Causal FIR symmetric paraunitary matrix extension and construction of symmetric tight M-dilated framelets

Abstract

We consider a causal FIR symmetric paraunitary (PU) matrix extension with parameters, and the constructions of symmetric orthogonal wavelets and symmetric tight framelets with an integer dilation factor M ⩾ 2 . Firstly, we propose an algorithm for factorizing a causal FIR symmetric Laurent polynomial vector into the product of order-one paraunitary matrices and constant vector. Secondly, based on the factorization algorithm, we propose a method for a causal symmetric PU extension with parameters of a Laurent polynomial vector. This method enables us to parameterize polyphase matrix whose first row is equal to the Laurent polynomial vector composed of polyphase components of given lowpass filter. And this symmetric PU extension provides a minimal factorization structure. Thirdly, we consider the constructions of symmetric orthogonal wavelets and symmetric tight framelets with integer dilation factor by the causal symmetric PU extension. Finally, several examples are provided to illustrate the construction methods proposed in this paper.