Commutativity of block decimators and expanders with arbitrary rational sampling ratios and block lengths

It is well known that samplers are linear time varying systems, so in general, the commutativity of samplers does not hold. There are some existing results on the commutativity of conventional decimators and expanders, block samplers with the same integer block lengths but different integer sampling ratios, and block samplers with different integer block lengths and integer sampling ratios. This paper extends the existing results to a necessary and sufficient condition for the commutativity of block decimators and expanders with arbitrary rational sampling ratios and block lengths.