STATISTICAL CHARACTERISTICS INVESTIGATION OF PREDICTION ERRORS FOR INTERFEROMETRIC SIGNAL IN THE ALGORITHM OF NONLINEAR KALMAN FILTERING

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STATISTICAL CHARACTERISTICS INVESTIGATION OF PREDICTION ERRORS

FOR INTERFEROMETRIC SIGNAL IN THE ALGORITHM OF NONLINEAR

KALMAN FILTERING

1

E.L. Dmitrieva , I.P. Gurov

ITMO University, Saint Petersburg, Russia, [email protected]

Basic peculiarities of nonlinear Kalman filtering algorithm applied to processing of interferometric signals are considered. Analytical estimates determining statistical characteristics of signal values prediction errors were obtained and analysis of errors histograms taking into account variations of different parameters of interferometric signal was carried out. Modeling of the signal prediction procedure with known fixed parameters and variable parameters of signal in the algorithm of nonlinear Kalman filtering was performed. Numerical estimates of prediction errors for interferometric signal values were obtained by formation and analysis of the errors histograms under the influence of additive noise and random variations of amplitude and frequency of interferometric signal. Nonlinear Kalman filter is shown to provide processing of signals with randomly variable parameters, however, it does not take into account directly the linearization error of harmonic function representing interferometric signal that is a filtering error source. The main drawback of the linear prediction consists in non-Gaussian statistics of prediction errors including cases of random deviations of signal amplitude and/or frequency. When implementing stochastic filtering of interferometric signals, it is reasonable to use prediction procedures based on local statistics of a signal and its parameters taken into account.

Keywords: interferometric signal, nonlinear Kalman filtering, prediction error, Gaussian noise, histogram.

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155 Reference

1. Akhmanov S.A., D'yakov Yu.E., Chirkov A.S. Vvedenie v statisticheskuyu radiofiziku i optiku [Introduction to Statisti-cal Radiophysics and Optics]. Moscow, Nauka Publ., 1981, 640 p.

2. Malacara D. Optical Shop Testing. 2nd ed. NY, John Wiley & Sons, Inc., 1992, 792 p.

3. Vasiliev V.N., Gurov I.P. Komp'yuternaya obrabotka signalov v prilozhenii k interferometricheskim sistemam [Comput-er processing of signals in the application to the int[Comput-erf[Comput-erometric system]. St. Pet[Comput-ersburg, BHV Publ., 1998, 240 p. 4. Takeda M. Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review. Applied

Optics, 2013, vol. 52, no. 1, pp. 20–29. doi: 10.1364/AO.52.000020

5. Kalman R.E. A New approach to linear filtering and prediction problems. Journal of Basic Engineering, 1960, vol. 82, no. 1, pp. 35–45. doi: 10.1115/1.3662552

6. Balakrishnan A.V. Kalman Filtering Theory. NY, Optimization Software Inc., 1984, 222 p.

7. Gurov I., Ermolaeva E., Zakharov A. Analysis of low-coherence interference fringes by the Kalman filtering method.

Journal of the Optical Society of America A: Optics and Image Science, and Vision, 2004, vol. 21, no. 2, pp. 242–251.

doi: 10.1364/JOSAA.21.000242

8. Gurov I., Volynsky M., Zakharov A. Evaluation of multilayer tissues in optical coherence tomography by the extended Kalman filtering method. Proc. SPIE - The International Society for Optical Engineering, 2007, vol. 6734, pp. 67341P. 9. Volynshy M.A., Gurov I.P., Zakharov A.S. Dynamic analysis of the signals in optical coherent tomography by the

method of nonlinear Kalman filtering. Journal of Optical Technology, 2008, vol. 75, no. 10, pp. 682–686.

10. Gurov I., Volynsky M. Dynamic fringe analysis in spectral interferometry and optical coherence tomography based on recurrence computational algorithms. Fringe 13, 7th International Workshop on Advanced Optical Imaging and

Me-trology. Springer-Verlag, 2013, pp. 313–316.

11. Gurov I.P., Zakharov A.S. Analysis of characteristics of interference fringes by nonlinear Kalman filtering. Optics and

Spectroscopy, 2004, vol. 96, no. 2, pp. 175–181. doi: 10.1134/1.1651239

12. Gurov I., Volynsky M., Vorobeva E. Dynamic wavefront evaluation in phase shifting interferometry based on recur-rence fringe processing. AIP Conference Proceedings, 2010, vol. 1236, pp. 479–484.

13. Park T.-J., Choi H.-S., Han C.-S., Lee Y.-W. Real-time precision displacement measurement interferometer using the robust discrete time Kalman filter. Optics and Laser Technology, 2005, vol. 37, no. 3, pp. 229–234. doi: 10.1016/j.optlastec.2004.03.009

14. Nies H., Loffeld O., Wang Y. Two dimensional Kalman filter approach for phase unwrapping of TerraSAR-X data2008.

7th European Conference on Synthetic Aperture Radar, EUSAR 2008. Friedrichshafen, Germany, 2008, p. 1–4.

15. Cherdyntsev V.A. Radiotekhnicheskie sistemy [Radio Engineering Systems]. Minsk, Vysshaya shkola, 1988, 372 p.

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