The role of integer matrices in multidimensional multirate systems

The basic building blocks in a multidimensional (MD) multirate system are the decimation matrix M and the expansion matrix L. For the D-dimensional case these a re D X D nonsingular integer matrices. When these matrices a re diagonal, most of the one-dimensional (ID) results can be extended automatically. However, for the nondiagonal case, these extensions are nontrivial. Some of these extensions, e.g., polyphase decomposition and maximally decimated perfect reconstruction systems, have already been successfully made by some authors. However, there exist several I D results in multirate processing, for which the multidimensional extensions are even more difficult. An example is the development of polyphase representation for rational (rather than integer) sampling rate alterations. In the ID case, this development relies on the commutativity of decimators and expanders, which is possible whenever M and L a r e relatively prime (coprime). The conditions for commutativity in the two-dimensional (2D) case have recently been developed successfully in [l]. In the MD case, the results a re more involved. In this paper we formulate and solve a number of problems of this nature. O u r discussions are based on several key properties of integer matrices, including greatest common divisors and least common multiples, which we first review. These properties a re analogous to those of polynomial matrices, some of which have been used in system theoretic work (e.g., matrix fraction descriptions, coprime matrices, Smith form, and so on).