# A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms
## 4 Simulation Results
Consequently, we can use a performance measure based on the regular normalized misalignment (NM):  
\[\mathrm{NM}\left[\mathbf{h},\widehat{\mathbf{h}}(n)\right]=\frac{\left\| \mathbf{h}-\widehat{\mathbf{h}}(n)\right\|_{2}^{2}}{\left\|\mathbf{h}\right\| _{2}^{2}}, \tag{58}\]  
which is also equivalent to \(\left\|\boldsymbol{\mathcal{H}}-\widehat{\boldsymbol{\mathcal{H}}}(n)\right\| _{\mathrm{F}}^{2}/\left\|\boldsymbol{\mathcal{H}}\right\|_{\mathrm{F}}^{2}\), where \(\widehat{\boldsymbol{\mathcal{H}}}(n)=\widehat{\mathbf{h}}_{1}(n)\circ \widehat{\mathbf{h}}_{2}(n)\circ\widehat{\mathbf{h}}_{3}(n)\) and \(\left\|\cdot\right\|_{\mathrm{F}}\) denotes the Frobenius norm (the Frobenius norm of a third-order tensor \(\boldsymbol{\mathcal{A}}\) is defined as \(\left\|\boldsymbol{\mathcal{A}}\right\|_{\mathrm{F}}=\sqrt{\left\langle \boldsymbol{\mathcal{A}},\boldsymbol{\mathcal{A}}\right\rangle}=\left\|\mathbf{ A}_{[1]}\right\|_{\mathrm{F}}=\left\|\mathbf{A}_{[2]}\right\|_{\mathrm{F}}=\left\|\mathbf{A}_{[3]} \right\|_{\mathrm{F}}\)). Figure 2: The components of the third-order system used in simulations: (**a**) \(\mathbf{h}_{1}\) is the first impulse response (of length \(L_{1}=64\)) from the G168 Recommendation [26]; (**b**) \(\mathbf{h}_{2}\) is a randomly generated impulse response (of length \(L_{2}=8\)), with Gaussian distribution; (**c**) the impulse response \(\mathbf{h}_{3}\) (of length \(L_{3}=4\)), with the coefficients computed as \(h_{3,l_{3}}=0.5^{l_{3}-1}\), with \(l_{3}=1,2,\ldots,L_{3}\); and (**d**) the global impulse response (of length \(L=L_{1}L_{2}L_{3}=2048\)) results based on (17), \(\mathbf{h}=\mathbf{h}_{3}\otimes\mathbf{h}_{2}\otimes\mathbf{h}_{1}\). The simulation results should provide answers to several important questions, as follows. (i) What is the influence of the forgetting factors on the performance of the proposed RLS-TF algorithm? (ii) What are the advantages of the RLS-TF algorithm over the previously developed NLMS-TF counterpart [17]? (iii) What are the advantages of the RLS-TF algorithm over the conventional RLS benchmark? The following three experiments are designed to address these issues. In the first experiment, the performance of the proposed RLS-TF algorithm was analysed with respect to its main parameters, i.e., the forgetting factors \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\). In the case of an RLS-based algorithm, the value of the forgetting factor is usually related to the filter length, following a well-known rule of thumb, as shown in Table 1 (see \"Initialization\"). In our case, the forgetting factors of the RLS-TF algorithm were set to \(\lambda_{1}=1-1/(KL_{1})\), \(\lambda_{2}=1-1/(KL_{2})\), and \(\lambda_{3}=1-1/(KL_{3})\). As we can notice, the value of each forgetting factor depended on the length of its related filter (i.e., \(L_{1}\), \(L_{2}\), or \(L_{3}\)), but also on the constant \(K\). This tuning parameter could be used to adjust the values of the forgetting factors, as indicated in Figures 3 and 4. Clearly, a higher value of \(K\) would result in a higher value of the forgetting factor (i.e., closer to one). We could expect that a higher value of the forgetting factor would reduce the misalignment, but slowing down the convergence/tracking [29]. On the other hand, reducing the forgetting factor improves the convergence/tracking, but increasing the misalignment. This behaviour was supported by the results depicted in Figures 3 and 4, in terms of NPM and NM, respectively. As we can notice, the value \(K=20\) lead to a good compromise between the performance criteria, so that it would be used in the following experiments.