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Continued Fractions and Szegö Polynomials in Frequency Analysis and Related Topics

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Abstract

This paper is an expository survey of recent research on the application of Szegö polynomials and PPC-continuedfractions to the frequency analysis problem described as follows: We want to determine the unknown frequencies ω1, ω2, ..., ω I from a sample of N observed values x N (m), m = 0, 1, ..., N− 1, arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with frequencies ω1,ω2, ..., ω I . The method is based on the property that certain zerosof the Szegö polynomials (and poles of the PPC-fraction approximants) converge (as N → ∞) to the frequency points ei ω j , j = ±1, ± 2, ..., ± I. The remaining zeros are bounded away from the unit circle |z|=1, asN → ∞. The Levinson algorithm is used to construct the Szegö polynomials and PPC-fractions from the values x N (m). A discussion is given on connections between the topics: Carathéodory functions,the trigonometric moment problem, Szegö polynomials and PPC-fractions. We also describe applications to Doppler radar, medicine, speech processing, speech therapy, meteorology and ocean tides.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Boulder, CO, 80309-0395, U.S.A.

    William B. Jones

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7034, Trondheim, Norway. e-mail

    Vigdis Petersen

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  1. William B. Jones
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Jones, W.B., Petersen, V. Continued Fractions and Szegö Polynomials in Frequency Analysis and Related Topics. Acta Applicandae Mathematicae 61, 149–174 (2000). https://doi.org/10.1023/A:1006454131615

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