Skip to main content
SpringerLink
Log in

Time–Frequency Analysis of Multichannel Signals Using Two-Sided Autoregressive Modeling

  • Published:
Circuits, Systems & Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, we propose a time-varying vector two-sided autoregressive (VTAR) model for time–frequency analysis of multichannel signals. The multichannel approach is suitable in many applications, such as electroencephalogram analysis and spatial data processing, where the signals are recorded from several sensors, giving rise to vector or multichannel processes. In VTAR modeling, the current sample of the signal in some channel is estimated by a symmetrically weighted sum of past and future samples of this channel as well as of the other channels. The multidimensional VTAR parameters are assumed to be time varying and they are modeled as a linear combination of a set of basis functions. The recursive least-squares algorithm is used to estimate the coefficients of the linear combination. The VTAR model requires a smaller order than the conventional vector autoregressive (VAR) model to achieve better resolution in the time–frequency plane. Numerical examples are given in order to compare the VTAR-based time–frequency distribution with the conventional VAR-based time–frequency distribution and the Choi–Williams distribution.

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. M. Arnold et al., Adaptive AR Modeling of Nonstationary Time Series by Means of Kalman Filtering, IEEE Trans. Biomed. Eng., 45(5), pp. 553–562, 1998.

    Article  Google Scholar 

  2. B. Boashash, Time-frequency analysis, in: S. Haykin (ed.), Advances in Spectrum Estimation, Prentice-Hall, Englewood Cliffs, NJ, 1990, pp. 418–517.

    Google Scholar 

  3. B. Boashash and V. Sucic, Resolution Measure Criteria for the Objective Assessment of the Performance of Quadratic Time-Frequency Distributions, IEEE Trans. Signal Processing, 51(5), pp. 1253–1263, 2003.

    Article  MathSciNet  Google Scholar 

  4. R. Charbonnier, M. Barlaud, G. Alengrin, and J. Menez, Results on AR-modeling of Nonstationary Signals, Signal Processing, 12(2), pp. 143–151, 1987.

    Article  Google Scholar 

  5. H. I. Choi and W. J. Williams, Improved Time-Frequency Representation of Multicomponent Signals Using Exponential Kernels, IEEE Trans. Acoust. Speech Signal Processing, 37(6), pp. 862–871, 1989.

    Article  Google Scholar 

  6. L. Cohen, Time-Frequency Distribution – A Review, Proceedings of the IEEE, 77(7), pp. 941–981, 1989.

    Article  Google Scholar 

  7. S. Conforto and T. D’alessio, Spectral Analysis of Nonstationary Signals from Mechanical Measurements: A Parametric Approach, Mec. Sys. Signal Processing, 13(3), pp. 395–411, 1999.

    Article  Google Scholar 

  8. S. Conforto and T. D’alessio, Optimal Estimation of Power Spectral Density by Means of a Time-Varying Autoregressive Approach, Signal Processing, 72(1), pp. 1–14, 1999.

    Article  MATH  Google Scholar 

  9. S. D. Cranstoun et al., Time-Frequency Spectral Estimation of Multichannel EEG using the Auto-SLEX Method, IEEE Trans. Biomed. Eng., 49(9), pp. 988–996, 2002.

    Article  Google Scholar 

  10. M. Elling and P. Sherman, A Kalman Filter Based Approach for Estimating Nonstationary VAR Models via Pole Tracking, Proceedings of the IEEE ICASSP, pp. 685–688, 2000.

  11. K. B. Eom, Analysis of Acoustic Signatures from Moving Vehicles Using Time-Varying Autoregressive Models, Multidimensional Systems and Signal Processing, 10, pp. 357–378, 1999.

    Article  MATH  Google Scholar 

  12. B. Friedlander and B. Porat, Multichannel ARMA Spectral Estimation by the Modified Yule-Walker Method, Signal Processing, 10(1), pp. 49–59, 1986.

    Article  Google Scholar 

  13. W. Gersch, A. Gevins, and G. Kitagawa, A Multivariate Time-Varying Autoregressive Modeling of Nonstationary Covariance Time Series, Proceedings of the 22nd IEEE Conf. Decis. Cont., pp. 579–584, 1983.

  14. W. Gersch and D. Stone, Multichannel Time-Varying Autoregressive Modelling: A Circular Lattice-Smoothness Priors Realization, Proceedings of the 29th IEEE Conf. Decis. Cont., pp. 859–860, 1990.

  15. Y. Grenier, Time-Dependent ARMA Modeling of Nonstationary Signals, IEEE Trans. Acoust. Speech Signal Processing, 31(4), pp. 899–911, 1983.

    Article  Google Scholar 

  16. Y. Grenier, Modèles ARMA à coefficients dépendant du temps: estimateurs et applications, Traitement du Signal, 3(4–5), pp. 219–233, 1986.

    Google Scholar 

  17. M. G. Hall, A. V. Oppenheim, and A. S. Willsky, Time-Varying Parametric Modeling of Speech, Signal Processing, 5(3), pp. 267–285, 1983.

    Article  Google Scholar 

  18. M. K. Hasan, M. J. Hossain, and M. A. Haque, Parameter Estimation of Multichannel Processes in Noise, Signal Processing, 83(3), pp. 603–610, 2003.

