Parallel and Pipelined Architectures for Cyclic Convolution by Block Circulant Formulation Using Low-Complexity Short-Length Algorithms

Abstract

Fully pipelined parallel architectures are derived for high-throughput and reduced-hardware realization of prime-factor cyclic convolution using hardware-efficient modules for short-length rectangular transform (RT). Moreover, a new approach is proposed for the computation of block pseudocyclic convolution using a block cyclic convolution of equal length along with some correction terms, so that the block pseudocyclic representation of cyclic convolution for non-prime-factor-length ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rP</i> , when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> are not mutually prime) could be computed efficiently using the algorithms and architectures of short-length cyclic convolutions. Low-complexity algorithms are derived for efficient computation of those error terms, and overall complexities of the proposed technique are estimated for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> =2, 3, 4, 6, 8 and 9. The proposed algorithms are used further to design high-throughput and reduced-hardware structures for cyclic convolution where the cofactors are not relatively prime. The proposed structures for high-throughput implementation are found to offer a reduction of nearly 50%-75% of area-delay product over the existing structures for several convolution-lengths. Low-complexity structures for input/output addition units of short length convolutions are derived and used them along with high-throughput modules for hardware-efficient realization of multifactor convolution, which offers nearly 25%-75% reduction of area-delay complexity over the existing structures for various non-prime-factor length convolutions.