ACCURACY OF THE DISCRETE FOURIER TRANSFORM AND THE FAST FOURIER TRANSFORM

Transcription

1 SIAM J. SCI. COMPUT. c 1996 Society for Industrial and Alied Matheatics Vol. 17, o. 5, , Seteber ACCURACY OF THE DISCRETE FOURIER TRASFORM AD THE FAST FOURIER TRASFORM JAMES C. SCHATZMA Abstract. Fast Fourier transfor FFT)-based coutations can be far ore accurate than the slow transfors suggest. Discrete Fourier transfors couted through the FFT are far ore accurate than slow transfors, and convolutions couted via FFT are far ore accurate than the direct results. However, these results deend critically on the accuracy of the FFT software eloyed, which should generally be considered susect. Poular recursions for fast coutation of the sine/cosine table or twiddle factors) are inaccurate due to inherent instability. Soe analyses of these recursions that have aeared heretofore in rint, suggesting stability, are incorrect. Even in higher diensions, the FFT is rearkably stable. Key words. fast Fourier transfor FFT), discrete Fourier transfor DFT) AMS subject classifications. 65T0, 65Y5, 68Q5, 4A65, 4C10 1. Introduction. The Fourier transfor is one of the ost iortant fundaental atheatical tools in existence today. While the discrete Fourier transfor DFT) and fast Fourier transfor FFT) are generally considered to be stable algoriths, reorted quantifications of the stability have been inconsistent and confusing. Soe analyses neglect the effect of errors in the coefficients also called the sine/cosine table or twiddle factors), which turns out to be otentially the largest source of error. Gentlean and Sande [4] reort an analysis of FFT errors for the Cooley Tukey algorith) in which the root ean square RMS) relative error is 1) K E RMS = 1.06 j ) 3 ǫ, j=1 where ǫ is the achine esilon and = 1... K. For = K, and using the radix- algorith, this becoes ) E RMS = 8.48 log ǫ. Their corresonding forula for the slow DFT is 3) E RMS = 1.06) 3 ǫ. The results of Gentlean and Sande are uer bounds. Our own tyical results are quite different, being asytotically of far better saller) order when the twiddle factors are accurate. Kaneko and Liu [7] give an analysis for the Cooley Tukey FFT with several tyes of inut data sequences. Their results are rather colex, but their conclusion that errors in the twiddle factors have virtually no effect on the results coare their Figures 4 and 7) is isleading. Calvetti [] gives an involved analysis for the Cooley Tukey algorith and the slow transfor. She searates the effects of roundoff errors in addition and ultilication. The results are Received by the editors Aril 1, 1993; acceted for ublication in revised for) May 1, Deartent of Matheatics, University of Wyoing, P.O. Box 3036, Laraie, WY 8071 jcs@uwyo.edu). This research was suorted artly by a University of Wyoing Faculty Grant-in-Aid. 1150

2 ACCURACY OF THE DFT AD FFT 1151 E RMS = log σ a, addition errors, FFT; E RMS = 1 log σ a, ultilication errors, FFT; 4) n 1 E RMS = σ, n addition errors, slow DFT; E RMS = 1 n σ, ultilication errors, slow DFT; where σ a, σ are the standard deviations of the assued indeendent rando errors in addition and ultilication. These errors are relative to the axiu nor of the inut data. Calvetti clais, For very sall exected value of the relative error for addition and ultilication, the traditional [slow] algorith will roduce ore accurate results. She further suggests, The FFT can be considered ore accurate than the TFT [slow DFT] only if the exected value of the relative error for addition is of the sae size or larger than the exected value of the relative error for ultilication. I do not believe that the analysis for the slow algorith is correct, nor is the general sense of Calvetti s conclusion that the slow DFT is accurate coared to the FFT) correct. I do agree with Calvetti s forulas for FFT errors, in the case where the twiddle factors are accurate. Calvetti s aer gives a useful bibliograhy of earlier work. According to Gottlieb and Orszag [5], Transfor ethods norally give no areciable alification of roundoff errors. In fact, the evaluation of convolution-like sus using FFTs often gives results with uch saller roundoff error than would be obtained if the convolution sus were evaluated directly. See Van Loan [1] for a ore recent ublication on this toic. Our own conclusions concerning accuracy are that the FFT is rearkably stable, when ileented roerly. The FFT is vastly ore accurate than the slow DFT. However, the FFT is very sensitive to accuracy of the internal sine/cosine table twiddle factors). A oular technique for calculating the sine/cosine table by recursion is ill-advised for this reason.. The DFT..1. Proerties of the standard DFT. The square DFT ight be written c = G x or 5) where ost coonly 1 c k = e ıω kt n x n, n=0 6) 7) 8) ω k = kω, t n = nt, ωt = π. 0 k 1, 0 n 1,.1.1. uerical errors. Errors in the coutation of the DFT are liited to errors in the aroxiations of the sine/cosine hase factors and roundoff errors in ultilication and addition. To aid in characterizing these errors, we define the achine ǫ in the usual way as the sallest ositive nuber such that 1 + ǫ is distinguishable fro unity in the floating oint reresentation eloyed. For tests with 3- and 64-bit IEEE floating oint, we use the corresonding ǫ 3 = and ǫ 64 = A useful exact characterization of exected error with floating oint calculations is difficult or iossible, articularly in the case of addition, because of the nature of the floating oint

