Index Transforms for Multidimensional Discrete Fourier Transforms

Abstract

Index transforms of m-dimensional arrays into n-dimensional arrays play a significant role in many fast algorithms of multivariate discrete Fourier transforms (DFT’s) and cyclic convolutions. Indeed, they provide one of the foundations on which row-column methods or very efficient nesting methods for the fast computation of DFT’s or cyclic convolutions are based (such as the prime factor algorithm [6, pp. 127–133], the Winograd algorithm [8; 6, pp. 133–145] and the Agarwal-Cooley algorithm [1; 6, pp. 43–52]). The general computing scheme for many fast m-dimensional DFT algorithms (convolution methods) is based on the following three essential steps:

• (1) By an index transform of the input data, the m-dimensional DFT (convolution) is transfered into an n-dimensional DFT (convolution) of “short lengths” (n > m).

• (2) By efficient algorithms for one-dimensional DFT’s (convolutions) of short lengths, the n-dimensional DFT (convolution) is computed in parallel (cf. [6]).

• (3) By an index transform of the output data, the desired result of the m-dimensional DFT (convolution) is obtained.