A numerically stable, finite memory, fast array recursive least squares filter for broadband active noise control

Abstract

For broadband active noise control applications with a rapidly changing primary path, it is desirable to find algorithms with a rapid convergence, a fast tracking performance, and a low computational cost. Recently, a promising algorithm has been presented, called the fast-array Kalman filter, which uses rotation matrices to calculate the filter parameters. However, when this algorithm is implemented, it can show unstable behavior because of finite precision error propagation. In this paper, a novel algorithm is presented, which exhibits the fast convergence and tracking properties and the linear calculation complexity of the fast-array Kalman filter but does not suffer from the mentioned numerical problems. This is accomplished by running two finite length growing memory recursive least squares filters in parallel and using a convex combination of the two filters when the control signal is calculated. A reset of the filter parameters with proper re-initialization is enforced periodically. The mixing parameters will be chosen in such a way that the total available information used for the calculation of the control signal will be approximately equal at every time instance. The performance of the filter is shown in numerical simulations and real-time lab experiments. The numerical experiments show that the algorithm performs better numerically than the fast-array sliding window recursive least squares filter, while achieving a comparable convergence rate and tracking performance. The real-time lab experiments confirm the behavior shown in the simulations. 32S. VAN OPHEM AND A. P. BERKHOFF fxLMS [2], fast affine projections, and preconditioned LMS [1]. Another way to improve the rate of convergence can be obtained by reformulating the ANC problem as a state estimation problem, as proposed by Sayyarrodsari et al. [3].The assumption of the linear time-invariant adaptive filter and secondary path also potentially influences the tracking performance of primary path changes. These changes will occur when the primary noise source is moving relative to the ANC system or the reference microphone is moving. Some examples of moving noise sources are airplanes and cars. With such noise sources, the primary path can change rapidly, violating the assumption of a system with slowly variating dynamics. Some examples of ANC with moving noise sources are given by Omoto and Fujiwara [4], Berkhoff [5], and Van Ophem and Berkhoff [6].The availability of affordable, high-performance digital signal processors makes it possible to implement more advanced signal processing algorithms with a potentially better performance than the fxLMS algorithms. Therefore, recent research has been focused on alternatives, such as the modified filtered-reference Recursive Least Squares (RLS) algorithm. This algorithm has a high rate of convergence, but is computationally demanding. It was shown by Fraanje et al.[7] that the modified filtered-RLS algorithm is a special case of a Kalman filter, which allows an efficient implementation in the form of a fast...

Filtered-error RLS for self-tuning disturbance feedforward control with application to a multi-axis vibration isolator

“…Various methods for mitigating numerical instabilities have been proposed, e.g. [12,29]. Numerical stability and mitigation of any instabilities are subject of further research.…”