3 115 J. C. SCHATZMA RELATIVE RMS ERROR e-05 1e-06 a) b) 1e-07 1e e+06 LEGTH OF TRASFORM FIG. 1. Relative RMS errors in the couted DFT for the 3-bit IEEE floating oint slow Fourier transfor using a) the IBM RS6000 library 3-bit sine and cosine functions and the CAR FFTPAK table calculation code, and b) an accurate sine/cosine table.,, +, and denote lengths that are owers of, 3, 4, and 5, resectively. The interolating error functions sooth curves) are e 1 ) = ǫ 3 1) and e ) = 0.3ǫ 3 1. The horizontal dashed line reresents the constant ǫ 3 = reresentation. Error bounds tend to lead to excessively essiistic error estiates. Roughly, for data vectors exhibiting a reasonable degree of stationarity, if the errors associated with each coefficient, ultilication, and addition are uncorrelated not a very reasonable assution) the RMS error of the DFT would be exected to be aroxiately 1 ties the error standard deviation associated with each ter. If the errors are correlated, an RMS error roortional to 1 would be exected. Assuing that relative ultilication errors are uniforly distributed on [ ǫ/, ǫ/] a reasonable aroxiation if full rounding is used), the corresonding error standard deviation is ǫ/ 1. Assuing that relative addition errors are uniforly distributed on [ ǫ, ǫ] a oor aroxiation), the corresonding error standard deviation is ǫ/ 3. Putting these ideas together, we obtain a rough estiate of the overall RMS error in the case of uncorrelated individual errors: 9) E RMS = 1) ) ǫ 3 + 1)σ C, where σ C is the error standard deviation of the coefficients. If σ C is ǫ/ 1, 9) reduces to E RMS = ǫ/ ) 1. Figures 1 and show a coarison of 18-bit floating oint DFT results taken to be truth) with 1a) IEEE 3-bit DFT coutations using the anufacturer s 3-bit sine/cosine functions, 1b) IEEE 3-bit DFT coutations odified to use an accurate table of hase factors,

4 ACCURACY OF THE DFT AD FFT e-11 RELATIVE RMS ERROR 1e-1 1e-13 1e-14 a) b) 1e-15 1e e+06 LEGTH OF TRASFORM FIG.. Relative RMS errors in the couted DFT for the 64-bit IEEE floating oint slow Fourier transfor using a) the IBM RS6000 library 64-bit sine and cosine functions and the CAR FFTPAK table calculation code, and b) an accurate sine/cosine table., +,, and denote lengths that are owers of, 3, 4, and 5, resectively. The interolating error functions sooth curves) are e 1 ) = 1.3ǫ 64 1) and e ) = 0.4ǫ The horizontal dashed line reresents the constant ǫ 64 = a) IEEE 64-bit DFT coutations using the anufacturer s 64-bit sine/cosine functions, b) IEEE 64-bit DFT coutations odified to use an accurate table of hase factors. The tests were erfored on an IBM RS/6000 couter. The inut data was a series of indeendent, identically distributed Gaussian rando sequences for the real and iaginary coonents). The RMS difference between the two coutations, scaled by the RMS value of the true result, is dislayed as a function of transfor length. As shown in the figures, fits to the easured RMS errors were obtained for the following error functions: 1a) 3-bit: ǫ 3 1), 1b) iroved 3-bit: 0.3ǫ 3 1, a) 64-bit: 1.3ǫ 64 1), b) iroved 64-bit: 0.4ǫ When accurate hase factors are used Figures 1b and b) square root error growth is observed; results are obtained that are ore accurate than redicted by 9) by about a factor of two. When inaccurate hase factors are used Figures 1a and a), linear growth is observed, suggesting that individual errors are correlated. Fro these results we conclude the following: 1) Accuracy of the sine/cosine table is critical to overall erforance of the nuerical DFT. When the sine/cosine table of the FFT is inaccurate, errors in the DFT ay be correlated and the overall RMS DFT errors are then roughly roortional to the length of the transfor.

