Matrix Extension with Symmetry and Its Application to Filter Banks

TL;DR

This paper completely solves the matrix extension problem with symmetry, providing algorithms to construct symmetric paraunitary matrices and filter banks, with applications to wavelets and filter design.

Contribution

It introduces a complete solution and algorithms for matrix extension with symmetry, enabling construction of symmetric paraunitary matrices and filter banks.

Findings

Provides a step-by-step algorithm for matrix extension with symmetry.

Develops a cascade structure for representing paraunitary matrices with symmetry.

Offers algorithms for constructing symmetric filter banks and multiwavelets.

Abstract

In this paper, we completely solve the matrix extension problem with symmetry and provide a step-by-step algorithm to construct such a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Furthermore, using a cascade structure, we obtain a complete representation of any $r\times s$ paraunitary matrix $\mathsf{P}$ having compatible symmetry, which in turn leads to an algorithm for deriving a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Matrix extension plays an important role in many areas such as electronic engineering, system sciences, applied mathematics, and pure mathematics. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric paraunitary filter banks and symmetric orthonormal multiwavelets by deriving high-pass filters with symmetry from any given low-pass filters with symmetry. Several examples are provided to illustrate the proposed algorithms and results in this paper.