A multivariate polynomial matrix-factorization algorithm is introduced and discussed. This algorithm and another algorithm for computing a globally minimal generating matrix of the syzygy of solutions associated with a polynomial matrix are bothassociated with a zero-coprimeness constraint that characterizes perfect-reconstruction filter banks. Generalizations, as well as limitations of recent results which incorporate the perfect reconstruction as well as the linear-phase constraints, arediscussed with several examples and counterexamples. Specifically, a Grobner basis-based proof for perfect reconstruction with linear phase is given for the case of two-band multidimensional filter banks, and the algorithm is illustrated by a nontrivialdesign example. Progress and bottlenecks in the multidimensional multiband case are also reported.
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