the fast fourier transform its applications

IEEE TRANSACTIONS ON EDUCATION, VOL. 12, NO. 1, MARCH 1969

The Fast Fourier Transform and

Its Applications

JAMES W. COOLEY, PETER A. W. LEWIS, AND PETER D. WELCH

Abstract-The advent of the fast Fourier transform method hasgreatly extended our ability to implement Fourier methods on digitalcomputers. A description of the alogorithm and its programming isgiven here and followed by a theorem relating its operands, the finitesample sequences, to the continuous functions they often are in-tended to approximate. An analysis of the error due to discrete sam-pling over finite ranges is given in terms of aliasing. Procedures forcomputing Fourier integrals, convolutions and lagged products areoutlined.

HISTORICAL BACKGROUNDT HE FAST Fourier transform algorithm has an

interesting history which has been described in[3]. Time does not permit repeating this history

here in detail. The essentials are, however, that untilthe recent publication of fast Fourier transformmethods, computer programs were using up hundredsof hours of computer time with procedures requiringsomething proportional to N2 operations to computeFourier transforms of N data points. It is not surprisingthen that the "new" methods requiring a number ofoperations proportional to N log N received consider-able attention and led to revisions in computer programsand in problemn-solving techniques using Fouriermethods. It was discovered later that the base 2 formof the fast Fourier transform algorithm had been pub-lished many years ago by Runge and Konig [10] andby Stumpff [12], [13]. These are authors whose worksare widely read and their papers certainly were used bythose computing Fourier series. How then could theseimportant algorithms have gone unnoticed? The answeris that the papers of Runge, K6nig, and Stumpif de-scribed primarily how one could use symmetries of thesine-cosine functions to reduce the amount of computa-tion by factors of 4, 8, or even more. Relatively smallportions of these papers mentioned the successive dou-bling algorithm which permitted one to take two Fourieranalyses of N-point samples of data and combine themin N operations to obtain an analysis of a 2N-pointsampling of the same data. Successive application ofthis algorithm obviously yields an N-point Fourieranalysis in 10g2 N doublings, and therefore, takes N log2N operations. Thus, while the computational methodusing symmetries reduced the proportionality factor inthe KN2 operations required to transform an N-pointsequence, the method based on the doubling algorithmtook a number of operations proportional to N log2 N.

Manuscript received August 16, 1968.The authors are with the IBM Watson Research Center, York-

town Heights, N.Y.

It is most likely that with the relatively small values ofN used in preelectronic computer days, the formermethods were easier to use and took fewer operations.Consequently, the methods requiring N log N operationswere neglected. With the arrival of electronic computerscapable of doing calculations of Fourier transforms withlarge values of N, the N log N methods were overlookedand the well-known hand calculator methods were pro-grammed for the computers. Perhaps there is somethingto be learned from this experience, namely, that theremay exist numerical methods in the older literaturewhich should be reappraised whenever computing de-vices undergo radical changes.

FAST FOURIER ALGORITHMSIn the interest of coherent presentation, the defini-

tions and procedures utilized herein will be developedin blocks and shown as figures. Thus, the discreteFourier series is defined in Fig. 1 with A (n) being asequence which gives the complex Fourier amplitudesas a function of frequency n. The X(j), j=0, 1, * * *,N-1 are regarded here as a complex sequence, and ina problem may represent a sampling of a signal at Nsampling points. Also,

(2iri' 2r 2rWN= exp .- = cos-+ isin-

N1 N N

is the principle Nth root of unity and if we substitutethe expression for WN in terms of sines and cosines, weobtain the perhaps more familiar sine-cosine Fourierseries. Complex Fourier series is employed for ease ofnotation and derivation of formulas. One should notenext the inversion formula in Fig. 1, giving the A (n)'sin terms of the X(j)'s. Since A (n) is also a Fourier series,an algorithm or a program for computing the A (n)'sfrom the X(j)'s can be used to compute the X(j)'s fromthe A(n)'s. Since WNN= 1, the exponent of WN is tobe interpreted modulo N. This leads to an essentialproperty of the sequences X(j) and A (n), i.e., that theyare periodic functions of j and n, respectively, withperiod N.