5 1154 J. C. SCHATZMA ) The IBM RS/6000 library sine/cosine functions are not very accurate; we do not recoend their use excet for noncritical alications. A siilar observation has been ade for the Cray CFT77 library routines. The coutation of long slow DFTs with IEEE 3-bit floating oint ay be exected to be of questionable accuracy unless ore accurate sine/cosine aroxiations are used. Using the RS/6000 library routines, only about five significant figures should be exected for about 100 oints ranging down to two significant figures for about 100k oints. 3) When the sine/cosine table is accurate, erforance is draatically suerior for both 3- and 64-bit coutations, and the RMS DFT errors grow roortionally to the square root of the length of the transfor. In this case, the DFT ay be regarded as a stable rocess. However, there still ay be significant error for very long transfors; five to six significant figures of accuracy ay be exected for transfors in the 100k+ oint range using 3-bit floating oint. There is obviously soe sensitivity of the errors to the data. An extree exale is the case of data consisting of ostly zeros. The above results should be interreted as tyical rather than definitive. 3. The FFT Sources of error. It is helful to list the theoretical sources of error: 1) instability of the FFT coutations associated with the factorization, ) instability of the underlying DFT blocks, 3) errors in the sine/cosine table twiddle factors), 4) roundoff error in all of the coutations, coounded randoly. While all errors could be considered roundoff errors, the above breakdown is reasonable. In articular, the idea of aroxiating roundoff by an instability effect and a rando effect is a owerful tool. The validity of this odel for general nuerical coutations reains to be deonstrated; there is good agreeent between theoretical and eirical results in this analysis. We will show that sources 1) and ) ay be discounted the associated coutations are extreely stable). Sources 3) and 4) are the rincial sources of error; in a roerly designed code, source 4) will doinate. 3.. FFT forulation. We forulate the conventional version of the FFT to concretize the differences between it and the slow DFT. Suose is even. We observe 10) 1 c k = w kn x j = w kj x j + w k w kj x j+1 j=0 = 1 j=0 1 j=0 c 0) k + w k c 1) k, 0 k 1, c 0) k wk c 1) k, k 1, where k = k, w = e ı π, c 0) is the length DFT of the even ters x 0, x,..., and c 1) is the length DFT of the odd ters x 1, x 3,.... This is the well-known result that a DFT of even length can be couted as the cobination of two DFTs of length. The coefficients w i and w i are called the twiddle factors. In atrix notation 10) ay be written I 1 c 0)) 11) c =, I 1 c 1)

6 ACCURACY OF THE DFT AD FFT 1155 where 1) q = diagz 0, z 1, z,..., z q 1 ), z = e i π q, and I 13) where 14) is the -by- identity atrix. Reeating this rocess for a ower of, we obtain F = S )... S) P,,,...,) S ) S ) S ) =.. S ) I 1 I 1 S ) 4 S ) 4 S ) 4 S ) 4 P,,,...,),, = bit-reversal ordering erutation atrix. For ore general, if = 1 K, where j are natural nubers, the FFT forula ay be written S ) S 3) 1 1 ) F = S 1) S ) S 3) 1 1 ) 15) where 16)... S K) K S K) K F ) 1,1 I F ) 1, S ) = F ),1 I F ),.. F ),1 I F ), = F ) Q). P 1,,..., K ), F ) 1, F ), F ),

7 1156 J. C. SCHATZMA Here F ) is the -by- atrix constructed fro the -by- DFT atrix F by relacing each eleent of F by the roduct of the eleent and the /-by-/ identity atrix. Also, 17) Q ) = diag I, 1,,..., 1 ) is the diagonal atrix of twiddle factors, and P 1,,..., K ) is the ixed-radix digit-reversal ordering erutation atrix, defined as 18) P 1,,..., K ) j,k = δ j,k, 0 j, k 1, k = k k) = d K + K d K 1 + K 1 d K + + d 1 )), k = d d + d d K 1 d K ))), where the ixed-radix digits d l satisfy 0 d l l 1. The forula 15) reresents the so-called ixed-radix tie-deciation FFT algorith. A atheatically equivalent forula that leads to the frequency-deciation algorith follows fro the observation that 19) F = 1 F ) 1 =... P 1,,..., K ) S ) 1 S 3) 1 ) ) 1 S K) K S 3) 1 ) S ) 1 S K) K 1 1 S 1) ) 1. 1 Then 0) P 1,,..., K ) ) 1 = P K, K 1,..., 1 ) and 1) S ) ) 1 = 1, R)

