What is the fast Fourier Transform

7/29/2019 What is the fast Fourier Transform

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IEEE T R A N S B C T I O N S O N A U D I O A N D E L E C T R O A C O U S T I C S , VOL. AU-15, O. 2, J UNE 1967 45

What is the Fast Fourier Transform?G-AE SubcommitteeonMeasurement Concepts

WILLIAM T. COCHRANJAMES W. COOLEYDAVIb L. FAVIN, MEMBER, B E EHOWARD D. HELMS, MEMBER, IEEEREGINALD A. KAENEL, SENIOR MEMBER, IEEEWILLIAM w. LANG, SENIOR MEMBER, E E EGEORGE C. ML4L1NG,JE., ASSOCIATE BB~~~BE R,EEEDAVIb E. NELSON, MEMBER, BEE,CHARLES M. RADER, MEMBER, IEEEPETER D. WELCH

Absfracf-The fastFourier ransform is a computational toolwhich facilitates signal analysissuchaspower spectrum analysis andfilter simulation by means of digital computers. It is a method forefficiently computing the discrete Fourier transform of a series ofdata samples (referred tos a time series). I n this paper, the discreteFourier transfomof a time seriesis defined, someof its propertiesare discussed, the associated fast method (fast Fourier transform)for computingthis transformisderived, and sameof &e compai.a-tional aspectsof the method arepresented. Examples are included todemonstratetheconcepts involved.

A INTRODUCTIONN ALGORITHM for the computation of Fouriercoefficients which requires much ess computa-tional effort than was required n the past wasreported by Cooley and Tukey[l ] n 1965. This methodis now widely known as the "fast Fourier transform,"and has produced major changes in computational tech-niques used in digital spectral analysis, fi lter simulation,andrefated fields. The echniquehasa Iong and n-teresting history that has been summarized by Cooley,Lewis, and Llielch in this issue2].

The astFourier ransform FFT) is a method orefficiently computing hediscreteFourier ransform(DFT)of a timeseries discrete data samples). Theefficiency of this method is such that solutions to manyproblems can now be obtained substantially more eco-nomically than n the past. This s the reason for thevery great current interest in this technique.

The discrete Fourier transform (DFT) is a transformi n its own rightsuch as the Fourier integra,' traxwformor the Fourier series transform. I t is a pourerful revers-

W. T. Cochran, D. L. Favip,and R. A. Kaenel are with BellH. D. Helms iswith Bell Telephone Laboratories, IK., Whippany,Manuscript received March 10, 1967.

Telephone Laboratories, Inc., Murray Hill, N. J .N. T ... 2 -Yorktown Heights, N. Y .J .W. Cooley andP. D. Welch are with the IBM Research Center,W. 1%. Lann and G. C. Malina are with the I BM CorDoration,Poughkeepsie, iiJ. Y .C. M. Rader is with Lincoln Laboratory, Massachusetts Instituteof Technology, Lexington, Mass. (Operated with support from theU. S.Air Force.)Dynamics Corporation, Rochester, N. Y .D. E. Nelson is with he Electronics Division of the General

I

iblemappingoperation or ime series. As thenameimplies, it has mathematical properties that are entirelyanalogous to those of the Fourier integral transform. Inparticular, i t defines aspectrumof a time series; multi-plicationof the ransform of two imeseriescorre-sponds to convolving the time series.I f digitalanalysisechniques are obe usedoranalyzing a corltinuous waveform then it is necessarythat the data be sampled usually at equallyspacedintervals of time) n order to produce a time series ofdiscrete samples which can be fed into a digital com-puter. As is w ~ l l nown [ 6 ] ,such a timeseriescom-pletely represents the continuous waveform, providedthis waveform isrequencyand-limited andhesamples are taken at a rate that s at least twice thehighest frequency present in the waveform. When thesesamples are eqeally spaced they are known as Nyquistsa ~~pk ~.t ~ 3 1eshown that the DFT of such a timeseries is closely related to the Fourier transform of thecontinuouswaveform fromwhich sampleshavebeentaken o orm he ime series. Thismakes heDFTparticularly useful orpower spectrumanalysisandfi lter simulation on digital computers.

