**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 1*<sup>=</sup>* <sup>1</sup> � <sup>0</sup>*:*99*z*�<sup>1</sup> ð Þ. The noise is white and Gaussian, having the variance equal to *σ*<sup>2</sup> *<sup>w</sup>* ¼ 0*:*01. We use two different orders of the system (*N* ¼ 4 and *N* ¼ 5). The first individual impulse response, **h**1, of length *L*<sup>1</sup> ¼ 16, is formed using the first 16 coefficients of the first network echo path from the ITU-T G168 Recommendation [42]. The second impulse response, **h**2, of length *L*<sup>2</sup> ¼ 8, is randomly generated from a Gaussian distribution, whereas the other three impulse responses, **h**3, **h**4, and **h**5, of lengths *L*<sup>3</sup> ¼ *L*<sup>4</sup> ¼ 4 and *L*<sup>5</sup> ¼ 2, respectively, are obtained using an exponential decay based on the rule *h <sup>j</sup>*,*<sup>l</sup> <sup>j</sup>* ¼ *a l <sup>j</sup>*�1 *<sup>j</sup>* , with *j* ¼ 3, 4, 5, where *a <sup>j</sup>* takes the values 0.9, 0.5, and 0.1, respectively. Hence, the global impulse responses **g**, computed using Eq. (11), have lengths 2048, when *N* ¼ 4, and 4096, when *N* ¼ 5. **Figure 1** illustrates the individual impulse responses **h**1, **h**2, **h**3, and **h**4, as well as the resulting global impulse response **g** ¼ **h**<sup>4</sup> ⊗ **h**<sup>3</sup> ⊗ **h**<sup>2</sup> ⊗ **h**1, in the case when *N* ¼ 4.

The measure of performance is the normalized misalignment (in dB) for the identification of the global impulse response, computed as 20 log <sup>10</sup> **<sup>g</sup>** � **<sup>g</sup>**^ð Þ *<sup>n</sup>* � � � � <sup>2</sup>*=* **g** � � � � 2 h i.

#### **Figure 1.**

*Impulse responses used in simulations of the multiple-input/single-output (MISO) system identification scenario (for N* ¼ 4*): (a)* **h**<sup>1</sup> *(of length L*<sup>1</sup> ¼ 16*) contains the first 16 coefficients of the first impulse response from G168 recommendation [42], (b)* **h**<sup>2</sup> *(of length L*<sup>2</sup> ¼ 8*) is a randomly generated impulse response, (c)* **h**<sup>3</sup> *(of length <sup>L</sup>*<sup>3</sup> <sup>¼</sup> <sup>4</sup>*) has the coefficients computed as h*3,*l*<sup>3</sup> <sup>¼</sup> <sup>0</sup>*:*9*<sup>l</sup>*3�<sup>1</sup>*, with l*<sup>3</sup> <sup>¼</sup> 1, 2, … , *<sup>L</sup>*3*, (d)* **<sup>h</sup>**<sup>4</sup> *(of length L*<sup>4</sup> <sup>¼</sup> <sup>4</sup>*) has the coefficients computed as h*4,*l*<sup>4</sup> <sup>¼</sup> <sup>0</sup>*:*5*<sup>l</sup>*4�<sup>1</sup>*, with l*<sup>4</sup> <sup>¼</sup> 1, 2, … , *<sup>L</sup>*4*, and (e)* **<sup>g</sup>** *(of length L* <sup>¼</sup> *<sup>L</sup>*1*L*2*L*3*L*<sup>4</sup> <sup>¼</sup> <sup>2048</sup>*) is the global impulse response, which results based on Eq. (11).*

A sudden change in the sign of the coefficients of **h**<sup>1</sup> is introduced in the middle of each experiment, with the goal of observing the tracking capabilities of the proposed algorithms.

