# A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms
## 4 Simulation Results  
Simulations were performed in the framework of a tensor-based system identification problem, which resulted following the MISO model defined by (12) and (20) and was similar to the setup used in [17]. The input signals that form the third-order tensor \(\mathbf{\mathcal{X}}(n)\) are AR(1) processes; they are generated by filtering white Gaussian noises through a first-order system \(1/\left(1-0.9z^{-1}\right)\). The additive noise \(w(n)\) is white and Gaussian; its variance was set to \(\sigma_{w}^{2}=0.01\). The third-order system used in the simulations and its components (\(\mathbf{h}_{1}\), \(\mathbf{h}_{2}\), and \(\mathbf{h}_{3}\)) are depicted in Figure 2. First, the component \(\mathbf{h}_{1}\) is an impulse response from the G168 Recommendation [26], of length \(L_{1}=64\); it is provided in Figure 2a. Second, in Figure 2b, the component \(\mathbf{h}_{2}\) is a random impulse response (with Gaussian distribution) of length \(L_{2}=8\). Third, the coefficients of the last component, i.e., the impulse response \(\mathbf{h}_{3}\), are depicted in Figure 2c; those were evaluated as \(h_{3,l_{3}}=0.5^{l_{3}-1},\;l_{3}=1,2,\ldots,L_{3}\), using the length \(L_{3}=4\). Therefore, the global impulse response from Figure 2d resulted as \(\mathbf{h}=\mathbf{h}_{3}\otimes\mathbf{h}_{2}\otimes\mathbf{h}_{1}\), and its length was \(L=L_{1}L_{2}L_{3}=2048\). This global impulse response resembled a channel with echoes, e.g., like an acoustic echo path [27]. Finally, the third-order (rank-1) tensor \(\mathbf{\mathcal{H}}\) of dimension \(L_{1}\times L_{2}\times L_{3}\) could be formed according to (13). In order to test the tracking capabilities of the algorithms, an abrupt change of the system was introduced in the middle of each experiment, by changing the sign of the coefficients of each impulse response. Figure 1: Computational complexity of the proposed RLS-TF algorithm, as compared to the conventional RLS and NLMS algorithms, as a function of \(L_{1}\); the other dimensions are set to \(L_{2}=8\) and \(L_{3}=4\): (**a**) number of multiplications per iteration and (**b**) number of additions per iteration. As shown in Section 3, the proposed RLS-TF algorithm was designed to estimate the individual components of the global system. However, we could identify \(\mathbf{h}_{1}\), \(\mathbf{h}_{2}\), and \(\mathbf{h}_{3}\) up to some scaling factors, as explained using (23). Therefore, to evaluate the identification of these individual impulse responses, a proper performance measure is the normalized projection misalignment (NPM) [28]:  
\[\mathrm{NPM}\left[\mathbf{h}_{1},\widehat{\mathbf{h}}_{1}(n)\right] =1-\left[\frac{\mathbf{h}_{1}^{T}\widehat{\mathbf{h}}_{1}(n)}{ \left\|\mathbf{h}_{1}\right\|_{2}\left\|\widehat{\mathbf{h}}_{1}(n)\right\|_{ 2}}\right]^{2}, \tag{55}\] \[\mathrm{NPM}\left[\mathbf{h}_{2},\widehat{\mathbf{h}}_{2}(n)\right] =1-\left[\frac{\mathbf{h}_{2}^{T}\widehat{\mathbf{h}}_{2}(n)}{ \left\|\mathbf{h}_{2}\right\|_{2}\left\|\widehat{\mathbf{h}}_{2}(n)\right\|_{ 2}}\right]^{2},\] (56) \[\mathrm{NPM}\left[\mathbf{h}_{3},\widehat{\mathbf{h}}_{3}(n)\right] =1-\left[\frac{\mathbf{h}_{3}^{T}\widehat{\mathbf{h}}_{3}(n)}{ \left\|\mathbf{h}_{3}\right\|_{2}\left\|\widehat{\mathbf{h}}_{3}(n)\right\|_{ 2}}\right]^{2}, \tag{57}\]  
where \(\left\|\cdot\right\|_{2}\) denotes the Euclidean norm. On the other hand, the global impulse response \(\mathbf{h}\) results without any scaling ambiguity.