8 ACCURACY OF THE DFT AD FFT 1157 where ) R ) = F ) 1,1 I F ) 1, I F ) 1, I F ),1. F ),1 1 1 F ),. F ), 1 1 F ),. F ),. 1 1 = Q ) F ) so that 3) F = P K, K 1,..., 1 )... R 3) 1 ) R K) K R K) K R 3) 1 ) R ) 1 R ) 1 R 1). These forulas ay be silified by use of the Kronecker atrix roduct which is defined so that A B is the LI-by-MJ atrix 4) A B = A 1,1 B A 1, B A 1,M B A,1 B A, B A,M B... A L,1 B A L, B A L,M B for an L-by-M atrix A and an I-by-J atrix B. Then F ) = F I, 5) S ) = F ) = Q ) F I ) Q ), R ) = Q ) = Q ) F ) F I ),

9 1158 J. C. SCHATZMA and 6) F = S 1) = = = ) I 1 S ) 1 I 1 S 3) 1 ) )... I 1... K 1 S K) K P 1,,..., K ) ) F 1 I Q 1) 1 { [ ) I 1 F I ) { [ ) I 1 F I 1 ]} Q 3) 1 ) { [ ]}... I 1... K 1 F K Q K) K P 1,,..., K ) { [ F1 ) ]} I q1 I Q 1 ) 1 { [ F ) ]} I q I 3 Q ) { [ F3 ) ]} I q3 I 4 Q 3 ) 3 { [... I qk 1 FK 1 I ) ]} K Q K 1 ) K 1 { [ FK ) ]} I qk I K+1 Q K ) K P 1,,..., K ) K j=1 { [ Fj ) ]} I qj I j+1 Q j ) j P 1,,..., K ) ]} Q ) 1 for the tie-deciation algorith, and 7) F = P K, K 1,..., 1 ) 1 j=k { [ I qj Q j) ) ]} j Fj I j+1 for the frequency-deciation algorith, where j / j 1 ) andq j 1 j 1. A odification of 6) and 7) is ossible where the ixed-radix erutation is distributed through the calculation as a series of two-factor erutations. The result is 8) F = K j=1 A j P q j, j ) q j+1 I j+1 ) = 1 j=k P j,q j ) q j+1 I j+1 ) B j, where 9) 30) A j = B j = { [ Fj ) ]} I qj I j+1 Q j ) j, { [ I qj Q j) ) ]} j Fj I j+1. More details are given in Schatzan [9]. We note that the erutations in 8) can be accolished without an exlicit erutation, but by aroriate indexing in each block

