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On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal Processing Applications

  • Chapter
Orthogonal Polynomials

Part of the book series: NATO ASI Series ((ASIC,volume 294))

Abstract

The aim of this contribution is first to show how the Szegö theory of orthogonal polynomials on the unit circle is intimately related to the celebrated Levinson algorithm, which is commonly used in digital signal processing (DSP) applications to solve various least-squares problems. A computationally more efficient substitute for the Levinson algorithm, termed the split Levinson algorithm, has recently been proposed in the DSP literature. In the case of real data, this new algorithm can be interpreted naturally in the framework of a well-defined one-to-one correspondence between the families of real Szegö polynomials and the families of polynomials orthogonal on the interval [-1, 1] with respect to a symmetric measure. More generally, the philosophy underlying the split Levinson algorithm opens the door to an interesting “tridiagonal approach” to the theory of complex Szegö polynomials, nonnegative definite Hermitian Toeplitz matrices, and related algebraic and function theoretic questions. Some of the main topics of this new mathematical framework are briefly reviewed and are shown on specific examples to be of particular interest in DSP applications.

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Authors and Affiliations

  1. Philips Research Laboratory Brussels, Box 8, Av. Van Becelaere 2, B-1170, Brussels, Belgium

    P. Dedsarte & Y. Genin

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  1. P. Dedsarte
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  2. Y. Genin
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  1. Department of Mathematics, The Ohio State University, Columbus, Ohio, USA

    Paul Nevai

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© 1990 Kluwer Academic Publishers

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Dedsarte, P., Genin, Y. (1990). On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal Processing Applications. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_5

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  • DOI: https://doi.org/10.1007/978-94-009-0501-6_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6711-9

  • Online ISBN: 978-94-009-0501-6

  • eBook Packages: Springer Book Archive

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