalgorithmometimes alledthe "four tepFFT" (see [1,p. 150],[4],[9, p. 569],and[17,pp. 202-203]). This FFT algorithm as received enewednterestately due toits suitability or ome advancedcomputer rchitectures,articularlyystemswithhierarchicalmemories. ormula 25) is also related o what s known s the "discreteZak transform"3].This same generalmethodcan be applied to compute n m-long egment fann-pointFRFT:

m-1 q-1(27) Gk(X,E ) = E E Xj+1qe-27i(j+1q)kce1=0 j=0q-1 m-1(28) = E e-27ijkc E Xj+le -27rilqkej=0 1=0q-1(29) = Ee-27iikeGk(yj,qc), O k


7/17

394 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBERwherezj = (xj+lqe-27ilqsa, 0 < 1 m). Note thattheexponential actorsn thesezj vectors an be incorporatedntothe factorsn (20) and (22) and thus do not needto be counted. Hence the dominant ost ofthis scheme s 20nlog2m floating ointoperations, he same as above.

4. Computing DFTs of sequences with prime lengths. It is popularlybelieved hat DFTs can onlybe evaluatedbya fast lgorithmwhenthe sizem ofthetransforms a highly actorablenteger,uchas a powerof 2. But since theDFT ismerely special case of theFRFT, and since thesize 2p ofFFTs used in evaluatingtheFRFTs maybe chosento be anyconvenient alue (suchas a powerof2) thatisat least 2m 2, it follows hat the fastfractional ourier ransformlgorithman beused to computem-pointDFTs even whenm is a prime.The computational ost isonly bout four imes hecostofa power f2 FFT ofcomparable ize. This algorithmwas first resentednBluestein'spaper [6].It should be emphasizedthat Bluestein'salgorithm ermitsDFTs of any size(not just primes) to be computed usingpowerof 2 FFTs, which are generally hemost efficientFTs availableon computer ystem ibraries.Thus, depending n thecomputer ystem nd implementation, luestein's algorithmmaybe more efficientforcomputingDFTs of certainsizes than conventional FTs. On some computersystems, uch as hypercubes, luestein's lgorithms preferableor omputing FTsofanysize that is not a powerof2 [15].Perhaps one reasonthatBluestein's lgorithm as not receivedmore ttentionsthat anotherFFT algorithm orprimem was discovered t about the same timebyRader [13]. This algorithmeduces DFT ofprime ize m to a circular onvolution

of size m - 1, which n theory s onlyabout half as costly s Bluestein'salgorithm.UnlikeBluestein'salgorithm, owever, ader's algorithm oes notgeneralize o thecase ofarbitrary a, nd there s littleflexibilitynthepermissibleizes ofFFTs thatmaybe used for fficientvaluation.There are a number fothermportantpplications fthe basictechnique ehindBluestein'salgorithm;we have termedthis techniquethe FRFT. Several of theseapplicationswill be presented ere, ncluding ome that wereoriginally resented n[12].5. Computing DFTs of sparse sequences. In thissectionwe willdescribeefficientlgorithms or omputing FTs ofsequences with argenumbers f zeroel-ements. DFTs ofsuchsequencesarise n trigonometricnterpolation,s we shall seein ?7. Another ommon pplication s indigitalfiltering, herehigh-orderarmonicsare often eroedbeforeperformingn inverse FT. A third nstance s in the com-putationof the "linear"or "aperiodic"convolution,which s defined n the n-longsequencesx andy as the2n-long equence = (Zj=0 XjYk-j, 0 < k < 2n-1), wherethesubscriptk - j is restricted o therange0 < k - j < n - 1. Linear convolutionsarisein applications s diverse s multiprecisionrithmetic nd thenumerical olu-tionofpartialdifferentialquations. This typeofconvolution s normally valuatedby paddingthe n-long nput sequencesx and y withzeros to length2n, and thencomputingheir2n-point ircular onvolution singFFTs.Three casesof parseDFTs willbe considered. irstwe willcompute hecompleteDFT of an input equencewhere nlyan initial egment s possiblynonzero.Second,we willcompute segments fsuch a DFT. Third, we will compute egments ftheDFT ofan input sequencethathas onlya fewnonzero egments.Let n = qm. Supposewe wishto compute heDFT ofan n-long equencex thatis known o be zeroexceptfor he firstm entries.This can be done by employing


