Notes on the FFT

This is a note describing results on eecient algorithms to calculate the discrete Fourier transform (DFT). The purpose is to report work done at Rice University, but other contributions used by the DSP research group at Rice are also cited. Perhaps the most interesting is the discovery that the Cooley-Tukey FFT was described by Gauss in 1805 1]. That gives some indication of the age of research on the topic, and the fact that a recently compiled bibliography 2] on eecient algorithms contains over 3400 entries indicates its volume. Three IEEE Press reprint books contain papers on the FFT 3, 4, 5]. An excellent general purpose FFT program has been described in 6, 7] and is available over the internet. that give a good modern theoretical background for this work, one book 17] that gives the basic theory plus both FORTRAN and TMS 320 assembly language programs, and other books 18, 19, 20] that contains a chapter on advanced FFT topics. The history of the FFT is outlined in 21, 1] and excellent survey articles can be found in 22, 23]. The foundation of much of the modern work on eecient algorithms was done by S. Winograd. This can be found in 24, 25, 26]. An outline and discussion of his theorems can be found in 18] as well as 8, 9, 10, 11]. EEcient FFT algorithms for length-2 M were described by Gauss and discovered in modern times by Cooley and Tukey 27]. These have been highly developed and good examples of FORTRAN programs can be found in 17]. Several new algorithms have been published that require the least known amount of total seems to have the best structure for programming, and an eecient program has been written 35] to implement it. A mixture of decimation-in-time and decimation-in-frequency with very good eeciency is given in 36]. Theoretical bounds on the number of multiplications required for the FFT based on Winograd's theories are given in 11, 37]. Schemes for calculating an in-place, in-order radix-2 FFT are A discussion of the relation of the computer architecture, algorithm and compiler can be found in 50, 51]. The \other" FFT is the prime factor algorithm (PFA) which uses an index map originally developed by Thomas and by Good. The theory of the PFA was derived in 52] and further developed and an eecient in-order and in-place program given in 53, 17]. More results …