The Kalman Filter in Active Noise Control

Most Active Noise Control (ANC) systems use some form of the LMS [5][7] algorithm due to its reduced computational complexity. However, the problems associated with it are well-known, namely slow convergence and high sensitivity to the eigenvalue spread [3][7]. To overcome this problems the RLS algorithm is often used, but it is now widely known, that the RLS loses many of its good properties for a forgetting factor lower than one. Namely, it has been shown that in some application the LMS algorithm is actually better in tracking non-stationary signals than the RLS algorithm [2][3]. One approach, which is works well with non-stationary signals, is to use some specialized form of the Kalman filter, which can be interpreted as a generalization of the RLS algorithm [1][3][4]. The Kalman filter has a high computational complexity, similar to that of the RLS algorithm, which can make it costly for some applications. Nevertheless for narrowband ANC, the number of taps is not very large [7] and the application of the Kalman filter in ANC may be easily handled by today DSP's. In fact, in multi-channel applications the Kalman gain can be computed only once for all the error signals, so the complexity hardly increases with a moderated number of those. Even in the cases where it can not be easily implemented in real time, it can still be used as a benchmark. In fact the Kalman filter is optimal for a given system model [1]. In this paper, specialized version of the Kalman filter fitted to ANC is developed both for primary and secondary path modeling. It is shown throw computer experiments that a large reduction in the residual noise can be achieved in non-stationary environments, compared with the LMS and RLS based algorithms, specially with on-line secondary path modeling.