# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## Appendix A
Next, we should concentrate on the last expectation term in (62), which can be expressed as  
\[\mathbb{E}\left[\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)\mathbf{\mathrm{x}} _{\widehat{\mathbf{\mathrm{h}}}}^{\mathrm{T}}(n)\right]=\mathbb{E}\left\{\Big{[} \mathbf{I}_{M}\otimes\widehat{\mathbf{\mathrm{h}}}(n-1)\Big{]}^{T}\widetilde{\mathbf{ \mathrm{x}}}(n)\widetilde{\mathbf{\mathrm{x}}}^{\mathrm{T}}(n)\left[\mathbf{I}_{M} \otimes\widehat{\mathbf{\mathrm{h}}}(n-1)\right]\right\}. \tag{63}\]  
The main diagonal terms of this matrix are \(\mathbb{E}\left[\widehat{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{x}}_{m} (n)\mathbf{\mathrm{x}}_{m}^{\mathrm{T}}(n)\widehat{\mathbf{\mathrm{h}}}(n-1)\right],\ m=1,2,\ldots,M\). In the following, we consider the assumption that the input signals are independent and have the same power, while their covariance matrices are close to a diagonal matrix [27; 28]. Consequently,  
\[ \begin{split}\mathbb{E}\left[\widehat{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n -1)\mathbf{\mathrm{x}}_{m}(n)\mathbf{\mathrm{x}}_{m}^{\mathrm{T}}(n)\widehat{\mathbf{ \mathrm{h}}}(n-1)\right]&=\mathrm{tr}\left\{\mathbb{E}\left[ \mathbf{\mathrm{x}}_{m}(n)\mathbf{\mathrm{x}}_{m}^{\mathrm{T}}(n)\widehat{\mathbf{\mathrm{h}}}(n -1)\widehat{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n-1)\right]\right\}\\ &=\mathrm{tr}\left\{\mathbb{E}\left[\mathbf{\mathrm{x}}_{m}(n)\mathbf{ \mathrm{x}}_{m}^{\mathrm{T}}(n)\right]\mathbb{E}\left[\widehat{\mathbf{\mathrm{h}}}(n -1)\widehat{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n-1)\right]\right\}\\ &=\sigma_{\mathbf{\mathrm{x}}}^{2}\mathbb{E}\left[\left\|\widehat\mathbf{{\mathrm{h}}}(n-1)\right\|^{2}\right].\end{split}  \tag{64}\]Finally, using (A3) and (A4) in (A2), we obtain  
\[A_{\mathbf{g}} =p_{\mathbf{g}}(n)+\sigma_{\mathbf{x}}^{2}\mathbb{E}\left[\left\| \widehat{\mathbf{n}}(n-1)\right\|^{2}\right]\left[m_{\mathbf{g}}(n-1)+M\sigma_ {\overline{\mathbf{n}}_{\mathbf{g}}}^{2}\right].\] (A5)  
In a similar manner, the corresponding term from (55) is derived as  
\[A_{\mathbf{h}} =p_{\mathbf{h}}(n)+\sigma_{\mathbf{x}}^{2}\mathbb{E}\left[\left\| \widehat{\mathbf{g}}(n-1)\right\|^{2}\right]\left[m_{\mathbf{h}}(n-1)+L\sigma_ {\overline{\mathbf{n}}_{\mathbf{h}}}^{2}\right].\] (A6)  
The terms \(p_{\mathbf{g}}(n)\) and \(p_{\mathbf{h}}(n)\) are given in (61) and (62), respectively. Further, we detail the evaluation of the expectation term from (54). To begin, let us focus on the product \(\mathbf{x}_{\widehat{\mathbf{h}}}^{T}(n)\mathbf{x}_{\widehat{\mathbf{h}}}(n)\). Relying on the same considerations and assumptions from (A3) and (A4), we obtain  
\[\mathbf{x}_{\widehat{\mathbf{h}}}^{T}(n)\mathbf{x}_{\widehat{ \mathbf{h}}}(n) =\widetilde{\mathbf{x}}^{T}(n)\left[\mathbf{I}_{M}\otimes \widehat{\mathbf{h}}(n-1)\right]\left[\mathbf{I}_{M}\otimes\widehat{\mathbf{h} }(n-1)\right]^{T}\widetilde{\mathbf{x}}(n)\] \[=\text{tr}\left\{\left[\mathbf{I}_{M}\otimes\widehat{\mathbf{h} }(n-1)\right]^{T}\widetilde{\mathbf{x}}(n)\widetilde{\mathbf{x}}^{T}(n) \left[\mathbf{I}_{M}\otimes\widehat{\mathbf{h}}(n-1)\right]\right\}\] \[\approx M\sigma_{\mathbf{x}}^{2}\left\|\widehat{\mathbf{h}}(n-1) \right\|^{2}.\] (A7)  
Similarly,  
\[\mathbf{x}_{\widehat{\mathbf{g}}}^{T}(n)\mathbf{x}_{\widehat{ \mathbf{g}}}(n)\approx L\sigma_{\mathbf{x}}^{2}\left\|\widehat{\mathbf{g}}(n- 1)\right\|^{2}.\] (A8)  
Hence, considering some degree of stationarity of the input signals, (A7) can be seen as a deterministic quantity, yielding  
\[B_{\mathbf{g}} =\mathbb{E}\left[\mathbf{x}_{\widehat{\mathbf{h}}}^{T}(n) \mathbf{x}_{\widehat{\mathbf{h}}}(n)e^{2}(n)\right]\approx M\sigma_{\mathbf{ x}}^{2}\mathbb{E}\left[\left\|\widehat{\mathbf{h}}(n-1)\right\|^{2}\right] \mathbb{E}\left[e^{2}(n)\right].\] (A9)  
Let us focus on the computation of the expectation term \(\mathbb{E}\left[e^{2}(n)\right]\).