It is shown in Fig. 2 that when N is a product, N= rs,the Fourier series can be calculated in a two-stage pro-cess. This is done just as though the sequences A (n) andX(j) were defined on two-dimensional rXs arrays withthe array indices (ji, jo) and (n1, no) being defined asshown. When we substitute for j and n in WNin, and jnis reduced modulo N, it is found that the series can be

27

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IEEE TRANSACTIONS ON EDUCATION, MARCH 1969

Discrete Fourier Series:N-1

X(j) - E A(n)wjnn,O N

where WN exp(27riIN)Inversion Formula:

N-1

N j.0 N

Let us write: X(tjt4-A(n)Periodicity:

WNN * I, Wj'N+N.

w/-w mod0'j<N

Xtj) Xtj mod N)

Atnt = Atn mod N)

Orthogonality:N-1 N if n-mmodNEWnj W-mj,

j 0 0 otherwise.

Fig. 1. Definition of discrete Fourier transform.

Mapping ot Indices:

Assume N = r s

i-- (il' io)jlr +jo

n-+lnl, no)n * n1s + no

0,.r-1jl * O,1, ..., s-l

nO = 0, .s-1n1 0,1.r-I

wlin 5wjlnlr - WWjNnls X Wiln°r x Wono

I1 x WjronIX WilO X Wjonor jN

s-1 r-1 j0n1 j0no jln0X(j1, jo) E E Aln1n0)Wr WN WS

no,On00nf0O)

jono r-i1ljono) WO° E 0WOn

njgo

ns1

Fig. 2. Factorization into subseries.

Number of operations (multiply-adds) for N-pointFourier analysis is:

Number of operations N2

Byfactoring, N r x s

Number of operations r2s + s2r N K (r+s).

Byfactoring N r1, r2 x .., x rm:

Number o operations- N x (r1 + r2 + + rm).

Forminimum, r1 ...- rm r, N rm, i.e.,

m bogrN:

Number of operations N r- IogrN N * o12N r

r r/1o92r2 2.003 1.884 2.005 2.156 2.317 2.498 2.679 2.8210 3.01

Fig. 3. Minimization of number of operations.

Let:XjvAn j, n = 0,1, ..,2N-1

X2j+1)A, } ij, n 0,1,. N-1

I 2N-1 jnAn = 2N I Xj W2Nj10

A5=2 {E X2 nW2465 + X2w,+lq2 1)n}

But W2 =W2N N

1 1N-i ,,In Ni1

An = L X2j NW2+ j+1 W2N

An 2 An2N

An+N 2 An 2N

Fig. 4. Successive doubling method.

calculated by first summing over n1 to form an inter-mediate array Ai(jo, no), and then summing over no.This gives Al(jo, no) as a phase factor WNiono times a setof r-term Fourier series with coefficients A (n1, no). Thefinal result is then a set of s-term Fourier series withAiU(ono) as coefficients. It is thus seen how the amountof computation is reduced from N2 for the original seriesto r2s+s2r = N(r+s) for the two-stage process. By iterat-ing on this procedure, the amount of calculation for amore composite N is reduced as shown in Fig. 3 to be Ntimes the sum of the factors of N. If N is arbitrary, it isseen here by this counting that the lowest number ofcalculations is obtained if all factors are equal to 3.However, by avoiding multiplications when the powersof WN are simple numbers like + 1 or + i, one can reducethe calculations even further for r =2, 4, 8, and 16.The successive doubling algorithm is the special case

when all factors of N are equal to 2, i.e., N=2M. Aseparate derivation is given for this algorithm in Fig. 4.The basic formula which can be copied into a programwith some simple logic for generating the indices ap-pears on the bottom two lines.To demonstrate the simplicity of the basic procedure,

a complete operable program in FORTRAN is given inFig. 5. The first half of the program including the"DO 7" loop performs a reordering of the data. Thesecond half including the "DO 20" loop actually doesthe calculation. This program requires log2N evalua-tions of the sine and cosine and N multiplications by W.For N=1024, this represents about 15 percent moreoperations ("operation" being defined here as a complexaddition or multiplication) than would be required bythe simple expedient of storing tables of powers of WN.Then, after that, a more complicated logic, as used inone of the available SHARE programs, would yield anadditional saving of 10 percent in arithmetic operations.However, bookkeeping takes about as much time as thearithmetic, so the present simple program is estimatedto be only about 12 percent less efficient than a moreoptimally programmed FORTRAN program.

