Convergence Behavior of NLMS Algorithm for Gaussian Inputs: Solutions Using Generalized Abelian Integral Functions and Step Size Selection

Abstract

This paper studies the mean and mean square convergence behaviors of the normalized least mean square (NLMS) algorithm with Gaussian inputs and additive white Gaussian noise. Using the Price's theorem and the framework proposed by Bershad in IEEE Transactions on Acoustics, Speech, and Signal Processing (1986, 1987), new expressions for the excess mean square error, stability bound and decoupled difference equations describing the mean and mean square convergence behaviors of the NLMS algorithm using the generalized Abelian integral functions are derived. These new expressions which closely resemble those of the LMS algorithm allow us to interpret the convergence performance of the NLMS algorithm in Gaussian environment. The theoretical analysis is in good agreement with the computer simulation results and it also gives new insight into step size selection.

On the Performance Analysis of the Least Mean M-Estimate and Normalized Least Mean M-Estimate Algorithms with Gaussian Inputs and Additive Gaussian and Contaminated Gaussian Noises

“…Therefore, under the stated assumptions, the maximum convergence rate of the normalized algorithms using ATS is faster than the LMS-based algorithms if the eigenvalues are unequal. Similar conclusion is obtained for the conventional NLMS algorithm [37,46].…”

“…(43) and (45) will reduce to the EMSE (∞) and stability bound for the conventional NLMS algorithm derived in [37]. For the NLMM algorithm using MHnonlinearity with a practical value of k x ¼ 2:576, A y , S y , and C y are quite close to one, and its performance is therefore similar to that of the conventional NLMS algorithm.…”

“…This approximation is more accurate when the step size is small. This applies to the LMM algorithm with [37], the performance of the NLMS algorithm in Gaussian noise was analyzed and it is shown that 1 2 ms 2 g serves as a useful lower bound for estimating the EMSE (misadjustment) of the NLMS algorithm. It is attractive because it does not require the knowledge of the eigenvalues or eigenvalue spread of R XX .…”

“…Particularly, these analytical results can be reduced to those for the conventional NLMS algorithms in Gaussian additive noise which agree with [17,18], except that new expressions for the excess mean square error (EMSE), stability bound and difference equations in terms of the generalized Abelian integrals are obtained. For clarity of presentation and its practical importance, this particular case is separately treated in the companion paper [37]. All the above results can also be readily generalized to the simpler LMS case.…”

“…Another key to the above analysis is the introduction of certain special functions called the generalized Abelian integral functions [37], which are generalizations of the Abelian integral functions [38]. They allow us to obtain decoupled analytical formulas by evaluating the expectations involved in the difference equations and help us interpret the convergence behaviors of the NLMS algorithms with M-nonlinearity.…”