A novel development for parallel cyclic convolution: The super block pseudocirculant matrix

As opposed to prime factor type algorithms, the only requirement made by parallel cyclic convolution techniques based on block pseudocirculant matrices is that the convolution length be composite. Highly composite lengths, in particular, give a larger variety of implementation choices. In this paper we offer an introduction to new mathematical constructs, the super block pseudocirculant matrix, and the block pseudocyclic shift operator, as a base to derive further structures for this important class of parallel, one dimensional, cyclic convolution algorithms based on block pseudocirculant matrices. Their modular composition makes them suitable for implementation in VLSI, FPGA or multiprocessor computers in either a pipelined or a parallel fashion. Block pseudocirculants appear in fields such as precoding systems, transmultiplexers, polyphase networks, block filtering, QMF banks, and others, therefore the new mathematical constructs introduced in this paper may have an impact that transcend its sole applications to parallel cyclic convolution and its related applications.