Mean-square Analysis of the NLMS Algorithm

Abstract

This work presents a novel approach to the mean-square analysis of the normalized least mean squares (NLMS) algorithm for circular complex colored Gaussian inputs. The analysis is based on the derivation of a closed-form expression for the Cumulative Distribution Function (CDF) of random variables of the form $(||\mathbf{u}|{|}_{{\mathbf{D}}_1}^2)(||\mathbf{u}|{|}_{{\mathbf{D}}_2}^2{)}^{-1}$ where $\mathbf{u}$ is an isotropic vector and ${\mathbf{D}}_1$ and ${\mathbf{D}}_2$ are diagonal matrices and using that to derive some moments of these variables. These moments in turn completely characterize the mean-square behavior of the NLMS algorithm in explicit closed-form expressions. Specifically, the transient, steady-state, and tracking mean-square behavior of the NLMS algorithm is studied.