Fig. 6 shows a slightly different fast Fourier transform

'28


COOLEY et al.: FAST FOURIER TRANSFORM

SUBROUTINE FFT(A,M)COPMLEX A(1024),U,W,TNI= 2*wwNV2 = N/2NFl = N-1J=1DO 7 l=1,NM1IF(I.GE.J) GO TO 5T = A(J)A(J) AMI)A(M) = T

5 K=NV26 IF(K.GE.J) GO TO 7

J = J-KK=K/2GO TO 6

7 J= J+KPI = 3.14159265358979D0 20 L=1,MLE = 2**LLE1 = LE/2U = (1.0,0.)W=CMPLX(COS(PI/LEl),SIN(PI/LEl)'DO 20 J-1,LE1DO 10 I=J,N,LEIP = I+LE1T=A( IP)*UA( I P)=A( )-T

10 A(I)=A(I)+T20 U=U*W

RETURNEND

FiG. 5. Program for computing DFT by FFT method.

Mappings --> 1j1,jo)n Inl, no)

defined by

n'rno+sni (modN, 0'n < N)

and

jo j (mod r)

jl a j (mod s)Then, j s * (s) Jo + r1r)-1 j,

where l; denotqs reciprocal mod r.

Then wh *n°j Wisnj noj nlj nol Wn1jo

r-I nj

Let A1tj0, no) - E A(n1, no) Wr10nlo

s-1I jn

X(jj, jo) * E Al(jono)l °no0O

Fig. 6. Prime factor algorithm.

2.5

2.0

1.5

E

ConventionalMethod

I.oi

.5

FastMethod

*1O 1024 2048 4096 8192

N = No. Real Data Points

Fig. 7. Time required for calculation of Fourier transform ofreal data on IBM 7094 using FORTRAN with conventional

and fast methods.

algorithm. This one requires that the factors of N bemutually prime and uses a different mapping of theone-dimensional indices j and n into the index pairs(Jl jo) and (ni, no), respectively, from that given in Fig.2. The result is a procedure quite similar to the one givenabove, except that the phase factor is missing in thecalculation of Al(jo, no). This can be used to advantagein combination with the previous algorithm. For ex-ample, to introduce an odd factor s, let N = s r where ris a power of two. Then one can program the primefactor algorithm using the power of two subroutine forthe r-point subseries.The time required for computing Fourier series by a

conventional program and by the FFT algorithm isillustrated in Fig. 7. It is seen here how quickly one canarrive at values of N which are so large as to make con-ventional methods unfeasible. The rate of growth withN of the fast methods shows that as technological ad-vances permit the measurement and collection of data atincreasing rates, the demands for computer speed riseonly slightly more than linearly. Since advances incomputer technology can be expected to increase at thesame rate as the data collection, this indicates that thedigital Fourier methods can be expected to continue tobe feasible with technological advances. Such a predic-tion would not be possible if N2 operation methods werenecessary. Perhaps the future will even see radios withdigital tuners.The errors found in the calculation of a Fourier trans-

form of a single complex exponential function are givenin Table I. It can be shown that if one has data whichare roughly evenly distributed, the FFT algorithm se-quences the calculations in such a way that the numbersadded on each successive intermediate step are approxi-mately of the same expected value. This minimizes theerror introduced by the shifting and chopping done infloating point calculations. A more careful error analysisof a sequentially ordered sum of products calculationwould disclose an error whose expected value could beproportional to N, while the FFT method by the sameestimation would give an error proportional to 10g2 N.To test this, the maximum error is divided by log2 Nin the third column of Table I and is found to follow thisrule. Table II shows the results of a test on randomdata where the transform to frequency and back waseffected and checked for accuracy. The same rate ofgrowth of error with log2 N instead of N is found.To demonstrate a simple application of the FFT pro-

