Root Locus Analysis of the Adaptation Process in Active Noise Control for Repetitive Impulses

The popular filtered-x least-mean squares (FxLMS) algorithm has been widely adopted in active noise control (ANC) for relatively stationary disturbances. The convergence behavior of the FxLMS algorithm has been well understood in the adaptation process for stationary sinusoidal or stochastic white noises. Its behavior for transient impulses has not received as much attention. This paper employs the root locus theory to develop a graphical tool for the analysis and design of the adaptive ANC system for repetitive impulses. It is found that there is a dominant pole controlling the stability of the adaptation process, in which the maximum step size can be determined. The analysis also observes a transient adaptation behavior in the FxLMS algorithm for repetitive impulses. In this case, the predicted step-size bound decreases as the number of repetitive impulses increases for a general secondary path. Furthermore, the dominant root tuning process is applied by incorporating a digital filter after the output of the adaptive controller, which significantly increases the step-size bound. The accuracy of the analysis was extensively validated by numerical simulation studies by assuming various secondary path models. The simulated results show an excellent agreement with analytical predictions.