8/17

FRACTIONAL FOURIER TRANSFORM 395conventional FT-based decimation cheme, s follows:

m-1(31) Fk+lq (X) = E X,e-27ij(k+lq)/nj=Om-1(32) = E xie-2rijk/n e-2rijl/mj=O(33) = Fl(yk), 0


9/17

396 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBERq-1(38) = ye-2xi(k+s)11qGk+ (Yj, 1/n), 0 < k< ml1=0

whereYi = (xj+lm, 0 < j < m). Withtheassumption hatonlya fewm-long nputsegments re zero, t followshatmostofthe terms f thissummation re zero. Thusonlya fewof theseFRFTs need to be actuallycomputed.We can assume thattheexponential actors n (38) have been incorporatedntothe exponential actors f formula19). Supposethatonlyt ofthe nput egments 1containpossiblynonzero lements.Then the total computational ost ofcomputingan m-long egment ftheDFT is on the order f20tmog2m operations.This is lessthanthe costofcomputing segment f a DFT usingtheconventional cheme 33)whenever < q/4 (for argen).As indicated n ?3,multiple egments fresults an be evenmore fficientlyom-puted by repeating 38) for everalvalues of s. The dominant ost ofcomputingvaluesoftheDFT in thisway s lOtrlog2m floating ointoperations whenr is largecompared om). This is less thanconventionalmethodswheneverr< n/2 for argen). Thus this method s not appropriate nlessonlya few egments ftheinputarepossiblynonzero nd onlya few egments fthe result re desired.

6. Analyzing sequences with non-integer periodic components. Inmanypractical pplications f DFTs and FFTs, notably nsignalprocessing,t is not reallytruethat theunderlyingime eriesbeingstudied s exactlyperiodicwithperiod m,wherem is the size of the data sequence. Instead,thesampling nterval nd samplesize are usually selectedarbitrarilyor the convenience fhardware nd software.Thus in suchapplicationstis the rulerather han theexception hata data sequencecontains omponentswithperiods hat are notexactlywhole numbers.One instanceofsuch an application s in computing olarpositions.Daily solaraltitude and declination nglesare readily vailablefrom lmanacs. Such positionsare notperiodicwith a periodof365 days,but theyare verynearlyperiodicwith afractional eriodof pproximately65.2422days. Inotherwords, heapparentmotionof the sun has a fairly impleFourierrepresentationased on a period of365.2422days, but the representations not simplebased on a periodof 365 days. In thisexamplewe knowfrom priori nformation hat thetrueperiod s. But in generalwe usuallydo noteven knowthat. Thus we seektechniques hatcan determine nunknown eriod ndcompute hespectrum asedonthe trueperiodonce tisknown.Let x be them-long equence e27iij3/m, 0 < j < m) where/ is unknown ndnot an integer.Then the DFT ofx is

m-1(39) Fk(X) = E e27iji/me-2rijk,,'mj=0m-1(40) = E e2rij(,3-k)/mj=O

_e27ri(,3-k)(41)~~~~41) 1 - e27i(f-k)/m(42) e7i(3k)(ml)/m sin[7r(o k)] 0 < k < m.sin[r(/3 k)/m]' -

Thus thespectrumf ucha sequence snot a "spike" t a single ndexbut nsteadisspreadover ll indices,withmagnitudeshat decline apidlywith ncreasing istance


10/17

FRACTIONAL FOURIER TRANSFORM 397from3. It is fairly asy to determine he unknown requency to within ne unitby examining hesemagnitudes. Suppose that this has been done and that b is thegreatest nteger ess than 3.One way to determine3more ccurately s to apply the following ormula, hichwas brought o theauthors'attention y Oscar Buneman of StanfordUniversity:

m _ sin(,7r/m)(43) / = b+ - tan- cosQr/m) in(m)7r cos7r/m)+ Fb(X)I/I1Fb+1(X)The derivation f thisformula s straightforwardrom 42). It is exact for puresinusoid ignalwithfrequency3n therangeb < 3 < b+ 1. In thepresence fnoiseor higher-orderarmonics,ts accuracy degrades,but it is nonetheless uite usableformany pplications.An alternatemethod,which s somewhatmorerobust n the presence f noise, s

based on the fractional ourier ransform.n particular, he FRFT can be used toefficientlyompute high-resolutionpectrum n the nterval b, + 1]. The m spectravalues starting t index b and increasingwith nterval are givenbym-1(44) Fb k6(X) = xj -2.rij(b+k6)/mj=Om-1

