MSE Analysis of the Krylov-proportionate NLMS and NLMF Algorithms

This paper proposes a novel adaptive filtering algorithm which converges faster than the Krylov proportionate normalized least mean square (KPNLMS) algorithm. KPNLMS is known to exhibit faster convergence than the standard NLMS algorithm for all unknown systems. Our algorithm is named Krylov proportionate normalized least mean fourth (KPNLMF) and it deals with mean fourth minimization of the error. In this paper, first the KPNLMF algorithm is derived from the KPNLMS algorithm and then steady-state mean square error (MSE) performance of both algorithms are analyzed. MSE analysis showed that both KPNLMS and KPNLMF converge to the desired solution with small excess mean square errors. Both algorithms enjoy the fast convergence behavior of the proportionate NLMS algorithm not only for sparse systems but also for dispersive (non-sparse) systems thanks to the Krylov subspace projection technique. In the simulations part, the KPNLMF algorithm is shown to converge faster than the KPNLMS algorithm when both algorithms converge to the same system mismatch value. The KPNLMF algorithm achieves this without any increase in the computational complexity. Further numerical examples comparing KPNLMF with NLMF and KPNLMS support the fast convergence of the KPNLMF algorithm.