AUTHORS: Oscar Ibarra-Manzano, Jose A. Andrade-Lucio, Yuriy Shmaliy

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ABSTRACT: To meet the needs of the IEEE Standard 1139-2008, this paper suggests using the recently designed iterative unbiased finite impulse response (UFIR) filtering algorithm to estimate clock state via measurement of the time interval error (TIE) on an interval of N most recent past points. The algorithm has the Kalman filter (KF) form. But, unlike the KF, requires neither the noise statistics nor the initial values. Because noise in clocks has colored Gaussian components (not white, as required by the KF), the UFIR filter demonstrates higher robustness and becomes optimal when N ≫ 1. Applications are given for state estimation in an ovenized crystal clock and error prediction in a masted clock. Based upon extensive experimental investigations, we show that the UFIR algorithm outperforms the KF, because the clock covariance matrix cannot be specified correctly in view of the colored noise.

KEYWORDS: Clock errors, unbiased FIR filter, Kalman filter, robustness.

REFERENCES:

[ 1] IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology – Random Instabilities, IEEE Standard 1139-2008; IEEE: Piscataway, NJ, 2009, pp. 1-45.

[2] ITU-T Recommendation G.810. Definitions and terminology fir synchronization networks, 1996.

[3] A. H. Jazwinski, Stochastic Processes and Filtering Theory, New York: Academic Press, 1970.

[4] Y. S. Shmaliy, GPS-based Optimal FIR Filtering of Clock Models, Nova Science Publ., New York, 2009.

[5] W. H. Kwon and S. Han, Receding Horizon Control: Model Predictive Control for State Models, London: Springer, 2005.

[6] Y. S. Shmaliy, F. Lehmann, S. Zhao, and C. K. Ahn, “Comparing robustness of the Kalman, H∞, and UFIR filters,” IEEE Trans. Signal Process., vol. 66, no. 13, pp. 3447–3458. Jul. 2018.

[7] Y. S. Shmaliy and O. Ibarra-Manzano, “Optimal FIR filtering of the clock time errors,” Metrologia, vol. 45, no. 5, pp. 571–576, Sep. 2008.

[8] Y. S. Shmaliy, “Linear optimal FIR estimation of discrete time-invariant state-space models,” IEEE Trans. on Signal Process., vol. 58, no. 6, pp, 3086– 3096, Jun. 2010.

[9] J. W. Choi, S. Han, and J. M. Cioffi, “An FIR channel estimation filter with robustness to channel mismatch condition,” IEEE Trans. Broadcast., vol. 54, no. 1, pp. 127–130, Mar. 2008.

[10] Y. S. Shmaliy, “An unbiased FIR filter for TIE model of a local clock in applications to GPS-based timekeeping,” IEEE Trans. on Ultrason., Ferroel. and Freq. Contr., vol. 53, no. 5, pp. 862–870, May 2006.

[11] Y. S. Shmaliy, “An iterative Kalman-like algorithm ignoring noise and initial conditions,” IEEE Trans. on Signal Process., vol. 59, no. 6, pp, 2465–2473, Jun. 2011.

[12] Y. S. Shmaliy, S. Zhao, and C. K. Ahn, “Unbiased FIR filtering: an iterative alternative to Kalman filtering ignoring noise and initial conditions,” IEEE Control Syst. Mag., vol. 37, no. 5, pp. 70–89, Oct. 2017.

[13] Y. Kou, Y. Jiao, D. Xu, M. Zhang, Y. Liu, and X. Li, “Low-cost precise measurement of oscillator frequency instability based on GNSS carrier observation,” Adv. Space Res., vol. 51, no. 6, pp. 969–977, Mar. 2013.

[14] J. H. Lee, S. Hwang, D.-H. Yu, C. Park, and S. J. Lee, “Software-based performance analysis of a pseudolite time synchronization method depending on the clock source,” J. Position. Navig. Timing, vol. 3, no. 4, pp. 163–170, 2014.

[15] Y. S. Shmaliy and O. Ibarra-Manzano, “Noise power gain for discrete-time FIR estimators,” IEEE Signal Process. Lett., vol. 18, no. 4, pp. 207–210, Apr. 2011.

[16] Y. Zhang, W. Lu, D. Lei, Y. Huang, and D. Yu, “Effective PPS signal generation with predictive synchronous loop for GPS,” IEICE Trans. Commun., vol. E97-B, no. 8, pp. 1742–1749, Aug. 2014.

[17] Y. Chen, S. Ding, Z. Xie, Z. Qi, and X. Liang, “Design study for a quasisynchronous CDMA sensor data collection system: an LEO satellite uplink access technique based on GPS,” Int. J. Distrib. Sens. Netw., vol. 2015, no. ID 421745, pp. 1–15, 2015.

[18] Y. S. Shmaliy, “On real-time optimal FIR estimation of linear TIE models of local clocks,” IEEE Trans. Usltrason. Ferroel. Freg. Contr., vol. 54, no. 11, pp. 2403–2406, Nov. 2007.

[19] R. Y. Ramlall, “Method for Doppler-aided GPS carrier-tracking using p-step ramp unbiased finite impulse response predictor,” U.S. Patent 8 773 305, July 8, 2014.

[20] J. B. Fu, J. Sun, S. Lu, and Y. Zhang, “Maneuvering target tracking with modified unbiased FIR filter,” J. Beijing Univ. Aeronaut. Astronaut., vol. 41, no. 1, pp. 77–82, Jan. 2015.

[21] Y. S. Shmaliy and L. Arceo-Miquel, “Efficient predictive estimator for holdover in GPS-based clock synchronization,” IEEE Trans. Ultrason. Ferroelec. Freq. Contr., vol. 55, no. 10, pp. 2131–2139, Oct. 2008

[22] S. Yin, J. Wang, and T. Liu, “Improved UFIR tracking algorithm for maneuvering target,” Indonesian J. Electr. Eng. Comput. Sci., vol. 2, no. 2, pp. 344–350, May 2016.

[23] J. Sun, J. B. Fu, and J. Wang, “Improved maneuvering target tracking method based on unbiased finite impulse response (UFIR) filter,” U.S. Patent 103 500 455 A, Jan. 8, 2014.

[24] R. H. Jones and P. V. Tryon, “Continuous time series models for unequally spaced data applied to modeling atomic clocks,” SIAM J. Sci. Stat. Comput., vol. 8, no. 1, pp, 71–81, Jan 1987.

[25] S. R. Stein and R. L. Filler, “Kalman filter analysis for real time applications of clocks and oscillators,” Proc. 42th Annual Frequency Control Symp., pp, 447–452, 1988.

[26] J. W. Chaffee, “Relating the Allan variance to the diffusion coefficients of a linear stochastic differential equation model for precision oscillators,” IEEE Trans. on Ultrason., Ferroel. and Freq. Contr., vol. 34, no. 6, pp, 655–658, Nov. 1987.

[27] Y. S. Shmaliy, J. Mu˜noz-Diaz, and L. Arceo-Miquel, and “Optimal horizons for a one-parameter family of unbiased FIR filters,” in Digital Signal Process., vol. 18, no. 5, pp. 739–750, Sep. 2008.

[28] Y. S. Shmaliy, A. V. Marienko, and A. V. Savchuk, “GPS-based optimal Kalman estimation of time error, frequency offset, and aging, 31st Precise Time and Time Interval (PTTI) Systems and Application Mtg., Dana Point, California, pp. 431–440, 1999