Recursive Least-Squares Algorithm Based on a Fourth-Order Tensor Decomposition for Acoustic Echo Cancellation

Adaptive filtering algorithms based on tensor decomposition represent appealing choices for system identification problems, especially when dealing with the estimation of long-length impulse responses, like in acoustic echo cancellation. The topic has recently been addressed in the literature, showing that the gain (compared to the conventional approach) is twofold in terms of both better performance and lower complexity. The main idea is that a system identification problem with a large parameter space (i.e., a long-length filter) is reformulated based on a group of shorter filters, while their coefficients are combined using the Kronecker product. Nevertheless, one of the main challenges is related to handling the tensor rank, which is particularly addressed for each specific decomposition order. Previous solutions have been designed for second-order (matrix case) and third-order tensorial decompositions. In this paper, we develop a recursive least-squares adaptive filtering algorithm that exploits a fourth-order tensor decomposition, aiming for further performance improvements compared to the existing solutions. In this framework, the influence of the decomposition setup is investigated, which is also related to the main parameters of the algorithm, i.e., the forgetting factors. Simulations performed in the context of acoustic echo cancellation support the theoretical findings and indicate the good performance of the proposed algorithm.