Exact Performance Analysis of the $\epsilon$-NLMS Algorithm for Colored Circular Gaussian Inputs

Abstract

This paper presents exact mean-square analysis of the -NLMS algorithm for circular complex correlated Gaussian input. The analysis is based on the derivation of a closed form expression for the cumulative distribution function (CDF) of random variables of the form ∥ui∥D12/(ϵ + ∥ui∥D12) and using that to derive the first and second moments of such variables. These moments in turn completely characterize the mean square (MS) behavior of the ϵ-NLMS in explicit closed form expressions. Both transient and steady-state behavior are analyzed. Consequently, new explicit closed-form expressions for the mean-square-error (MSE) behavior are derived. Our simulations of the transient and steady-state behavior of the filter match the expressions obtained theoretically for various degrees of input correlation and for various values of ϵ.