Parallel Cyclic Convolution Based on Recursive Formulations of Block Pseudocirculant Matrices

Abstract

In this paper, we show how novel recursive formulations of block pseudocirculant matrices lead to a new class of parallel cyclic convolution algorithms that exhibit a high degree of regularity and modularity and are suitable for parallel or pipelined implementation into today's very large scale integration (VLSI) circuits, multifield-programmable gate arrays (multi-FPGAs) systems, and multiprocessor architectures. In addition to the architectural advantages, the proposed formulations offer a comparable number of parallel subsections, and a reduction in the number of pre/postprocessing vector operations, for the same range of decimation rates proposed by the most efficient alternative algorithm. The proposed algorithms do not impose any of the traditional constraints, such as the demand that the convolution length be factorable into mutually prime factors. The use of recursion results in the definition of two new mathematical constructs, which are intrinsic to these novel architectures, the higher order block pseudocirculant or superblock pseudocirculant matrix and the block pseudocyclic shift operator that leads to unfolded data-flow graphs of cyclic shifts.Index Terms-Block pseudocirculants, cyclic shift, fast convolution, higher order block pseudocirculants, parallel cyclic convolution, superblock pseudocirculant matrices, unfolded cyclic shift operators.

An operator-based approach to the unfolding transformation

“…The recursive use of stride permutations was further developed in [4] in order to obtain reduced complexity 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,…”

“…By using the block pseudocirculant H p2 instead of the circulant H N different parallel architectures can be defined [4]. If we apply this procedure, in a recursive fashion, to circulant matrices of highly composite lengths we obtain, after two iterations, block pseudocirculants with pseudocirculant sub-blocks.…”

“…If we apply this procedure, in a recursive fashion, to circulant matrices of highly composite lengths we obtain, after two iterations, block pseudocirculants with pseudocirculant sub-blocks. Such constructs are termed Higher Order Block Pseudocirculants or Super Block Pseudocirculants [4] and they lead to reduced complexity parallel cyclic convolution. The constructs introduced in [4] can also be used to obtain reduced complexity parallel FIR filtering architectures.…”

“…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.…”