Construction of conjugate quadrature filters with specified zeros
type: Item
instance of: scholarly article
title: Construction of conjugate quadrature filters with specified zeros
zbMATH Open document ID: 0902.65008
DOI: 10.1023/A:1019137702681
author: Charles A. Micchelli
author: Wayne Lawton
published in: Numerical Algorithms
publication date: 1998-12-02T00:00:00Z
review text: Let \({\mathcal L}\) denote the ring of complex-valued Laurent polynomial functions on \(\mathbb{C} \backslash \{0\}\). Let \({\mathcal L}_R \subset {\mathcal L}\) denote the set of Laurent polynomials whose restriction to the unit circle is real valued and let \({\mathcal L}_N \subset {\mathcal L}\) be the set of Laurent polynomials whose restriction to the unit circle is non-negative. The authors prove that for any two Laurent polynomials \(P_1\), \(P_2\) \(\in {\mathcal L}_N\), which have no common zeros in \(\mathbb{C} \backslash \{0\}\) there exists a pair of Laurent polynomials \(Q_1\), \(Q_2 \in {\mathcal L}_N\) satisfying the equation \(Q_1 P_1 + Q_2 P_2 = 1\). The authors provide some information about the minimal length Laurent polynomials \(Q_1\) and \(Q_2\) with this property and describe an algorithm to compute them. The authors apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset \(\Lambda \subset C \backslash \{0\}\) with the property that \((\Lambda \cup \bar {\Lambda}^{-1}) \cap (\Lambda \cup \bar {\Lambda}^{-1}) = \emptyset\).
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 1078346
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: conjugate quadrature filter
zbMATH Keywords: spectral factorization
zbMATH Keywords: polynomial approximation
zbMATH Keywords: algorithm
reviewed by: José L. Fernández Muñiz
MaRDI profile type: Publication
full work available at URL: a:1019137702681
OpenAlex ID: W137532746
Recommended article: Q3538065
Recommended article: On the construction and frequency localization of finite orthogonal quadrature filters
Recommended article: Construction of Lanczos type filters for the Fourier series approximation
Recommended article: Design of FIR filters using quadratic programming approach
Recommended article: On the design of two channel perfect construction QMF filters
Recommended article: A simple method for designing quadrature mirror filter banks via quadratic constrained optimisation
Recommended article: Design of zero-phase recursive 2-D variable filters with quadrantal symmetries
Recommended article: An Arithmetic Characterization of the Conjugate Quadrature Filters Associated to Orthonormal Wavelet Bases
Recommended article: Optimum noise reduction of conjugate quadrature filter bank
Recommended article: Interpolation filters with prescribed zeros and their refinable functions
description: scientific article; zbMATH DE number 1078346