A Complete Parameterization of All Positive Rational Extensions of a Covariance Sequence

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A Complete Parameterization of All Positive Rational Extensions of a Covariance Sequence. / Byrnes, Christopher I.; Lindquist, Anders; Gusev, Sergei V.; Matveev, Alexei S.

In: IEEE Transactions on Automatic Control, Vol. 40, No. 11, 11.1995, p. 1841-1857.

Research output: Contribution to journal › Article › peer-review

Author

Byrnes, Christopher I. ; Lindquist, Anders ; Gusev, Sergei V. ; Matveev, Alexei S. / A Complete Parameterization of All Positive Rational Extensions of a Covariance Sequence. In: IEEE Transactions on Automatic Control. 1995 ; Vol. 40, No. 11. pp. 1841-1857.

BibTeX

@article{e765ee34c86242fa8d823fe54a7cc158,

title = "A Complete Parameterization of All Positive Rational Extensions of a Covariance Sequence",

abstract = "In this paper we formalize the observation that filtering and interpolation induce complementary, or “dual,” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou. Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics.",

author = "Byrnes, {Christopher I.} and Anders Lindquist and Gusev, {Sergei V.} and Matveev, {Alexei S.}",

year = "1995",

month = nov,

doi = "10.1109/9.471206",

language = "English",

volume = "40",

pages = "1841--1857",

journal = "IEEE Transactions on Automatic Control",

issn = "0018-9286",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "11",

}

RIS

TY - JOUR

T1 - A Complete Parameterization of All Positive Rational Extensions of a Covariance Sequence

AU - Byrnes, Christopher I.

AU - Lindquist, Anders

AU - Gusev, Sergei V.

AU - Matveev, Alexei S.

PY - 1995/11

Y1 - 1995/11

N2 - In this paper we formalize the observation that filtering and interpolation induce complementary, or “dual,” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou. Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics.

AB - In this paper we formalize the observation that filtering and interpolation induce complementary, or “dual,” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou. Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics.

UR - http://www.scopus.com/inward/record.url?scp=0029406314&partnerID=8YFLogxK

U2 - 10.1109/9.471206

DO - 10.1109/9.471206

M3 - Article

AN - SCOPUS:0029406314

VL - 40

SP - 1841

EP - 1857

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 11

ER -

ID: 50912078