On upsampling, downsampling, and rational sampling rate filter banks

Solutions to the problem of design of rational sampling rate filter banks in one dimension has previosly been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. The present paper develops a complete theory for the analysis of arbitrary combinations of upsamplers, downsamplers and filters in multiple dimensions. Although some of the simpler results are well known, the more difficult results concerning swapping upsamplers and downsamplers and variations thereof are new. As an application of this theory, the authors obtain algebraic reductions of the general multidimensional rational sampling rate problem to a multidimensional 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 matrices (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 downsampling operators are derived using the Aryabhatta/Bezout identity over integer matrices as a fundamental tool. A number of new results in the theory of integer matrices that a relevant to the filter bank problem 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