0.9 The prime factor and winograd fourier transform algorithms  (Page 5/5)

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There are other modifications of the basic structure of the Type-1 index map DFT algorithm. One is to use the same indexstructure and conversion of the short DFT's to convolution as the PFA but to use some other method for the high-speed convolution.Table look-up of partial products based on distributed arithmetic to eliminate all multiplications [link] looks promising for certain very specific applications, perhaps for specialized VLSIimplementation. Another possibility is to calculate the short convolutions using number-theoretic transforms [link] , [link] , [link] . This would also require special hardware. Direct calculation of short convolutions is faster on certainpipelined processor such as the TMS-320 microprocessor [link] .

Evaluation of the pfa and wfta

As for the Cooley-Tukey FFT's, the first evaluation of these algorithms will be on the number of multiplications and additionsrequired. The number of multiplications to compute the PFA in [link] is given by Multidimensional Index Mapping: Equation 3 . Using the notation that $T\left(N$ ) is the number of multiplications or additions necessary to calculate alength-N DFT, the total number for a four-factor PFA of length-N, where $N={N}_{1}{N}_{2}{N}_{3}{N}_{4}$ is

$T\left(N\right)={N}_{1}{N}_{2}{N}_{3}T\left({N}_{4}\right)+{N}_{2}{N}_{3}{N}_{4}T\left({N}_{1}\right)+{N}_{3}{N}_{4}{N}_{1}T\left({N}_{2}\right)+{N}_{4}{N}_{1}{N}_{2}T\left({N}_{3}\right)$

The number of multiplications necessary for the WFTA is simply the product of those necessary for the required modules,including multiplications by unity. The total number may contain some unity multipliers but it is difficult to remove them in apractical program. [link] contains both the total number (MULTS) and the number with the unity multiplies removed (RMULTS).

Calculating the number of additions for the WFTA is more complicated than for the PFA because of the expansion of the datamoving through the algorithm. For example the number of additions, TA, for the length-15 example in [link] is given by

$TA\left(N\right)={N}_{2}TA\left({N}_{1}\right)+T{M}_{1}TA\left({N}_{2}\right)$

where ${N}_{1}=3$ , ${N}_{2}=5$ , $T{M}_{1}$ = the number of multiplies for the length-3 module and hence the expansion factor. As mentionedearlier there is an optimum ordering to minimize additions. The ordering used to calculate [link] is the ordering used in [link] , [link] which is optimal in most cases and close to optimal in the others.

 Length PFA PFA WFTA WFTA WFTA N Mults Adds Mults RMults Adds 10 20 88 24 20 88 12 16 96 24 16 96 14 32 172 36 32 172 15 50 162 36 34 162 18 40 204 44 40 208 20 40 216 48 40 216 21 76 300 54 52 300 24 44 252 48 36 252 28 64 400 72 64 400 30 100 384 72 68 384 35 150 598 108 106 666 36 80 480 88 80 488 40 100 532 96 84 532 42 152 684 108 104 684 45 190 726 132 130 804 48 124 636 108 92 660 56 156 940 144 132 940 60 200 888 144 136 888 63 284 1236 198 196 1394 70 300 1336 216 212 1472 72 196 1140 176 164 1156 80 260 1284 216 200 1352 84 304 1536 216 208 1536 90 380 1632 264 260 1788 105 590 2214 324 322 2418 112 396 2188 324 308 2332 120 460 2076 288 276 2076 126 568 2724 396 392 3040 140 600 2952 432 424 3224 144 500 2676 396 380 2880 168 692 3492 432 420 3492 180 760 3624 528 520 3936 210 1180 4848 648 644 5256 240 1100 4812 648 632 5136 252 1136 5952 792 784 6584 280 1340 6604 864 852 7148 315 2050 8322 1188 1186 10336 336 1636 7908 972 956 8508 360 1700 8148 1056 1044 8772 420 2360 10536 1296 1288 11352 504 2524 13164 1584 1572 14428 560 3100 14748 1944 1928 17168 630 4100 17904 2376 2372 21932 720 3940 18276 2376 2360 21132 840 5140 23172 2592 2580 24804 1008 5804 29100 3564 3548 34416 1260 8200 38328 4752 4744 46384 1680 11540 50964 5832 5816 59064 2520 17660 82956 9504 9492 99068 5040 39100 179772 21384 21368 232668

from [link] we see that compared to the PFA or any of the Cooley-Tukey FFT's, the WFTA has significantly fewer multiplications. For the shorterlengths, the WFTA and the PFA have approximately the same number of additions; however for longer lengths, the PFA has fewer and theCooley-Tukey FFT's always have the fewest. If the total arithmetic, the number of multiplications plus the number of additions, iscompared, the split-radix FFT, PFA and WFTA all have about the same count. Special versions of the PFA and WFTA have been developed forreal data [link] , [link] .

The size of the Cooley-Tukey program is the smallest, the PFA next and the WFTA largest. The PFA requires the smallest numberof stored constants, the Cooley-Tukey or split-radix FFT next, and the WFTA requires the largest number. For a DFT of approximately1000, the PFA stores 28 constants, the FFT 2048 and the WFTA 3564. Both the FFT and PFA can be calculated in-place and the WFTA cannot.The PFA can be calculated in-order without an unscrambler. The radix-2 FFT can also, but it requires additional indexing overhead [link] . The indexing and data transfer overhead is greatest for the WFTA because the separate preweave and postweave sections eachrequire their indexing and pass through the complete data. The shorter modules in the PFA and WFTA and the butterflies in the radix2 and 4 FFT's are more efficient than the longer ones because intermediate calculations can be kept in cpu registers rathergeneral memory [link] . However, the shorter modules and radices require more passes through the data for a given approximatelength. A proper comparison will require actual programs to be compiled and run on a particular machine. There are many openquestions about the relationship of algorithms and hardware architecture.

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