Mathematical Control and Related Fields (MCRF)

Sign-error adaptive filtering algorithms involving Markovian parameters

Pages: 781 - 806, Volume 5, Issue 4, December 2015      doi:10.3934/mcrf.2015.5.781

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Araz Hashemi - Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States (email)
George Yin - Department of Mathematics, Wayne State University, Detroit, Michigan 48202, United States (email)
Le Yi Wang - Department of Electrical and Computer Engineering, Wayne State University, MI 48202, United States (email)

Abstract: 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:  Sign-error algorithm, regime-switching model, stochastic approximation, mean squares error, convergence, tracking property.
Mathematics Subject Classification:  Primary: 93E10, 93E11, 93E24, 93E35.

Received: February 2015;      Revised: March 2015;      Available Online: October 2015.