gram (Fig. 8), data from a strain seismograph of theRat Island, Alaska, earthquake were Fourier-analyzedby Dr. L. Alsop of IBM in 1966. The data were recordedat 2048 points over a 131-hour period after transientsresulting from the initial shock had died out. A conven-tional program took 1567.8 seconds to compute theperiodogram or estimated power spectrum (the modulussquared of the Fourier transform), which is shown inFig. 9. The same task took 2.4 seconds with the FFTprogram and gave more accurate results. The result of

s

29

----------9-



TABLE ICALCULATED MAXIMUM AND Rms ERROR IN COMPUTING THE,FOURIER TRANSFORM OF A SINGLE COMPLEx ExPONENTIALFUNCTION EXP (27ri.jk/N),j=0, 1, 2, - -, N-i1, k=2*

MaximuError

Errors in units of 10-8

Max

10g2N Eins,rror

Time(minutes)

512 4.47 0.50 0.0133 0.0081024 5.22 0.52 0.0072 0.0172048 5.22 0.47 0.0036 0.0374096 5.96 0.50 0.0019 0.0858192 6.71 0.52 0.0010 0.175

* The program was compiled by IBSYS, FORTRAN iv and runon the IBM 7094. The time for calculating one Fourier transform isgiven in minutes.

I-zidz0

3:0

-j

z

C3I

1-

z0

-4

z

z2

291.97

242.17

169.37

136.56

63.78

30.36

-21 .80

-74 50

-127.10

-190.19

-232.99

TABLE IIRESULTS OF TRANSFORMING RANDom DATA DISTRIBUTED EVENLY

BETWEEN 0 AND 1 To FREQUENcy DATA AND BACK*

Errors in Units of 10-8

N Maximum Maxrms rms___

Error lOg2N Error lOgIN256 13.4 1.68 7.7 0.96512 14.9 1.66 8.6 0.961024 17.8 1.78 9.6 0.962048 20.9 1.90 10.8 0.984096 24.5 2.04 13.0 1.088192 29.8 2.29 14.6 1.12

* The maximum and rms differences between the starting andresulting arrays are listed. These quantities, divided by log2N, showthe errors to be proportional to log2N.

30

N

TIME IN HOURS

Fig. 8. Strain seismograph of Rat Island earthquake. N=2048; T= 131 hours.

32.34

29.-11

16.17 U

12e.93

9.70 95 PER CENT CONrIDENCE LEVEL

6.46

3.23

23.00

PERIOO IN MINUTES

Fig. 9. Power spectrum of strain seismograph of Rat Island earthquake. N=2048; T= 13' hours.



the calculation is a set of possibly significant sharp peaksin the periodogram (power spectrum) corresponding tothe natural modes of vibration of the earth. It shouldbe noted that the 0.95 confidence band given is for indi-vidual values of A (n) 2.

SAMPLED REPRESENTATION OF DATA

The data to be treated in Fourier analysis are ofteninfinite in extent and/or continuous in the time or spacedomain and/or in the frequency domain. Of necessityone must represent and perhaps approximate such datain terms of the finite discrete sequences and theirFourier transforms, as was just discussed. Of funda-mental importance, then, is the relationship between the"true" data and its sampled representation [4]. Thisrelationship is expressed in the form of a theorem inFig. 10. This theorem says that if x(t) and a(f) are anintegral transform pair, one must first construct fromthem periodic aliased functions xp(t) and ap(f) withperiod T and F, respectively. The theorem furtherstates that these periodized functions, each evaluated atN discrete points in its period, form a discrete finiteFourier transform pair.As an illustration, a numerical example is worked out

