# 8.5: Evaluation of the Cooley-Tukey FFT Algorithms

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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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