The fast Fourier transform (FFT), then, i s a highlyefficient procedure or computing the DFT of a timeseries. I t takes advantage of the fact that the calcula-tion of the coefficients of the DFT can be carried outiteratively, which results n a considerablesavingsofcomputation time, This manipulation isnot intuitivelyobvious,perhapsexplainingwhyhisapproachwasoverlooked for such a long ime. Specifically, f thetime eries onsists of N=2" samples, henabout2nN=2N*10g2N arithmetic operations will be shownto be required to evaluate all N associated DFT CO-efficients. In comparison with the number of operationsrequired for the calculationf the DFTcoefficients withstraightforwardprocedures (W), thisnumber is sosmall whenN is largeas to completely change the com-putationally economical approach to various problems.Forexample, it has been eported that for N=8192samples, the computations require about fiveseconds


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46 IEEE TRANSBCTIOKSNUDIONDLECTROACOUSTICS,UNE 19for the evaluation ofall 8192 DFT coefficients on anI BM 7094 computer. Conventional procedures take onthe order of half an hour.

The known applications where substantial reductionincomputationimehasbeen achieved include: 1)computation of the power spectra and autocorrelationfunctions of sampled data [4]; 2) simulation of filters[ 5 ] ; 3j pattern recognition by using a two-dimensionalform of the DFT; 4) computation of bispectra, cross-covariance unctions,cepstraand elated unctions;and 5) decomposing of convolved unctions.

THEDISCRETE OURIERRANSFORMDFT)Definition of theD F T and i t : , Inverse

Since the FFT is an efficient method for computingthe DFT t is appropriate to begin by discussing theDFT and someof the properties that make it so usefula transformation. The DFT s defined by1

N-1A , = X kexp ( -2~rj rk I N) r =0, . . . ,M - 1 (1)where A, s the rth coefficient of the DFT and X k de-notes the kth sample of the time series which consistsof N samples and j =47. he Xks can be complexnumbers and the A,s are almost always complex. Fornotationalconvenience (1) isoftenwrittenas

k=O

N- 1A, = (X k )W , k Y =0, . . ,N - 1 (2)k=O

whereW = exp ( -2~rj lN ). (3)

Since theXksare often values of a function at discretetime points, the ndex r issometimescalled the fre-quency of theDFT. The DFT haslso been called thediscrete ourierransform rhe discreteime,finite range Fourier transform.There exists the usual nverse of the DFT and, be-cause the form s very similar to that of the DFT, theFFT may be used to compute it.

The inverseof ( 2) isN-1X =( l / N ) A,W4 1 = 0, 1, . . ,N - 1. (4)r=O

This relationship s called the nverse discrete Fouriertransform (IDFT). I tseasy to show that this inversionis valid by inserting (2) into (4)

x1= (Xk/*W)W(k-l). (5)N -1 AT-1r=o k =OInterchanging n (5) theorder of summingover heindices r and k, and using the orthogonality relation

1 The definitionof the DFT s not uniform in the literature.Someauthors use A, /N as the DFT coefficients, others useA,/dT, stillothers use a positive exponent.

N- 1exp (2~rj (n-m)r /N) =LIT,if n =mmod A7r =O

=0, otherwise (establishes thattherightside of (5) is in actequalto Xk.

