Analysis of the LMS and NLMS Algorithms Using the Misalignment Norm

Abstract

This work describes the convergence of the misalignment square norm (MSN) of the NLMS and LMS algorithms. It is shown that the MSN decrease is almost proportional to the mean square error (MSE). This allows obtaining simple expressions for the steady-state MSE. Also, it allows limiting the amount of time that the MSE takes large values and a curve that limits the MSE of LMS at any given time, independent of the input and background noise signals' properties. Finally, it is also shown that many complications in the analysis of the LMS and NLMS algorithms can come from variations in the input vector square norm. The proposed analysis becomes very simple for long filters or constant power signals.

Improved LMS Performance for System Identification Application

“…This table also includes a column showing the value used by a number of references for LMS step size. Small +ve value Not given [3] 0< µ < 1/10NP x * Not given [31], [32], [33] 0 < µ < 1/λ max ** 0.0625 and 0.08 [34] 0 < µ < 2/λ max from 0.001 to 1 [35] 0 < µ < 0.2 0.01 and 0.004 [36] 0 < µ < 1 from 0.001 to 0.1 [37] 0 < µ < 2 from 0.005 to 2 * P x denotes the power included in the input signal x(n). ** λ max denotes the maximum eigenvalue of the covariance matrix of the input signal x(n).…”

“…It can be noticed that the equations given in table 1 can only be used to specify the limits of the step size, leaving a wide range of feasible values to select the step size from. The optimum step size within a given range has been the material of many studies [34]- [37], where LMS behavior is assessed at a range of step sizes and based on the results, the most suitable value is recommended. Nevertheless, this value is vulnerable to change when the filter order, weight values or input signal properties is changed, limiting the efficiency of such studies.…”