Linear system identification is a key problem in many important applications, among which echo cancellation is a very challenging one. Due to the long length impulse responses (i.e., echo paths) to be identified, there is always room (and needs) to improve the performance of the echo cancellers, especially in terms of complexity, convergence rate, robustness, and accuracy. In this paper, we propose a new way to address the system identification problem (from the echo cancellation perspective), by exploiting an optimal approximation of the impulse response based on the nearest Kronecker product decomposition. Also, we make a first step toward this direction, by developing an iterative Wiener filter based on this approach. As compared to the conventional Wiener filter, the proposed solution is much more attractive, since its gain is twofold. First, the matrices to be inverted (or, preferably, linear systems to be solved) are smaller as compared to the conventional approach. Second, as a consequence, the iterative Wiener filter leads to a good estimate of the impulse response, even when a small amount of data is available for the estimation of the statistics. Simulation results support the theoretical findings and indicate the good results of the proposed approach, for the identification of different network and acoustic impulse responses.

The recursive least-squares (RLS) adaptive filter is an appealing choice in many system identification problems. The main reason behind its popularity is its fast convergence rate. However, this algorithm is computationally very complex, which may make it useless for the identification of long length impulse responses, like in echo cancellation. Computationally efficient versions of the RLS algorithm, like those based on the dichotomous coordinate descent (DCD) iterations or QR decomposition techniques, reduce the complexity, but still have to face the challenges related to long length adaptive filters (e.g., convergence/tracking capabilities). In this paper, we focus on a different approach to improve the efficiency of the RLS algorithm. The basic idea is to exploit the impulse response decomposition based on the nearest Kronecker product and low-rank approximation. In other words, a high-dimension system identification problem is reformulated in terms of low-dimension problems, which are combined together. This approach was recently addressed in terms of the Wiener filter, showing appealing features for the identification of low-rank systems, like real-world echo paths. In this paper, besides the development of the RLS algorithm based on this approach, we also propose a variable regularized version of this algorithm (using the DCD method to reduce the complexity), with improved robustness to double-talk. Simulations are performed in the context of echo cancellation and the results indicate the good performance of these algorithms.

System identification problems are always challenging to address in applications that involve long impulse responses, especially in the framework of multichannel systems. In this context, the main goal of this review paper is to promote some recent developments that exploit decomposition-based approaches to multiple-input/single-output (MISO) system identification problems, which can be efficiently solved as combinations of low-dimension solutions. The basic idea is to reformulate such a high-dimension problem in the framework of bilinear forms, and to then take advantage of the Kronecker product decomposition and low-rank approximation of the spatiotemporal impulse response of the system. The validity of this approach is addressed in terms of the celebrated Wiener filter, by developing an iterative version with improved performance features (related to the accuracy and robustness of the solution). Simulation results support the main theoretical findings and indicate the appealing performance of these developments.

The identification of room acoustic impulse responses represents a challenging problem in the framework of many important applications related to the acoustic environment, like echo cancellation, noise reduction, and microphone arrays, among others. The main issues are related to the long length of such impulse responses and their time-variant nature. Recently, a decomposition-based approach was developed for the identification of low-rank systems, which can also be applied (to some extent) for the identification of acoustic impulse responses. This approach exploits the nearest Kronecker product decomposition of the impulse response and solves a high-dimension system identification problem using a combination of low-dimension solutions (provided by shorter filters). Nevertheless, it does not consider the intrinsic nature of the room acoustic impulse responses, which contain specific components (e.g., early reflections and late reverberation) that can be very different in nature. In this article, we propose an improved decomposition-based method (via the Kronecker product) that takes into account these specific components and processes them separately. Following this approach, an iterative Wiener filter is firstly developed, followed by a recursive least-squares (RLS) algorithm. Both solutions outperform the conventional benchmarks. Moreover, they achieve superior performances as compared to the recently developed versions based on the nearest Kronecker product decomposition, also owning lower computational complexities than their previous counterparts. Simulations are performed in the framework of acoustic echo cancellation and the obtained results support the performance features of the proposed algorithms.

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.

In the context of adaptive filtering, the recursive least-squares (RLS) is a very popular algorithm, especially for its fast convergence rate. The most important parameter of this algorithm is the forgetting factor. It is well-known that a constant value of this parameter leads to a compromise between misadjustment and tracking. In this paper, we present a variable forgetting factor approach, aiming to better compromise between the performance criteria of the RLS algorithm. Also, we propose a practical solution to estimate the power of the system noise (in a system identification scenario), which is required within the algorithm. Experiments performed in the context of network echo cancellation support the advantages of the proposed approach.