Calculating the n-dimensional fast Fourier transform

Abstract

The one-dimensional fast Fourier transform (FFT) is the most popular tool for calculating the multidimensional Fourier transform. As a rule, to estimate the n-dimensional FFT, a standard method of combining one-dimensional FFTs, the so-called "by rows and columns" algorithm, is used in the literature. For fast calculations, different researchers try to use parallel calculation tools, the most successful of which are searches for the algorithms related to the computing device architecture: cluster, video card, GPU, etc. [1, 2]. The possibility of paralleling another algorithm for FFT calculation, which is an n-dimensional analog of the Cooley-Tukey algorithm [3, 4], is studied in this paper. The focus is on studying the analog of the Cooley-Tukey algorithm because the number of operations applied to calculate the n-dimensional FFT is considerably less than in the conventional algorithm nN n log2 N of addition operations and 1/2N n + 1log2 N of multiplication operations of addition operations and $$\frac{{2^n - 1}} {{2^n }}N^n \log _2 N$$ of multiplication operations against: N n + 1log2 N of addition operations and 1/2N n + 1log2 N of in combining one-dimensional FFTs.