Stability of adaptive filters with linearly interfering update errors

We provide a time-domain analysis of the stability for two adaptive algorithms of gradient type that interfere with each other via their update errors. Such coupling can occur naturally as well as by desire of the designer. Especially, system identification algorithms that combine two adaptive schemes can often be described by such a structure. We derive precise statements on local contracting/expanding behaviour that in turn allow to deduce bounds ensuring Lyapunov stability. The application of our findings to a specific example shows how these bounds are obtained and how they outperform our previous results that were based on the small gain theorem.