Stable condition considering modeling error in the filtered-x LMS algorithm

When an ANC system is operated with the filtered-x LMS algorithm, the step size parameter needs to be determined. If this parameter is too small, the convergence of the filter coefficient becomes slow. If, on the other hand, it is increased too much to speed up the convergence, the system becomes unstable. It is necessary to find the upper limit for stable operation so that system stability is satisfied and convergence speed is fast. Such a value has hitherto been unknown. The step size parameter, therefore, has been determined by trial and error. For the filtered-x LMS algorithm, we need a model of the secondary path between the secondary source and the error sensor in the ANC. Naturally, modeling error is contained in this model. In this paper, a theoretical equation for the upper limit of the step size parameter for stable operation is derived. We also attempt to represent this theoretical equation in terms of information obtainable at the time of system operation