Analysis of the filtered-X LMS algorithm

The presence of a transfer function in the auxiliary-path following the adaptive filter and/or in the error-path, as in the case of active noise control, has been shown to generally degrade the performance of the LMS algorithm. Thus, the convergence rate is lowered, the residual power is increased, and the algorithm can even become unstable. To ensure convergence of the algorithm, the input to the error correlator has to be filtered by a copy of the auxiliary-error-path transfer function. This paper presents an analysis of the filtered-X LMS algorithm using stochastic methods. The influence of off-line and on-line estimation of the error-path filter on the algorithm is also investigated. Some derived bounds and predicted dynamic behavior are found to correspond very well to simulation results.