A Fourth-Order Decomposition-Based RLS Algorithm with Variable Forgetting Factors

The recursive least-squares (RLS) algorithm is an appealing choice in many adaptive filtering applications, especially due to its fast convergence rate. In the context of echo cancellation, the implementation of the conventional RLS algorithm is a challenging task, mainly due to the long length of the impulse response to be identified, which corresponds to the echo path. The computational complexity of this algorithm is proportional to the square of the filter length, which becomes prohibitive since the lengths of such impulse responses could be on the order of hundreds/thousands of coefficients. Recently, an efficient solution has been developed, which exploits a fourth-order (tensor-based) decomposition of the filter impulse response, using the nearest Kronecker product. This approach fits very well for low-rank impulse responses, like in echo cancellation. The resulting RLS-type algorithm designed in this context combines four (much shorter) adaptive filters, which work in parallel and are connected via their coefficients. Thus, the system owns an intrinsic symmetry-related structure. The overall behavior of this decomposition-based RLS algorithm is controlled by the forgetting factors associated to the four component filters. In order to achieve a proper balance between the main performance criteria (e.g., tracking versus accuracy), the current paper proposes a version of this algorithm with variable forgetting factors (VFFs). The design of these VFFs results in a practical manner, without requiring other additional control mechanisms or extra tuning parameters. Extensive simulation results obtained in the framework of echo cancellation support the performance features of the proposed decomposition-based RLS-type algorithm with VFFs.