Skip to main content
SpringerLink
Log in

Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals Part Three: Low-Complexity Staircase Solution

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

In the first part [2] of this set of papers we stated the multidimensional nonlinear Schur parametrization problem for higher-order stochastic sequences. In the second part [1] we proposed the recursive solution to this problem, deriving the general multidimensional nonlinear Schur parametrization algorithm. The goal of this paper is to introduce a low-complexity solution to the nonlinear Schur parametrization problem, following from a multidimensional generalization of the two-indexed matrix extension problem. To obtain the solution, we derive a global multidimensional nonlinear Schur algorithm, then formulate and prove generalized staircase extension and interpolation theorems. The obtained results allow to achieve a considerable complexity reduction of the Schur parametrization algorithms for higher-order stochastic sequences as well as of orthogonal nonlinear approximate filters (of␣ band-structure) of the Volterra–Wiener class.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Zarzycki, Multidimensional Nonlinear Schur Parametrization of Non-Gaussian Stochastic Signals-Part Two: Generalized Schur Algorithm, MSSP Journal, vol. 15, No. 3, 2004, pp. 243–275.

    Article  Google Scholar 

  2. J. Zarzycki, Multidimensional Nonlinear Schur Parametrization of Non-Gaussian Stochastic Signals-Part One: Statement of the Problem, MSSP Journal, vol. 15, No. 3, 2004, pp. 217–241.

    Article  Google Scholar 

  3. P. Dewilde,A Course on the Algebraic Schur and Nevanlinna-Pick Interpolation Problems, in Algorithms and Parallel VLSI Architectures, vol.A: Tutorials, eds. E.F. Deprettere and A.-J. van der Veen, Elseviere Science Publ., 1991, pp. 13–69.

  4. P. Dewilde and E.F.A. Deprettere, Approximative Inversion of Positive Matrices with Applications to Modelling, in Modelling, Robustness and Sensitivity Reduction in Control Systems, NATO ASI Series, vol. F34, ed. R.F. Curtain, Berlin-Heidelberg: Springer-Verlag 1987, pp. 212–238.

    Google Scholar 

  5. P. Dewilde and H. Dym, ''Lossless Inverse Scattering for Digital Filters,'' IEEE Transactions Information Theory, vol. IT-30, 1984, pp. 644–662.

    Article  Google Scholar 

  6. E.F.A. Deprettere, ''Mixed Form Time-Variant Lattice-Recursions,'' in Outils et modeles mathematiques pour l'automatique, l'analyse de systemes et le traitement du signal, Paris: CNRS (ed.), 1982, pp. 545–562.

  7. J. Schur, ''Uber Potenzreichen, die im Innern des Einheitskreises beschrankt sind,'' Journal für die Reine und Angewandte Math., vol. 147, 1917, pp. 205–232.

    Google Scholar 

  8. P. Dewilde and E.F.A. Deprettere, ''The Generalized Schur Algorithm: Approximation and Hierarchy,'' in Operator Theory: Advances and Applications, vol. 29, Basel: Birkhäuser Verlag, 1988, pp. 97–116.

    Google Scholar 

  9. P. Dewilde and H. Dym, ''Schur Recursions, Error Formulas and Convergence of Rational Estimators for Stationary Stochastic Sequences,'' IEEE Transactions on Information Theory, vol. IT-27(4), July 1981, pp. 446–461.

    Article  Google Scholar 

  10. P. Dewilde and Zhen-Qiu Ning, Models for Large Integrated Circuits, Boston/Dordrecht/London: Kluwer Academic Publishers, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Signal Theory Section, Department Electrical Engineering, Wroclaw University of Technology, W.Wyspianskiego 27, 50-370, Wroclaw, Poland

    Jan Zarzycki

Authors
  1. Jan Zarzycki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zarzycki, J. Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals Part Three: Low-Complexity Staircase Solution. Multidimensional Systems and Signal Processing 15, 313–340 (2004). https://doi.org/10.1023/B:MULT.0000037344.44914.9b

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:MULT.0000037344.44914.9b

Search

Navigation