A New Approach to Spectral Factorization of a Class of Matrix-Valued Spectral Densities

In this paper we propose a new approach to spectral factorization of a class of matrix-valued spectral densities. Our results are based on a recent necessary and sufficient uniform log-integrability condition for the canonical spectral factorization mapping to be sequentially continuous. In particular, we derive a new set of easily verifiable sufficient conditions for uniform log-integrability to hold. The proposed approach does not require the spectral density to be coercive, and the class to which it is applicable is reasonably large as to include many spectral densities which are of interest in applications. We also present a new spectral factorization algorithm for scalar analytic spectral densities along with a numerical example.

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

[2]  G. Heinig.  Inversion of Toeplitz-plus-Hankel matrices with arbitrary rank profile , 2001 .

[3]  Dante C. Youla,et al.  A new technique for ARMA-system identification and rational approximation , 1993, IEEE Trans. Signal Process..

[4]  Ali H. Sayed,et al.  A survey of spectral factorization methods , 2001, Numer. Linear Algebra Appl..

[5]  Peter E. Caines,et al.  On the L ∞ consistency of L 2 estimators , 1989 .

[6]  A. Balakrishnan.  Introduction to Random Processes in Engineering , 1995 .

[7]  Anders Lindquist,et al.  Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach , 2003, IEEE Trans. Autom. Control..

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

[9]  Peter E. Caines,et al.  Corrigendum to “On the L∞ consistency of L2 estimators”☆☆☆ , 1989 .

[10]  Steven Barclay,et al.  Continuity of the Spectral Factorization Mapping , 2004 .

[11]  Mohsen Pourahmadi,et al.  Foundations of Time Series Analysis and Prediction Theory , 2001 .

[12]  T. Kailath,et al.  On a generalized Szegö- Levinson realization algorithm for optimal linear predictors based on a network synthesis approach , 1978 .

[13]  N. Wiener,et al.  The prediction theory of multivariate stochastic processes , 1957 .

[14]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[15]  Y. Kamp,et al.  Orthogonal polynomial matrices on the unit circle , 1978 .

[16]  Tryphon T. Georgiou,et al.  The interpolation problem with a degree constraint , 1999, IEEE Trans. Autom. Control..

[17]  Anders Lindquist,et al.  From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach , 2001, SIAM Rev..

[18]  Anders Lindquist,et al.  A covariance extension approach to identification of time series , 2000, Autom..

[19]  C. Byrnes,et al.  A Convex Optimization Approach to the Rational Covariance Extension Problem , 1999 .

[20]  H.I. Nurdin,et al.  Computation of Degree Constrained Rational Interpolants with Non-Strictly Positive Parametrizing Functions via Homotopy Continuation , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[21]  Dante C. Youla,et al.  Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle , 1978 .

[22]  P. Enqvist.  A homotopy approach to rational covariance extension with degree constraint , 2001 .

[23]  H. Helson,et al.  Prediction theory and Fourier Series in several variables , 1958 .

[24]  Georg Heinig,et al.  New Fast Algorithms for Toeplitz-Plus-Hankel Matrices , 2003, SIAM J. Matrix Anal. Appl..

[25]  Hendra Ishwara Nurdin,et al.  On the solutions of the rational covariance extension problem corresponding to pseudopolynomials having boundary zeros , 2006, IEEE Transactions on Automatic Control.

[26]  Hendra Ishwara Nurdin.  New results on the rational covariance extension problem with degree constraint , 2006, Syst. Control. Lett..

[27]  H. F. Davis.  Fourier series and orthogonal functions , 1965 .

[28]  D. Lieberman,et al.  Fourier analysis , 2004, Journal of cataract and refractive surgery.

[29]  Hendra I. Nurdin,et al.  Spectral Factorization of Non-Rational Matrix-Valued Spectral Densities , 2005 .