The geometry of positive real functions with applications to the rational covariance extension problem

In this paper we 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 bianalytic parameterization using familiar objects from systems theory. In particular, this proves a long-standing conjecture by Georgiou. The methodology is based on global analysis of the dynamics of certain fast algorithms for Kalman filtering.<<ETX>>

[1]  C. Carathéodory.  Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen , 1907 .

[2]  C. Carathéodory.  Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen , 1911 .

[3]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[4]  A. Lindquist.  A New Algorithm for Optimal Filtering of Discrete-Time Stationary Processes , 1974 .

[5]  J. Makhoul,et al.  Linear prediction: A tutorial review , 1975, Proceedings of the IEEE.

[6]  A. Lindquist.  Some reduced-order non-Riccati equations for linear least-squares estimation : the stationary, single-output case† , 1976 .

[7]  A.S. Willsky,et al.  Relationships between digital signal processing and control and estimation theory , 1978, Proceedings of the IEEE.

[8]  S. Haykin.  Nonlinear Methods of Spectral Analysis , 1980 .

[9]  Albert Benveniste,et al.  AR and ARMA identification algorithms of Levinson type: An innovations approach , 1981 .

[10]  S.M. Kay,et al.  Spectrum analysis—A modern perspective , 1981, Proceedings of the IEEE.

[11]  Patrick Dewilde,et al.  Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences , 1981, IEEE Trans. Inf. Theory.

[12]  R. Kálmán.  Realization of Covariance Sequences , 1982 .

[13]  J. Cadzow,et al.  Spectral estimation: An overdetermined rational model equation approach , 1982, Proceedings of the IEEE.

[14]  Paul Van Dooren,et al.  Speech modelling and the trigonometric moment problem , 1982 .

[15]  Paul Van Dooren,et al.  On the role of the partial trigonometric moment problem in AR speech modelling , 1983, ICASSP.

[16]  I. Schur,et al.  On Power Series Which are Bounded in the Interior of the Unit Circle II , 1986 .

[17]  Tryphon T. Georgiou,et al.  Realization of power spectra from partial covariance sequences , 1987, IEEE Trans. Acoust. Speech Signal Process..

[18]  Masanao Aoki,et al.  State Space Modeling of Time Series , 1987 .

[19]  Anders Lindquist,et al.  Predictability and unpredictability in Kalman filtering , 1991 .

[20]  G. Picci,et al.  On “Subspace Methods” Identification and Stochastic Model Reduction , 1994 .

[21]  C. Byrnes,et al.  Toward a solution of the minimal partial stochastic realization problem , 1994 .

[22]  Anders Lindquist,et al.  On the Nonlinear Dynamics of Fast Filtering Algorithms , 1994 .

[23]  C. Byrnes,et al.  A complete parameterization of all positive rational extensions of a covariance sequence , 1995, IEEE Trans. Autom. Control..