Enumeration of Downsampling Lattices in Two-Dimensional Multirate Systems

Lattices are essential for resampling in two-dimensional multirate systems. However, the resampling matrix for a given lattice is not unique. For a given resampling ratio, how many different lattices exist and how to parameterize them have not been addressed before. In this correspondence, such enumeration and parametrization are presented for the cases where the resampling ratio is prime or a composite number. The set of nonseparable resamplers is also studied. Examples and an application of the proposed enumeration and parametrization are included for demonstration.