ON UPSAMPLING, DOWNSAMPLING, AND RATIONAL SAMPLING RATE FILTER BANKS

Recently, solutions to the problem of design of rational sampling rate filter banks in one dimension has been proposed. The ability to inter change the operations of upsampling, downsampling, and filtering plays an important role in these solutions. This paper develops a complete theory for the analysis of arbitrary combinations of upsamplers, downs amplers and filters in multiple dimensions. Although some of the simpl er results are well known, the more difficult results concerning swapp ing upsamplers and downsamplers and variations thereof are new. As an application of this theory, we obtain algebraic reductions of the gene ral multidimensional rational sampling rate problem to a multidimensio nal uniform filter bank problem. However, issues concerning the design of the filters themselves are not addressed. In multiple dimensions, upsampling and downsampling operators are determined by integer matric es (as opposed to scalars in one dimension), and the noncommutativity of matrices makes the problem considerably more difficult. Cascades of upsamplers and downsamplers in one dimension are easy to analyze. The new results for the analysis of multidimensional upsampling and downs ampling operators are derived using the Aryabhatta/Bezout identity ove r integer matrices as a fundamental tool. A number of new results in t he theory of integer matrices that a relevant to the filter bank probl em are also developed. Special cases of some of the results pertaining to the commutativity of upsamplers/downsamplers have been obtained in parallel by several authors.