ABSTRACT
This chapter illustrates some considerations behind time-varying recursive filters, with a more detailed analysis into filter stability beginning. It suggests slow variation analyses typically argue that a given time-varying system, if slowly varying, should not be too far from a related time-invariant system. The chapter presents an informal approach, along with some inherent limitations of slow variation analyses. Stability of the time-varying infinite impulse response (IIR) filter which results during adaptation is a generic necessary condition for any candidate parameter adaptation law to converge. The chapter introduces thus the notion of exponential stability for the homogeneous portion, and then shows how this property implies bounded-input-bounded-ouput (BIBO) stability of the system. Lyapunov stability methods have the virtue of being equally lucid in both time-varying and time-invariant settings. The stability properties of time-varying lattice filters first surfaced in the work of Gray and Markel, who showed that the zero input state recursion of the filter could never go unstable.