    Article  MATH  Google Scholar 

  19. S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, NJ, 1991, Chapter 13, pp. 477–507.

    MATH  Google Scholar 

  20. F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and Quadratic Time-Frequency Signal Representations, IEEE Signal Processing Magazine, 9(2), pp. 21–67, 1992.

    Article  Google Scholar 

  21. J. Hsu and A. E. Yagle, Fast Algorithms for Close-to-Toeplitz-Plus-Hankel Systems and Two-Sided Linear Prediction, IEEE Trans. Signal Processing, 41(7), pp. 2349–2361, 1993.

    Article  Google Scholar 

  22. J. Hsu and A. E. Yagle, Blind Deconvolution of Symmetric Noncausal Response Using Two-Sided Linear Prediction, IEEE Trans. Signal Processing, 42(6), pp. 1509–1517, 1994.

    Article  Google Scholar 

  23. J. Hsu and A. E. Yagle, Similarities and Differences Between One-Sided and Two-Sided Linear Prediction, IEEE Trans. Signal Processing, 43(1), pp. 345–349, 1995.

    Article  Google Scholar 

  24. A. Isaksson, A. Wennberg, and L. H. Zetterberg, Computer Analysis of EEG Signals with Parametric Models, Proceedings of the IEEE, 69(4), pp. 451–461, 1981.

    Article  Google Scholar 

  25. X. Q. Jiang and G. Kitagawa, A Time-Varying Coefficient Vector AR Modeling of Nonstationary Covariance Time Series, Signal Processing, 33(3), pp. 315–331, 1993.

    Article  MATH  Google Scholar 

  26. R. H. Jones, Identification and Autoregressive Spectrum Estimation, IEEE Trans. Aut. Cont., 19(6), pp. 894–898, 1974.

    Article  Google Scholar 

  27. A. Kacha, F. Grenez, and K. Benmahammed, Time-Frequency Analysis and Instantaneous Frequency Estimation Using Two-Sided Linear Prediction, Signal Processing, 85(3), pp. 491–503, 2005.

    Article  MATH  Google Scholar 

  28. S. M. Kay, Modern Spectral Estimation: Theory and Application, Prentice-Hall, Englewood Cliffs, NJ, 1988, Chapter 14, pp. 446–478.

    MATH  Google Scholar 

  29. F. Kozin and F. Nakajima, The Order Determination Problem for Linear Time-Varying AR Models, IEEE Trans. Aut. Cont., vol. 25(2), pp. 250–257, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. C. Lee, A New Autoregressive Method for High-Performance Spectrum Analysis, J. Acoust. Soc. Amer., 86(1), pp. 150–157, 1989.

    Article  Google Scholar 

  31. T. Leou and J. K. Aggarwal, Recursive Implementation of LTV Filters—Frozen-Time Transfer Function Versus Generalized Transfer Function, Proceedings of the IEEE, 72(7), pp. 980–981, 1984.

    Article  Google Scholar 

  32. S. L. Marple Jr, Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, NJ, 1987, Chapter 15, pp. 386–431.

    Google Scholar 

  33. E. Möller, B. Schack, M. Arnold, and H. Witte, Instantaneous Multivariate EEG Coherence Analysis by Means of Adaptive High-Dimensional Autoregressive Models, J. Neuroscience Methods, 105(2), pp. 143–158, 2001.

    Article  Google Scholar 

  34. M. Morf, A. Vieira, D. T. L. Lee, and T. Kailath, Recursive Multichannel Maximum Entropy Spectral Estimation, in: D. Childers (ed.), Modern Spectral Analysis, IEEE Press, San Diego, CA, 1978, pp. 172–181.

    Google Scholar 

  35. M. Niedzwiecki, Identification of Time-Varying Processes, Wiley, New York, 2000.

    Google Scholar 

  36. M. J. Paulik, N. Mohankrishnan, and M. Nikiforuk, A Time-Varying Vector Autoregressive Model for Signature Verification, Proceedings of the IEEE ICASSP, pp. 1395–1398, 1995.

  37. B. G. Quinn, Order Determination for a Multivariate Autoregression, J. Royal Stat. Soc. (B), 42(2), pp. 182–185, 1980.

    MathSciNet  MATH  Google Scholar 

  38. G. Schwarz, Estimating the Dimension of a Model, The Annals of Statistics, 6(2), pp. 461–464, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. K. Tugnait, Modeling and Identification of Symmetric Noncausal Impulse Responses, IEEE Trans. Acoust. Speech Signal Processing, 34(5), pp. 1171–1181, 1986.

    Article  Google Scholar 

  40. R. A. Wiggins and E. A. Robinson, Recursive Solution to the Multichannel Filtering Problem, J. Geophys. Res., 70(8), 1885–1891, 1965.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Service Ondes et Signaux, Faculté des Sciences Appliquées, Université Libre de Bruxelles, Av. F. D. Roosevelt 50, CP 165/51, 1050, Brussels, Belgium

    A. Kacha & F. Grenez

  2. Département Electronique, Faculté des Sciences de l’Ingénieur, Université de Sétif, Sétif, 19000, Algeria

    K. Benmahammed

Authors
  1. A. Kacha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Grenez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Benmahammed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Kacha.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kacha, A., Grenez, F. & Benmahammed, K. Time–Frequency Analysis of Multichannel Signals Using Two-Sided Autoregressive Modeling. Circuits Syst Signal Process 27, 309–330 (2008). https://doi.org/10.1007/s00034-008-9032-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-008-9032-0

Keywords

Search

Navigation