Analysis of the normalized LMS algorithm with Gaussian inputs

The LMS adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter μ for stability and convergence. Since the input power level is usually one of the statistical unknowns, it is normally estimated from the data prior to beginning the adaptation process. It is then assumed that the estimate is perfect in any subsequent analysis of the LMS algorithm behavior. In this paper, the effects of the power level estimate are incorporated in a data dependent μ that appears explicitly within the algorithm. The transient mean and second-moment behavior of the modified LMS (NLMS) algorithm are evaluated, taking into account the explicit statistical dependence of μ upon the input data. The mean behavior of the algorithm is shown to converge to the Wiener weight. A constant coefficient matrix difference equation is derived for the weight fluctuations about the Wiener weight. The equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single-frequency input in steady state for small μ. Expressions for the misadjustment error are also presented. It is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the LMS algorithm. In the ALE application, the steady-state weight fluctuations are shown to be mode dependent, being largest at the frequency of the input.