# Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response
## 3 Proposed Kalman Filter Based on TOT Decomposition  
The proposed Kalman filter based on TOT decomposition, which will be further developed in this section, considers simplified first-order Markov models for the impulse responses from (13)-(15), such that  
\[\underline{\mathbf{h}}_{2}(n) =\underline{\mathbf{h}}_{2}(n-1)+\mathbf{w}_{2}(n), \tag{18}\] \[\overline{\mathbf{h}}_{12}(n) =\overline{\mathbf{h}}_{12}(n-1)+\mathbf{w}_{12}(n),\] (19) \[\overline{\mathbf{h}}_{11}(n) =\overline{\mathbf{h}}_{11}(n-1)+\mathbf{w}_{11}(n), \tag{20}\]  
where \(\mathbf{w}_{2}(n)\), \(\mathbf{w}_{12}(n)\), and \(\mathbf{w}_{11}(n)\) are three vectors of lengths \(L_{2}^{2}\), \(PL_{12}L_{2}\), and \(PL_{11}L_{2}\), respectively, which contain samples of zero-mean white Gaussian noises. Their associated covariance matrices are \(\mathbf{R}_{\mathbf{w}_{2}}(n)=\sigma_{\mathbf{w}_{2}}^{2}(n)\mathbf{I}_{L_{2} ^{2}}\), \(\mathbf{R}_{\mathbf{w}_{12}}(n)=\sigma_{\mathbf{w}_{12}}^{2}(n)\mathbf{I}_{PL_ {12}L_{2}}\), and \(\mathbf{R}_{\mathbf{w}_{11}}(n)=\sigma_{\mathbf{w}_{11}}^{2}(n)\mathbf{I}_{PL_ {11}L_{2}}\), where \(\sigma_{\mathbf{w}_{12}}^{2}(n)\), \(\sigma_{\mathbf{w}_{12}}^{2}(n)\), and \(\sigma_{\mathbf{w}_{11}}^{2}(n)\) are the corresponding variances (i.e., level of uncertainties), while \(\mathbf{I}_{*}\) generally denotes the identity matrix of size indicated in subscript (\(\star\times\star\)). As outlined at the end of Section 2.3, the goal is to find the estimates of the component impulse responses from (13)-(15), which are denoted by \(\widehat{\mathbf{h}}_{2}(n)\), \(\widehat{\overline{\mathbf{h}}}_{12}(n)\), and \(\widehat{\overline{\mathbf{h}}}_{11}(n)\), respectively. Their components can be obtained similar to (13)-(17), such that the estimate of the global impulse response is similar to (12), i.e.,  
\[\widehat{\mathbf{h}}(n)=\sum_{l=1}^{L_{2}}\sum_{p=1}^{P}\widehat{\mathbf{h}}_{ 2}^{l}(n)\otimes\widehat{\mathbf{h}}_{12}^{lp}(n)\otimes\widehat{\mathbf{h}}_ {11}^{lp}(n), \tag{21}\]  
where \(\widehat{\mathbf{h}}_{2}^{l}(n)\), \(\widehat{\mathbf{h}}_{12}^{lp}(n)\), and \(\widehat{\mathbf{h}}_{11}^{lp}(n)\) denote the estimates of \(\mathbf{h}_{2}^{l}(n)\), \(\mathbf{h}_{12}^{lp}(n)\), and \(\mathbf{h}_{11}^{lp}(n)\), respectively (with \(l=1,2,\ldots,L_{2}\) and \(p=1,2,\ldots,P\)). At this point, it is important to express the a priori error from (3) in such a way as to separate (or extract) the component impulse responses. To this purpose, we take advantage of the properties of the Kronecker product [50], which allow us to equivalently express the term (of the sum) from the right-hand side of (21) in three different ways, i.e.,  
\[\widehat{\mathbf{h}}_{2}^{l}(n)\otimes\widehat{\mathbf{h}}_{12}^{lp }(n)\otimes\widehat{\mathbf{h}}_{11}^{lp}(n) = \Big{[}\mathbf{I}_{L_{2}}\otimes\widehat{\mathbf{h}}_{12}^{lp}(n) \otimes\widehat{\mathbf{h}}_{11}^{lp}(n)\Big{]}\widehat{\mathbf{h}}_{2}^{l}(n) = \widehat{\mathbf{H}}_{12,11}^{lp}(n)\widehat{\mathbf{h}}_{2}^{l}(n) \tag{22}\] \[= \Big{[}\widehat{\mathbf{h}}_{2}^{l}(n)\otimes\mathbf{I}_{L_{2}} \otimes\widehat{\mathbf{h}}_{11}^{lp}(n)\Big{]}\widehat{\mathbf{h}}_{12}^{lp}( n) = \widehat{\mathbf{H}}_{2,11}^{lp}(n)\widehat{\mathbf{h}}_{12}^{lp}(n)\] (23) \[= \Big{[}\widehat{\mathbf{h}}_{2}^{l}(n)\otimes\widehat{\mathbf{h} }_{12}^{lp}(n)\otimes\mathbf{I}_{L_{11}}\Big{]}\widehat{\mathbf{h}}_{11}^{lp} (n) = \widehat{\mathbf{H}}_{2,12}^{lp}(n)\widehat{\mathbf{h}}_{11}^{lp}(n), \tag{24}\]  
where \(\widehat{\mathbf{H}}_{12,11}^{lp}(n)\), \(\widehat{\mathbf{H}}_{2,11}^{lp}(n)\), and \(\widehat{\mathbf{H}}_{2,12}^{lp}(n)\) denote the terms (i.e., matrices) that multiply the \"extracted\" components from (22)-(24), respectively.