Improved Steady State Analysis of the Recursive Least Squares Algorithm

Abstract

This paper presents a new approach for studying the steady state performance of the Recursive Least Square (RLS) adaptive filter for a circularly correlated Gaussian input. Earlier methods have two major drawbacks: (1) The energy relation developed for the RLS is approximate (as we show later) and (2) The evaluation of the moment of the random variable $\Vert \mathbf{u}_{i}\Vert_{\mathrm{P}_{\mathrm{i}}}^{2}$ , where $\mathrm{u}_{i}$ is input to the RLS filter and $\mathbf{P}_{i}$ is the estimate of the inverse of input covariance matrix by assuming that $\mathbf{u}_{i}$ and $\mathbf{P}_{i}$ are independent (which is not true). These assumptions could result in negative value of the stead-state Excess Mean Square Error (EMSE). To overcome these issues, we modify the energy relation without imposing any approximation. Based on modified energy relation, we derive the steady-state EMSE and two upper bounds on the EMSE. For that, we derive closed from expression for the aforementioned moment which is based on finding the cumulative distribution function (CDF) of the random variable of the form $\displaystyle \frac{1}{\gamma+\Vert \mathbf{u}\Vert _{\mathrm{D}}^{2}}$ , where $\mathrm{u}$ is correlated circular Gaussian input and $\mathrm{D}$ is a diagonal matrix. Simulation results corroborate our analytical findings.