# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 4 Simulation Results  
The experimental setup considers an echo cancellation scenario, which basically represents a system identification problem, where the main goal is to model (or estimate) the impulse response of an echo path [45; 47; 48]. In our experimental framework, this impulse response is obtained based on the first cluster from G.168 ITU-T recommendation [55], which is padded with zeros up to the length \(L=512\); the sampling rate is 8 kHz. As indicated by the example provided in Section 2.3, this length can be factorized using \(L_{11}=L_{12}=16\) and \(L_{2}=2\), which represents the TOT decomposition setup. In order to assess the tracking behavior of the algorithms, an echo path change scenario is considered in some of the experiments, by changing the sign of the coefficients (of the impulse response) at a certain moment of time. The performance of the algorithms is evaluated in terms of the normalized misalignment (in dB), which is defined as \(20\text{log}_{10}\left\|\mathbf{h}(n)-\mathbf{\hat{h}}(n)\right\|_{2}/\left\| \mathbf{h}(n)\right\|_{2}\), where the notation \(\left\|\cdot\right\|_{2}\) stands for the Euclidean norm. When the echo path \(\mathbf{h}(n)\) abruptly changes, the reference/desired signal also changes according to (2), which results in a sudden increase in the error signal and, consequently, in the normalized misalignment (due to the bias of the filter estimate). Therefore, it is desirable that the algorithm should recover (as fast as possible) after this change, which depends on its tracking capability. The input signal is either a stationary sequence obtained by filtering a white Gaussian noise through a first-order autoregressive (AR) model with the pole at 0.8, thus resulting in an AR(1) process, or a nonstationary sequence obtained from a recorded speech signal of a female voice. The AR(1) process has a high correlation degree, since the pole is set closer to 1; similar results can be obtain using higher-order AR processes. Also, using speech as input is a challenging case (specific to echo cancellation), since this is a nonstationary and highly correlated signal. The output of the echo path is corrupted by a white Gaussian noise, \(v(n)\), with different levels of the signal-to-noise ratio (SNR). This is defined as \(\sigma_{y}^{2}/\sigma_{r}^{2}\), where \(\sigma_{y}^{2}=E\big{[}y^{2}(n)\big{]}\) stands for the variance in the output signal, \(y(n)=\mathbf{h}^{T}(n)\mathbf{x}(n)\). In echo cancellation, \(v(n)\) is the background noise, which biases the adaptive filter estimate [48]. Three scenarios are considered in the experiments, using \(\text{SNR}=20\) dB, \(10\) dB, and \(0\) dB, which correspond to mild, moderate, and heavy noise conditions, respectively. An SNR level of 20 dB or higher corresponds to favorable conditions, with low background noise, so that a good accuracy of the adaptive filter (i.e., echo canceler) is expected. When the SNR is decreased to 10 dB, the background noise already has a noticeable influence, which is reflected in higher misalignment. It should be outlined that in echo cancellation applications the background noise could be significant, thus influencing the accuracy of the estimates, while the algorithms should be robust in different noisy environments. This is the reason for also including in the experiments a heavy noise scenario with \(\text{SNR}=0\) dB, in order to test the robustness of the algorithms under this challenging condition. As explained in Section 3, the KF-TOT requires \(\sigma_{v}^{2}\) within the computation of its Kalman gain vectors. This is also valid for the conventional Kalman filter, according to (9). While there are different approaches for estimating this parameter in practice [7; 54], the influence of these methods on the overall performance of the KF-TOT is beyond the scope of this paper. Consequently, we consider that \(\sigma_{v}^{2}\) is available in all the experiments reported in this section.