in Fig. 11 for the function x(t) = 0 for t < 0 and x(t) = e-tfor t>0. In this example, T=8 and N=16 giving asampling integral of DT= 2. Since T is so large, x(t) isindistinguishable from the aliased function xp(t), whichis plotted in the upper curve. Sample values suppliedto the discrete Fourier transform program are indicatedby dots. Note that at the discontinuity at t=0, onemust define x,(t) as the average of x(-0) and x(+0),since this is the value given by the inverse Fouriertransform at a discontinuity. The correct transforma(f) = (1 +2rif)-l is plotted in the solid line in the twolower graphs, the real part in one and the imaginary partin the other. The dots represent the computed values ofap(f) at N= 16 points in the frequency period F= 1/DT=N/T=2. Here, the effect of "aliasing" in the fre-quency domain is evident. The computed values followa curve which is the sum of a(f) and a(f) displaced tothe right by F. One can see that the error due to thisaliasing is significant for f near F/2. If one seeks in-creased accuracy, it is obviously necessary to push thealiasing curve to the right by increasing F. To do this,DT is decreased by increasing N. The results for N= 32are given in Fig. 12. Here, the effect of the doubled F isshown to decrease the error in the range 0.f< F/2.The above theorem and computation also show how

to compute Laplace transforms by using the Fouriertransform. The example can be regarded as the casewhere one is computing the Laplace transform a*(s) ofthe step function u(t)=0 for t<0 and u(t)=1 for t>0.Then, letting the transform variable be s = 1+ 2rif, theLaplace transform is the integral treated above and

1a(f) = a*(s) = -

S

Theorem: SupposeJCO

x(t) J aff)e27jift df

We write: xit) 4-> a(f) .

Assume values of T, At, F, Atf, N such that

T NAt - 1/AfF - N-Af = 1/At

Construct periodic functions:

xp(t) x(t) +xl t + T) + x(t - T) + x(t + 2 T) + x(t - 2 T) +...

for 0 t T

ap(f)- af) + a(f+ F) + a(f - F) + alf + 2xF) + alf -2 x F) +

for 0' f F

Let:An - apn-Atf)x1 T xp(j-At)

Then:

Xi <-v.An

Fig. 10. Relation between discrete finite Fourier transformand integral Fourier transform.

1.0

1.

.5

.5

0

-.5

x(t) = e-tT=8 F=2DT= 1/2 DF= 1/8

- 4i oI -40 1 2 3 4 5 6 7 8

t t

I~~~~~~~~~~~~~~~~~a(f)= 1

1+2rif

Re a(l *

,10 1 2 3 4

0

Im alf)

t tF/2 F

f.

0

Li, * 1 _ I* I

_,,*

42

F/2 F

Fig. 11. Calculation of DFT of e-I with N=16.

It should be noted that there is a serious problem inthe above example. This is that the function a.(f) doesnot exist if the a,(f) is defined by the sum formula; thesum does not converge for a*(s) = l/s. There is, how-ever, a way of defining ap(s) so that it turns out to be thefunction computed here. This will not be discussed itidetail now; we will just mention what one must do in thecalculation to invert the Laplace transform, i.e., tocompute x(t) from a(f). For this, instead of treating thetransform pair x(t), a(f) as defined above, one can treatthe pair x(t) +x(-t) and 2 - Re a(f). The latter is a func-tion for which the aliased function does exist. We areactually using the fact here that a causal function, i.e.,one for which f(t) = 0 for t <0, is completely determined

I. .I

31

T

II

t



4'dxlt a e-tT 8 F 4DTI 1/4 DF ' 1/8

0 1 2 3 4 5 6 7 8

I ~~~~~~~T~0

a(f) 11+2vif S

5-Re a(f)

o 1; *b' 2 3 4t I

F/2 F

tf m a(fM

0of _^**

-.5

Convolution:

N-1

i E Xk Yj-ki.e.,

ZO\ YO YN-1 YN-2 ...Y / Xo

I1 N0 1- 2 X

Z2 _ Y2 Y1 YO Y3 X2

ZNO 'YN-1 YN-2 YN-3 *.. YO XN-1

Theorem: IfXj AnYj B

z; C

then

Cn - N An Rn

Fig. 13. Convolution theorem.

