Design of a Simple Orthogonal Multiwavelet Filter by Matrix Spectral Factorization

Abstract

We consider the design of an orthogonal symmetric/antisymmetric multiwavelet from its matrix product filter by matrix spectral factorization (MSF). As a test problem, we construct a simple matrix product filter with desirable properties, and factor it using Bauer's method, which in this case can be done in closed form. The corresponding orthogonal multiwavelet function is derived using algebraic techniques which allow symmetry to be considered. This leads to the known orthogonal multiwavelet SA1, which can also be derived directly. We also give a lifting scheme for SA1, investigate the influence of the number of significant digits in the calculations, and show some numerical experiments.Keywords orthogonal multiwavelets · matrix spectral factorization · Bauer's algorithm · Youla and Kazanjian algorithm · lifting scheme · PLUS matrices · Alpert multiwavelet 1 Introduction and Problem Formulation A digital filter bank is a collection of r filters F k (z), k = 0, . . . , r − 1, which split a discretetime signal into r subbands. These subband signals are usually decimated by a factor of r.