Second moment analysis of the filtered-X LMS algorithm

Active noise and vibration control (ANC) has become an important application area for adaptive filters. The most popular adaptive algorithm used for ANC is the Filtered-X LMS (FXLMS) algorithm [1,2]. This algorithm is a modification of the well known LMS algorithm. The reference signal is filtered so as to compensate for filters inherent in the electro-acoustic adaptation loop. These additional filters in the adaptation system significantly complicate the behavior and analysis of the adaptive algorithm. Conventional LMS analysis is not applicable to the FXLMS algorithm. Also, simplifying LMS assumptions cannot easily be extended to FXLMS. This applies to the well known independence theory (IT), in which time-lagged input vectors are assumed independent. The additional loop filters generate signal correlations. These correlations invalidate the IT for simplifying the statistical analysis of the FXLMS. Hence, the exact analysis of FXLMS becomes very complex. This complexity holds even for the exact analysis of the conventional LMS algorithm [3]. Most of the FXLMS stochastic analyses in the open literature concentrate upon algorithm stability [2-5]. Stability results are useful for algorithm design. However, transient and steady-state weight statistical behavior under different implementation conditions is necessary for a better understanding of the algorithm’s properties.