An in-place and in-order WFTA

TLDR: A hybrid of the nested and the prime factor approach is presented to compute DFT using WFTA (Winograd Fourier transform algorithm) and is highly suitable for a two-level memory hierarchy where most of the computing takes place with a very small data set and the accesses to the main memory are minimized.

Abstract: In this paper, a hybrid of the nested and the prime factor approach is presented to compute DFT using WFTA (Winograd Fourier transform algorithm). A one-dimensional DFT is written as a two-dimensional DFT where DFT along each dimension is computed using the nested form of Winograd. Compared to the nested form, the number of multiplications increase only marginally and there is a corresponding decrease in the number of additions. The algorithm is in-place and in-order requiring very little storage other than the data. The coefficient tables required for this approach are very small. The approach suggested is highly suitable for a two-level memory hierarchy where most of the computing takes place with a very small data set and the accesses to the main memory are minimized. This approach is also highly suitable for a vector processor or a hardware implementation. A general-N FORTRAN program has been developed to compute DTFs of lengths up to 5040 (59 different values) using this approach. For this, the computing times are almost proportional to the transform lengths.