(45) = S Xje-2wijb/me-27rijkb/mj=O

(46) =Gk(y,6/m), O< k


11/17

398 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBERr- 1(50) = E32riij/me-27rijk/rj=Or-1

(51) = E e-2rij(b-k)/rj=O(52) =r6(k, b)where&(k, ) = 1 ifk = b,but zero otherwise. hus inthis case we havepreciselyhedesiredresult.When mca s not an integer,t is still true that Xb = r, but theoff-peakaluesare not zero. However, heirmagnitude s nonethelessmuch maller hanr. To makethisprecise, et s = r - mc, and note that Isl < 1. Then we can write

r-1(53) |Xk| = Zxje-2rijk/(ma)j=O

(54) | -e2rij(b-k)/(moe)-e27rir(b-k)/(ma)55) 1 e27ri(b-k)/(me)12- 7ris(b-k)/(mce)(56) 1 e-e27ri(b-k)/(me)

(57) _ sin[irs(b k)/(mc)]( sin[7r(b k)/(mc)]When k is close to b,we find hat to a first-orderpproximationXkl is equalto 2sl,which s bounded by 1 and thereforeuite small comparedwithXb = r.When k is large nearr) and b is small,or the reverse, he situation s not quite asfavorable,ince thedenominator f (57) mightbecomesmall. However,we still findthat themagnitudes f the off-peakpectralvaluesXk are bounded by m/(7r3)orm/[ir(m p)], which are small comparedwithr provided hat 3 is neither verysmall nor a very argefractionfm.Ifthe original ignal s notmonochromatic ithfrequency , but insteadconsistsof a componentwithfrequency plus severalharmonics f this frequency,hesehigher-orderarmonicswillbe evident n the adjustedspectrum t k = 2b, 3b, etc.In thiscase an analysisof off-peakmplitudes ivesresults hat are somewhat argerthan above but still quitesmall comparedwithr for ransformsfreasonably argesizes. The formula48) also givesgoodresultswith headdition fmoderateGaussiannoise.7. Performing high-resolution trigonometric interpolation. The FFT is

frequentlysed to performrigonometricnterpolation. orexample,given sequencex in tabular form,we maywishto determinepproximatemidpoint aluesXj+l/2.In theprevious ectionswehaveused the "aliased" form ftheDFT:m-1(58) Fk(X) = j xjeI2iik/m 0K k m.j=O


12/17

FRACTIONAL FOURIER TRANSFORM 399In this sectionwe will use the "unaliased" form f theDFT, which s defined ni = (xj, -m/2 < j < m/2) as

m/2-1(59) Fk(z) = xie-2i jk/r mrm/2


13/17

400 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBER(67) YM/2= 2 -m/2(:)l(68) Yk= ?, -n/2 < k


14/17

FRACTIONAL FOURIER TRANSFORM 401(79) = exii/qGj(Z, 1/n), 0 < j


15/17

402 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBERotherwordswhere heamplitude fthesignal s much ess than theamplitude f thenoise.The usual approachto detecting nd analyzing uch signals s to consider hen-long ime eries s a q x m array nrowmajororder. We firstompute hem-pointDFT ofeach row n this rray i.e.,the DFT of uccessive ontiguousm-long egmentsofy), andthen ompute quaredmagnitudes feachresulting rray ntry see [8]and[10]):(85) Xj,k = IFk(Zj)12, O < j


16/17

FRACTIONAL FOURIER TRANSFORM 403m/2

(89) Xj,k+(a+&r)j EXi1e-2ri3k+(a+6r)j]1/m1=-m/2Thus q-1 m/2

(90) Sr,k = > E xj,1e-27i[k+(a+6r)i]1/mi=0 1=-m/2m/2 q-1

(91) = Exj,3e2riaj1/me21ri6rij/m e-27rikl/m1=-m/2 [j=O

m/2(92) = Srle-2rikl/ml1=-m/2

where(93) Sr,l = Gi zi, br/m),(94) Zl = (Xj 1e-2riaijl/m 0 < j < q).

