Transient Analysis of the Conventional Filtered-x Affine Projection Algorithm for Active Noise Control

Affine projection (AP) algorithms have been proposed in recent years for use in active noise control systems. This is due to their potential high convergence speed along with their robustness and moderate computational cost. However, these algorithms can exhibit an excessive computational cost for high projection orders (just when higher convergence speed is achieved). Thus, computationally efficient versions of these algorithms have been proposed. For the particular case of the AP algorithms applied to active noise control, the use of the conventional filtered-x structure instead of the commonly used modified filtered-x method can be understood as an efficient strategy, since it needs fewer operations to update the adaptive filter coefficients. However, the use of this structure implies different algorithm behavior for the following two reasons: the signals needed in the coefficient updates do not correspond exactly to the AP algorithm and this structure introduces a delay between the update of the adaptive filter coefficients and its effect on the noise signal. In practice, this dual effect mainly affects convergence of the algorithms in the transient regime. This correspondence presents a mathematical model so that the transient behavior of the conventional filtered-x AP algorithm can be predicted from the reference signal statistics and algorithm parameters.