Stability of FX-LMS algorithm for short adaptive filters

Active Noise Control (ANC) is still receiving considerable attention among research fields in modem science. Adaptive ANC solutions use almost exclusively Least Mean Squares (LMS) algorithm, or actually one of its modifications, known as FX-LMS algorithm. The major problem with FX-LMS algorithm is the same as with other LMS family members: it is necessary to chose the step size very carefully. Too large step size results in large excess mean square error and may cause loss of stability, to small step size results in small adaptation speed and degrades ANC performance, especially in in nonstationary environments. This paper tries to find the FX-LMS sufficient stability condition in sense of the step size. Knowledge of such step size is helpful during the phase of rapid adaptation, just after the ANC system starts to operate, or after detecting substantial change in the environment. Rapid adaptation is crucial in many ANC applications, e.g. in large industrial halls, where the silence zone should follow persons moving around. Therefore, FX-LMS algorithms should operate with large step sizes during such phase. However, it is even more crucial not to make the step size too large and risk the lose of stability. The paper presents derivation of the stability condition in case of short filters as well as computer simulations that help to reinforce the developed theory.