10 diagonal atrix-vector ultilication ste 31) 1 l,=0 ACCURACY OF THE DFT AD FFT 1159 A j ) k,l P q j, j ) q j+1 I j+1 )l, z 1 = A j ) k, z, where the index aing ) is couted according to =0 3) = d 0 + d j+1 + d 1 j+1 j, = d 0 + d 1 j+1 + d j+1 q j, where the three-radix digits d 0, d 1, and d satisfy 0 d 0 j+1 1, 0 d 1 q j 1, and 0 d j 1. This constitutes a block two radix reversed-digit aing with a block size of j+1. These results are siilar to those of Teerton [11]. Finally, the rie factor algorith version of the FFT is nearly identical to the above, excet that inuts and oututs are reordered, generally to effect the eliination of the twiddle factors Burrus [1]; Schatzan [10]). Although the coutations are reordered, and in soe versions the underlying FFT blocks are odified by raising each eleent to a fixed integer ower, the arguents above about stability of the stages of the algorith still aly. The absence of twiddle factors could result in soe iroveent in accuracy; to this author s knowledge no results on this question have been ublished Stability of the FFT decoosition. If we exaine 6) 8) we see that the DFT is accolished through a series of atrix-vector roducts. The erutation atrices have eigenvalues on the unit circle, with hases equal to ultiles of π, where L is the length of L a erutation cycle associated with the atrix. The atrices A j and B j are ore colex. The atrix F j I j+1 has the sae sectru asf j, naely eigenvalues± j and±ı j, but ultilied in ultilicity by the factor j+1. Likewise, the reultilication I qj C only ultilies the ultilicity of the eigenvalues of an arbitrary atrix C. Thus, the eigenvalues of A j and B j are odified only by the effect of ostultilying and reultilying F j I j+1 by Q j) j, resectively. Because the atrices are unitary or scaled unitary, the eigenvalues of A j and B j all have agnitude j ; however, the hases of the eigenvalues are not the sae as F j. For exale, the eigenvalues of S ) = ) F I / Q ) are λ j = 1 1 w j ± ) 33) w j 1) + 8w j, 0 j 1, where w = e ıπ/, which are irregularly distributed around the circle of radius in the colex lane. Fro this eleentary analysis, we see that no colications, such as increased sensitivity to nuerical error due to increased condition nuber of the coefficient atrices, are to be exected fro the FFT in any for. In fact, as we will see below, the FFT is very stable in ost resects uerical errors. Like the slow DFT, errors in the coutation of the FFT consist of errors in the aroxiations of the sine/cosine hase factors and roundoff errors in ultilication and addition. To test these redictions, slow DFTs were couted with single and double recision IEEE floating oint. Figures 3 and 4 show a coarison of 18-bit floating oint FFT results taken to be truth) with 3a) IEEE 3-bit FFT coutations using the anufacturer s 3-bit sine/cosine functions and the FFTPAK recursion see coents below), 3b) IEEE 3-bit FFT coutations odified to use accurate sine/cosine hase factors,

11 1160 J. C. SCHATZMA c) RELATIVE RMS ERROR e-05 1e-06 a) d) 1e-07 b) 1e e+06 LEGTH OF TRASFORM FIG. 3. Relative RMS errors in the couted DFT for the 3-bit IEEE floating oint FFT using a) the IBM RS6000 library 3-bit sine and cosine functions and the CAR FFTPAK table calculation code, b) an accurate sine/cosine table, c) the Schatzan algorith [9] with inaccurate tables, and d) the Schatzan algorith [9] with accurate tables.,+,, and denote lengths that are owers of, 3, 4, and 5, resectively; and denote the results for the Schatzan algorith [9] with inaccurate and accurate tables, resectively. The interolating error functions sooth curves) are e 1 ) = 1 10 ǫ 3 1) for the conventional FFT with inaccurate tables, e ) = 0.6ǫ 3 log with accurate tables, e 3 ) = 1 ǫ 3 1) for the Schatzan algorith [9] with inaccurate tables, and e 4 ) = 0.9ǫ 3 log with accurate tables. The horizontal dashed line reresents the constant ǫ 3 = a) IEEE 64-bit FFT coutations using the anufacturer s 3-bit sine/cosine functions and the FFTPAK recursion see coents below), 4b) IEEE 64-bit FFT coutations odified to use accurate sine/cosine hase factors. The CAR FFTPAK Version, February 1978) FFT code was used as the basis for these results. However, the ackage was odified in several ways: 1) Minor odifications to the code were ade to bring it into conforance with the ASI Fortran 77 standard. ) Power-of-two transfors were evaluated using =, not a ix of = and = 4 as in the original code. This change was ade so that the erforance of the = code could be easured indeendently of the = 4 code. 3) For the accurate table tests, the 3- and 64-bit sine/cosine table coutations were relaced with 64- or 18-bit recision coutations, resectively. Inut data and analysis of the results were as above for the slow DFT. Fits to the easured RMS errors were obtained for the following error functions: 3a) 1 10 ǫ 3 1), 3b) 0.6ǫ 3 log, 4a) 1 0 ǫ 64 1), 4b) 0.6ǫ 64 log. Also shown as Figures 3c, 3d, 4c, and 4d are results for the new algorith of Schatzan [9] for rie lengths. Conclusions fro these results follow:

