  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.