I t is useful to extend the range of definitionof A,all integers positiveandnegative).Within his deinition it follows that

A, =A.V+~=A,hr+ = . . . (Similarly, x1=XW+l =N,hT+ = . . . . (Relationshipsbetween he DFT and heFourierTrans-fo r m of a Continuous Waveform

A n importantproperty hatmakes heDFTeminently useful s the relationship between the DFof asequenceof Nyquist samples and the Fourier trform of a continuous waveform, that is represented btheNyquist samples. T o recognize this elationship,consider a frequency band-limited waveform g( t ) whosNyquist samples, X k , vanish outside the time nterval0 5 t l N T

where T is the imespacingbetween he samples.periodic repetition of g ( t ) can be constructed that haidentically the same Nyquist samples n the time n-terval 0< _


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G-AE SUBCOMMITTEE : T HE FAST FOURIERRANSFORM 47

wheren =r forn =0 , 1) ,q - N/2 (13)and

G(n/NT)=D, =1V.A,/2 for n =N/2 . (14)N TEquations (13) and (14) give a direct relationship be-tween the DFT coefficients and the Fourier transformat discrete frequencies for the waveform stipulated by(9). A one-to-one correspondence could have been ob-tained if the running variable r had been bounded by& N /2 . This, however, would have required distinguish-

L= N T

ing between even and odd values of N , a distinctionavoided by keeping r positive.A waveform of the type considered by (9) is shownin Fig. l (e). I t s usually obtained as an approximationof a frequency band-limited source waveform [such asthe one sketched in Fig.1 a)] by truncating the Nyquistsample series of this waveform, and reconstructing thecontinuouswaveformcorresponding o he runcatedNyquistsampleseries Fig.1 (b), (d), and e)].Not-withstanding the identity f the Nyquist samples f thisreconstructedwaveformndherequencyand-limited source waveform, these waveforms differ in thetruncation nterval [Fig. (c) and (e)]. The differenceis usually referred to asaliasing distortion; the mechan-ics of this distortion is most apparent in the frequencydomain [Fig. l(c)-(e)]. I t can be made negligibly smallby choosing a sufficiently largeproduct of the fre-quencybandwidth of the sourcewaveform and heduration of theruncationnterval [6] (eg., N isgreater than ten).

I \ * f(Frequency)4

(a) Frequency-band-limited source waveform.(b)Nyquist samples of the frequencyband-limitedsource waveform.(c) Truncated source waveform.(d) Truncated series o Nyquist samples of the sourcewaveform.(e) Frequency-band-limited waveform whose Nyquistsamples are identical to he runcated series ofNyquist samples o the source waveform.( f ) Periodic continuation of theruncated source(g) Periodic continuation of the runcated series of(h) DFT coefficients interpreted as Fourier series co-

waveform.Nyquist samples of the source waveform.efficients producing complex waveform.

Fig. 1. Related waveforms and their corresponding spec-forms for energy-limited waveforms; series transformtra asdefined by theFourier transforms (integral trans-for periodic waveforms).


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48 IEEE TRANSACTIONS ON AUDIO A ND ELECTROACOUSTICS,GNE 196These aliasing distortions are carried over directly tothe discrete spectra of the periodically repeated wave-

forms[Fig. (f)and(g)],andappearcorrespondinglyin the DFT of the truncated series of Nyquist samples[Fig. (h)]. t may be of interest to observe that thewaveform corresponding to the DFT coefficients inter-pretedasFourierseries coefficients is complex Fig.1 h)1.Some Useful Properties of the D F T

Anotherproperty hatmakes he DFT eminentlyuseful is the convolution relationship. hat s, the IDFTof the productof two DFTs s the periodic mean convo-lution of the two time series f the DFTs. This relation-ship proves very useful when computing the filter out-putas a result of an nputwaveform; t becomesespeciallyeffective whencomputedby he FFT. Aderivation of this property is given in Appendix.

Other properties of the DFT are in agreement withthecorrespondingproperties of theFourier ntegraltransform,perhapswithslight modifications. For ex-ample, the DFTf a time series circularly shifted by isthe DFTof the time series multiplied byWmrh.urther-more, the DFTf the sumof two functions is the sumfthe DFT of the wo unctions.Thesepropertiesarereadily derived using the definition of the DFT. Theseand other properties have beencompiled by Gentlemanand Sande [7].

