Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables

In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral FejerRiesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parametrized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem.

[1]  Michael G. Strintzis,et al.  Tests of stability of multidimensional filters , 1977 .

[2]  H. Lev-Ari,et al.  Multidimensional maximum-entropy covariance extension , 1985, ICASSP '85. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[3]  Marinus A. Kaashoek.  Metric constrained interpolation and control theory , 2005 .

[4]  G. Szegő.  Beiträge zur Theorie der Toeplitzschen Formen , 1920 .

[5]  Y. Kamp,et al.  Counterexample in the least-squares inverse stabilisation of 2D recursive filters , 1975 .

[6]  G. Szegő.  Beiträge zur Theorie der Toeplitzschen Formen , 1921 .

[7]  M. Zwaan.  An introduction to hilbert space , 1990 .

[8]  M. A. Kaashoek,et al.  Equivalence, linearization, and decomposition of holomorphic operator functions , 1978 .

[9]  Hugo J. Woerdeman,et al.  Minimal rank completions of partial banded matrices , 1993 .

[10]  N. K. Bose,et al.  Positivity an d Stability Tests For Multidirnensiona I Filters (Discrete-Continuous) , 1974 .

[11]  M. A. Kaashoek,et al.  Unique minimal rank extensions of triangular operators , 1988 .

[12]  Arthur E. Frazho,et al.  Metric Constrained Interpolation, Commutant Lifting and Systems , 1998 .

[13]  B. Dickinson,et al.  Two-dimensional Markov spectrum estimates need not exist , 1980, IEEE Trans. Inf. Theory.

[14]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[15]  M. G. Kreĭn,et al.  Some questions in the theory of moments , 1962 .

[16]  Hugo J. Woerdeman,et al.  A numerical algorithm for stable 2D autoregressive filter design , 2003, Signal Process..

[17]  John W. Woods,et al.  Two-dimensional Markov spectral estimation , 1976, IEEE Trans. Inf. Theory.

[18]  J. A. Ball,et al.  I. Schur methods in operator theory and signal processing , 1987 .

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

[20]  P. Whittle.  ON STATIONARY PROCESSES IN THE PLANE , 1954 .

[21]  M. Morf,et al.  Inverses of Toeplitz operators, innovations, and orthogonal polynomials , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[22]  L. Rodman,et al.  Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band , 1999 .

[23]  H. Dym,et al.  Extensions of matrix valued functions with rational polynomial inverses , 1979 .

[24]  Y. Kamp,et al.  Planar least squares inverse polynomials: Part I-Algebraic properties , 1979 .

[25]  H. Dym.  J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation , 1989 .

[26]  Tiberiu Constantinescu.  Schur Parameters, Factorization and Dilation Problems , 2002 .

[27]  H. Helson,et al.  Prediction theory and fourier series in several variables. II , 1961 .

[28]  Hugo J. Woerdeman,et al.  The lower order of lower triangular operators and minimal rank extensions , 1987 .

[29]  T. Constantinescu,et al.  Schur Analysis of Positive Block-Matrices , 1986 .

[30]  J. Geronimo.  Matrix orthogonal polynomials on the unit circle , 1981 .

[31]  I Gohberg.  I. Schur methods inoperator theory and signal processing , 1986 .

[32]  W. Rudin.  The extension problem for positive-definite functions , 1963 .

[33]  M. A. Kaashoek,et al.  Minimal Factorization of Matrix and Operator Functions , 1980 .

[34]  T. Marzetta.  Two-dimensional linear prediction: Autocorrelation arrays, minimum-phase prediction error filters, and reflection coefficient arrays , 1980 .

[35]  William H. Gustafson.  A note on matrix inversion , 1984 .

[36]  C. Foias,et al.  The commutant lifting approach to interpolation problems , 1990 .

[37]  Glaysar Castro.  Coefficients de réflexion généralisés : extension de covariances multidimensionnelles et autres applications , 1997 .

[38]  E. Hewitt,et al.  Abstract Harmonic Analysis , 1963 .

[39]  J. Cadzow.  Maximum Entropy Spectral Analysis , 2006 .

[40]  M. Ekstrom,et al.  Two-dimensional spectral factorization with applications in recursive digital filtering , 1976 .

[41]  L. Silverman.  Realization of linear dynamical systems , 1971 .

[42]  P. Delsarte,et al.  Half-plane Toeplitz systems , 1980, IEEE Trans. Inf. Theory.

[43]  B. Francis,et al.  A Course in H Control Theory , 1987 .

[44]  Y. Kamp,et al.  A simple proof of Rudin's multivariable stability theorem , 1980 .

[45]  Freidrich Riesz,et al.  Über die Randwerte einer analytischen Funktion , 1923 .

[46]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[47]  Michael A. Dritschel,et al.  On Factorization of Trigonometric Polynomials , 2004 .

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

[49]  Jae Lim,et al.  A new algorithm for two-dimensional maximum entropy power spectrum estimation , 1981 .

[50]  Israel Gohberg,et al.  The band method for positive and strictly contractive extension problems: An alternative version and new applications , 1989 .

[51]  H. Dym,et al.  Extensions of band matrices with band inverses , 1981 .

[52]  H. Dym,et al.  Extensions of kernels of Fredholm operators , 1982 .

[53]  Leiba Rodman,et al.  Abstract Band Method Via Factorization, Positive and Band Extensions of Multivariable Almost Periodic Matrix Functions, and Spectral Estimation , 2002 .

[54]  J. Doob.  Stochastic processes , 1953 .

[55]  Murray Rosenblatt,et al.  A multi-dimensional prediction problem , 1958 .

[56]  Prediction and the inverse of Toeplitz matrices , 1994 .

[57]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[58]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .