**1. Introduction**

System identification is an important topic nowadays since it can be used in solving numerous problems [1]. The aim of system identification is to estimate an unknown model using the available and observed data, namely the input and output of the system. In this context, the well-known Wiener filter is a popular solution, along with the adaptive filters which can be derived starting from this approach.

In multilinear system identification, dealing with a large parameter space represents an important challenge [2, 3]. The huge length of the filter (hundreds or thousands of coefficients) is also a serious problem [4, 5]. The methods used for addressing these issues usually rely on tensor decomposition and modeling [2, 6–17], meaning that a high-dimension problem is rewritten as a combination of lower-dimension structures, using the Kronecker product decomposition [18].

In the context of multilinear forms identification, a few approaches were proposed recently, addressing the cases when the large system is decomposed into two or three

smaller components (i.e., bilinear and trilinear forms, respectively) [10, 17–23]. The aforementioned solutions outperform their conventional counterparts, offering at the same time a lower computational complexity.

Motivated by the appealing performance of these previous developments, we extended the tensor decomposition technique to higher-order systems, and in this framework, this chapter presents a part of the work and results obtained recently by the authors in the context of multilinear system identification. An iterative Wiener filter and a family of LMS-based algorithms tailored for multilinear forms are presented. For more details on the results summarized here, the works [24–26] can be consulted.

Related to the work presented here, several other tensor-based solutions relying on the recursive least-squares (RLS) algorithm were also developed recently [27, 28]. Possible applications of such system identification frameworks can be encountered in topics such as big data [29], machine learning [14], but they may be also useful in nonlinear acoustic echo cancelation [30, 31], source separation [13, 32, 33], channel equalization [12, 34], array beamforming [16, 35], blind identification [36], object recognition [37, 38], and cardiac applications [39].

The rest of this chapter is organized in the following way. In Section 2, we introduce the system model for the multiple-input/single-output (MISO) system identification problem. In this context, Section 3 presents an iterative Wiener filter tailored for the identification of multilinear systems. Next, in Section 4, an LMS-based algorithm is presented, together with its normalized version, and then, in Section 5, the performance of these algorithms is proved through simulations. Finally, conclusions are drawn in Section 6.