THE AST OURIERRANSFORMGeneral Description of the F F T

As mentioned n the ntroduction, he FFT is analgorithm that makes possible the computation of the

DFT of a time series more rapidly than do other algorithms available. Thepossibility of computing the DFbysuch a fastalgorithmmakes heDFT echniqueimportant. A comparison of the cornputational savingthat may be chieved through use of theFFT issummarized in Table I for various computations that are frequently performed. t simportant to add that theomputationalefforts isted epresentcomparableupperbounds; the actual efforts dependon the number N anthe programming ingenuity applied7].

I t may beuseful to point out that theFFT not onlreduces hecomputation ime; i t also ubstantiallyreduces round-off errors associated with these computtions. In act,bothcomputation imeand round-oferror essentially are reduced by a factor of (logzN ) / Nwhere N is the number of data samples n the timeseries. Forexample, if N=1024=21, then N . ogN=10240 [7], [9]. Conventional methods for computing (1) for N=1024would require an effort proportionto N 2=1048 576, more than50times that required wthe FFT.

The FFT is a clever computational technique of sequentiallyombining progressively largerweightedsums of data samples so as to produce the DFT coefficients as defined by (2). The technique can be nter-preted in termsof combining the DFTsof the individuadata samples such that the occurrence times of thesesamples are taken into account sequentially and apto the DFTs of progressively larger mutually exclusivesubgroups of data samples, which are combined to ultimately produce the DFT of the complete seriesof datasamples. The explanation of theFFT algorithm adoptein this papersbelieved to be particularly descriptiveorprogramming purposes.

TABLE ICOMPARISON OF THE xU MB ER OF hI ULTI PLI CATI ONSREQUIREDSING DIREcr AND FFT METHODS_ _ _Operation

Discrete Fourier Transform (DFT)

Filtering (Convolution)

Autocorrelation Functions

Two-Dimensional FourierTransform Pattern Analysis:

Two-Dimensional Filtering


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G-AE SUBCOMMITTEE:HEAST 49Conventional Forms of the F F T

Decimation in Time: The DF T [asper (2)] and tsinverse [see (4)] are of the same form so that a proce-dure, machine, or sub-routine capableof computing onecan be used for computing the other by simply exchang-ing he roles of X k and A,, andmakingappropriatescale-factor and sign changes. The two basic orms ofthe FFT, each with its several modifications, are there-foreequivalent.However, i t isworthdistinguishingbetween them and discussing them separately. Let usfirst consider the form used by Cooley and Tukey [l ]which shall be called decimation in time. Reversing theroles of A , andXk ives the form called decimation infrequency, which will be considered afterwards.

Supposea time series having N samples [such asXkshown in Fig. 2(a)] s divided into two functions, k andzk, each of which has only half as many points (N /2 ) .The function Y k iscomposed of the even-numberedpoints (Xo, 2,X g . ),and Z k is composed of the oddnumbered points (XI , p,X5 . - ). These functions areshown nFig. 2(b) and (c), and we maywrite hemformally as

yk =X 2 kA TK = 0 , 1 , 2 , . . . - -' 2 1. (15)zk =X2k+

Since Y k andZA are sequences of N / 2 points each, theyhave discrete Fourier transforms efined by

h=O Nr = 0 , 1 , 2 ; . . --' 2 1. (16)(N/2) - -1C, = zk xp (-4rjrK/;V)

k= O

The discreteFourier ransform that we want is A,,whichwecanwrite in terms of theodd-andeven-numbered points

+2, exp(-25 [2K+11))NY =0 , 1, 2, * . - ,N - 1 (17)

or(N/2)- -1A , = Y k exp(- 4.-jrk/Ar)

k=O(Ni2)-1+exp(- 2 ~ j r / N ) Zkexp( -4nj rk /N) (18)

k=O

which, using(16), may be written in theollowing form:A,. =B, +exp (- 27rjr/N)C, O l r


