H_∞ bounded optimal updating - down-dating algorithm

The LMS algorithm, which is widely used in the adaptive filtering community, has been proved to be H_∞ optimal. We (Kothari et al. (2002)) have analyzed the other performance measures in the H_∞ setting which are of direct relevance to adaptive filtering and system identification. In that paper we considered the system identification and estimation employing exponential window problems. This problems are basically of rank I updating class, where we have to update the estimation as the new information comes into picture, while reducing the effect of the past data with a predefined factor. Due to this the effect of past data is not removed completely. The H_∞ performance measure in the situation of removing the past data effect completely and optimal H_∞ filter in this situation was still an open problem. In this paper we examine the performance measure in the H_∞ setting employing a sliding window. We present explicit algorithms and the achievable bound in this case.