# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 4 Simulation Results
### Parameters of KF-TOT  
In the first set of experiments, we focus on the main parameters of KF-TOT in order to outline their influence on the overall performance of the algorithm. In these simulations, the input signal is an AR(1) process and \(\text{SNR}=20\) dB. First, we assess the influence of the decomposition parameter \(P\). As shown in [36], its values are usually much smallerthan \(L_{12}\), while for network impulse responses (like the one involved in the experiments), the value of \(P\) is in the vicinity of \(L_{12}/8\). In Figure 1, three values of \(P\) are considered within the KF-TOT, while using null uncertainty parameters, i.e., \(\sigma_{w_{2}}^{2}(n)=\sigma_{w_{12}}^{2}(n)=\sigma_{w_{11}}^{2}(n)=0\). As explained in Section 3, this setting targets the best accuracy of the algorithm, i.e., the lowest misalignment, while sacrificing in terms of tracking (as shown in the next experiment). First, we can notice in Figure 1 that the KF-TOT using \(P=1\) leads to a reasonable attenuation of the misalignment, with the smallest parameter space, i.e., using only 68 coefficients, according to the example provided in Section 2.3, related to (12). Increasing the value of \(P\) improves the performance of the KF-TOT, but up to a certain limit, which is related to the low-rank features of the impulse response to be identified. As we can notice in Figure 1, using \(P=2\) and 3 leads to similar performances. Consequently, increasing the value of \(P\) beyond this limit will not bring additional gains. Moreover, two of the three component filters of KF-TOT have \(PL_{11}L_{2}\) and \(PL_{12}L_{2}\) coefficients, respectively. Hence, increasing the value of \(P\) also leads to a larger parameter space, which is not justified in terms of better performance. In our scenario, a value of \(P=2\) represents a reasonable choice, taking into account the compromise between performance and complexity. In this case, the parameter space of KF-TOT comprises 132 coefficients (as also indicated in Section 2.3), still representing a significant reduction as compared to the length \(L=512\) of the full impulse response.  
Next, in Figure 2, we analyze the tracking capabilities of the KF-TOT (with \(P=2\)) when using constant values of the uncertainties parameters. For the sake of simplicity, we generally denote these parameters by \(\sigma_{*}^{2}(n)\), where \(*\in\{2,\ 12,\ 11\}\). The echo path changes after 15 s from the start of the experiment. As outlined in the previous simulation (related to Figure 1), the values \(\sigma_{*}^{2}(n)=0\) lead to good accuracy, but poor tracking, as we can also notice in Figure 2. Reducing the values of \(\sigma_{*}^{2}(n)\) improves the tracking but increases the misalignment (i.e., reduces the accuracy of the estimate). Consequently, a compromise should be considered between these main performance criteria.  
For this purpose, it would be useful to evaluate the uncertainties parameters within the algorithm, using a certain measure that captures the variation from \(\widehat{\mathbf{h}}(n-1)\) to \(\widehat{\mathbf{h}}(n)\). A simple and practical way to assess this issue was proposed in [7], by computing in each iteration the quantity \(\left\|\widehat{\mathbf{h}}(n)-\widehat{\mathbf{h}}(n-1)\right\|_{2}^{2}/L\). Setting the values of \(\sigma_{*}^{2}(n)\) in this manner allows a better compromise between convergence/tracking and misalignment, as outlined in the rest of the experiments from this section. In other words, a small difference between \(\widehat{\mathbf{h}}(n)\) and \(\widehat{\mathbf{h}}(n-1)\) indicates that the algorithm is in steady-state, where small values of  
Figure 1: Normalized misalignment of the KF-TOT using different values of \(P\) and null uncertainty parameters. The input signal is an AR(1) process and SNR \(=20\) dB.  
\(\sigma_{*}^{2}(n)\) should be used. On the other hand, a large difference between the estimates from two consecutive time indices requires a fast reaction of the algorithm, by increasing the values of \(\sigma_{*}^{2}(n)\).