An optimal variable step-size affine projection algorithm for the modified filtered-x active noise control

•This paper derives the recursion form of the mean square deviation (MSD) of the basic coefficient update equation of the modified filtered-x affine projection (MFxAP) algorithm, which is different from that of APA.•This paper considered the difference between the APA and the MFxAP algorithm, including the pre-filtered input from the estimate of the secondary path.•Full theoretical justification is provided for the superior performance of the proposed algorithm.•An optimal step size is proposed according to the recursion form of the MSD.•The non-stationary condition and the efficiency consideration is considered and applied to the proposed algorithm.

This paper introduces an optimal variable step-size affine projection algorithm for the modified filtered-x active noise control systems. First, the recursion form of the error covariance from the tap weight update equation is constructed, not ignoring the dependency between the estimation error and the secondary noise signal. Such consideration has not been concerned previously for the analysis of the modified filtered-x affine projection algorithm. Second, a recursion form of the mean square deviation is derived from that of the error covariance. From the recursion form, an optimal step size is decided to get the fastest convergence rate. Both the recursion forms of the mean square deviation and the optimal step size require scalar additions and multiplications that do not contribute to the overall complexity seriously. The simulation results on the active noise control environments show both fast convergence rate and low steady-state error.