Stochastic analysis of FXLMS-based internal model control feedback active noise control systems

It is well-known that feedback active noise control (ANC) systems are not able to control stochastic white noise, which is completely unpredictable. However, in practical applications, the undesired primary noise is usually not purely white, but band-limited, and can still be controlled by a feedback ANC system. Nevertheless, theoretical analysis of feedback ANC performance is lacking in the literature, especially for band-limited white noise and imperfect secondary-path modeling. In this paper, a stochastic analysis of a filtered-X least-mean-square (FXLMS)-based internal model control (IMC) feedback ANC system is conducted when the primary noise is band-limited white noise. As a result, a mathematical model is developed for the adaptation process in the FXLMS-based IMC feedback ANC system, and based on this, a step-size upper bound for maintaining stability and an optimum step size for fastest convergence are derived. Furthermore, it is found that noise bandwidth affects the stability and convergence performance, which is similar to but different than its impact on a feedforward ANC system. Extensive computer simulations are carried out to verify the theoretical analysis results under different noise bandwidths and secondary-path modeling errors. Band-limited white Gaussian noise, FIR-type secondary path, and imperfect secondary path model are considered.The step-size upper bound and optimum step size in the FXLMS-based IMC feedback ANC system are derived.Increase in bandwidth gives increase in step-size upper bound and optimum step size for IMC feedback and feedforward ANC.Imperfect secondary-path model has more impact on IMC feedback ANC than feedforward ANC.Delay error degrades the step-size upper bound and optimum step size much more severely than magnitude error.