Matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields

Abstract

Algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field Q. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. In this paper, we properly formulate the matrix splitting problem over any general subfield of it C, including an algebraic number field as a special case, and provide a simple necessary and sufficient condition for a 2×2 matrix of Laurent polynomials with symmetry to be able to be factorized by a 2×2 matrix of Laurent polynomials with certain symmetry structure. We propose an effective algorithm on how to obtain the factorization matrix step by step. As an application, we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper.