# A Connection Between the Kalman Filter and an Optimized LMS Algorithm for Bilinear Forms
## 4 Optimized LMS Algorithm for Bilinear Forms  
In this section, we approach the system identification problem based on the LMS algorithm, aiming to optimize its step-size parameter in order to address the compromise between the main performance criteria, i.e., convergence rate versus misadjustment [27]. In the following, the proposed optimized LMS algorithm for bilinear forms (namely OLMS-BF) is derived based on the same system model given in Section 2. As will be explained in Section 5, this algorithm has striking resemblances with the SKF-BF, even if their derivations follow different patterns. Let us consider the two estimated impulse responses \(\widehat{\mathbf{h}}(n)\) and \(\widehat{\mathbf{g}}(n)\), such that the estimated signal is given by (13). As a consequence, the a priori error  
can be seen as an additional \"noise\" term, of variance \(\sigma_{v_{\mathbf{g}}}^{2}(n)\), introduced by the system \(\mathbf{g}\). The terms \(\mathbf{c_{h}}\) and \(\mathbf{c_{h_{3}}}\) were previously defined in (19) and (21), respectively. For the sake of simplicity, the scaling factor \(\eta\) does not appear explicitly in the following. As explained before, this parameter is included in the expression of the uncertainty parameter, thus leading to the notation from (23) and (24). In a similar way, for the second system we have  
\[d(n) =\mathbf{h}^{T}(n)\mathbf{x_{g}}(n)+\mathbf{h}^{T}(n)\mathbf{x_{ \tilde{g}}}(n)-\mathbf{h}^{T}(n)\mathbf{x_{\tilde{g}}}(n)+v(n)\] \[=\mathbf{h}^{T}(n)\mathbf{x_{\tilde{g}}}(n)+v(n)+v_{\mathbf{h}}( n), \tag{45}\]  
where  
\[v_{\mathbf{h}}(n) =\mathbf{h}^{T}(n)\left[\mathbf{x_{g}}(n)-\mathbf{x_{\tilde{g}}}( n)\right]\] \[=\mathbf{c}_{\mathbf{g}_{\mathbf{a}}}^{T}(n)\mathbf{x_{h}}(n)\] \[=\left[\mathbf{c_{g}}(n-1)+\overline{\mathbf{w}}_{\mathbf{g}}(n )\right]^{T}\mathbf{x_{h}}(n) \tag{46}\]  
can be interpreted as an additional \"noise\" term, of variance \(\sigma_{v_{\mathbf{h}}}^{2}(n)\), related to the system \(\mathbf{h}\). Here, the a posteriori and a priori misalignments corresponding to the system \(\mathbf{g}\) were defined in (20) and (22), respectively. In Figures 1 and 2, the equivalent system identification scheme is represented in terms of the two components, \(\mathbf{g}(n)\) and \(\mathbf{h}(n)\), respectively; it can be observed that each system influences the other one through the additional \"noise\" term. Figure 1: Equivalent system identification scheme when considering the system \(\mathbf{g}(n)\) and the input \(\mathbf{x_{\tilde{h}}}(n)\). Figure 2: Equivalent system identification scheme when considering the system \(\mathbf{h}(n)\) and the input \(\mathbf{x_{\tilde{g}}}(n)\). In the framework of the LMS algorithm for bilinear forms, namely LMS-BF [12], the updates are the following:  
\[\widehat{\mathbf{g}}(n) =\widehat{\mathbf{g}}(n-1)+\mu_{\widehat{\mathbf{g}}}\mathbf{x}_{ \widehat{\mathbf{h}}}(n)e(n), \tag{47}\] \[\widehat{\mathbf{h}}(n) =\widehat{\mathbf{h}}(n-1)+\mu_{\widehat{\mathbf{h}}}\mathbf{x}_{ \widehat{\mathbf{g}}}(n)e(n), \tag{48}\]  
where \(\mu_{\widehat{\mathbf{g}}}\) and \(\mu_{\widehat{\mathbf{h}}}\) are the step-size parameters. In this context, the vectors corresponding to the a posteriori misalignments become  
\[\mathbf{c}_{\mathbf{g}}(n) =\mathbf{g}(n)-\widehat{\mathbf{g}}(n-1)-\mu_{\widehat{\mathbf{g} }}\mathbf{x}_{\widehat{\mathbf{h}}}(n)e(n)\] \[=\mathbf{h}^{T}(n)\mathbf{x_{\tilde{g}}}(n)+v(n)+v_{\mathbf{h}}( n), \tag{45}\]0 \[=\mathbf{h}^{T}(n)\mathbf{x_{\tilde{g}}}(n)+v(n)+v_{\mathbf{h}}( n), \tag{45}\]1 \[=\mathbf{h}^{T}(n)\mathbf{x_{\tilde{g}}}(n)+v(n)+v_{\mathbf{h}}( n), \tag{45}\]2  
At this point, let us introduce the notation \(m_{\mathbf{g}}(n)=\mathbb{E}\left[\|\mathbf{c}_{\mathbf{g}}(n)\|^{2}\right]\) and \(m_{\mathbf{h}}(n)=\mathbb{E}\left[\|\mathbf{c}_{\mathbf{h}}(n)\|^{2}\right]\).