Abstract Algebraic Construction of Biorthogonal Multiwavelets

摘要

Polyphase matrix extension is a fundamental approach for the construction of compactly supported biorthogonal multiwavelets, but there are no explicit formulas available so far. In this paper, based on the canonical forms of polyphase matrices of scaling vectors, a novel abstract algebraic approach over the Laurent polynomial ring (denoted as R[z]) has been developed so that closed-form solution can be obtained for the construction of compactly supported biorthogonal multiwavelets. Moreover, via the multiplications of unimodular square matrices over R[z], the relationship between any two different extensions for the same scaling vectors can be obtained from one to another, which leads to a complete solution set for the polyphase matrix extension problem.