## 1 Introduction  
Linear system identification problems represent key issues in many important applications [1]. In this context, one of the main challenges is related to the long length of the impulse response to be identified, which is usually the case in applications related to the acoustic environment [2], like echo cancellation, noise reduction, microphone arrays, etc. For example, in echo cancellation [3], such a network/acoustic impulse response can have hundreds/thousands of coefficients (even for a sampling rate of 8 kHz), which significantly increases the difficulty of the system identification problem, especially in terms of the complexity and accuracy of the solution.  
An efficient approach that can be exploited in the framework of such high-dimension system identification problems is based on the Kronecker product decomposition of the impulse response, e.g., see [4, 5], and the references therein. Such an approach fits very well for the identification of linearly separable (or multilinear) systems, because it involves tensor modeling techniques. In other words, a high-dimension system identification problem is solved based on a combination of low-dimension solutions (i.e., shorter impulse responses),which are tensorized together. This approach can find applications in many important fields, like wireless communication systems [6, 7], nonlinear acoustic echo cancellation [8, 9], source separation [10, 11], array beamforming [12, 13], and object recognition [14, 15]. Furthermore, by using the decomposition based on the nearest Kronecker product, followed by low-rank approximations, more general forms of impulse responses can be identified, which are not linearly separable [16, 17, 18, 19]. Recent studies on this topic have targeted specific applications, like echo cancellation, noise reduction, adaptive beamforming, microphone arrays, linear prediction, and speech dereverberation, e.g., see [20, 21, 22, 23, 24, 25, 26, 27, 28] and references therein.  
Recently, the identification of linearly separable systems that own particular intrinsic symmetric/antisymmetric properties has been addressed in [29, 30]. As compared to the previous approaches, these very recent works exploit a particular \"hidden\" symmetry feature of the system to be identified, thus leading to improved solutions as compared to their conventional counterparts. In [29], the solution was formulated in terms of an iterative Wiener filter, whereas [30] developed a family of least-mean-square (LMS) adaptive algorithms tailored for such systems.  
In this paper, we follow the idea of using such a particular symmetric filter for linear system identification problems. As compared to the works reported in [29, 30], we do not limit our solution only to the identification of linearly separable systems, but for more general forms of impulse responses. The resulting contribution is basically twofold. First, an iterative Wiener filter is developed, with improved performance features as compared to the conventional Wiener approach, especially when a small amount of data is available for the estimation of the required statistics. Furthermore, we propose a recursive least-squares (RLS) adaptive algorithm based on a similar framework, which represents a more practical solution (as compared to the Wiener filter) for real-world/real-time applications.  
Following this introduction, the system identification problem using the specific model that involves the symmetric filter is presented in Section 2. Next, the solution based on the Wiener filter is developed in Section 3. In Section 4, we propose the RLS-based algorithm following a similar approach. Simulation results are provided in Section 5, in the framework of network and acoustic echo cancellation, thus supporting the theoretical findings and the performance gain of the proposed solutions. Finally, a summarized discussion is included in Section 6, while Section 7 concludes this work, outlining the main ideas and results, together with several perspectives for future works.