# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 3 Proposed Kalman Filter Based on TOT Decomposition
Similar to (7), in the framework of Kalman filtering and the linear sequential Bayesian approach [49], the updates of the three components result from using the associated expressions of the error signal from (25)-(27), respectively, i.e.,  
\[\widehat{\overline{\mathbf{h}}}_{2}(n) = \widehat{\overline{\mathbf{h}}}_{2}(n-1)+\mathbf{k}_{2}(n)e(n),\] (28) \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\] (29) \[\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:  
\[\widetilde{\mathbf{m}}_{2}(n) =\mathbf{h}_{2}(n)-\widehat{\mathbf{h}}_{2}(n), \tag{31}\] \[\widetilde{\mathbf{m}}_{12}(n) =\mathbf{\overline{h}}_{12}(n)-\widehat{\mathbf{h}}_{12}(n),\] (32) \[\widetilde{\mathbf{m}}_{11}(n) =\mathbf{\overline{h}}_{11}(n)-\widehat{\mathbf{h}}_{11}(n), \tag{33}\]  
with the covariance matrices \(\mathbf{R}_{\widetilde{\mathbf{m}}_{2}}(n)=E\big{[}\widetilde{\mathbf{m}}_{2 }(n)\widetilde{\mathbf{m}}_{2}^{\top}(n)\big{]}\), \(\mathbf{R}_{\widetilde{\mathbf{m}}_{12}}(n)=E\big{[}\widetilde{\mathbf{m}}_{1 2}(n)\widetilde{\mathbf{m}}_{12}^{\top}(n)\big{]}\), and \(\mathbf{R}_{\widetilde{\mathbf{m}}_{11}}(n)=E\big{[}\widetilde{\mathbf{m}}_{ 11}(n)\widetilde{\mathbf{m}}_{11}^{\top}(n)\big{]}\), respectively. Also, using the estimates from time index \(n-1\), the a priori misalignments can be evaluated as  
\[\mathbf{m}_{2}(n) =\mathbf{h}_{2}(n)-\widehat{\mathbf{h}}_{2}(n-1), \tag{34}\] \[\mathbf{m}_{12}(n) =\mathbf{\overline{h}}_{12}(n)-\widehat{\mathbf{h}}_{12}(n-1),\] (35) \[\mathbf{m}_{11}(n) =\mathbf{\overline{h}}_{11}(n)-\widehat{\mathbf{h}}_{11}(n-1), \tag{36}\]  
which are associated with the covariance matrices \(\mathbf{R}_{\mathbf{m}_{2}}(n)=E\big{[}\mathbf{m}_{2}(n)\mathbf{m}_{2}^{\top }(n)\big{]}\), \(\mathbf{R}_{\mathbf{m}_{12}}(n)=E\big{[}\mathbf{m}_{12}(n)\mathbf{m}_{12}^{ \top}(n)\big{]}\), and \(\mathbf{R}_{\mathbf{m}_{11}}(n)=E\big{[}\mathbf{m}_{11}(n)\mathbf{m}_{11}^{ \top}(n)\big{]}\), respectively. The Kalman gain vectors required within the updates (28)-(30) result from minimizing the cost functions  
\[I\big{[}\mathbf{k}_{2}(n)\big{]} =\frac{1}{L_{2}^{2}}\text{tr}\big{[}\mathbf{R}_{\widetilde{ \mathbf{m}}_{2}}(n)\big{]}, \tag{37}\] \[J\big{[}\mathbf{k}_{12}(n)\big{]} =\frac{1}{PL_{12}L_{2}}\text{tr}\big{[}\mathbf{R}_{\widetilde{ \mathbf{m}}_{12}}(n)\big{]},\] (38) \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]0  
In each case, the optimization criterion is applied based on a multilinear strategy [51; 52; 53] by considering that two of the component filters are fixed within the optimization criterion of the third one. Consequently, following (37) and similar to (8) [49], we obtain  
\[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]1 \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]2  
Similarly, following (38) and (39), the corresponding Kalman vectors result in  
\[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]3 \[\widehat{\overline{\mathbf{h}}}_{12}(n) = \widehat{\overline{\mathbf{h}}}_{12}(n-1)+\mathbf{k}_{12}(n)e(n),\]4  
respectively.