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50 IEEE TRANSACTIONS ON AUDIO AND ELECTROA COUSTICS, UNE 19

1 0 00 0 0

0 0 0

0 0

kC

0 2 4 6 8 1 0 1 2 1 4

1 0 0( b)Y k 1 o 0

0 0

0

t k0 1 2 3 4 5 6 7

( C ) 0 0

zk 00 0 0

0

0 1 2 3 4 5 6 7Fig. 2. Decomposition of a time series into two part-timeseries, each of which consists of half the samples.

c3w7= A7

Fig. 3. Signal flow graph llustrating the reduction of endpointDF T totwo DFTs of N/2 points each, using decimation in time.The signal flow graph may be unfamiliar to some readers. Basi-cally i t is composed of dots (or nodes) and arrows transmis-sions). Each node represents a variable, and thearrows terminat-ingat that node originateat the nodes whose variables contributeto the value of the variable at that node. The contributions areadditive, and the eightof each contribution, if other than unity,is indicated by the constant written close to the arrowhead ofbottom right node is equal to B3fW7XCa. Operations other thanthe transmission. Thus, in this example, the quantity A7 at theaddition and constantmultiplication must be clearly indicated bysymbols other than * or --+

0 . 0

0 0 .

0 . .

x7-

Fig. 4. Signal flow graph l lustrating further reduction ofthe DFT computationsuggested by Fig. 3.

Fig. 5. Signal flow graph illustrating the computation of the DFwhen the 0perations:involved are completely reduced to multiplcations and additions


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G-AE SUBCOMMITTEE:HEST 51plications are required for computation of the discreteFourier transform of an N point sequence, whereN is apower of 2.

When N is not a power of 2, but has a factor p , thedevelopment of equationsanalogous o (15) through(22) is possible byorming p differentequences,Yk( i )Xpk+;, ach having N / p samples. Each of thesesequences has aDFT Br(i),nd the DFTf the sequenceXk an be computed from the simpler DFTs with pNcomplex multiplications and additions. Thats,

AT+rn(N lP ) = &(i)WiI+(/P)Ip-1

i= Om = 0 , 1 , 2 , . . . , p - l

N9 .= 0 ,1 ,2 , . . . - - P 1. (23)The computation of the DFTs can be further simplifiedif N has additional prime factors.

Further information about theastFourier transformcan be extracted from Fig. 5 . For example, if the inputsequenceXk is stored in computer memory in the order

xo,x4,x 2 , X6, XI, x5, 3 , ? , (24)as inFig. 5, ihe computation of the discrete Fouriertransform may be done in place,hat is, by writingallintermediate results over the original data sequence, andwriting the final answer over the ntermediate results.Thus, no storageisneeded beyond that required for theoriginal N complex numbers. To see this, suppose thateachnodecorresponds o womemory egisters thequantities to be stored are complex). The eight nodesfarthest to the leftn Fig. 5 then represent the registerscontaining the shuffled order input data. The first stepin the computation is to compute the contents of theregisters represented by the eight nodesust to the rightof the input nodes. But each pair of input nodes affectsonly the corresponding pairof nodes immediately to theright, and if the computation deals with two nodes at atime, the newly computed quantities may be writteninto he egisters fromwhich the nputvalues weretaken, since the nput values are no longer needed forfurther computation. The second step, computation ofthe quantities associated with the next vertical array fnodes to the right, also involves pairs of nodes althoughthese pairs are now two locations apart instead of one.This fact does not change the property of in placecomputation, since each pair of nodes affects only thepair of nodes immediately to the right. Afteranew pairof results is computed, it may be stored n the registerswhich held the old results that are no onger needed. Inthecomputation or he inalarray ofnodes,corre-sponding to the values of the DFT, the computationinvolves pairs of nodes separated by four locations,butthe in place property still holds.

