Evaluation of two-dimensional discrete Fourier transforms via generalized FFT algorithms

In this paper two-dimensional fast Fourier transforms (FFT´s) are expressed as special cases of a generalization of the one-dimensional Cooley-Tukey algorithm. This generalized algorithm allows the efficient evaluation of discrete Fourier transforms (DFT´s) of rectangularly sampled sequences, hexagonally sampled sequences and arbitrary periodically sampled sequences. Significant computational savings can be realized using this generalized algorithm when the periodicity matrix of the sequence is highly composite. Alternate factorizations of the periodicity matrix lead to different FFT algorithms, including the row-column decomposition and the vector-radix algorithm. This paper will present a generalized DFT, derive the general 2-D Cooley-Tukey algorithm and conclude by interpreting several 2-D FFT algorithms in terms of the generalized one.