0

K K K K . - -

3 4

Ff2

Fig. 12. Calculation of DFT of e-t with N=32.

by either the real or imaginary parts of its Fouriertransform. The real part of a(f) happens to be summablewhen sampled.

USE OF THE CONVOLUTION THEOREM

Most of the very important applications of the FFTinvolve the use of the convolution theorem. Thistheorem for discrete finite Fourier transforms is definedin Fig. 13. This simply states that the finite Fouriertransform of the convolution is the product of the finiteFourier transforms of the two functions. A similartheorem holds for continuous functions, and in such a

case, if the convolution integral were approximated by a

sum formula, one would indeed obtain approximatelythe expression given in Fig. 13. However, as in the case

of the inversion formula, these formulas are exact andare not simply numerical approximations of integrals.An important aspect to be noted in connection with

the convolution theorem is that while the convolutiontakes a number of operations proportional to N2 by thedirect method, the FFT method requires a number ofoperations proportional to N log N. The latter methodconsists in Fourier-transforming the two sequences

Xj and Yj to the frequency domain, multiplying thefrequency functions A. and B, and transforming theproduct back. Note that, from here on, subscripts willbe used on sequence elements.

For the convolution theorem to hold, it is necessary

that sequences be periodic and of period N. Therefore,the convolution obtained is defined as if one is to defineY as the periodic repetition of its values on 0, N-1

N-1

Zj E XkYj+kk10

Then Cn -N A-n Bn

zo\N'0 N1 N-3 N-2 YN x0\

Z N'21 N2.3* NN-2 n-1 X1

Z2 | Y2 Y3 ... YN-1 YO Yi

ZN 1 YN-1 y" "I YN-4 YN-3 YN-2 XN-

For Zo. Z -° ZL tobe correct, X must have L

trailing O's. If N' is length of original sequence,

Ng, OpS. in sum opf prgouts method (N'-U2)(L+1l)No. Ops. in F. T. rim thod 3(N'+L)l102lN'+L)

Fig. 14. Lagged products.

when the subscript of YJ-k goes out of the range 0 toN-1. In many actual situations, however, one reallywants to assume the data to be zero or equal to theiraverage value outside the region in which they are given.One then uses the simple expedient of lengthening thearray by appending zeros or an average value and let-ting N of the present discussion be the length of theaugmented array. The number of appended zeros, ofcourse, depends upon how many lags or values of Zj are

wanted, and the number of nonzero values in the arrays.Fig. 14 illustrates a slight variation on the convolution

theorem obtained by replacing k with -k. As describedin the figure, the last L values of the Xk vector will bemultiplied by values in the periodic repetition of the Ykvector in the calculation of Zo, Z1, *, ZL. Therefore,if the last L values of Xk are zero, the convolution willnot be affected by the periodic nature of the sequences.The formula for the speed ratio is given at the bottom

of the figure; this is seen to be the number of operationsin evaluating the convolution by the conventional

1.

32

1



TABLE IIINUMBER OF OPERATIONS REQUIRED TO COMPUTE LAGGED PRODUCTS BY FOURIER TRANSFORM METHOD AND SPEED RATIO, I.E.,

THE NUMBER OF OPERATIONS BY THE CONVENTIONAL DIVIDED BY FOURIER TRANSFORM METHODS

N+L*

1282565121024204840968192163843276865536

FT Method

2688614413824307206758414745631948868812814745603145728

L=32

1.01.11.11.11.00.90.80.80.70.7

L =64

0.81.72.02.01.91.81.71.51.41.4

* N= original record length. L = number of lags.

L= 128

1.33.03.53.53.43.23.02.92.7

Speed Ratio

L=256

2.45.46.36.56.36.05.65.3

L=512

4.39.711.611.911.611.110.6

L= 1024

7.817.821.422.121.720.9

L = 2048

14.232.839.641.340.7

L

£j= Xkyj.kk-0

N- I3 .

