The techniques of the generalized fast Fourier transform algorithm

A general method of deriving DFT (discrete Fourier transform) algorithms, generalised fast Fourier transform algorithms, is presented. It is shown that a special case of the method is equivalent to nesting of FFTs. The application of the method to the case where N has mutually prime factors results in a new interpretation of the permutations characteristic of this class of algorithms. It is shown that the equalization of FFTs leads to results which are different from the widely used intuitive ones. The high efficiency of split-radix FFTs is explained. It is shown that the formulae of the method can be easily adapted for deriving algorithms for the cosine/sine DFT. A set of FFTs that has smaller arithmetical and/or memory complexities than any algorithm known is presented. In particular, a method of deriving split-radix-2/sup s/ FFTs requiring N log/sub 2/ N-3N+4 real multiplications and 3N log/sub 2/ N-3N+4 additions for any s>1 is presented.