Commutativity of D-dimensional decimation and expansion matrices, and application to rational decimation systems

Multidimensional (MD) multirate systems, which find applications in the coding and compression of image and video data, have attracted much attention. The basic building blocks in a MD multirate system are the decimation matrix M, the expansion matrix L, and MD digital filters. With D denoting the number of dimensions, M and L are D×D nonsingular integer matrices. When these matrices are diagonal, most of the one-dimensional multirate results can be extended automatically. However, for the nondiagonal case, these extensions are nontrivial. One example of this nature is the commutativity of MD decimation and expansion matrices. Using the concepts of coprimeness and least common right/left multiples of integer matrices, a set of necessary and sufficient conditions is derived for a decimation matrix and an expansion matrix to commute. This commutativity is also used to derive an efficient polyphase implementation of an MD decimation system with rational decimation matrix