A unified treatment of Cooley-Tukey algorithms for the evaluation of the multidimensional DFT

In this paper the Cooley-Tukey fast Fourier transform (FFT) algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete Fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or Fourier domain. This general algorithm incorporates both the traditional rectangular row-column and vector-radix algorithms as special cases. This FFT algorithm is shown to result from the factorization of an integer matrix; for each factorization of that matrix, a different algorithm can be developed. This paper presents the general algorithm, discusses its computational efficiency, and relates it to existing multi-dimensional FFT algorithms.