Lattice factorization based symmetric PMI paraunitary matrix extension and construction of symmetric orthogonal wavelets

Abstract

In this paper, we propose a lattice factorization based matrix extension method for constructing the causal FIR symmetric paraunitary filter banks (PUFBs) whose filters Hkz,k=0,1,…,M−1 satisfy the pairwise mirror image (PMI) property, i.e. the condition Hkz=HM−1−k−z,k=0,1,…,M−1. And, based on the extension method, we provide a method for constructing compactly supported symmetric orthogonal wavelets. Firstly, for a given symmetric real-valued M-orthogonal filter H0(z), we propose an algorithm for factorizing a Laurent polynomial matrix composed of polyphase components of the filter pair {H0(z),H0(−z)} into the product of lattice factors and constant matrix. Secondly, based on the lattice factorization algorithm, we propose a method for the causal symmetric PU extension with PMI property of the given Laurent polynomial matrix. This method provides a lattice structure for fast implementation of the resulting symmetric PMI PUFB. Thirdly, we provide a method for constructing compactly supported symmetric orthogonal wavelets by the causal symmetric PMI PU extension. Lastly, several examples are provided to illustrate the construction method proposed in this paper.