## Abstract

Nonlinear systems have been studied for a long time and have applications in numerous research fields. However, there is currently no global solution for nonlinear system identification, and different used approaches depend on the type of nonlinearity. An interesting class of nonlinear systems, with a wide range of popular applications, is represented by multilinear (or multidimensional) systems. These systems exhibit a particular property that may be exploited, namely that they can be regarded as linearly separable systems and can be modeled accordingly, using tensors. Examples of well-known applications of multilinear forms are multiple-input/single-output (MISO) systems and acoustic echo cancellers, used in multi-party voice communications, such as videoconferencing. Many important fields (e.g., big data, machine learning, and source separation) can benefit from the methods employed in multidimensional system identification. In this context, this chapter aims to briefly present the recent approaches in the identification of multilinear systems. Methods relying on tensor decomposition and modeling are used to address the large parameter space of such systems.

### Keywords

- nonlinear systems
- tensor decomposition
- multilinear forms
- Wiener filter
- adaptive filters
- system identification

## 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, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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, 18, 19, 20, 21, 22, 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, 25, 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.

## 2. System model in the multilinear framework

Let us consider a MISO system, whose output signal at the time index

where the individual channels are modeled by the vectors:

the superscript denotes the transpose operator, and the input signals may be expressed in a tensorial form as

^{T}

where

where

where

and so on, where

where

with

Here, an observation can be made: the solution of the decomposition in Eq. (11) is not unique [17, 24]. Despite this, no scaling ambiguity occurs in the identification of the global impulse response,

Using Eqs. (9)–(11), we may rewrite

We aim to identify the global impulse response,

where

with

where

The optimization criterion is the minimization of the mean-squared error (MSE), which can be defined using Eq. (15):

where

Relation (17) provides the global impulse response. In order to obtain the

## 3. Multilinear iterative Wiener filter

It can be easily checked that

where

Hence, the cost function given by Eq. (16) may be expressed in

where

If all coefficients except

The minimization of this convex cost function with respect to

Using this result, an iterative approach can be derived. A set of initial values

The minimization of the cost function yields

Using

where

The minimization of the cost function yields

All the other estimates

## 4. LMS and NLMS algorithms for multilinear forms

The limitations of the Wiener filter (e.g., matrix inversion, statistics estimation) can restrict the applicability of the previously presented approach in real-world situations (for example, in nonstationary conditions, or when real-time processing is needed). Therefore, a better approach may be represented by adaptive filters. In this context, the well-known least-mean-square (LMS) algorithm is among the most popular solutions, due to its simplicity. In the following, a family of LMS-based algorithms for multilinear forms identification is presented.

By using the estimated impulse responses

where

We can easily check that

where

The global filter estimate is obtained as

We may also identify the global impulse response using the classical LMS algorithm:

where

When choosing the constant values of the step-size parameters from Eqs. (39)–(41), we need to consider the compromise between convergence rate and steady-state misadjustment. In certain cases, it can be more useful to have variable step-size parameters. Hence, the update equations become

Then, the a posteriori error signals can be defined as

After replacing Eq. (39) in Eq. (50), Eq. (40) in Eq. (51), and Eq. (41) in Eq. (52), and then canceling the a posteriori error signals, we get

We assume that

In the numerators of Eqs. (56)–(58), the normalized step-size parameters

The initialization of the individual impulse responses can be done using Eqs. (42, 43). We may also identify the global impulse response using the regular NLMS algorithm:

where

## 5. Experimental results

The purpose of this section is to illustrate through simulations the improved performance of the proposed solutions for the identification of multilinear forms. We performed experiments involving MISO system identification. As input signals, we used white Gaussian noises and AR(1) processes, obtained by filtering white Gaussian noises through a first-order system with the transfer function

The measure of performance is the normalized misalignment (in dB) for the identification of the global impulse response, computed as

First, we aim to show comparatively the performances of the LMS-MF and LMS algorithms. When choosing the step-size parameter values, we need to take into account the theoretical upper bound, which for the conventional LMS is

Figure 2 shows the case when

Next, in Figure 3,

When the system order increases, the improvement in performance brought by the LMS-MF is even more apparent. This can be seen in Figure 4, where

In the following, we aim to illustrate the performance of the NLMS-MF and NLMS algorithms in the identification of the global system. Since the step-size parameter does no longer have a constant value, the normalized algorithms can work better in nonstationary environments. The fastest-convergence bound for the value of the normalized step-size parameter of the conventional NLMS algorithm is 1 [40].

Figure 5 illustrates the case when the inputs are white Gaussian noises and

The improvement offered by the proposed approach is even more significant for correlated inputs. In Figure 6, the input signals are AR(1) processes. It is noticed that even when the NLMS-MF algorithm uses lower values for the normalized step-sizes, it can still outperform the NLMS algorithm working in the fastest convergence mode.

The same conclusion applies when the order

Next, we aim to show the influence of the normalized step-size values on the performance of the proposed algorithm. In Figure 8, the order of the system is

The last experiment involving the NLMS-MF algorithm aims to show the performance in the case when the normalized step-size parameters

Due to the important improvement in performance brought by the adaptive tensor-based LMS algorithms, observed through experiments, these algorithms may represent appealing solutions for the identification of long-length separable system impulse responses.

## 6. Conclusions

In this chapter, we have presented a decomposition-based approach for dealing with the identification of high-dimension MISO systems. Unlike the conventional method, which is based on the identification of the global system impulse response, our solution focuses on regarding the system as an

## Acknowledgments

This work was supported by a grant from the Romanian Ministry of Education and Research, CNCS–UEFISCDI, Project Number PN-III-P1-1.1-PD-2019-0340, within PNCDI III.

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