8.5: Evaluation of the Cooley-Tukey FFT Algorithms

The evaluation of any FFT algorithm starts with a count of the real (or floating point) arithmetic. The Table 8.5.1 below gives the number of real multiplications and additions required to calculate a length-N FFT of complex data. Results of programs with one, two, three and five butterflies are given to show the improvement that can be expected from removing unnecessary multiplications and additions. Results of radices two, four, eight and sixteen for the Cooley-Tukey FFT as well as of the split-radix FFT are given to show the relative merits of the various structures. Comparisons of these data should be made with the table of counts for the PFA and WFTA programs in The Prime Factor and Winograd Fourier Transform Algorithms . All programs use the four-multiply-two-add complex multiply algorithm. A similar table can be developed for the three-multiply-three-add algorithm, but the relative results are the same.

From the table it is seen that a greater improvement is obtained going from radix-2 to 4 than from 4 to 8 or 16. This is partly because length 2 and 4 butterflies have no multiplications while length 8, 16 and higher do. It is also seen that going from one to two butterflies gives more improvement than going from two to higher values. From an operation count point of view and from practical experience, a three butterfly radix-4 or a two butterfly radix-8 FFT is a good compromise. The radix-8 and 16 programs become long, especially with multiple butterflies, and they give a limited choice of transform length unless combined with some length 2 and 4 butterflies.

N	M1	M2	M3	M5	A1	A2	A3	A5
2	4	0	0	0	6	4	4	4
4	16	4	0	0	24	18	16	16
8	48	20	8	4	72	58	52	52
16	128	68	40	28	192	162	148	148
32	320	196	136	108	480	418	388	388
64	768	516	392	332	1152	1026	964	964
128	1792	1284	1032	908	2688	2434	2308	2308
256	4096	3076	2568	2316	6144	5634	5380	5380
512	9216	7172	6152	5644	13824	12802	12292	12292
1024	20480	16388	14344	13324	30720	28674	27652	27652
2048	45056	36868	32776	30732	67584	63490	61444	61444
4096	98304	81924	73736	69644	147456	139266	135172	135172
4	12	0	0	0	22	16	16	16
16	96	36	28	24	176	146	144	144
64	576	324	284	264	1056	930	920	920
256	3072	2052	1884	1800	5632	5122	5080	5080
1024	15360	11268	10588	10248	28160	26114	25944	25944
4096	73728	57348	54620	53256	135168	126978	126296	126296
8	32	4	4	4	66	52	52	52
64	512	260	252	248	1056	930	928	928
512	6144	4100	4028	3992	12672	11650	11632	11632
4096	65536	49156	48572	48280	135168	126978	126832	126832
16	80	20	20	20	178	148	148	148
256	2560	1540	1532	1528	5696	5186	5184	5184
4096	61440	45060	44924	44856	136704	128514	128480	128480
2	0	0	0	0	4	4	4	4
4	8	0	0	0	20	16	16	16
8	24	8	4	4	60	52	52	52
16	72	32	28	24	164	144	144	144
32	184	104	92	84	412	372	372	372
64	456	288	268	248	996	912	912	912
128	1080	744	700	660	2332	2164	2164	2164
256	2504	1824	1740	1656	5348	5008	5008	5008
512	5688	4328	4156	3988	12060	11380	11380	11380
1024	12744	10016	9676	9336	26852	25488	25488	25488
2048	28216	22760	22076	21396	59164	56436	56436	56436
4096	61896	50976	49612	48248	129252	123792	123792	123792

Table 8.5.1: Number of Real Multiplications and Additions for Complex Single Radix FFTs

In Table 8.2.1 Mi and Ai refer to the number of real multiplications and real additions used by an FFT with i separately written butterflies. The first block has the counts for Radix-2, the second for Radix-4, the third for Radix-8, the fourth for Radix-16, and the last for the Split-Radix FFT. For the split-radix FFT, M3 and A3 refer to the two- butterfly-plus program and M5 and A5 refer to the three-butterfly program.

The first evaluations of FFT algorithms were in terms of the number of real multiplications required as that was the slowest operation on the computer and, therefore, controlled the execution speed. Later with hardware arithmetic both the number of multiplications and additions became important. Modern systems have arithmetic speeds such that indexing and data transfer times become important factors. Morris has looked at some of these problems and has developed a procedure called autogen to write partially straight-line program code to significantly reduce overhead and speed up FFT run times. Some hardware, such as the TMS320 signal processing chip, has the multiply and add operations combined. Some machines have vector instructions or have parallel processors. Because the execution speed of an FFT depends not only on the algorithm, but also on the hardware architecture and compiler, experiments must be run on the system to be used.

In many cases the unscrambler or bit-reverse-counter requires 10% of the execution time, therefore, if possible, it should be eliminated. In high-speed convolution where the convolution is done by multiplication of DFT's, a decimation-in-frequency FFT can be combined with a decimation-in-time inverse FFT to require no unscrambler. It is also possible for a radix-2 FFT to do the unscrambling inside the FFT but the structure is not very regular.

Although there can be significant differences in the efficiencies of the various Cooley-Tukey and Split-Radix FFTs, the number of multiplications and additions for all of them is on the order of \(N\log N\). That is fundamental to the class of algorithms.

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