Recursive least-squares algorithm based on a third-order tensor decomposition for low-rank system identification

A recently developed third-order tensor (TOT) decomposition-based method has proved to be working very well in linear system identification problems that target the estimation of long length impulse responses. This technique exploits the decomposition of the impulse response based on the nearest Kronecker product and low -rank approximations, but it also owns a specific advantage. Its main feature is related to the way of handling the rank of a third-order tensor, which is controlled in terms of the rank of a matrix and, as a result, is limited to small values. In this paper, we further develop a recursive least-squares (RLS) algorithm based on the TOT decomposition technique. The resulting solution combines the estimates provided by three shorter adaptive filters, thus gaining in terms of both performance and complexity. The three component filters are updated in parallel, which represents an important practical advantage, in terms of the modularity of implementation. Also, as compared to the conventional RLS algorithm, a significant reduction of the computational complexity can be achieved for the common setup of the decomposition parameters. Besides, since the length of the adaptive filter highly influences the main performance criteria (e.g., convergence, tracking, and accuracy), the proposed RLS version based on the TOT decomposition outperforms both the conventional algorithm and a previously developed RLS-based solution that involves a second-order decomposition. These performance gains mainly reflect in a faster tracking capability and a better accuracy of the estimate (i.e., lower misalignment). Simulations performed in the framework of echo cancellation support the performance features of the proposed tensorial RLS-based algorithm.