Designing commutative cascades of multidimensional upsamplers and downsamplers

In multiple dimensions, the cascade of an upsampler by L and a downsampler by M commutes if and only if the integer matrices L and M are right coprime and LM=ML. This letter presents algorithms to design L and M that yield commutative upsampler/dowsampler cascades. We prove that commutativity is possible if the Jordan canonical form of the rational (resampling) matrix R=LM/sup -1/ is equivalent to the Smith-McMillan form of R. A necessary condition for this equivalence is that R has an eigendecomposition and the eigenvalues are rational.