OSA's Digital Library

Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 73, Iss. 12 — Dec. 1, 1983
  • pp: 1799–1811

State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem

S. Y. Kung, K. S. Arun, and D. V. Bhaskar Rao  »View Author Affiliations


JOSA, Vol. 73, Issue 12, pp. 1799-1811 (1983)
http://dx.doi.org/10.1364/JOSA.73.001799


View Full Text Article

Acrobat PDF (1593 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We present new high-resolution methods for the problem of retrieving sinusoidal processes from noisy measurements. The approach taken is by use of the so-called principal-components method, which is a singular-value-decomposition- based approximate modeling method. The low-rank property and the algebraic structure of both the data matrix and the covariance matrix (under noise-free conditions) form the basis of exact modeling methods. In a noisy environment, however, the rank property is often perturbed, and singular-value decomposition is used to obtain a low-rank approximant in factored form. The underlying algebraic structure of these factors leads naturally to least-squares estimates of the state-space parameters of the sinusoidal process. This forms the basis of the Toeplitz approximation method, which offers a robust Pisarenko-like spectral estimate from the covariance sequence. Furthermore, the principle of Pisarenko's method is extended to harmonic retrieval directly from timeseries data, which leads to a direct-data approximation method. Our simulation results indicate that favorable resolution capability (compared with existing methods) can be achieved by the above methods. The application of these principles to two-dimensional signals is also discussed.

© 1983 Optical Society of America

Citation
S. Y. Kung, K. S. Arun, and D. V. Bhaskar Rao, "State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem," J. Opt. Soc. Am. 73, 1799-1811 (1983)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-73-12-1799


Sort:  Author  |  Year  |  Journal  |  Reset

References

  1. O. S. Halpeny and D. G. Childers, "Composite wavefront decomposition vai multidimensional digital filtering of array data," IEEE Trans. Circuits Syst. CAS-22, 552–562 (1975).
  2. S. M. Kay and S. L. Marple, Jr., "Spectrum analysis—a modern perspective," Proc. IEEE 69, 1380–1418 (1981).
  3. J. A. Cadzow, "Spectral estimation: an overdetermined rational model equation approach," Proc. IEEE 70, 907–939 (1982).
  4. D. W. Tufts and R. Kumaresen, "Estimation of frequencies of multiple sinusoids: making linear prediction perform like maximum likelihood," Proc. IEEE 70, 975–989 (1982).
  5. G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, Calif., 1966).
  6. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  7. N. Levinson, "The Wiener rms (root mean square) error criterion in filter design and prediction," J. Math. Phys. 25, 261–278 (1947).
  8. J. P. Burg, "Maximum entropy spectral analysis," Ph.D Thesis (Stanford University, Stanford, Calif., 1975).
  9. W. Gersh, "Estimation of the autoregressive parameters of mixed autoregressive moving-average time series," IEEE Trans. Autom. Control AC-15, 583–588 (1970).
  10. T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N.J., 1980).
  11. L. Kronecker, "Zur Theorie der Elmination einer Variabeln aus zwei Algebraischen Gleichungen," Trans. Royal Prussian Academy of Science (see collected works, Vol. 2), 1881.
  12. B. L. Ho and R. E. Kalman, "Effective construction of linear state variable models from input/output data," in Proceedings of the 3rd Allerton Conference on Circuits and System Theory (U. Illinois Press, Urbana, Ill., 1965), pp. 449–459.
  13. J. Rissanen, "Recursive identification of linear systems," J. SIAM Control 9, 420–430 (1971).
  14. P. Faurre, "Stochastic realization algorithms" in System Iden-tification: Advances and Case Studies, R. K. Mehra and D. G. Lainiotis, eds. (Academic, New York, 1976).
  15. V. C. Klemma and A. J. Laub, "The singular value decomposition: its computation and some applications," IEEE Trans. Autom. Control AC-25, 164–176 (1980).
  16. S. Y. Kung, "A new identification and model reduction algorithm via singular value decomposition", in Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers, (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 705–714.
  17. B. C. Moore, "Principal component analysis in linear systems: controllability, observability, and model reduction," IEEE Trans. Autom. Control AC-26, 17–32 (1981).
  18. C. T. Mullis and R. A. Roberts, "Synthesis of minimum round-off noise fixed point digital filters," IEEE Trans. Circuits Syst. CAS-23, 551–562 (1976).
  19. S. Y. Kung and K. S. Arun, "A novel Hankel approximation method for ARMA pole-zero estimation from noisy covariance data," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical Society of America, Washington, D.C., 1983), pp. WA19-1–WA19-5.
  20. U. B. Desai and D. Pal, "A realization approach to stochastic model reduction and balanced stochastic realizations," in Proceedings of the Twenty-First IEEE Conference on Decision and Control (Institute of Electronics and Electrical Engineers, New York, 1982), pp. 1105–1112.
  21. S. Y. Kung and K. S. Arun, "Approximate realization methods for ARMA spectral estimation," in Proceedings of the IEEE International Symposium on Circuits and Systems (Institute of Electrical and Electronics Engineers, New York, 1983).
  22. V. F. Pisarenko, "The retrieval of harmonics from a covariance function," Geophys. J. R. Astron. Soc. Can. 33, 347–366 (1973).
  23. D. V. Bhaskar Rao, "Adaptive notch filtering for the retrieval of harmonics," Ph.D Thesis (University of Southern California, Los Angeles, Calif., 1983).
  24. G. R. B. Prony, "Essai experimental et analytique, etc.," J. Ec. Polytech. 1, 24–76 (1795).
  25. T. J. Ulyrch and R. W. Clayton, "Time series modeling and maximum entropy," Phys. Earth Planet. Inter. 12, 188–200 (1976).
  26. S. Y. Kung, "A Toeplitz approximation method and some applications," in Proceedings of the International Symposium on Mathematical Theory of Networks and Systems (Western, North Hollywood, Calif., 1981), pp. 262–266.
  27. J. P. Burg, "A new analysis technique for time series data," presented at NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics, August 12–23, 1968.
  28. G. H. Golub and C. Reinsch, "Singular value decomposition and least squares solutions," Numer. Math. 14, 403–420 (1970).
  29. S. Attasi, "Modelling and recursive estimation for double indexed sequences," in System Identification: Advances and Case Studies, R. K. Mehra and D. G. Lainiotis, eds. (Academic, New York, 1976).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited