# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 3 Kalman Filter for Bilinear Forms
For the sake of simplicity of the upcoming developments, let us multiply \(\mathbf{w}_{\mathbf{h}}(n)\) and \(\mathbf{w}_{\mathbf{g}}(n)\) by \(1/\eta\) and \(\eta\), respectively, and introduce the notation:  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]0 \[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]1  
These new terms are also zero-mean white Gaussian noise vectors, having the correlation matrices \(\mathbf{R}_{\overline{\mathbf{w}}_{\mathbf{h}}}(n)=\sigma_{\overline{\overline{ \mathbf{w}}}_{\mathbf{h}}}^{2}\mathbf{I}_{L}\) and \(\mathbf{R}_{\overline{\mathbf{w}}_{\mathbf{g}}}(n)=\sigma_{\overline{\overline {\mathbf{w}}}_{\mathbf{g}}}^{2}\mathbf{I}_{M}\), respectively. Clearly, we have \(\sigma_{\overline{\overline{\mathbf{w}}}_{\mathbf{h}}}^{2}=\sigma_{ \overline{\overline{\mathbf{w}}}_{\mathbf{h}}}^{2}/\eta^{2}\) and \(\sigma_{\overline{\overline{\mathbf{w}}}_{\mathbf{g}}}^{2}=\eta^{2}\sigma_{ \overline{w}_{\mathbf{g}}}^{2}\). Consequently, using this notation in (21) and (22), we obtain  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]2 \[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]3  
In this context, the Kalman gain vectors are computed from the minimization of the criteria:  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]4 \[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]5  
with respect to \(\mathbf{k}_{\mathbf{h}}(n)\) and \(\mathbf{k}_{\mathbf{g}}(n)\), respectively, where \(\mathrm{tr}\left[\cdot\right]\) denotes the trace of a square matrix. From these minimizations, we find that  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]6 (29) \[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]7 \[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]8  
Hence, the Kalman gain vectors for the temporal and spatial impulse responses become  
\[\mathbf{x}_{\widehat{\mathbf{h}}}(n)=[\mathbf{I}_{M}\otimes\widehat{\mathbf{ h}}(n-1)]^{T}\widetilde{\mathbf{x}}(n). \tag{16}\]9 \[\widehat{\mathbf{h}}(n) =\widehat{\mathbf{h}}(n-1)+\mathbf{k_{h}}(n)e(n), \tag{17}\]0  
where \(\delta_{\mathbf{h}}(n)=\sigma^{2}_{v}/\sigma^{2}_{\mathbf{c_{h_{a}}}}(n)\) and \(\delta_{\mathbf{g}}(n)=\sigma^{2}_{v}/\sigma^{2}_{\mathbf{c_{ga}}}(n)\) can be seen as variable regularization parameters. Then, we use the Kalman vectors from (35) and (36) in the updates (17) and (18), respectively. Next, a new simplification can be made, by considering that the matrices appearing in the updates of \(\mathbf{R_{c_{h}}}(n)\) and \(\mathbf{R_{c_{g}}}(n)\) can be approximated as  
\[\widehat{\mathbf{h}}(n) =\widehat{\mathbf{h}}(n-1)+\mathbf{k_{h}}(n)e(n), \tag{17}\]1 \[\widehat{\mathbf{h}}(n) =\widehat{\mathbf{h}}(n-1)+\mathbf{k_{h}}(n)e(n), \tag{17}\]2  
We can perform these approximations because, as the filters start to converge, the misalignments of the individual coefficients tend to become uncorrelated; due to this fact, the matrices \(\mathbf{R_{c_{h}}}(n)\) and \(\mathbf{R_{c_{g}}}(n)\) tend to become diagonal.