12 ACCURACY OF THE DFT AD FFT e-11 1e-1 c) RELATIVE RMS ERROR 1e-13 1e-14 1e-15 a) d) 1e-16 b) 1e e+06 LEGTH OF TRASFORM FIG. 4. Relative RMS errors in the couted DFT for the 64-bit IEEE floating oint FFT using a) the IBM RS6000 library 64-bit sine and cosine functions and the CAR FFTPAK table calculation code, b) an accurate sine/cosine table, c) the Schatzan algorith [9] with inaccurate tables, and d) the Schatzan algorith [9] with accurate tables., +,, and denote lengths that are owers of, 3, 4, and 5, resectively; and denote the results for the Schatzan algorith [9] with inaccurate and accurate tables, resectively. The interolating error functions sooth curves) are e 1 ) = 1 0 ǫ 64 1) for the conventional FFT with inaccurate tables, e ) = 0.6ǫ 64 log with accurate tables, and e 3 ) = 1.5ǫ 64 1) for the Schatzan algorith [9] with inaccurate tables. The horizontal dashed line reresents the constant ǫ 64 = ) Coaring Figures 1a and 3a, 1b and b, etc., we conclude that FFTs tyically roduce aroxiate DFTs that are ore accurate than the slow algorith by a factor of at least 10. ) Accuracy of the sine/cosine hase factors are crucial to the overall accuracy of the DFT. However, in the CAR FFTPAK code, only art of the table is couted by exlicit sine and cosine evaluations. The reainder of the table is generated recursively. Relacing the exlicit sine/cosine evaluations with extreely accurate calculations 18-bit) ade very little difference. The resulting lots are nearly indistinguishable fro Figures 3a and 4a. The recursive evaluation of entries of the sine/cosine table, as ileented in the CAR FFTPAK code, is by far the largest single source of error. The articular recursion used in the CAR FFTPAK accentuates the errors of the initial sine/cosine aroxiations. 3) When the sine/cosine table of the FFT is inaccurate as in Figures 3a and 4a) the RMS DFT errors are roughly roortional to the length of the transfor. 4) When the sine/cosine table is accurate Figures 3b and 4b) the RMS DFT errors grow slowly, aarently ore slowly than the logarith of the length of the transfor. 5) The coutation of long DFTs with IEEE 3-bit floating oint is a rocess about which one should be cautious. If accuracy of better than 0.1% is required three digits) for transfors in the 10,000+ oint range, 64-bit floating oint or a very carefully ileented 3-bit FFT should be used. 6) The Schatzan [9] algorith is less accurate than the conventional FFT, but only odestly so.

13 116 J. C. SCHATZMA e-05 b) RELATIVE RMS ERROR 1e-06 1e-07 a) c) 1e-08 1e e+06 LEGTH OF COVOLUTIO FIG. 5. Relative RMS errors in the couted convolution for a) the 3-bit IEEE floating oint direct coutation, b) the FFT coutation with inaccurate sine/cosine tables, and c) with accurate tables., +,, and denote lengths that are owers of, 3, 4, and 5, resectively. The interolating error functions sooth curves) are a) e 1 ) = 0.3ǫ 3 1, b) e ) = 0.15ǫ 3 1), and c) e 3 ) = ǫ 3 log. The horizontal dashed line reresents the constant ǫ 3 = uerical errors for convolutions. We now study the coutation of a circular convolution or cross-correlation via Fourier transfor. Figures 5 and 6 show results for crosscorrelation of white Gaussian colex sequences using direct coutation and the coarable results using FFTs. We obtain fits to the data with the following functions: 5a) Direct coutation, 3-bit: 0.3ǫ 3 1; 5b) FFT coutation, 3-bit: 0.15ǫ 3 1); 5c) FFT coutation, 3-bit with accurate sine/cosine table: ǫ 3 log ; 6a) Direct coutation, 64-bit: 0.3ǫ 64 1; 6b) FFT coutation, 64-bit: 0.ǫ 64 1); 6c) FFT coutation, 64-bit with accurate sine/cosine table: ǫ 64 log. These errors are very siilar to the errors for the slow and fast DFT theselves, as reorted above. We recall the clai of Gottlieb and Orszag [5]: Transfor ethods norally give no areciable alification of roundoff errors. In fact, the evaluation of convolution-like sus using FFTs often gives results with uch saller roundoff error than would be obtained if the convolution sus were evaluated directly. We have quantitatively confired these observations; the FFT ethod gives ore accurate results for vector lengths of aroxiately 100 and greater with IEEE 3- and 64-bit floating oint. However, this conclusion deends on having accurate FFT software. For exale, the stock CAR FFTPAK software gives less accurate results for the convolution than direct coutation by arithetic of the sae recision. Asytotically for large vector lengths, the RMS error is roortional to the square root of the logarith of the vector length for the accurate) FFT ethod and roortional to

