Convergence of proportionate-type LMS adaptive filters and choice of gain matrix

Recently, it has been shown that the proportionate-type LMS adaptive litters are converging for the sufficiently small adaptation stepsize parameter and white input. In addition to this, a theoretical connection between proportionate-type steepest descent algorithms and proportionate-type stochastic algorithms for a constant gain matrix has been revealed. Motivated by this theoretical connection, we seek a connection between these types of algorithms for a time-varying gain matrix. To that end, we examine the feasibility of predicting the performance of a stochastic proportionate algorithm with a time-varying gain matrix by analyzing the performance of its associated deterministic steepest descent algorithm. In doing so we have found that this approach has merit. Using this insight, various steepest descent algorithms are studied and used to predict and explain the behavior of their counterpart stochastic algorithms. In particular, it is shown that the mu-PNLMS algorithm possesses robust behavior. In addition to this the epsilon-PNLMS algorithm is proposed and its performance is evaluated.