Convergence Properties of Adaptive Estimators of Time-Varying Linear Systems using Basis Functions

The estimation of time-varying linear systems using a basis function approach has been applied in various fields such as equalization of mobile radio channels and in estimation of dynamics in biological systems. Typically, time-varying finite impulse response system models have been employed with recursive least squares or least mean squares adaptation. In this paper the convergence properties of these and other adaptive algorithms employed in this setting are formulated. The use of time-varying ARMA models is also included in the framework that is examined. The relation of the prediction error with the parameter error and the system regressor is exposed, indicating that a previously analyzed class of adaptive algorithms is appropriate for these problems. The convergence of this class of algorithms is dependent on a persistent excitation condition on system signals and a passivity condition on a system operator. Requirements for system regressors to be persistently exciting are derived for the time-varying linear system identification using basis functions, and the relevant operator conditions are described