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.

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

In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices.

Fast algorithms for the multidimensional discrete Fourier transform

The split-radix approach for computing the discrete Fourier transform (DFT) is extended for the vector-radix fast Fourier transform (FFT) to two and higher dimensions. It is obtained by further splitting the (N/2*N/2) transforms with twiddle factors in the radix (2*2) FFT algorithm. The generalization of this split vector-radix FFT algorithm to higher radices and higher dimensions is also presented. By introducing a general approach for constructing the fast Hartley transform (FHT) from the corresponding FFT, new vector- and split-vector-radix FHT algorithms with the same desirable properties as their FFT counterparts are obtained. >

Split vector-radix fast Fourier transform

Methodologies for constructing fast algorithms to compute the discrete Fourier transform (DFT) of a 2-D real sequence are introduced. The resulting algorithms are shown to be in-place and butterfly-style as well as the usual 1-D FFT algorithms. Above all, the computational load of these algorithms is reduced to less than one-half of their complex counterparts. Due to the in-place property, the storage requirement is exactly halved. A comparison is made on the basis of arithmetic complexity, storage, and input/output requirements. >

In-place butterfly-style FFT of 2-D real sequences

Floating-point error is conducted for three algorithms commonly used for the calculation of two-dimensional fast Fourier transforms (FFTs), namely, the conventional row-column FFT, the vector-radix FFT, and the polynomial-transform FFT The respective errors are determined both analytically and on the basis of computer simulation Comparison shows that the vector-radix FFT and the polynomial-transform FFT, even though computationally more efficient than the row-column FFT, show approximately the same (and sometimes reduced) susceptibility to errors in floating-point arithmetic >

Floating point error analysis of two-dimensional, fast Fourier transform algorithms

Conventional two dimensional fast Fourier transforms become very slow if the size of the matrix becomes too large to be contained in memory. This is due to the transposition of the matrix that is required. This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process. This algorithm is extremely valuable on small disc based computers.

W. Berry

Papers

An algorithm for the two dimensional FFT