New Look on <inline-formula> <tex-math notation="LaTeX">$q2^r$</tex-math> </inline-formula>-Point Fast Fourier Transforms

Abstract

In this paper, the method of splitting orparallelization of calculation of the N-point discrete Fourier transform (DFT) by the DFTs of smaller orders is described. For that the concept of partitions revealing the one-dimensional (1-D) DFT of order q2 r , where r > 1and q > 1 is a positive odd number, is described. Two different partitions are considered and the corresponding effective algorithms of calculation of the q2 r -point 1-D DFT are described for the q = 3 case. The calculation of the transform is based on the paired representations, when the signal is represented as a set of 1-D signals which define the 1-D DFT in disjoint subsets of frequency-points which cover the set of all frequencies. The splittings of the q2 r -point 1-D DFT are performed by the 1-D discrete 2and qpaired transforms which allow for calculating with a minimum number of operations. The examples of the paired transforms and computational complexity of the proposed algorithms for the N = 6 and 12 cases are given.