The Krylov-proportionate normalized least mean fourth approach: Formulation and performance analysis

• Introducing novel algorithms named PNLMF, KPNLMF, and KPNLMMN that demonstrate faster convergence.
• The steady-state mean-square analysis of the KPNLMS, KPNLMF and KPNLMMN algorithms.
• The tracking performance analysis of the KPNLMS, KPNLMF and KPNLMMN algorithms in the non-stationary environment.
• Numerical simulations demonstrating the improved performance of the novel algorithms.

We propose novel adaptive filtering algorithms based on the mean-fourth error objective while providing further improvements on the convergence performance through proportionate update. We exploit the sparsity of the system in the mean-fourth error framework through the proportionate normalized least mean fourth (PNLMF) algorithm. In order to broaden the applicability of the PNLMF algorithm to dispersive (non-sparse) systems, we introduce the Krylov-proportionate normalized least mean fourth (KPNLMF) algorithm using the Krylov subspace projection technique. We propose the Krylov-proportionate normalized least mean mixed norm (KPNLMMN) algorithm combining the mean-square and mean-fourth error objectives in order to enhance the performance of the constituent filters. Additionally, we propose the stable-PNLMF and stable-KPNLMF algorithms overcoming the stability issues induced due to the usage of the mean fourth error framework. Finally, we provide a complete performance analysis, i.e., the transient and the steady-state analyses, for the proportionate update based algorithms, e.g., the PNLMF, the KPNLMF algorithms and their variants; and analyze their tracking performance in a non-stationary environment. Through the numerical examples, we demonstrate the match of the theoretical and ensemble averaged results and show the superior performance of the introduced algorithms in different scenarios.