A new method mathematically links fast Fourier transform algorithms with fast cyclic convolution algorithms

Parallel Cyclic Convolution Based on Recursive Formulations of Block Pseudocirculant Matrices

“…Our original technique, which splits a 1-D cyclic convolution into subconvolutions, was developed in [2] and [3] by introducing the block pseudocirculant matrix as its foundation.…”

“…In this paper, by formally introducing recursion in conjunction with the original formulation of block pseudocirculant matrices, the full potential of this technique can be developed. The use of recursive formulations was suggested in our original work [2], [3] but, at the time, it was neither formally nor comprehensively developed.…”

An operator-based approach to the unfolding transformation

“…Circulant matrices appear, among others, in the context of circular convolution and were shown in [2] and [3] to be expressible in blocked form by using stride permutations. The resultant matrices are termed Block Pseudocirculants and have circulant sub-blocks that lead to the formulation of parallel cyclic convolution.…”

“…We have shown that stride permutations acting on Circulant Matrices of composite size give Block Pseudocirculant Matrices, [2], [3], [4], [6], whose sub-blocks are themselves circulant and amenable to be processed in parallel. In mathematical notation, with N=r 0 s, it is as follows,…”

“…We do this by noting that matrices associated to cyclic shifts are circulant and, as such, they can be expressed in blocked form by using stride permutations [2], [3], [4]. This allows the circular shift to be realized in parallel.…”