For this version of the algorithm, the initial shufflingof thedata sequence, X,, wasnecessary or theinplacecomputation.This shuffling s due o he re-peated movement of odd-numbered members of a se-quence to the end of the sequence during each stage ofthe reduction, as shown in Figs. 3, 4, and 5. This shuf-fling has been called bit reversal2because the samples arestored in bit-reversedorder; .e., X4=X(100)2s storedinposition (011)z=3, etc. Note that the nitial datashuffling can also be done (in place.Variations of Decimation in Time: If one so desires,the signal flow graph shown n Fig. 5 can be manipu-lated to yield different forms of the decimation in timeversion of thelgorithm. If onemagines that inFig. 5 all the nodeson thesamehorizontal level asA1 are nterchangedwith all the nodeson thesamehorizontal level as A 4,and all the nodes on the level ofAs are interchanged with the nodes on the level of Ag,with hearrows carriedalong with henodes, then oneobtains a low graph like thatf Fig. 6.

For this rearrangement one need not shuffle the origi-nal data nto the bit-reversed order, but the resultingspectrum needs to be unshufled. An additional disad-vantage might be that the powers of W needed n thecomputation are n bit-reversed order. Cooleys originaldescription of the algorithm [l ]corresponds to the flowgraph of Fig. 6.

A somewhat more complicated rearrangementof Fig.5 yields the signal flow graph of Fig. 7. For this caseboth the input data and the resulting spectrum are innatural order, and the oefficients in the computationare also used in anatural order. However, the computa-tion may no longer be done in place. Therefore, atleastoneotherarray of registersmustbeprovided.This signal flow graph, and a procedure correspondingto it, are due to Stockham8].Decimation in Frequency: Let us now onsider asecond,quitedistinct, form of the ast Fourier trans-form algorithm, decimation in requency. This form wasfound ndependentlybySande [ 7 ] , and Cooley andStockham [SI. Let the time series X k have a DFT A,.The series and he DFT bothcontain N terms. Asbefore, we divide X k into two sequenceshaving N/2pointseach.However, he irstsequence, Yk, s nowcomposed of the first N/2 points inXL , nd the second,Zk , is composed of the lastN/2 points inXk. Formally,then

Yr,=X k

* This is a special case of digit reversal where the radix of theaddress is2;more general digit reversalsare available for transformswith other radices.


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52 IEEE TRANSSCTIONSN AUD IO AND ELECTROACOUSTICS, J U N E 19

Fig. 6. Rearrangement of the flow graph of Fig. 5 illustrating he Fig. 7. Rearrangement of the flow graph of Fig. 5 illustrating thDFT computation from naturally orderedime samples. DFT computation withoutit reversal.

Fig. 8. Signal flow graph illustrating the reduction of endpointDFT to two DFTs of Ar/2 points each, using decimation n fre-quency.


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G-AE SUBCOMMI TTEE: THE FAST FOURIER TRANSFORM 53

TRANSFORMX I A 4

x 2

x 3 *6

AIe . .

A5

A3

k- w = r" 7Fig. 9. Signal flow graph llustrating further reduction ofthe DFT computationsuggested by Fig. 8. Fig. 10. Signal flow graph llustrating the computation o the DFTplications and additions.when the operations involved are completely reduced to multi-

Fig. 11. Rearrangementof the flow graph of Fig. 10illustrating thecomputation o the DFT to yield naturally ordered DFT coeffi-cients.Fig. 12. Rearrangementof the flow graph of Fig. 10 illustratingthe DFT computation without bit eversal.


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54 IEEE TRANSACTIONS ON AUDIONDLECTROACOUSTICS, J U N E 196

( N / 2 ) - - 1A, = { Yk+[exp( - r j r ) ] Zk] exp (-2?rjrL/N). (27)k =O

Let us consider separately the even-numbered and odd-numbered points of the transform. Let the even-num-bered points be X, and the odd-numbered points beS,,where

Rr =A Zr0 5 r


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G-BE SUBCOMMITTEE: THE FAST FOURIER TRANSFORM 55A Useful Computational Vari ation: I t may be worth

pointing out here how some programming simplicity srealized when the factors p and p = N/p are relativelyprime. As described by Cooley, Lewis, and Welch, [2],the twiddle actor Wi of (23) can be eliminated bychoosingsubsequences of the Xks thataredifferentthan hose usedbefore. The DFT computationsarethen conveniently performed in two stages.

