Analysis of recursive least moduli algorithm with generalized error modulus

This paper first revisits least mean modulus (LMM) algorithm for complex-domain adaptive filters, presents a mathematical model for impulsive observation noise called CGN, and reviews recursive least moduli (RLM) algorithm that combines the LMM algorithm with recursive estimation of inverse covariance matrix of filter inputs. The RLM algorithm is effective in making the convergence of an adaptive filter with a strongly correlated filter input significantly faster, while preserving the robustness of the LMM algorithm against impulsive observation noise. Next, a generalized modulus of a complex number (“p-modulus”) is defined. We modify the RLM algorithm with p-modulus of the error. Analysis of the RLM algorithm is developed to derive a set of difference equations for calculating transient and steady-state behavior. Through experiment with simulations and theoretical calculations of filter convergence, we find that the filter convergence behavior does not critically depend on the value of p. We also demonstrate effectiveness of the RLM algorithm in improving the filter convergence speed and robustness against the CGN. Good agreement between simulated and theoretical convergence validates the analysis.