Combinations of Adaptive Filters: Performance and convergence properties

Adaptive filters are at the core of many signal processing applications, ranging from acoustic noise supression to echo cancelation [1], array beamforming [2], channel equalization [3], to more recent sensor network applications in surveillance, target localization, and tracking. A trending approach in this direction is to recur to in-network distributed processing in which individual nodes implement adaptation rules and diffuse their estimation to the network [4], [5].

Problem Formulation and Notation

We generally assume the availability of a reference signal and an input regressor vector, and , respectively, which satisfy a linear regression model of the form

$$\begin{equation*}d(n)=\mathrm{u}^{\top}(n)\mathrm{w}_{o}(n)+\upsilon(n),\tag{1}\end{equation*}$$ where represents the (possibly) time-varying optimal solution, and is a noise sequence, which is considered independent and identically distributed, and independent of for all . In this article, we will mostly restrict ourselves to the case in which all involved variables are real, although the extension to the complex case is straightforward, and has been analyzed in other works [6]. To estimate the optimal solution at time , adaptive filters typically implement a recursion of the form $$\begin{equation*}\mathrm{w}(n+1)=\mathrm{f}[\mathrm{w}(n), d(n),\mathrm{u}(n),\ \mathrm{s}(n)],\end{equation*}$$ where different adaptive schemes are characterized by their update functions and represents any other state information that is needed for the update of the filter.