Sign-Error Adaptive Filtering Algorithms for Markovian Parameters

TL;DR

This paper introduces sign-error adaptive filtering algorithms tailored for estimating parameters that change over time according to a Markov process, with analysis across multiple time scales and convergence properties.

Contribution

It develops a multi-time-scale framework for sign-error algorithms estimating Markovian parameters, extending prior work to time-varying systems with convergence analysis.

Findings

Algorithms converge to differential equations or stochastic models depending on time scales.

Convergence rates are established using weak convergence methods.

The approach reduces computational complexity for time-varying parameter estimation.

Abstract

Motivated by reduction of computational complexity, this work develops sign-error adaptive filtering algorithms for estimating time-varying system parameters. Different from the previous work on sign-error algorithms, the parameters are time-varying and their dynamics are modeled by a discrete-time Markov chain. A distinctive feature of the algorithms is the multi-time-scale framework for characterizing parameter varia- tions and algorithm updating speeds. This is realized by considering the stepsize of the estimation algorithms and a scaling parameter that defines the transition rates of the Markov jump process. Depending on the relative time scales of these two pro- cesses, 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 equa- tions with random switching. Using weak convergence methods, convergence and rates of convergence of the algorithms are obtained for all these cases.