A Stieltjes transform approach for analyzing the RLS adaptive Filter

Although the RLS filter is well-known and various algorithms have been developed for its implementation, analyzing its performance when the regressors are random, as is often the case, has proven to be a formidable task. The reason is that the Riccati recursion, which propagates the error covariance matrix, becomes a random recursion. The existing results are approximations based on assumptions that are often not very realistic. In this paper we use ideas from the theory of large random matrices to find the asymptotic (in time) eigendistribution of the error covariance matrix of the RLS filter. Under the assumption of a large dimensional state vector (in most cases n = 10-20 is large enough to get quite accurate predictions) we find the asymptotic eigendistribution of the error covariance for temporally white regressors, shift structured regressors, and for the RLS filter with intermittent observations.