,X2;t~~~~~=L- I XI

xlx3

j=L-I

Number of Operations per Filter Output:By Sum of Products:

By Fourier Transform Method: No. Operatiyms , 2Nlo2NNo.Filter uutputs N-1.

Fig. 15. Digital filtering.

method, divided by the number of operations taken bythe FFT method when N' is the length of the originalsequence and L is the number of lags required. Values ofthis ratio for various N' and L are given in Table III.

It should be pointed out that for short sequences andfew lags, the conventional method is more efficient. Aturnover point is at L = 32. For L less than or equal to32, one will at best only improve the speed by a factorof 1.1, and for some N' values the FFT method willtake longer. For L = 64, unless N is as small as 64, one

can obtain an improvement which as much as doublesthe speed. Then, for high L, one gets higlh speed ratios,such as the 41 for N' = 32768, L = 2048.An application to digital filtering is described in

Fig. 15. This is again an example of a convolution calcu-lation and corresponds to a situation where one may

have a signal whose sampled values are the values of Yj,and a filter with a point spread function described by a

shorter sequence Xj of its sampled values. The outputof such a filter is the sequence Zj, the convolution of X

TABLE IVTABLE OF VALUES OF 2Nlog2N/(N-L)*

NL

16 32 64 128 256 512 1024 2048 8192

8 16 13 14 15 17 1816 20 16 16 17 1924 19 17 18 1932 18 18 1964 21 21 2196 26 22 22 23128 32 24 23 24

1024 30

* 2Nlog2N/(N-L) is the approximate number of complexmultiply-adds required to comptute each digital filter output for seg-ments of length N'= N-L where L is the number of filter weights. Lis the number of operations per output by the direct method, and itis to be compared with table entries in evaluating the two methods.

and Y. Suppose the input signal is infinite in extent inboth directions in the time domain. One then divides theinput signal into time slices of N samples each, where Nis chosen at one's convenience. Each such N-pointsequence is Fourier-transformed and its transform ismultiplied by the transform of Xj; the result is thentransformed back to yield a filtered signal for that blockof data. Now, due to the periodic nature of the convolu-tion, the first L values of Zj will have beenconvolvedwith the periodic repetition of the Yj sequence andthereby contaminated. However, one can discard thesevalues and take the blocks of N values of Y1 so that theyoverlap by an amount L.To determine the block size N which minimizes the

amount of arithmetic, one writes the formula for thenumber of operations per filter output. Two Fouriertransforms are required, for which a conservative esti-mate of 2N log2 N operations is assumed. There will beN-L good filter outputs, so that the number of opera-tions per filter output is 2N log2 N/(N-L). This is tobe compared with L, the number of operations by theconventional method. From the numerical values givenin Table IV, it is seen that at L= 16, the best N is

-i ir

33

l


IEEE TRANSACTIONS ON EDUCATION, VOL. 12, NO. 1, MARCH 1969

N= 128; however, the amount of computation consistsof 16 operations, the same as that required by the con-ventional method. As one goes to higher L, the optimalN increases and the speedup ratio goes up. For example,at L = 128, a value of 1024 is obtained for N, so that theFFT method is almost six times as fast.

SUMMARY

It has already been found that the use of the FFTmethods has greatly increased the effectiveness of digitalmethods for a very wide range of problems such asspectral analysis, signal processing, Fourier spectro-scopy, image processing, and the solution of differentialequations.Most efforts of the past several years have involved

the reprogramming of previous procedures for the sakeof economy in computer usage. Recent developmentshave been in the application of Fourier methods toproblems which, due to computational effort, would notbe tractable were it not for the use of the FFT method.Among these are the real-time digital Fourier methods,for which special-purpose computers are now beingbuilt. There are a number of areas in which a largeamount of future development can be anticipated.Among these are the problems of the numerical solutionof differential equations, image processing, and multipletime series analysis and filtering.The preceding material has been a very rapid survey

of the fast Fourier transform algorithm and of some of

its applications. A much more detailed treatment isgiven in [5 ].

