## 3 Optimum Mixing Parameters and EMSE
### Formulation of the EMSE for the Combination
Indeed, if we replace \(\lambda_{k}^{\circ}\left[n\right]\) in (24), we achieve:  
\[\xi_{k}^{\circ}\left[n\right]\approx\xi_{k}^{(1)}\left[n\right].\]  
Case 2: \[=\lambda_{k}^{2}\left[n\right]\mathrm{E}\left\{\left|\mathbf{e}_{ ak}^{\left(1\right)}\left[n\right]\right|^{2}\right\}+\left(1-\lambda_{k}\left[n \right]\right)^{2}\mathrm{E}\left\{\left|\mathbf{e}_{ak}^{\left(2\right)} \left[n\right]\right|^{2}\right\}+2\lambda_{k}\left[n\right]\left(1-\lambda_{k }\left[n\right]\right)\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1\right)}\left[n \right]e_{ak}^{\left(2\right)}\left[n\right]\right\}\]0  
Here, we have that \(\Delta\xi_{k}^{(1)}\left[n\right]>0\) and \(\Delta\xi_{k}^{(2)}\left[n\right]\leq 0\). Again, the combined filter turns out to perform like the best individual filter, which in this case is the second one. As a matter of fact, the optimum mixing parameter is \(\lambda_{k}^{\circ}\left[n\right]=0\); thus, the optimum EMSE is:  
\[=\lambda_{k}^{2}\left[n\right]\mathrm{E}\left\{\left|\mathbf{e}_{ ak}^{\left(1\right)}\left[n\right]\right|^{2}\right\}+\left(1-\lambda_{k}\left[n \right]\right)^{2}\mathrm{E}\left\{\left|\mathbf{e}_{ak}^{\left(2\right)} \left[n\right]\right|^{2}\right\}+2\lambda_{k}\left[n\right]\left(1-\lambda_{k }\left[n\right]\right)\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1\right)}\left[n \right]e_{ak}^{\left(2\right)}\left[n\right]\right\}\]1  
Case 3: \[=\lambda_{k}^{2}\left[n\right]\mathrm{E}\left\{\left|\mathbf{e}_{ ak}^{\left(1\right)}\left[n\right]\right|^{2}\right\}+\left(1-\lambda_{k}\left[n \right]\right)^{2}\mathrm{E}\left\{\left|\mathbf{e}_{ak}^{\left(2\right)} \left[n\right]\right|^{2}\right\}+2\lambda_{k}\left[n\right]\left(1-\lambda_{k }\left[n\right]\right)\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1\right)}\left[n \right]e_{ak}^{\left(2\right)}\left[n\right]\right\}\]2  
In this case, considering (24), it is easy to conclude that the cross-EMSE is lower than both individual EMSEs; therefore, \(\Delta\xi_{k}^{(j)}\left[n\right]>0,j=1,2\), while \(\lambda_{k}^{\circ}\left[n\right]\in(0,1)\). This result can be justified by the fact that since the correlation between the a priori errors of both individual components is small, their weighted combination provides an estimation error of reduced variance [1]. In this case, the optimum EMSE is still represented by (24). * \(\xi_{k}^{(1)}\left[n\right]=\xi_{k}^{(2)}\left[n\right]=\xi_{k}^{(12)}\left[n\right]\). For completeness, we consider also this particular case, whose condition is rather rare to find in practice. In this case, we have \(\Delta\xi_{k}^{(j)}\left[n\right]=0,j=1,2\); thus, the optimum EMSE is: \[=\lambda_{k}^{2}\left[n\right]\mathrm{E}\left\{\left|\mathbf{e}_{ ak}^{\left(1\right)}\left[n\right]\right|^{2}\right\}+\left(1-\lambda_{k}\left[n \right]\right)^{2}\mathrm{E}\left\{\left|\mathbf{e}_{ak}^{\left(2\right)} \left[n\right]\right|^{2}\right\}+2\lambda_{k}\left[n\right]\left(1-\lambda_{k }\left[n\right]\right)\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1\right)}\left[n \right]e_{ak}^{\left(2\right)}\left[n\right]\right\}\]3  
From these cases, it can be gathered that when the cross-EMSE is comprised between the two individual EMSEs (i.e., Cases 1 and 2), the cross-correlation between the a priori errors of the two filters is high enough so that their combination can perform at most as the best individual filter. However, when the cross-EMSE is lower than the individual EMSEs (i.e., Case 3), a lower excess error is achieved for the combination. In order to achieve the expressions of the EMSEs resulting from the above cases, we need to derive the individual EMSEs of the two MISO filters and the cross-EMSE. To this end, we analyze the mean squared performance of the individual MISO filters in Section 4, from which we derive the expressions of \(\tilde{\xi}_{k}^{(j)}\left[n\right]=0,j=1,2\), and the mean squared performance of the combination in Section 5, from which we derive the cross-EMSE \(\xi_{k}^{(12)}\left[n\right]\).