Linear Bijections and the Fast Fourier Transform

Abstract.

 Fast Fourier transform algorithms rely upon the choice of certain bijective mappings between the indices of the data arrays. The two basic mappings used in the literature lead to Cooley–Tukey algorithms or to prime factor algorithms. But many other bijections also lead to FFT algorithms, and a complete classification of these mappings is provided. One particular choice leads to a new FFT algorithm that generalizes the prime factor algorithm. It has the advantage of reducing the floating point operation count by reducing the number of trigonometric function evaluations.

A certain equivalence relation is defined on the set of bijections that lead to FFT algorithms, and its connection with isomorphism classes of group extensions is studied. Under this equivalence relation every equivalence class contains bijections leading to an FFT algorithm of the new type.