Analysis of the normalized LMS algorithm with Gaussian inputs

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

“…In the conventional NLMS algorithm studied in [9] and [10] with α=1, similar difference equation for the mean weight-error vector (c.f. [10, Eq.…”

“…14]), where 1 i is the i-th eigenvalue of R XX . It was evaluated analytically in [9] for three important cases with different eigenvalue distributions (in [10], only the first case was elaborated): (1) white input signal with 1 1 =...=1 L ; (2) two signal subspaces with equal powers…”

“…On the other hand, the NLMS algorithm generally possesses an improved convergence speed over the LMS algorithm, but its analysis is more complicated due to the step size normalization. In [9] and [10], the mean and mean square behaviors of the NLMS algorithm for Gaussian inputs were studied. Analysis for independent Gaussian inputs in [11] also revealed the advantage of the NLMS algorithm over the LMS algorithm.…”

“…Due to the difficulties in evaluating the expectations involved in the difference equations for the mean weight-error vector and its covariance matrix, general closed-form expressions for these equations and the excess mean square error (EMSE) are in general unavailable. Consequently, the works in [9,10] only concentrated on certain special cases of eigenvalue distribution of the input autocorrelation matrix. In [12,13], particular or simplified input data model was introduced to facilitate the performance analysis so that useful analytical expressions can still be derived.…”

“…Using the Price's theorem [20] and the framework in [9,10], new decoupled difference equations describing the mean and mean square convergence behaviors of the NLMS algorithm using the generalized Abelian integral functions are derived. The final results closely resemble the classical results for LMS in [1].…”