Filtered-X Quasi Affine Projection Algorithm for Active Noise Control Networks

The affine projection (AP) algorithm enhances the performance of gradient-based adaptive algorithms when dealing with colored reference signals, which is typically the case with filtered-X type algorithms. This enhancement is achieved by using various delayed versions of the reference signal data vector, which are appropriately orthogonalized and normalized to optimize convergence performance. The number of these vectors, known as the projection order of the AP, increases the computational requirements, mainly due to the calculation of a matrix inversion whose dimensions are proportional to this projection order. When used in distributed systems, the AP algorithm typically requires each acoustic node in the system to compute the complete matrix inversion, even though they only need a specific set of data (a subblock) from it. This means that the AP does not offer much advantage in terms of computational savings when used in distributed collaborative networks. To address this issue, an approximate version of the filtered-X affine projection (FXAP) algorithm is introduced in this work. This approximate version avoids the matrix inversion computation in each iteration using a precalculated inverse matrix. This strategy provides computational savings and enables easy distribution of the algorithm. Additionally, a variable step-size approach is proposed to mitigate the deviation caused by a precalculated matrix, which provides good performance, high robustness, and cost-effective distribution.