Hence thissummation lgorithmonsists f performingnverse FTs on rowsoftheX array, ollowed yFRFTs on the resulting olumns with certain xponentialmultipliers),ollowed y forwardFTs on theresultingows.The computational ostof entire etection lgorithms approximatelyOn og2n floating ointoperations, rin otherwordsonlyabout twicethe cost of an n-point omplexFFT. Forcompletedetails, ee [14].

9. Conclusions. We have described number f applications f the FRFT andits associated fastalgorithm.The flexibilitynd efficiencyf these methodspermitsnew solutions o a variety f problems hat nvolve iscrete ourier nalysis. n manycases, theresulting lgorithms re faster hantheconventionalmethodsofsolutionthatutilizeordinary FTs. In fact, or ufficientlyargeproblem izes,thesemethodsare often aster hantheconventional lgorithms yarbitrarilyargefactors.Acknowledgments. The authorswishto thankKent Cullersof NASA AmesResearch Centerand IzidorGertner f the CUNY Graduate Centerfor nformativediscussions n this subject.

REFERENCES[1] R. C. AGARWAL AND J. W. COOLEY, Fourier transform nd convolution ubroutinesfor theIBM 3090 vectorfacility, BM J. Res. Devel., 30 (1986), pp. 145-162.[2] , New algorithms or digital convolution, EEE Trans. Acoust. Speech Signal Process.,25, (1977), pp. 392-410.[3] L. AUSLANDER, I. GERTNER, AND R. TOLIMIERI, The discrete Zak transform: Applications

to time-frequencynalysis and synthesis of non-stationary ignals, IEEE Trans. Acoust.Speech Signal Process., to appear.[4] D. H. BAILEY, FFTs in external or hierarchicalmemory,J.Supercomputing, (1990), pp. 23-35.[5] - , A high-performanceFT algorithm or vector supercomputers,nternat. J. Supercom-puter Appl., 2 (1988), pp. 82-87.[6] L. I. BLUESTEIN, A linearfiltering pproach to the computation of the discreteFourier trans-form, EEE Trans. Audio Electroacoust., 18 (1970), pp. 451-455.


17/17

404 DAVID H. BAILEY AND PAUL N. SWARTZTRAUBER[7] J. W. COOLEY AND J. W. TUKEY, An algorithm ormachine computationof complex Fourierseries,Math. Comp., 19 (1965), pp. 297-301.[8] D. K. CULLERS, I. R. LINSCOTT, AND B. M. OLIVER, Signal processing n SETI, Comm. ACM,28 (1985), pp. 1151-1163.[9] W. M. GENTLEMAN AND G. SANDE, Fast Fourier transforms-forfun and profit, FIPS Proc.,29 (1966), pp. 563-578.[10] I. GERTNER, Optimal detection and separation of chirp signals, Proc. ICASSP-90, 1990, toappear.[11] H. J.NUSSBAUMER, Fast Fourier Transform nd ConvolutionAlgorithms, pringer-Verlag, ewYork, 1981.[12] L. R. RABINER, R. W. SCHAFER, AND C. M. RADER, The chirp -transformlgorithmnd itsapplication,Bell SystemTech. J., 48 (1969), pp. 1249-1292.[13] C. M. RADER, Discrete fourier transformswhen the number of data samples is prime, Proc.IEEE, 56 (1968), pp. 1107-1108.[14] P. N. SWARZTRAUBER AND D. H. BAILEY, Efficientdetection of a continuous wave signal witha linearfrequencydrift,ubmitted forpublication.[15] P. N. SWARZTRAUBER, R. A. SWEET, W. L. BRIGGS, V. E. HENSON, AND J. OTTO, Bluestein's

FFT for arbitraryN on the hypercube, ubmitted forpublication.[16] P. N. SWARZTRAUBER, FFT Algorithmsfor vector computers, Parallel Comput., 1 (1984),pp. 45-63.[17] , Multiprocessor FTs, Parallel Comput., 5 (1987), pp. 197-210.

Fractional Fourier Transform and Application

Documents

Transcript of Fractional Fourier Transform and Application