Rules for multidimensional multirate structures

Abstract

Identifies a comprehensive set of compact rules and efficient algorithms for simplifying and rearranging structures common in multidimensional multirate signal processing. The authors extend the 1D rules reported by Crochiere and Rabiner (1983), especially the many equivalent forms of cascades of upsamplers and downsamplers. They also include rules reported by other authors for completeness. The extension to mD is based primarily on the Smith form decomposition of resampling (nonsingular integer square) matrices. The Smith form converts non-separable multidimensional operations into separable ones by means a shuffling of input samples and a reshuffling of the separable operations. Based on the Smith form, the authors have developed algorithms for 1) computing coset vectors 2) finding greatest common sublattices 3) simplifying cascades of up/downsampling operations. The algorithms and rules are put together in a form that can be implemented efficiently in a symbolic algebra package. The authors have encoded the knowledge in the commercially available Mathematica environment. >