14 ACCURACY OF THE DFT AD FFT e-11 1e-1 RELATIVE RMS ERROR 1e-13 1e-14 1e-15 b) a) c) 1e-16 1e e+06 LEGTH OF COVOLUTIO FIG. 6. Relative RMS errors in the couted convolution for a) the 64-bit IEEE floating oint direct coutation, b) the FFT coutation with inaccurate sine/cosine tables, and c) with accurate tables. The interolating error functions sooth curves) are a) e 1 ) = 0.3ǫ 64 1, b) e ) = 0.ǫ 64 1), and c) e 3 ) = ǫ 64 log. The horizontal dashed line reresents the constant ǫ 64 = the square root of the vector length for the direct ethod. The exlanation is that the FFT involves log) additions of uncorrelated errors for each coonent of the result, whereas the direct coutation involves 1 additions of uncorrelated errors. 5. Error in the twiddle factors. As observed above, the twiddle factors can be the doinant source of error. In soe cases the library software for the sine/cosine is inaccurate. For exale, on the Cray Y/MP, it aears that the decision was ade to rovide seed instead of accuracy. Accuracy of standard library routines should be tested; high-accuracy routines should be substituted if the library routines are inaccurate, keeing in ind that seed is generally irrelevant for this alication initiation of the FFT). How to design algoriths that roduce high-accuracy one-half achine esilon) sine/consine functions given floating oint arithetic of only achine esilon accuracy is beyond the scoe of this aer. I have found that it is difficult to obtain highly accurate sine and cosine functions using standard Taylor series or continued fraction exansions with floating hardware that does not rovide double recision interediary roducts. For exale, the IBM RS/6000 rocessors rovide high-accuracy rounded) roducts whereas oular Intel and MIPS rocessors do not. After considerable exerientation, I offer the following conclusions and advice: 1) Eirically, the error in accurate rounded 3- and 64-bit IEEE twiddle factor values as required by the FFT is essentially uniforly distributed on [ ǫ/, ǫ/] for all n between 16 and 131,07. In other words, there is nothing eculiar about the twiddle factors that would uset this standard assution. Therefore, the standard deviation of error in otial twiddle factors is ǫ/ 1.

15 1164 J. C. SCHATZMA ) The error obtained by direct library call to sine/cosine functions with the best libraries I tested is soewhat greater than the otiu nearly ties the otiu. With other libraries the error can be uch greater. Direct ileentation of Taylor series leads to errors with full-rounding floating oint) of nearly 0.7ǫ. Kahan s suation ethod Higha [6]) added to the Taylor series coutation reduces the error to the theoretically otial ǫ/ 1 or below. 3) Errors resulting fro the use of the recursion 34) are tyically all of one sign ore about this below). The oint is that the errors in the sine/consine table when couted this way are definitely not rando that is, not uniforly distributed over [ ǫ/, ǫ/]). A consequence is that overall FFT errors can be uch larger than anticiated. 4) If higher-recision arithetic is available, by all eans I recoend using that higher recision to coute the twiddle factors, which should then be rounded to the desired recision. Of course, arbitrary recision software is available ublic doain) but the slowness ay be objectionable. 5) The sine/cosine tables could be retabulated to high accuracy and recorded in binary for. Once this is done for a given floating oint forat, the code is ortable to achines using that forat. If ileented intelligently, the aount of disk sace required for a colete tabulation for odestly large is quite reasonable. Soe standard FFT ackages use the following recursion for the sine and cosine functions: 34) { Cn = cos θ C n 1 sin θ S n 1, S n = cos θ S n 1 + sin θ C n 1 for n = 1,,..., C 0 = cos θ 0, and S 0 = sin θ 0. Relacing cosθ by the aroxiate value c and sin θ by the aroxiate value s, we obtain 35) C S) n ) c s C = s c S) The eigenvalues of the coefficient atrix are λ = c ± ıs, and. n 1 36) λ = c + s ) e ı tan 1 s c. Alying asytotic analysis to 36), we see that errors in C n and S n grow at first linearly and later exonentially with the length of the recursion. This is highly undesirable; what is needed is a self-correcting recursion in which errors do not grow. While this analysis ignores roundoff error, actual calculations verify this exonential growth. Chu [3] analyses the recursion 34), including the effects of roundoff, and concludes that errors grow linearly. This analysis is erroneous, resulting fro ignoring errors in sines in the analysis of the cosines and vice versa. Oliver [8] gives an excellent review of recursions known at that tie, but does not include 34). Chu [3] describes 34) and any other algoriths; soe of the results of this aer are in error, however. Also, 34) can easily be iroved as follows: A n = cos θ C n 1 sin θ S n 1 + sin θ α, B n = cos θ S n 1 + sin θ C n 1, 37) C n = A n, A n+bn S n = B n, A n+b n

