A derivation of the stable condition for the filtered-x LMS algorithm in the case where (C)over-cap filter has modeling error

When an ANC system is operated with the Filtered-x LMS algorithm, the step size parameter constant needs to be determined. If this constant is too sma ll, the convergence of the filter coefficient becomes slow. If, on the othe r hand, it is increased too much to speed up the convergence, the system be comes unstable. It is necessary to find the upper limit for stable operatio n so that system stability is satisfied and convergence is fast. Such a val ue has hitherto been unknown. Therefore, the step size parameter has been d etermined by trial and error. For the Filtered-x LMS algorithm, we need a f ilter (C) over cap that identifies the error path C between the secondary s peaker in the ANC system and the error detection microphone. Naturally, mod eling error is contained in this filter. In this paper, a theoretical equat ion for the upper limit of the step size parameter for stable operation is derived from the phase error of the allowable C and (C) over cap. We also a ttempt to represent this theoretical equation in terms of information obtai nable at the time of system operation. The convergence characteristics in t he case using the step size parameter value obtained from the theoretical e quation and in the case using other values are compared, and the effectiven ess of the proposed theoretical equation is demonstrated. (C) 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(11): 65-73, 1999.