Advances on the analysis of the LMS algorithm with a colored measurement noise

Abstract

Due to the inherent feedback feature of adaptive filtering algorithms, a comprehensive theoretical understanding of their learning process is still challenging. In order to make the mathematics tractable, most stochastic analyses adopt simplifications. One of these is the independence assumption, which presumes statistical independence between adaptive weights and input signal samples. Furthermore, the additive noise is usually assumed to be white, although it may not be the case in practice. This paper advances a novel theoretical model of the least-mean-square adaptive filter that, under the independence assumption, does not presume a white noise signal. Additionally, a theoretical result is established which implies that the stability properties of such an adaptive filter are not influenced by noise coloring. Such a result does not employ the above-mentioned assumptions, i.e., it is valid for both colored noise and non-infinitesimally small step size values. An optimal step-size sequence that minimizes the mean-square deviation and takes into account the stochastic coupling of the excitation data and adaptive weights is proposed. It is noteworthy that such a design does not induce divergence in practice and does not assume, for the first time, a white measurement noise. The theoretical contributions are confirmed by simulations.