# A modified split-radix FFT with fewer arithmetic operations

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3 PUBLISHED I IEEE TRAS. SIGAL PROCESSIG 55 (1, ( but none of this affects the flop count. Although in this paper we consider only decimation-in-time (DIT decompositions, a dual decimation-in-frequency (DIF algorithm can always be obtained by networ transposition 4 (reversing the flow graph of the computation with identical operation counts. III. EW FFT: RESCALIG THE TWIDDLES The ey to reducing the number of operations is the observation that, in Algorithm 1, both and (the -th outputs of the sie-/4 sub-transforms are multiplied by a twiddle factor ω or ω before they are used to find y. This means that we can rescale the sie-/4 sub-transforms by any factor 1/s /4, desired, and absorb the scale factor into ω s /4, at no cost. So, we merely need to find a rescaling that will save some operations in the sub-transforms. (As is conventional in counting FFT operations, we assume that all data-independent constants lie ω s /4, are precomputed and are therefore not included in the flops. Moreover, we rely on the fact that and have conjugate twiddle factors in the conjugate-pair algorithm, so that a single rescaling below will simplify both twiddle factors to save operations this is not true for the ω and ω 3 factors in the usual split radix. Below, we begin with an outline of the general ideas, and then analye the precise algorithm in Sec. IV. Consider a sub-transform of a given sie that we wish to rescale by some 1/s, for each output y. Suppose we tae s, = s,+/4 = cos(2π/ for /8. In this case, y from Algorithm 1 becomes y u /s, + (t, + t,, where t, = 1 i tan(2π/ = ω / cos(2π/. (6 Multiplying ω requires 4 real multiplications and 2 real additions (6 flops for general, but multiplying t, requires only 2 real multiplications and 2 real additions (4 flops. (A similar rescaling was proposed [30] to increase the number of fused multiply-add operations, and an analogous rescaling also relates the Givens and fast Givens QR algorithms [31]. Thus, we have saved 4 real multiplications in computing t, ± t,, but spent 2 real multiplications in u /s, and another 2 for u +/4 /s,, for what may seem to be no net change. However, instead of computing u /s, directly, we can instead push the 1/s, scale factor down into the recursive computation of u. In this way, it turns out that we can save most of these extra multiplications by combining them with twiddle factors inside the /2 transform. Indeed, we shall see that we need to push 1/s, down through two levels of recursion in order to gain all of the possible savings. Moreover, we perform the rescaling recursively, so that the sub-transforms and are themselves rescaled by 1/s /4, for the same savings, and the product of the sub-transform scale factors is combined with cos(2π/ and pushed up to the top-level transform. The resulting scale factor s, is given by the following recurrence, where we let 4 = mod /4: 4 etwor transposition is equivalent to matrix transposition [29] and preserves both the DFT (a symmetric matrix and the flop count (for equal numbers of inputs and outputs, but changes DIT into DIF and vice versa. scale factor Fig. 1. Scale factor s,, from Eq. (7, vs. one period of for = 2 12 = s =2m, = 1 for 4 s /4,4 cos(2π 4 / for 4 /8 s /4,4 sin(2π 4 / otherwise, (7 which has an interesting fractal pattern plotted in Fig. 1. This definition has the properties: s,0 = 1, s,+/4 = s,, and s,/4 = s, (a symmetry whose importance appears in subsequent sections. We can now generally define: t, = ω s /4, /s,, (8 where s /4, /s, is either sec or csc and thus t, is always of the form ±1±i tan or ± cot ±i. This last property is critical because it means that we obtain t, ± t, in all of the scaled transforms and multiplication by t, requires at most 4 flops as above. Rather than elaborate further at this point, we now simply present the algorithm, which consists of four mutuallyrecursive split-radix lie functions listed in Algorithms 2 3, and analye it in the next section. As in the previous section, we omit for clarity the special-case optimiations for = 0 and = /8 in the loops, as well as the trivial base cases for = 1 and = 2. IV. OPERATIO COUTS Algorithms 2 and 3 manifestly have the same number of real additions as Algorithm 1 (for 4/2 mult/add complex multiplies, since they only differ by real multiplicative scale factors. So, all that remains is to count the number M( of real multiplications saved compared to Algorithm 1, and this will give the number of flops saved over Yavne. We must also count the numbers M S (, M S2 (, and M S4 ( of real multiplications saved (or spent, if negative in our three rescaled sub-transforms. In newfft (x itself, the number of multiplications is clearly the same as in splitfft (x, since all scale factors are absorbed into the twiddle factors note that s /4,0 = 1 so the = 0 special case is not worsened either and thus the savings come purely in the subtransforms: M( = M(/2 + 2M S (/4. (9