16 ACCURACY OF THE DFT AD FFT 1165 where α deends on and the tye of floating oint hardware. Thus, we correct both the hase and alitude of the sine/cosine air at each ste. This iroved ethod is based on the eirical observations that the errors in 34) are atterned. However, the results of the best recursions are still significantly worse than accurate direct calculations. My recoendation is that the twiddle factors be couted to high accuracy, either with high-recision arithetic or Kahan s suation ethod if Kahan s ethod works effectively on the given hardware). Truncated Taylor s series work very well for this urose. If seed in initialization is required, deending on the achine architecture, either twiddle factors should be recouted and stored on disk or faster converging aroxiations such as rational or continued fraction aroxiations should be used. Finally, if a recursive coutation ust be used for soe reason, 34) should never be used but a ore stable recursion Oliver [8] and Chu [3]) should be eloyed. 6. Two and higher diensions. I have not undertaken a thorough exaination of the two- and higher-diensional roble to verify that the one-diensional results extend in a natural way. However, cursory exaination of the roble suggests that higher-diensional FFTs are even ore rearkably stable, at least if the size of the diensions is large and the nuber of diensions is sall. For exale, 104-by-104 transfors were executed with white rando data using IEEE 64-bit floating oint arithetic. Using the anufacturer s sine/cosine functions and the usual recursions, RMS relative errors of 00ǫ 64 were observed. Using accurate sine/cosine tables, RMS relative errors of less than 3ǫ 64 were observed. These two-diensional transfors were done in the usual way through reeated alication of one-diensional transfors along the rows and coluns. Interestingly, the differences between the rows first/coluns last and coluns first/rows last results were observed to be aroxiately 4ǫ 64 regardless of the accuracy of the sine/cosine table. Finally, averaging the row/colun and colun/row results for the accurate sine/cosine table gave RMS relative errors of aroxiately ǫ 64. This is very little larger than the error for a one-diensional transfor of length ǫ 64 ). It is rearkable that the transfors along the second diension do not increase the error ore than this. To lose only one bit of accuracy after thousands of floating oint oerations draatizes the great stability of the FFT, when ileented carefully! 7. Conclusions. When an accurate sine/cosine table is used, the FFT is rearkably stable. However, the accuracy is greatly iaired by inaccurate sine/cosine tables. Inaccuracy in the sine/cosine table is coon and results fro inaccurate underlying software library functions and ill-advised recursions for table construction. Acknowledgents. The anonyous reviewers were very helful in drawing the author s attention to relevant literature. REFERECES [1] C. S. BURRUS, Index aings for ultidiensional forulation of the DFT convolution, IEEE Trans. Acoustics, Seech, Signal Processing, ), [] D. CALVETTI, A stochastic roundoff error analysis for the fast Fourier transfor, Math. Co., ), [3] C. Y. CHU, The Fast Fourier Transfor on Hyercube Parallel Couters, Technical Reort 87-88, Det. of Couter Science, Cornell University, Ithaca, Y, [4] W. M. GETLEMA AD G. SADE, Fast Fourier transfors For fun and rofit, in 1966 Fall Joint Couter Conference, AFIPS Conf. Proceedings #9, Sartan, Washington, D.C., [5] D. GOTTLIEB AD S. A. ORSZAG, uerical Analysis of Sectral Analysis: Theory and Alications, Society for Industrial and Alied Matheatics, Philadelhia, PA, [6]. J. HIGHAM, The accuracy of floating-oint suation, SIAM J. Sci. Cout., ),

17 1166 J. C. SCHATZMA [7] T. KAEKO AD B. LIU, Accuulation of round-off error in fast Fourier transfors, J. Assoc. Cout. Mach., ), [8] J. OLIVER, Stable ethods for evaluating the oints cosiπ/n), J. Inst. Math. Alic., ), [9] J. SCHATZMA, Fast Fourier transfor algoriths and colexity of the discrete Fourier transfor, Math. Co., subitted. [10], Index aings for the fast Fourier transfor, IEEE Trans. Signal Processing, ). [11] C. TEMPERTO, Self-sorting ixed-radix fast Fourier transfors, J. Cout. Phys., ), [1] C. VA LOA, Coutational Fraeworks for the Fast Fourier Transfor, Society for Industrial and Alied Matheatics, Philadelhia, PA, 199.