1) Compute the q-point transforms

of each of the p sequences

2) Compute, then, the p-point transforms

of the p sequences B,(%),here(33)

and the notation (p)pl is meant to representthe re-ciprocal of p, mod p, i.e., thesolution of ~( p ) ~- ~>1

CONCLUSION(mod (I).

The integral transform method has been one of thefoundations of analysis for many years because of theeasewithwhich he ransformedexpressionsmaybemanipulated, particularly n such diverse areassacous-tic wave propagation. speech transmission, inear net-work theory, transport phenomena, optics, and electro-magnetic theory. Many problems which are particularlyamenable osolutionby ntegral ransformmethodshave not been attacked by this method in the past be-cause of the high cost of obtaining numerical results thisway.

The fast Fourier transform has certainly odified theeconomics of solution by transform methods. Some newapplicationsarepresented in thisspecial ssue,andfurther interesting and profitable applications probablywill be found during the next ew years.

APPENDIXAs is well known, if the filter impulse response is fre-

quency-band-limited o1/2THzand is given by tsNyquist samplesYhspaced T second apart, and further-more, if the nput waveform is also requency-band-limited to 1/2T Hz andgiven by i ts Nyquist sampleskspaced T second apart, then the filter output waveform

is also requency-band-limited o l/2T Hz and com-pletely specified by tsNyquistsamples Z, spaced Tsecond apart

The convolution relationship facilitates computation ofthis equation.To prove the convolution relationship, let the FT of

the XLSbe A, andcorrespondingly the DFT of theYhs be B,. The I DFT of the product of A,.B, thenbecomes [see (4) ]

I .* I Ai-1

=(+)-Zs +perturbation term. (36)If the first N/2 samples of each of the two time series(X, ) nd ( Y h ) are assumed to be dentically zero, thenthe perturbation termof (36) is zeroso that the IDFTfthe productof the twoDFTsmultiplied by N isequal tothe convolution product Z, of (35). Since it is alwayspossible to select the time series to be convolved suchthat half of the samples are zero, the convolution rela-tionship for the DFT can be used to compute the con-volution product [see (35)] of two time series.

I t is useful to point out that if A,=B,, aperiodicautocorrelation function emerges.

REFERENCES[l ] J . W. Cpoley and J . W. Tukey, ILAnalgorithm for the machinecalculation of complex Fourier series, Math. o Comput., vol. 19,pp. 297-301, April 1965.[2] J .W. Cooley, P.A. M. Lewis, andP. D. Welch, Historical notes[3] R. B. Blackman and J . W. Tukey, TheMeasurement of Poweron the fast Fourier transform this ssue,p. 76-79.[4] C. Bingham, M . D. Godfrey, and J . W. Tukey, Modern tech-Spectra. New Y ork: Dover, 1959.[5] T. G. Stockham, High speed convolution and correlation, 1966niques o power spectrum estimation, this issue, p. 56-66.Spring oint ComputerConf., AF IPS Proc., vol. 28. Washing-[6] W.T. Cochran, J . J . Downing, D. L. Davin, H. D. Helms, R. A.ton, D. C.: Spartan, 1966, pp. 229-233.Kaenel, W. W. Lang, and D. E. Nelson, Burst measurements inthe frequency domain, Proc. IEEE, vol. 54, pp. 830-841, J une1966.[7] W. M. Gentleman and G. Sande, Fast Fourier transforms-forfun and profit, 1966Fall J oint ComputerConf. AF IPS Proc.,vol. 29. Washington, D. C.: Spartan, 1966, pp. 563-578.181 Private communication.[9] J .W. Cooley, Applications of the fast Fourierransform method,Proc. o the I BM Scientific Computing Symp., J une 1966.

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What is the fast Fourier Transform