Enumeration and Parametrization of Distinct Downsampling Patterns in Two-Dimensional Multirate Systems

Resampling matrices are essential in multidimensional multirate signal processing. However, different resampling matrices may correspond to the same resampling pattern. It is surely more effective to study such distinct patterns rather than study exhaustively all the possible cases, which is infinite. Motivated by this, we study the resampling patterns for a given decimation ratio in this paper. Our analysis enumerates the number of distinct resampling patterns for any prime decimation ratio, and provides an upper bound on the number for any composite decimation ratio. Parametrization of these distinct patterns is also proposed. Among these patterns, patterns which are nonseparable are also enumerated and parameterized separately