## 3 Optimum Mixing Parameters and EMSE
### Formulation of the EMSE for the Combination  
The mean squared performance of the combined MISO filter can be evaluated in terms of the excess mean squared error (EMSE) (see for example [1; 22; 41]), which can be defined according to the expression of the a priori error (19), for \(k=0,\ldots,K_{\max}-1\):  
\[\xi_{k}\left[n\right] =\mathrm{E}\left\{\left|\mathbf{e}_{ak}\left[n\right]\right|^{2}\right\} \tag{20}\] \[=\lambda_{k}^{2}\left[n\right]\mathrm{E}\left\{\left|\mathbf{e}_{ ak}^{\left(1\right)}\left[n\right]\right|^{2}\right\}+\left(1-\lambda_{k}\left[n \right]\right)^{2}\mathrm{E}\left\{\left|\mathbf{e}_{ak}^{\left(2\right)} \left[n\right]\right|^{2}\right\}+2\lambda_{k}\left[n\right]\left(1-\lambda_{k }\left[n\right]\right)\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1\right)}\left[n \right]e_{ak}^{\left(2\right)}\left[n\right]\right\}\] \[=\lambda_{k}^{2}\left[n\right]\xi_{k}^{\left(1\right)}\left[n \right]+\left(1-\lambda_{k}\left[n\right]\right)^{2}\xi_{k}^{\left(2\right)} \left[n\right]+2\lambda_{k}\left[n\right]\left(1-\lambda_{k}\left[n\right] \right)\xi_{k}^{\left(12\right)}\left[n\right]\]  
where \(\xi_{k}^{\left(j\right)}\left[n\right]\), \(j=1,2\), are the EMSEs of the \(k^{\text{th}}\) projection for the two individual MISO filters and:  
\[\xi_{k}^{\left(12\right)}\left[n\right]=\mathrm{E}\left\{\mathbf{e}_{ak}^{\left(1 \right)}\left[n\right]e_{ak}^{\left(2\right)}\left[n\right]\right\} \tag{21}\]represents the cross-EMSE, as defined in [6], which provides a measure of the ability of the combination to improve the performance with respect to both filter components [1]. The cross-EMSE satisfies the Cauchy-Schwartz inequality:  
\[\left|\xi_{k}^{(12)}\left[n\right]\right|^{2}\leq\xi_{k}^{(1)}\left[n\right] \xi_{k}^{(2)}\left[n\right] \tag{22}\]  
which implies that the magnitude of the cross-EMSE \(\xi_{k}^{(12)}\left[n\right]\) can never be simultaneously larger than the individual EMSEs \(\xi_{k}^{(1)}\left[n\right]\) and \(\xi_{k}^{(2)}\left[n\right]\). We want to minimize (20) to find the optimum mixing parameter for each projection. The minimization must take into account that \(\lambda_{k}\left[n\right]\) is contained in the interval \([0,1]\) and that (20) is nonnegative and quadratic in \(\lambda_{k}\left[n\right]\)[1]. Therefore, the optimum mixing parameter is given by:  
\[\lambda_{k}^{\circ}\left[n\right]=\left.\frac{\xi_{k}^{(2)}\left[n\right]-\xi _{k}^{(1)}\left[n\right]}{\xi_{k}^{(1)}\left[n\right]+\xi_{k}^{(2)}\left[n \right]-2\xi_{k}^{(12)}\left[n\right]}\right|_{0}^{1}=\left.\frac{\Delta\xi_ {k}^{(2)}\left[n\right]}{\Delta\xi_{k}^{(1)}\left[n\right]+\Delta\xi_{k}^{(2) }\left[n\right]}\right|_{0}^{1} \tag{23}\]  
where \(\Delta\xi_{k}^{(j)}\left[n\right]=\xi_{k}^{(j)}\left[n\right]-\xi_{k}^{(12)} \left[n\right]\), \(j=1,2\), and the vertical line denotes the limits of the interval for the optimum mixing parameters. We now substitute (23) in (20) to find the definition of the optimum EMSE for all the input projections:  
\[\xi_{k}^{\circ}\left[n\right] =\xi_{k}^{(1)}\left[n\right]-\left(1-\lambda_{k}^{\circ}\left[n \right]\right)\Delta\xi_{k}^{(1)}\left[n\right] \tag{24}\] \[\equiv\xi_{k}^{(2)}\left[n\right]-\lambda_{k}^{\circ}\left[n \right]\Delta\xi_{k}^{(2)}\left[n\right].\]  
The expressions of the optimum EMSE are valid for any time instant \(n\) and also at steady state, i.e., for \(n\rightarrow\infty\). It is worth noting from (23) and (24) that the combination depends on the value of the cross-EMSE. In particular, we can identify four different cases [1,6]:  
\[\xi_{k}^{(1)}\left[n\right]\leq\xi_{k}^{(12)}\left[n\right]<\xi_{k}^{(2)} \left[n\right].\]  
For this case, it is easy to verify that \(\Delta\xi_{k}^{(1)}\left[n\right]\leq 0\) and \(\Delta\xi_{k}^{(2)}\left[n\right]>0\). In this situation, the optimum mixing parameter is \(\lambda_{k}^{\circ}\left[n\right]=1\), and the combined scheme turns out to perform like the best individual filter, i.e., the one with the lower EMSE, which is the first filter.