# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## Appendix A  
We detail here the evaluation of the expectation terms from (53)-(56), which are required in the development of (51) and (52). First, based on (43), we can express the error as  
\[e(n) =\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{ x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)+\mathbf{\mathrm{w}}_{\mathbf{\mathrm{g}}}^{ \mathrm{T}}(n)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)+v(n)+v_{\mathbf{ \mathrm{g}}}(n)\] \[=\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{ x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)+\mathbf{\mathrm{w}}_{\mathbf{\mathrm{g}}}^{ \mathrm{T}}(n)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)+v(n)+\mathbf{\mathrm{ c}}_{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{x}}_{\mathbf{\mathrm{g}}}(n)+ \mathbf{\mathrm{w}}_{\mathbf{\mathrm{h}}}^{\mathrm{T}}(n)\mathbf{\mathrm{x}}_{\mathbf{\mathrm{g }}}(n). \tag{61}\]  
Using this relation, together with (14) and (16), and taking into account that \(\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}(n-1)\) and \(\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)\) are uncorrelated, the term from (53) results in  
\[ \begin{split} A_{\mathbf{\mathrm{g}}}&=\mathbb{E}\big{\{} \big{[}\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}^{\mathrm{T}}(n-1)+\mathbf{\overline{ \mathrm{w}}}_{\mathbf{\mathrm{g}}}^{\mathrm{T}}(n)\big{]}\mathbf{\mathrm{x}}_{\widehat {\mathbf{\mathrm{h}}}}(n)e(n)\big{\}}\\ &=\mathbb{E}\left\{\Big{[}\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}^{ \mathrm{T}}(n-1)+\mathbf{\overline{\mathrm{w}}}_{\mathbf{\mathrm{g}}}^{\mathrm{T}}(n) \Big{]}\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}^{\mathrm{T}}(n)\mathbf{\mathrm{ y}}_{\widehat{\mathbf{\mathrm{h}}}}(n)+v_{\mathbf{\mathrm{g}}}(n)+\mathbf{\mathrm{c}}_{\mathbf{ \mathrm{g}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}^{ \mathrm{-}}(n)\Big{]}\right\}\\ &=p_{\mathbf{\mathrm{g}}}(n)+\mathbb{E}\left[\mathbf{\mathrm{c}}_{\mathbf{ \mathrm{g}}}^{\mathrm{T}}(n-1)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n) \mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}^{\mathrm{T}}(n)\mathbf{\mathrm{c}}_{ \mathbf{\mathrm{g}}}(n-1)+\mathbf{\overline{\mathrm{w}}}_{\mathbf{\mathrm{g}}}^{\mathrm{T} }(n)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)\mathbf{\mathrm{x}}_{\widehat{ \mathbf{\mathrm{h}}}}^{\mathrm{T}}(n)\mathbf{\overline{\mathrm{w}}}_{\widehat{\mathbf{ \mathrm{g}}}}(n)\right]\\ &=p_{\mathbf{\mathrm{g}}}(n)+\mathbb{tr}\left\{\mathbb{E}\left[\mathbf{ \mathrm{c}}_{\mathbf{\mathrm{g}}}(n-1)\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}^{\mathrm{ T}}(n-1)+\mathbf{\overline{\mathrm{w}}}_{\mathbf{\mathrm{g}}}(n)\mathbf{\overline{\mathrm{w}}}_{ \widehat{\mathbf{\mathrm{g}}}}^{\mathrm{T}}(n)\right]\mathbb{E}\left[\mathbf{\mathrm{x}} _{\widehat{\mathbf{\mathrm{h}}}}(n)\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}^{ \mathrm{T}}(n)\right]\right\}\\ &=p_{\mathbf{\mathrm{g}}}(n)+\left[m_{\mathbf{\mathrm{g}}}(n-1)+M\sigma_{ \overline{\mathbf{\mathrm{g}}}_{\mathbf{\mathrm{g}}}_{\mathbf{\mathrm{g}}}}^{2}\right] \mathbb{E}\left[\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)\mathbf{\mathrm{x}} _{\widehat{\mathbf{\mathrm{h}}}}^{\mathrm{T}}(n)\right],\end{split}  \tag{62}\]  
where \(p_{\mathbf{\mathrm{g}}}(n)\) is given in (61) and we took into account that \(\mathbf{\mathrm{c}}_{\mathbf{\mathrm{g}}}(n-1)\) and \(\mathbf{\mathrm{x}}_{\widehat{\mathbf{\mathrm{h}}}}(n)\) are uncorrelated.