# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 3 Kalman Filter for Bilinear Forms  
In this section, we address the previously described system identification problem in terms of the Kalman filter, summarizing the findings from [13]. In this context, the signal model from (1) may be interpreted as the observation equation, while the system impulse responses can be considered as state equations. Given the two adaptive filters \(\widehat{\mathbf{h}}(n)\) and \(\widehat{\mathbf{g}}(n)\), the estimated signal is given by  
\[\widehat{y}(n)=\widehat{\mathbf{h}}^{\mathcal{T}}(n-1)\mathbf{X}(n)\widehat{ \mathbf{g}}(n-1). \tag{13}\]  
As a result, the a priori error signal between the desired and estimated signals can be defined as  
\[ \begin{split} e(n)&=d(n)-\widehat{y}(n)\\ &=d(n)-\widehat{\mathbf{h}}^{\mathcal{T}}(n-1)\mathbf{X}(n) \widehat{\mathbf{g}}(n-1)\\ &=d(n)-\left[\widehat{\mathbf{g}}(n-1)\otimes\widehat{\mathbf{h} }(n-1)\right]^{T}\widetilde{\mathbf{x}}(n)\\ &=d(n)-\widehat{\mathbf{f}}^{\mathcal{T}}(n-1)\widetilde{ \mathbf{x}}(n)\\ &=d(n)-\widehat{\mathbf{h}}^{\mathcal{T}}(n-1)\mathbf{x}_{ \widehat{\mathbf{g}}}(n)\\ &=d(n)-\widehat{\mathbf{g}}^{\mathcal{T}}(n-1)\mathbf{x}_{ \widehat{\mathbf{h}}}(n),\end{split}  \tag{14}\]  
where  
\[\mathbf{x}_{\widehat{\mathbf{g}}}(n)=[\widehat{\mathbf{g}}(n-1)\otimes \mathbf{I}_{L}]^{T}\widetilde{\mathbf{x}}(n), \tag{15}\]  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]  
In the context of the linear sequential Bayesian approach, the optimal estimates of the state vectors have the forms [25]:  
\[\widehat{\mathbf{h}}(n) =\widehat{\mathbf{h}}(n-1)+\mathbf{k_{h}}(n)e(n), \tag{17}\] \[\widehat{\mathbf{g}}(n) =\widehat{\mathbf{g}}(n-1)+\mathbf{k_{g}}(n)e(n), \tag{18}\]  
where \(\mathbf{k_{h}}(n)\) and \(\mathbf{k_{g}}(n)\) are the Kalman gain vectors. Next, we can define the a posteriori misalignments (which represent the state estimation errors) related to the temporal and spatial impulse responses:  
\[\mathbf{c_{h}}(n)=\frac{1}{\eta}\mathbf{h}(n)-\widehat{\mathbf{h}}(n), \tag{19}\]  
\[\mathbf{c_{g}}(n)=\eta\mathbf{g}(n)-\widehat{\mathbf{g}}(n), \tag{20}\]  
for which their correlation matrices are \(\mathbf{R_{c_{h}}}(n)=\mathbb{E}[\mathbf{c_{h}}(n)\mathbf{c_{h}^{T}}(n)]\) and \(\mathbf{R_{c_{g}}}(n)=\mathbb{E}[\mathbf{c_{g}}(n)\mathbf{c_{g}^{T}}(n)]\), respectively, where \(\mathbb{E}[\cdot]\) denotes mathematical expectation. As mentioned in Section 2, we can only identify the impulse responses up to this arbitrary scaling factor \(\eta\); however, the pair \(\mathbf{h}(n)/\eta\) and \(\eta\mathbf{g}(n)\)is equivalent to the pair \(\mathbf{h}\) and \(\mathbf{g}\) in the bilinear form. Let us also define the a priori misalignments related to the two impulse responses:  
\[\mathbf{c}_{\mathbf{h}_{a}}(n) =\frac{1}{\eta}\mathbf{h}(n)-\widehat{\mathbf{h}}(n-1)\] \[=\mathbf{c}_{\mathbf{h}}(n-1)+\frac{1}{\eta}\mathbf{w}_{\mathbf{h }}(n), \tag{21}\] \[\mathbf{c}_{\mathbf{g}_{a}}(n) =\eta\mathbf{g}(n)-\widehat{\mathbf{g}}(n-1)\] \[=\mathbf{c}_{\mathbf{g}}(n-1)+\eta\mathbf{w}_{\mathbf{g}}(n), \tag{22}\]  
whose correlation matrices are \(\mathbf{R}_{\mathbf{c}_{\mathbf{h}_{a}}}(n)=\mathbb{E}\left[\mathbf{c}_{ \mathbf{h}_{a}}(n)\mathbf{c}_{\mathbf{h}_{a}}^{\mathrm{T}}(n)\right]\) and \(\mathbf{R}_{\mathbf{c}_{\mathbf{g}_{a}}}(n)=\mathbb{E}\left[\mathbf{c}_{ \mathbf{g}_{a}}(n)\mathbf{c}_{\mathbf{g}_{a}}^{\mathrm{T}}(n)\right]\), respectively.