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 2*= Lσ*<sup>2</sup> *x* , where *σ*<sup>2</sup> *<sup>x</sup>* is the input signal's variance [40]. In practical scenarios, this limit may not be usable, due to stability issues. The step-size parameters for the LMS-MF and LMS algorithms were chosen in our simulations in such a manner that similar misalignment values are obtained, in order to compare their convergence rate and tracking. The largest value of the step-size parameter for the conventional LMS algorithm is chosen close to its stability limit, such that the fastest convergence possible is obtained.

**Figure 2** shows the case when *N* ¼ 4 and the input signals are white Gaussian noises. The resulting global filter length is *L* ¼ 2048. It can be seen that the LMS-MF achieves a higher convergence rate and faster tracking as compared to the conventional LMS algorithm. Of course, when the value of the step-size parameter decreases, the misalignment decreases, but at the same time, the convergence becomes slower.

Next, in **Figure 3**, *N* ¼ 4 and the input signals are highly-correlated AR(1) processes. The performance gain of the LMS-MF with respect to the conventional LMS algorithm in terms of both convergence rate/tracking and steady-state misalignment is even higher in this scenario.

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 *N* ¼ 5 and the input signals are AR(1) processes. The resulting global impulse response length is *L* ¼ 4096. It is easily observed that the proposed algorithm is superior to the conventional LMS in terms of both convergence rate/tracking and misalignment.

#### **Figure 2.**

*Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response* **g***. The input signals are white Gaussian noises, N* ¼ 4*, and L* ¼ 2048*.*

**Figure 3.**

*Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 4*, and L* ¼ 2048*.*

## **Figure 4.**

*Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 5*, and L* ¼ 4096*.*

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 *N* ¼ 4, leading to a length of the global system of *L* ¼ 2048. Similar to the case of the LMS-MF and LMS algorithms from **Figure 2**, it can be concluded that the performance of the NLMS-MF algorithm is significantly better than the one of its conventional counterparts. This is even more apparent in the case of smaller normalized step-size values.

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 *N* is increased. **Figure 7** shows the case when *N* ¼ 5 and, hence, the length of the global impulse response is *L* ¼ 4096. Again, the NLMS-MF algorithm achieves a significantly better convergence rate and tracking with respect to the conventional NLMS algorithm.

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 *N* ¼ 4 and the input signals are AR(1) processes. As it was expected, lower values of the normalized step-sizes improve the misalignment, but at the cost of a slower convergence and tracking.

The last experiment involving the NLMS-MF algorithm aims to show the performance in the case when the normalized step-size parameters *α<sup>i</sup>* (*i* ¼ 1, 2, … , *N*) take different values for each individual filter. In **Figure 9**, the value of the normalized

#### **Figure 5.**

*Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response* **g***. The input signals are white Gaussian noises, N* ¼ 4*, and L* ¼ 2048*.*

**Figure 6.**

*Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 4*, and L* ¼ 2048*.*

**Figure 7.**

*Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 5*, and L* ¼ 4096*.*

**Figure 8.**

*Performance of the NLMS-MF algorithm using different normalized step-size parameters (with equal values of αi*, *i* ¼ 1, 2, … , *N), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 4*, and L* ¼ 2048*.*

#### **Figure 9.**

*Performance of the NLMS-MF algorithm using different normalized step-size parameters (with different values of αi*, *i* ¼ 1, 2, … , *N), for the identification of the global impulse response* **g***. The input signals are AR(1) processes, N* ¼ 4*, and L* ¼ 2048*.*

step-size parameter for the first filter, of highest length, *L*1, is *α*<sup>1</sup> ¼ 0*:*25, whereas the other normalized step-sizes *α <sup>j</sup>*, *j* ¼ 2, 3, 4 are varied. The system order is *N* ¼ 4 and the input signals are AR(1) processes. The compromise between convergence rate and misalignment can be seen again. Since the convergence is mostly influenced by the individual filter with the highest length [20], different values of the normalized step-sizes may be used in different scenarios.

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.
