# A Recursive Least-Squares Algorithm for the Identification of Trilinear Forms
## 4 Simulation Results
Figure 3: Normalized projection misalignment (NPM) evaluated based on (55)–(57), in dB, for the identification of the individual impulse responses from Figure 2a–c, using the RLS-TF algorithm with different values of the forgetting factors \(\lambda_{i}=1-1/(KL_{i})\), \(i=1,2,3\) (varying the value of \(K\)): (a) NPM \(\left[\mathbf{h}_{1},\widehat{\mathbf{h}}_{1}(n)\right]\), (b) NPM \(\left[\mathbf{h}_{2},\widehat{\mathbf{h}}_{2}(n)\right]\), and (c) NPM \(\left[\mathbf{h}_{3},\widehat{\mathbf{h}}_{3}(n)\right]\). Next, we compare the performance of the proposed RLS-TF algorithm with its previously developed counterpart based on the NLMS algorithm, i.e., NLMS-TF [17]. The overall performance of this algorithm is mainly controlled by its normalized step-sizes, which are positive constants smaller than one. Using notation similar to that involved in [17], we set equal values for these parameters, i.e., \(\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha\). In the case of the NLMS-TF algorithm, the fastest convergence mode was obtained when \(\alpha_{1}+\alpha_{2}+\alpha_{3}\approx 1\), so that we could use \(\alpha=0.33\). Smaller values of the normalized step-sizes (e.g., \(\alpha=0.1\)) reduced the convergence/tracking, but led to a lower misalignment. As shown in Figures 5 and 6 (in terms of NPM and NM, respectively), the RLS-TF algorithm clearly outperformed the NLMS-TF counterpart, achieving a faster convergence rate and tracking, together with low misalignment. Finally, in the last experiment, we investigated the comparison between the RLS-TF solution and the conventional RLS algorithm. As explained in the last part of Section 3 (related to the computational complexity), the conventional RLS algorithm could also be used for the identification of the global impulse response of length \(L\), based on the cost function from (36). This algorithm uses a single forgetting factor, which can also be set as \(\lambda=1-1/(KL)\), where \(K\) is the same tuning constant. The influence of the value of \(\lambda\) on the performance of the algorithm is also related to the well-known compromise between low misalignment and fast tracking. In the experiment reported in Figure 7, the conventional RLS algorithm uses two values of the forgetting factor, which were set by varying the tuning constant to \(K=1\) and \(K=20\). As we can notice, even if the largest value of the forgetting factor (obtained for \(K=20\)) led to a lower misalignment, the tracking capability of the conventional RLS algorithm was significantly reduced. Clearly, the tracking was improved when using a smaller forgetting factor (corresponding to \(K=1\)), but the misalignment of the conventional RLS algorithm was much higher in this case. On the other hand, the RLS-TF algorithm outperformed by far its conventional counterpart, in terms of both performance criteria. Moreover, the complexity of the conventional RLS algorithm, i.e., \(\mathcal{O}(L^{2})\), was prohibitive for practical implementations, due to the long length of the global filter (\(L=2048\)). On the other hand, the RLS-TF algorithm worked on the individual components and combined the solutions of three shorter filters, of lengths \(L_{1}\), \(L_{2}\), and \(L_{3}\) (with \(L_{1}L_{2}L_{3}=L\)); thus, it was much more computationally efficient. As a trivial example related to the last experiment given in Figure 7, we could mention that the simulation time (using MATLAB) of  
Figure 4: Normalized misalignment (NM) evaluated based on (58), in dB, for the identification of the global impulse response \(\mathbf{h}\) (of length \(L=2048\)) from Figure 2d, using the RLS-TF algorithm with different values of the forgetting factors \(\lambda_{i}=1-1/(KL_{i}),\ i=1,2,3\) (varying the value of \(K\)). the RLS-TF algorithm was less than one minute, while the conventional RLS algorithm took hours to reach the final result.