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Tensor-based signal processing methods are usually employed when dealing with multidimensional data and/or systems with a large parameter space. In this paper, we present a family of tensor-based adaptive filtering algorithms, which are suitable for high-dimension system identification problems. The basic idea is to exploit a decomposition-based approach, such that the global impulse response of the system can be estimated using a combination of shorter adaptive filters. The algorithms are mainly tailored for multiple-input/single-output system identification problems, where the input data and the channels can be grouped in the form of rank-1 tensors. Nevertheless, the approach could be further extended for single-input/single-output system identification scenarios, where the impulse responses (of more general forms) can be modeled as higher-rank tensors. As compared to the conventional adaptive filters, which involve a single (usually long) filter for the estimation of the global impulse response, the tensor-based algorithms achieve faster convergence rate and tracking, while also providing better accuracy of the solution. Simulation results support the theoretical findings and indicate the advantages of the tensor-based algorithms over the conventional ones, in terms of the main performance criteria.

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High-dimensional system identification problems can be efficiently addressed based on tensor decompositions and modelling. In this paper, we design a recursive least-squares (RLS) algorithm tailored for the identification of trilinear forms, namely RLS-TF. In our framework, the trilinear form is related to the decomposition of a third-order tensor (of rank one). The proposed RLS-TF algorithm acts on the individual components of the global impulse response, thus being efficient in terms of both performance and complexity. Simulation results indicate that the proposed solution outperforms the conventional RLS algorithm (which handles only the global impulse response), but also the previously developed trilinear counterparts based on the least-mean- squares algorithm.