American Institute of Mathematical Sciences

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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

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 & 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, (1990).  doi: 10.1007/978-3-642-75894-2.  Google Scholar

[2]

P. Billingsley, Convergence of Probability Measures,, J. Wiley, (1968).   Google Scholar

[3]

H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering,, IEEE Trans. Automat. Control, 48 (2003), 1545.  doi: 10.1109/TAC.2003.816967.  Google Scholar

[4]

E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data,, IEEE Trans. Inform. Theory, 37 (1991), 1450.   Google Scholar

[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.   Google Scholar

[6]

A. Gersho, Adaptive filtering with binary reinforcement,, IEEE Trans. Inform. Theory, 30 (1984), 191.   Google Scholar

[7]

L. Guo, Stability of recursive stochastic tracking algorithms,, SIAM Journal on Control and Optimization, 32 (1994), 1195.  doi: 10.1137/S0363012992225606.  Google Scholar

[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), (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TIT.1984.1056897.  Google Scholar

[11]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, 2nd ed., (2003).   Google Scholar

[12]

L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications,, Birkhäuser, (2010).  doi: 10.1007/978-0-8176-4956-2.  Google Scholar

[13]

B. Widrow and S. D. Stearns, Adaptive Signal Processing,, Prentice-Hall, (1985).   Google Scholar

[14]

G. Yin, Adaptive filtering with averaging,, in Adaptive Control, (1995), 375.  doi: 10.1007/978-1-4419-8568-2_18.  Google Scholar

[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.  doi: 10.1007/BF02714090.  Google Scholar

[16]

G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters,, Asian J. Control, 16 (2014), 95.  doi: 10.1002/asjc.678.  Google Scholar

[17]

G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit,, IEEE Trans. Automat. Control, 50 (2005), 577.  doi: 10.1109/TAC.2005.847060.  Google Scholar

[18]

G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications,, Springer, (2005).   Google Scholar

[19]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

show all references

References:
[1]

A. Benveniste, M. Metivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-642-75894-2.  Google Scholar

[2]

P. Billingsley, Convergence of Probability Measures,, J. Wiley, (1968).   Google Scholar

[3]

H.-F. Chen and G. Yin, Asymptotic properties of sign algorithms for adaptive filtering,, IEEE Trans. Automat. Control, 48 (2003), 1545.  doi: 10.1109/TAC.2003.816967.  Google Scholar

[4]

E. Eweda, Convergence of the sign algorithm for adaptive filtering with correlated data,, IEEE Trans. Inform. Theory, 37 (1991), 1450.   Google Scholar

[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.   Google Scholar

[6]

A. Gersho, Adaptive filtering with binary reinforcement,, IEEE Trans. Inform. Theory, 30 (1984), 191.   Google Scholar

[7]

L. Guo, Stability of recursive stochastic tracking algorithms,, SIAM Journal on Control and Optimization, 32 (1994), 1195.  doi: 10.1137/S0363012992225606.  Google Scholar

[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), (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TIT.1984.1056897.  Google Scholar

[11]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, 2nd ed., (2003).   Google Scholar

[12]

L. Y. Wang, G. Yin, J.-F. Zhang and Y. L. Zhao, System Identification with Quantized Observations: Theory and Applications,, Birkhäuser, (2010).  doi: 10.1007/978-0-8176-4956-2.  Google Scholar

[13]

B. Widrow and S. D. Stearns, Adaptive Signal Processing,, Prentice-Hall, (1985).   Google Scholar

[14]

G. Yin, Adaptive filtering with averaging,, in Adaptive Control, (1995), 375.  doi: 10.1007/978-1-4419-8568-2_18.  Google Scholar

[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.  doi: 10.1007/BF02714090.  Google Scholar

[16]

G. Yin, A. Hashemi and L. Y. Wang, Sign-regressor adaptive filtering algorithms for Markovian parameters,, Asian J. Control, 16 (2014), 95.  doi: 10.1002/asjc.678.  Google Scholar

[17]

G. Yin and V. Krishnamurthy, Least mean square algorithms with Markov regime switching limit,, IEEE Trans. Automat. Control, 50 (2005), 577.  doi: 10.1109/TAC.2005.847060.  Google Scholar

[18]

G. Yin and Q. Zhang, Discrete-time Markov Chains: Two-time-scale Methods and Applications,, Springer, (2005).   Google Scholar

[19]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

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