Sign-error adaptive filtering algorithms involving Markovian parameters

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December 2015, 5(4): 781-806. doi: 10.3934/mcrf.2015.5.781

Sign-error adaptive filtering algorithms involving Markovian parameters

Araz Hashemi ^1, , George Yin ^2, and Le Yi Wang ^3,

1.	Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States
2.	Department of Mathematics, Wayne State University, Detroit, Michigan 48202
3.	Department of Electrical and Computer Engineering, Wayne State University, MI 48202

Received February 2015 Revised March 2015 Published October 2015

Abstract
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Motivated by reduction of computational complexity, this work develops sign-error adaptive filtering algorithms for estimating randomly time-varying system parameters. Different from the existing work on sign-error algorithms, the parameters are time-varying and their dynamics are modeled by a discrete-time Markov chain. Another distinctive feature of the algorithms is the multi-time-scale framework for characterizing parameter variations and algorithm updating speeds. This is realized by considering the stepsize of the estimation algorithms and a scaling parameter that defines the transition rate of the Markov jump process. Depending on the relative time scales of these two processes, suitably scaled sequences of the estimates are shown to converge to either an ordinary differential equation, or a set of ordinary differential equations modulated by random switching, or a stochastic differential equation, or stochastic differential equations with random switching. Using weak convergence methods, convergence and rates of convergence of the algorithms are obtained for all these cases. Simulation results are provided for demonstration.

Keywords: tracking property., regime-switching model, convergence, mean squares error, stochastic approximation, Sign-error algorithm.

Mathematics Subject Classification: Primary: 93E10, 93E11, 93E24, 93E3.

Citation: Araz Hashemi, George Yin, Le Yi Wang. Sign-error adaptive filtering algorithms involving Markovian parameters. Mathematical Control and Related Fields, 2015, 5 (4) : 781-806. doi: 10.3934/mcrf.2015.5.781

References:

[1]	A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-75894-2.
[2]	P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, 1968.
[3]	H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering, IEEE Trans. Automat. Control, 48 (2003), 1545-1556. doi: 10.1109/TAC.2003.816967.
[4]	E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data, IEEE Trans. Inform. Theory, 37 (1991), 1450-1457.
[5]	J. Fang and H. Li, Adaptive distributed estimation of signal power from one-bit quantized data, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 1893-1905.
[6]	A. Gersho, Adaptive filtering with binary reinforcement, IEEE Trans. Inform. Theory, 30 (1984), 191-199.
[7]	L. Guo, Stability of recursive stochastic tracking algorithms, SIAM Journal on Control and Optimization, 32 (1994), 1195-1225. doi: 10.1137/S0363012992225606.
[8]	M. L. Honig and H. V. Poor, Adaptive interference suppression in wireless communication systems, in Wireless Communications: Signal Processing Perspectives (eds. H. V. Poor and G. W. Wornell), Prentice Hall, 1998.
[9]	V. Krishnamurthy, G. Yin and S. Singh, Adaptive step size algorithms for blind interference suppression in DS/CDMA systems, IEEE Trans. Signal Processing, 49 (2001), 190-201.
[10]	H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains, IEEE Trans. Inform. Theory, 30 (1984), 177-182. doi: 10.1109/TIT.1984.1056897.
[11]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed., Springer-Verlag, New York, NY, 2003.
[12]	L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4956-2.
[13]	B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood, Cliffs, NJ, 1985.
[14]	G. Yin, Adaptive filtering with averaging, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. R. Kumar), IMA Volumes in Mathematics and Its Applications, 74, Springer-Verlag, New York, 1995, 375-396. doi: 10.1007/978-1-4419-8568-2_18.
[15]	G. Yin and H.-F. Chen, On asymptotic properties of a constant-step-size sign-error algorithm for adaptive filtering, Scientia Sinica, 45 (2002), 321-334. doi: 10.1007/BF02714090.
[16]	G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters, Asian J. Control, 16 (2014), 95-106. doi: 10.1002/asjc.678.
[17]	G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit, IEEE Trans. Automat. Control, 50 (2005), 577-593. doi: 10.1109/TAC.2005.847060.
[18]	G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications, Springer, New York, NY, 2005.
[19]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

show all references

References:

[1]	A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-75894-2.
[2]	P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, 1968.
[3]	H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering, IEEE Trans. Automat. Control, 48 (2003), 1545-1556. doi: 10.1109/TAC.2003.816967.
[4]	E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data, IEEE Trans. Inform. Theory, 37 (1991), 1450-1457.
[5]	J. Fang and H. Li, Adaptive distributed estimation of signal power from one-bit quantized data, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 1893-1905.
[6]	A. Gersho, Adaptive filtering with binary reinforcement, IEEE Trans. Inform. Theory, 30 (1984), 191-199.
[7]	L. Guo, Stability of recursive stochastic tracking algorithms, SIAM Journal on Control and Optimization, 32 (1994), 1195-1225. doi: 10.1137/S0363012992225606.
[8]	M. L. Honig and H. V. Poor, Adaptive interference suppression in wireless communication systems, in Wireless Communications: Signal Processing Perspectives (eds. H. V. Poor and G. W. Wornell), Prentice Hall, 1998.
[9]	V. Krishnamurthy, G. Yin and S. Singh, Adaptive step size algorithms for blind interference suppression in DS/CDMA systems, IEEE Trans. Signal Processing, 49 (2001), 190-201.
[10]	H. J. Kushner and A. Shwartz, Weak convergence and asymptotic properties of adaptive filters with constant gains, IEEE Trans. Inform. Theory, 30 (1984), 177-182. doi: 10.1109/TIT.1984.1056897.
[11]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed., Springer-Verlag, New York, NY, 2003.
[12]	L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4956-2.
[13]	B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood, Cliffs, NJ, 1985.
[14]	G. Yin, Adaptive filtering with averaging, in Adaptive Control, Filtering and Signal Processing (eds. K. Aström, G. Goodwin and P. R. Kumar), IMA Volumes in Mathematics and Its Applications, 74, Springer-Verlag, New York, 1995, 375-396. doi: 10.1007/978-1-4419-8568-2_18.
[15]	G. Yin and H.-F. Chen, On asymptotic properties of a constant-step-size sign-error algorithm for adaptive filtering, Scientia Sinica, 45 (2002), 321-334. doi: 10.1007/BF02714090.
[16]	G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters, Asian J. Control, 16 (2014), 95-106. doi: 10.1002/asjc.678.
[17]	G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit, IEEE Trans. Automat. Control, 50 (2005), 577-593. doi: 10.1109/TAC.2005.847060.
[18]	G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications, Springer, New York, NY, 2005.
[19]	G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

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