Stability analysis of adaptation process in FxLMS-based active noise control

• Main Author: Tabatabaei Ardekani, Iman
• Other Authors: Abdulla, Waleed H
• Published: ResearchSpace@Auckland 2012
• Online Access: http://hdl.handle.net/2292/12430

This thesis is concerned with the Filtered-x Least Mean Square (FxLMS) adaptation algorithm and its applications in Active Noise Control (ANC). Generally, this algorithm is used in system identification problems in which a physical channel, called the secondary path, follows the adaptive filter. The FxLMS algorithm compensates for the secondary path effect by filtering the input signal (training data sequence) using an available estimate of the secondary path, called the secondary path model. However, this filtering causes the analytical model of the adaptation process to become highly complex. Because of this complexity, this model has to be simplified when it is desired to derive closed-form expressions for formulating different behaviours of the adaptation process. Usually, this simplification has been carried out by using unrealistic assumptions of pure delay secondary paths, broad-band acoustic noise, and perfect secondary path models. The first contribution of this thesis is the derivation of a set of closed-from mathematical expressions for formulating behaviours of FxLMS-based ANC systems in steady-state and transient conditions. This derivation is carried out without using any simplifying assumption regarding the secondary path. Consequently, the obtained expressions extend the available knowledge on FxLMS-based ANC systems. The second contribution is formulating influences of acoustic noise band-width on the newly-derived expressions. In the analysis of ANC systems with stochastic noise, it is usual to assume a broad-band acoustic noise with a flat frequency spectrum (which is usually referred to as a white signal), in order to avoid mathematical complexity. However, even if the acoustic noise has a flat spectrum over a wide frequency range, the signal picked up and fed to the ANC system is required to be processed with a sampling frequency higher than the maximum frequency of the acoustic noise. For this reason, a realistic noise signal can only have a flat spectrum over a limited band-width. The third contribution is investigating influences of secondary path models on the newly-derived expressions. Usually, it is acceptable to assume a perfectly-accurate secondary path model; however, in order to generalise the obtained closed-form expressions, this assumption is also removed in this thesis. Consequently, the final closed-form expressions, proposed in this thesis, can apply to a relatively general case with an arbitrary secondary path, an acoustic noise with an arbitrary band-width, and an arbitrary (imperfect) secondary path model. Another contribution of this thesis is determining trajectories of the poles of the FxLMS adaptation process in the z-plane. This investigation leads to find the FxLMS adaptation process root locus. It is shown that the dominant pole of this locus always locates on a certain branch with a typical trajectory in the z-plane. A mechanism for localising this pole is then proposed in this thesis, resulting in a novel ANC algorithm, called the Filtered Weight FxLMS. In addition to several numerical analyses and computer simulations, a FPGA-based ANC setup, developed for this research, is used to study the validity of the theoretical results obtained in this thesis. This setup is developed by using a flexible FPGA programming structure which can be used for the implementation of other ANC algorithms. Different experiments with this setup confirms the validity of the theoretical results proposed in this thesis.