Stochastic Analysis of the Filtered-x LMS Algorithm for Active Noise Control

The filtered-x least-mean-square (FxLMS) algorithm has been widely used for the active noise control. A fundamental analysis of the convergence behavior of the FxLMS algorithm, including the transient and steady-state performance, could provide some new insights into the algorithm and can be also helpful for its practical applications, e.g., the choice of the step size. Although many efforts have been devoted to the statistical analysis of the FxLMS algorithm, it was usually assumed that the reference signal is Gaussian or white. However, non-Gaussian and/or non-white processes could be very widespread in practice as well. Moreover, the step-size bound that guarantees both of the mean and mean-square stability of the FxLMS for an arbitrary reference signal and a general secondary path is still not available in the literature. To address these problems, this article presents a comprehensive statistical convergence analysis of the FxLMS algorithm without assuming a specific model for the reference signal. We formulate the mean weight behavior and the mean-square error (MSE) in terms of an augmented weight vector. The covariance matrix of the augmented weight-error vector is then evaluated using the vectorization operation, which makes the analysis easy to follow and suitable for arbitrary input distributions. The stability bound is derived based on the first-order and second-order moments analysis of the FxLMS. Computer simulations confirmed the effectiveness of the proposed theoretical model.