REFERENCES[1] L. E. Alsop and A. A. Nowroozi, "Faster Fourier analysis," J.

Geophys. Res., vol. 70, no. 22, p. 5482, 1966.[2] W. T. Cochran et al., "What is the fast Fourier transform?"

Proc. IEEE, vol. 55, pp. 1664-1677, October 1967.[3] J. W. Cooley, P. A. W. Lewis, and P. D. Welch, Historical

notes on the fast Fourier transform," Proc. IEEE, vol. 55, pp.1675-1677, October 1967.

[4] to, "Application of the fast Fourier transform to computa-tion of Fourier integrals, Fourier series, and convolution inte-grals," IEEE Trans. Audio and Electroacoustics, vol. AU-15,pp. 79-84, June 1967.

[5] - , The Fast Fourier Transform and Its Applications. NewYork: Wiley, to be published.

[6] J. W. Cooley and J. W. Tukey, "An algorithm for the machinecalculation of complex Fourier series," Math. of Comp., vol. 19,no. 90, pp. 297-301, 1965.

[7] G. C. Danielson and C. Lanczos, "Some improvements in practi-cal Fourier analysis and their application to X-ray scatteringfrom liquids," J. Franklin Inst., vol. 233, pp. 365-380, 1942.

[8] W. M. Gentleman and G. Sande, "Fast Fourier transforms forfun and profit," Fall Joint Computer Conf. 1966, A FIPS Proc.,vol. 29. Washington, D.C.: Spartan, pp. 563-578.

[9] I. J. Good, "The interaction algorithm and practical Fourieranalysis," J. Roy. Statist. Soc., series B., vol. 20, pp. 361-371,1958; Addendum, vol. 22, pp. 372-375, 1960, MR 21 No. 1674;MR 23 No. A4231.

[10] C. Runge and H. Konig, "Die Grundlehren der MathematischenWissenschaften," Band XI, Vorlesungen uber NumerischesRechnen. Berlin: Springer, 1924.

[11] P. Rudnick, "Note on the calculation of Fourier series," Math.of Comp., vol. 20, no. 95, pp 429-430, 1966.

[12] K. Stumpif, Grundlagen und Methoden der Periodenforschung.Berlin: Springer, 1937.

[13] , Tafelin und Aufgaben zur Harmonischen Analyse undePeriodogrammrechnung. Berlin: Springer, 1939.

[14] L. H. Thomas, "Using a computer to solve problems in phvsics,"Applications of Digital Computers. New York: Ginn, 1963.

Methods for Processing Complex Waveforms Through

Filters for Use in Monte Carlo Noise Studies

MURLAN S. CORRINGTON, FELLOW, IEEE, T. C. HILINSKI, AND W. B. SCHAMING

Abstract-Any N-pole linear lumped-constant time-invariant net-work can be represented by a linear homogeneous difference equa-tion. If an arbitrary input waveform is sampled at equally spaced timeintervals, so that eight or more samples are available at the highestsignificant frequency present, the output waveform can be computedin a £imple manner from the difference equation. If K input samplesare to be filtered, the saving in computer time, when compared to theusage of the convolution integral, is equal to K. The method is illus-trated by the simulation of an FM receiver, operating at threshold, inthe presence of Gaussian noise.

Manuscript received August 14, 1968; revised September 4, 1968.M. S. Corrington and W. B. Shaming are with the Radio Corpor-

ation of America, Camden, N. J.T. C. Hilinski was with the Radio Corporation of America, Cam-

den, N. J. He is now with MITRE Corporation, McLean, Va.

INTRODUCTIONE ) IGITAL-COMPUTER simulation of electrical

networks responding to signals contaminated bynoise can be done by numerical integration using

the convolution integral. This method is not very usefulif long streams of sampled data are to be filtered, sincethe number of points required in the integrand increaseslinearly with time, and the computer time required istoo great. It is not possible to update the previous cal-culation after each new input sample; all of the ordinatesof the integrand must be recomputed and integratedeach time.

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the fast fourier transform its applications

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