A novel general formulation of up/downsampling commutativity

The paper establishes the commutativity conditions of upsampling and downsampling for multidimensional signals defined on discrete Abelian groups (lattices). The general condition includes results available in the literature as particular cases. Complete generality is achieved by giving an abstract definition of up/downsampling and by working in the signal domain instead of the traditional approach based on z and Fourier transforms. Examples of applications are outlined 1) for the one-dimensional (1-D) case verifying existing results, 2) for television scanning, 3) for degenerate (reduced-dimensionality) lattices, and 4) for multiplicative groups.