# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 3 Proposed Kalman Filter Based on TOT Decomposition
Also,  
\[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]5 \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]6Finally, based on (31)-(36), we obtain the following relations between the covariance matrices:  
\[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]7 \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]8 (47) \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]9  
The resulting Kalman filter based on TOT decomposition, namely, KF-TOT, is defined by the relations (28)-(30) and (40)-(48). Within the algorithm, we also need to evaluate the auxiliary matrices \(\widehat{\mathbf{H}}_{12,11}^{lp}(n)\), \(\widehat{\mathbf{H}}_{2,11}^{lp}(n)\), and \(\widehat{\mathbf{H}}_{2,12}^{lp}(n)\), according to (22)-(24), respectively. These auxiliary matrices are used to evaluate the \"inputs\" \(\mathbf{x}_{12,11}(n)\), \(\mathbf{x}_{2,11}(n)\), and \(\mathbf{x}_{2,12}(n)\), as defined after (25)-(27), respectively. The initialization setup of the KF-TOT is  
\[\widehat{\overline{\mathbf{h}}}_{11}(n) = \widehat{\overline{\mathbf{h}}}_{11}(n-1)+\mathbf{k}_where \(\mathbf{k}_{2}(n)\), \(\mathbf{k}_{12}(n)\), and \(\mathbf{k}_{11}(n)\) are the corresponding Kalman vectors. At this point, we can define the a posteriori misalignments related to the component filters:  
\[\widetilde{\mathbf{m}}_{2}(n) =\mathbf{h}_{2}(n)-\widehat{\mathbf{h}}_{2}(n), \tag{31}\]0 \[\widehat{\overline{\mathbf{h}}}_{11}(n) = \widehat{\overline{\mathbf{h}}}_{11}(n-1)+\mathbf{k}_where \(\mathbf{k}_{2}(n)\), \(\mathbf{k}_{12}(n)\), and \(\mathbf{k}_{11}(n)\) are the corresponding Kalman vectors. At this point, we can define the a posteriori misalignments related to the component filters:  
\[\widetilde{\mathbf{m}}_{2}(n) =\mathbf{h}_{2}(n)-\widehat{\mathbf{h}}_{2}(n), \tag{31}\]1 (50) \[\widehat{\overline{\mathbf{h}}}_{11}(n) = \widehat{\overline{\mathbf{h}}}_{11}(n-1)+\mathbf{k}_where \(\mathbf{k}_{2}(n)\), \(\mathbf{k}_{12}(n)\), and \(\mathbf{k}_{11}(n)\) are the corresponding Kalman vectors. At this point, we can define the a posteriori misalignments related to the component filters:  
\[\widetilde{\mathbf{m}}_{2}(n) =\mathbf{h}_{2}(n)-\widehat{\mathbf{h}}_{2}(n), \tag{31}\]2  
with \(l=1,2,\ldots,L_{2}\), \(p=1,2,\ldots,P\), and \(0<\xi\leq 1\); here, \(\mathbf{0}_{\bullet}\) generally denotes an all-zeros vector with the length indicated in subscript. In addition, we need to initialize \(\mathbf{R}_{\widehat{\mathbf{m}}_{2}}(0)=\zeta\mathbf{I}_{L_{2}^{2}}\), \(\mathbf{R}_{\widehat{\mathbf{m}}_{12}}(0)=\zeta\mathbf{I}_{PL_{12}L_{2}}\), and \(\mathbf{R}_{\widehat{\mathbf{m}}_{11}}(0)=\zeta\mathbf{I}_{PL_{11}L_{2}}\), where \(\zeta\) is a positive constant. The additive noise power \(\sigma_{\triangledown}^{2}\) is required within the evaluation of the Kalman gain vectors from (40), (42) and (43). This parameter could be a priori available or estimated within the algorithm, e.g., during the silence periods of a talker (in echo cancellation or noise reduction applications). Other methods for estimating \(\sigma_{\triangledown}^{2}\) can be found in [7; 54]. The other important parameters of the KF-TOT are the uncertainties \(\sigma_{w_{2}}^{2}(n)\), \(\sigma_{w_{12}}^{2}(n)\), and \(\sigma_{w_{11}}^{2}(n)\), which are required in (46)-(48). Similar to the conventional Kalman filter and the influence of its uncertainties [7], small values of these parameters lead to a good accuracy of the solution (i.e., low misalignment), but a slow tracking reaction in case of an abrupt change in the unknown system. On the other hand, large uncertainties keep the adaptive system alert (i.e., good tracking behavior), but sacrifice in terms of accuracy.