# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 4 Simulation Results
### RLS-TOT versus KF-TOT  
Finally, in the fourth set of experiments, the recently developed RLS-TOT [37] is involved in comparisons. It is known that there are inherent similarities between RLS-based algorithms and the Kalman filter [2; 7]. However, due to its specific parameters, the Kalman filter allows a better control, especially in nonstationary conditions. In the case of the RLS-TOT, its overall performance is controlled in terms of the forgetting factors, denoted by \(\lambda_{*}\), where \(*\in\{2,\ 12,\ 11\}\). These are positive constants, which are smaller or equal to one. Larger values of \(\lambda_{*}\) (i.e., close or equal to one) lead to good accuracy of the filter estimate, thus reaching lower misalignment levels. On the other hand, the tracking reaction of the algorithm is reduced. This can be improved by decreasing the values of \(\lambda_{*}\), while sacrificing in terms of accuracy (i.e., increasing the misalignment). In this context, a reliable compromise between these main performance criteria was proposed in [37], by using \(\lambda_{2}=1\) (which corresponds to the shortest filter of length \(L_{2}^{2}\)), while the other two forgetting factors are set according to \(\lambda_{12}=1-1/(KPL_{12}L_{2})\) and \(\lambda_{11}=1-1/(KPL_{11}L_{2})\), with \(K>1\). In the scenario analyzed in Figure 7, different values of \(K\) are used for the RLS-TOT. The input signal is speech, \(\text{SNR}=20\) dB, the echo path changes after 3.75 s, and both TOT-based algorithms use the same value of the decomposition parameter, which is set to \(P=2\). First, we can notice the compromise between the main performance criteria of the RLS-TOT (i.e., tracking versus misalignment) when using different values of \(K\), i.e., different forgetting factors. On the other hand, the proposed KF-TOT achieves a much better tracking behavior, while still reaching a lower misalignment level.  
Figure 6: Normalized misalignment of the KF-NKP [27] (with \(P^{*}=3\)) and KF-TOT (using different values of \(P\)). The input signal is a speech sequence, the echo path changes after 3.75 s: (**a**) \(\text{SNR}=10\) dB, \(P=3\); and (**b**) \(\text{SNR}=0\) dB, \(P=2\).  
The accuracy of the solution provided by the KF-TOT can also be achieved in lower SNR environments, as supported in Figure 8a,b, where \(\text{SNR}=10\) dB and \(0\) dB, respectively. The other conditions are basically the same as in the previous experiment, but without changing the echo path, in order to focus only on the accuracy of the estimate. As expected, increasing the values of the forgetting factors of RLS-TOT (i.e., increasing the values of \(K\)) leads to lower misalignment levels. Nevertheless, the KF-TOT provides a smooth convergence behavior toward a low steady-state misalignment level. To reach this accuracy, the forgetting factors of the RLS-TOT should be further increased, but the price to pay will be a poor tracking reaction in nonstationary environments (as previously outlined in Figure 7). Concluding, taking into account its overall performance, the KF-TOT represents a more reliable choice.  
Figure 8: Normalized misalignment of the RLS-TOT [37] (using different forgetting factors) and KF-TOT, with \(P=2\). The input signal is a speech sequence, while (**a**) \(\text{SNR}=10\) dB and (**b**) \(\text{SNR}=0\) dB.  
Figure 7: Normalized misalignment of the RLS-TOT [37] (using different forgetting factors) and KF-TOT, with \(P=2\). The input signal is a speech sequence, \(\text{SNR}=20\) dB, and the echo path changes after 3.75 s.