4 PUBLISHED I IEEE TRAS. SIGAL PROCESSIG 55 (1, ( Algorithm 2 ew FFT algorithm of length (divisible by 4. The sub-transforms newffts /4 (x are rescaled by s /4, to save multiplications. The sub-sub-transforms of sie /8, in turn, use two additional recursive subroutines from Algorithm 3 (four recursive functions in all, which differ in their rescalings. function y =0.. 1 newfft (x n : {computes DFT} u 2=0.../2 1 newfft /2 (x 2n2 4=0.../4 1 newffts /4 (x 4n4+1 newffts 4=0.../4 1 /4 (x 4n4 1 for = 0 to /4 1 do y u + ( ω s /4, + ω s /4, y +/2 u ( ω s /4, + ω s /4, y +/4 u +/4 i ( ω s /4, ω s /4, y +3/4 u +/4 + i ( ω s /4, ω s /4, function y =0.. 1 newffts (x n : {computes DFT / s, } u 2=0.../2 1 newffts2 /2 (x 2n2 4=0.../4 1 newffts /4 (x 4n4+1 newffts 4=0.../4 1 /4 (x 4n4 1 for = 0 to /4 ( 1 do y u + t, + t, ( y +/2 u t, + t, ( y +/4 u +/4 i y +3/4 u +/4 + i t, t, ( t, t, In newffts (x, as discussed above, the substitution of t, for ω means that 2 real multiplications are saved from each twiddle factor, or 4 multiplications per iteration of the loop. This saves multiplications, except that we have to tae into account the = 0 and = /8 special cases. For = /8, t, = 1 i, again saving two multiplications (by 1/ 2 per twiddle factor. Since t,0 = 1, however, the = 0 special case is unchanged (no multiplies, so we only save 4 multiplies overall. Thus, M S ( = M S2 (/2 + 2M S (/ (10 At first glance, the newffts2 (x routine may seem to have the same number of multiplications as splitfft (x, since the 2 multiplications saved in each t, (as above are exactly offset by the s, /s 2, multiplications. (ote that we do not fold the s, /s 2, into the t, because we also have to scale by s, /s 2,+/4 and would thus destroy the t, common sub-expression. However, we spend 2 extra multiplications in the = 0 special case, which ordinarily requires no multiplies, since s,0 /s 2,/4 = 1/s 2,/4 ±1 (for > 2 appears in y /4 and y 3/4. Thus, M S2 ( = M S4 (/2 + 2M S (/4 2. (11 Algorithm 3 Rescaled FFT subroutines called recursively from Algorithm 2. The loops in these routines have more multiplications than in Algorithm 1, but this is offset by savings from newffts /4 (x in Algorithm 2. function y =0.. 1 newffts2 (x n : {computes DFT / s 2, } u 2=0.../2 1 newffts4 /2 (x 2n2 4=0.../4 1 newffts /4 (x 4n4+1 newffts 4=0.../4 1 /4 (x 4n4 1 for = 0 to /4 ( 1 do y u + + i t, + t, y +/2 u y +/4 ( u +/4 i t, t, y +3/4 ( u +/4 t, t, ( t, + t, (s, /s 2, (s, /s 2, (s, /s 2,+/4 (s, /s 2,+/4 function y =0.. 1 newffts4 (x n : {computes DFT / s 4, } u 2=0.../2 1 newffts2 /2 (x 2n2 4=0.../4 1 newffts /4 (x 4n4+1 newffts 4=0.../4 1 /4 (x 4n4 1 for = 0[ to /4 ( 1 do ] y u + t, + t, (s, /s 4, [ ( ] y +/2 u t, + t, [ (s, /s 4,+/2 ( ] y +/4 u +/4 i t, t, [ (s, /s 4,+/4 ( ] y +3/4 u +/4 + i t, t, (s, /s 4,+3/4 Finally, the routine newffts4 (x involves Θ( more multiplications than ordinary split-radix, although we have endeavored to minimie this by proper groupings of the operands. We save 4 real multiplications per loop iteration because of the t, replacing ω. However, because each output has a distinct scale factor (s 4, s 4,+/2 s 4,+/4 s 4,+3/4, we spend 8 real multiplications per iteration, for a net increase of 4 multiplies per iteration. For the = 0 iteration, however, the t,0 = 1 gains us nothing, while s 4,0 = 1 does not cost us, so we spend 6 net multiplies instead of 4, and therefore: M S4 ( = M S2 (/2 + 2M S (/4 2. (12 Above, we omitted the base cases of the recurrences, i.e. the = 1 or 2 that we handle directly as in Sec. II (without recursion. There, we find M( 2 = M S ( 2 = M S2 ( 2 = M S4 (1 = 0 (where the scale factors are unity, and M S4 (2 = 2. Finally, solving these recurrences

