Orthogonal rational functions and frequency analysis
type: Item
instance of: scholarly article
title: Orthogonal rational functions and frequency analysis
zbMATH Open document ID: 0968.42017
DOI: 10.1023/A:1006417408359
author: Haakon Waadeland
published in: Acta Applicandae Mathematicae
publication date: 2001-08-28T00:00:00Z
review text: Given a trigonometric signal \(x(m)=\sum_{j=-I}^I A_j \exp(i\omega_jm)\) with \(A_0\geq 0\), \(A_{-j}=\bar{A}_j\) complex and \(\omega_{-j}=-\omega_j\) real, one has to extract the frequencies \(\omega_j\) from the signal. Traditionally, this is done using orthogonal Szegő polynomials. These polynomials are orthogonal with respect to the spectral measures of the signal \(x(m)\), \(m=1,\ldots,N\). Als \(n\to\infty\), the zeros of the orthogonal polynomials approach the points on the circle corresponding to the desired frequencies. In this paper the analog in the case where the polynomials are replaced by orthogonal rational functions with prescribed poles is considered [see \textit{O.~Njåstad} and \textit{H.~Waadeland}, J. Math. Anal. Appl. 206, No. 1, 280-307 (1997; Zbl 0872.42006); J. Comput. Appl. Math. 77, No. 1-2, 255-275 (1997; Zbl 0864.42010)]. This paper reviews the method and its problems and gives explicit expressions for the moment integrals and the orthogonal rational functions, which is elaborated for the simple example \(x(m)=\exp(im\omega)+\exp(-im\omega)\).
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 94A11
zbMATH DE Number: 1546754
zbMATH Keywords: orthogonal rational function
zbMATH Keywords: trigonometric signal scheduling
MaRDI profile type: Publication
full work available at URL: a:1006417408359
OpenAlex ID: W115284690
description: scientific article; zbMATH DE number 1546754