7 PUBLISHED I IEEE TRAS. SIGAL PROCESSIG 55 (1, ( TABLE III FLOPS REQUIRED FOR THE DISCRETE COSIE TRASFORM (DCT BY PREVIOUS ALGORITHMS AD BY OUR EW ALGORITHM, DCT type Previous best ew algorithm 16, DCT-II , DCT-II , DCT-II , DCT-II , DCT-IV , DCT-IV , DCT-IV , DCT-IV , DCT-IV , DCT-I , DCT-I , DCT-I proved sufficiently powerful to automatically derive optimalarithmetic real-data FFTs from the corresponding optimal complex-data algorithm it merely imposes the appropriate input/output symmetries and prunes redundant outputs and computations [33]. Given our new FFT, we find that it can again automatically derive a real-data algorithm matching the predicted flop count of Eq. (15. VIII. DISCRETE COSIE TRASFORMS Similarly, our new FFT algorithm can be specialied for DFTs of real-symmetric data, otherwise nown as discrete cosine transforms (DCTs of the various types [40] (and also discrete sine transforms for real-antisymmetric data. Moreover, since FFTW s generator can automatically derive algorithms for types I IV of the DCT and DST [3], we have found that it can automatically realie arithmetic savings over the best-nown DCT/DST implementations given our new FFT, as summaried in Table III. 5 Although here we exploit the generator and have not derived explicit general- algorithms for the DCT flop count (except for type I, the same basic principles (expressing the DCT as a larger DFT with appropriate symmetries and pruning redundant computations from an FFT have been applied to manually derive DCT algorithms in the past and we expect that doing so with the new algorithm will be straightforward. Below, we consider types II, III, IV, and I of the DCT. A type-ii DCT (often called simply the DCT of length is derived from a real-data DFT of length 4 with appropriate symmetries. Therefore, since our new algorithm begins to yield improvements starting at = 64 for real/complex data, it yields an improved DCT-II starting at = 16. Previously, a 16-point DCT-II with 112 flops was reported [11] for the (unnormalied -point DCT-II defined as: y = 1 n=0 [ π(n x n cos ] { 1 for = 0 2 otherwise, (16 whereas our generator now produces the same transform with only 110 flops. In general, 2 lg flops were required for 5 The precise multiplication count for a DCT generally depends upon the normaliation convention that is chosen; here, we use the same normaliations as the references cited for comparison. this DCT-II [41] ( > 1, as can be derived from the standard split-radix approach [42] (and is also reproduced automatically by our generator starting from complex split radix, whereas the flop counts produced by our generator starting from our new FFT are given in Table III. The DCT-III (also called the IDCT since it inverts DCT-II is simply the transpose of the DCT-II and its operation counts are identical. It is also common to compute a DCT-II with scaled outputs, e.g. for the JPEG image-compression standard where an arbitrary scaling can be absorbed into a subsequent quantiation step [43], and in this case the scaling can save 6 multiplications [11] over the 40 flops required for an unscaled 8- point DCT-II. Since our newffts (x attempts to be the optimal scaled FFT, we should be able to derive this scaled DCT-II by using it in the generator instead of newfft (x indeed, we find that it does save exactly 6 multiplies over our unscaled result (after normaliing the DFT by an overall factor of 1/2 due to the DCT symmetry. Moreover, we can now find the corresponding scaled transforms of larger sies: e.g. 96 flops for a sie-16 scaled DCT-II, and 252 flops for sie 32, saving 14 and 30 flops, respectively, compared to the unscaled transform above. For the DCT-IV, which is the basis of the modified discrete cosine transform (MDCT [44], the corresponding symmetric DFT is of length 8, and thus the new algorithm yields savings starting at = 8: the best (split-radix methods for an 8-point DCT-IV require 56 flops (or 2 lg + [42] for the DCT-IV defined by y = 1 2 n=0 [ π(n + 1 x n cos 2 ( ], (17 whereas the new algorithm requires 54 flops for = 8 (as derived by our generator, with other sies shown in Table III. Finally, a type-i DCT of length (with + 1 data points defined as 1 y = x 0 + ( 1 x + 2 x n cos n=1 [ ] πn (18 is exactly equivalent to a DFT of length 2 where the input data are real-symmetric (x 2 n = x n, and the split-radix FFT adapted for this symmetry requires 2 lg lg +5 flops [38]. 6 Because the scale factors s, preserve this symmetry, one can employ exactly the same approach to save M(2/4 multiplications starting from our new FFT (proof is identical. Indeed, precisely these savings are derived by the FFTW generator for the first few, as shown in Table III. IX. COCLUDIG REMARKS The longstanding arithmetic record of Yavne for the powerof-two DFT has been broen, but at least two important questions remain unanswered. First, can one do better still? Second, will the new algorithm result in practical improvements to actual computation times for the FFT? 6 Our count is slightly modified from that of Duhamel [38], who omitted all multiplications by 2 from the flops.

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