## 5 Mean Squared Performance of the Combination of MISO Filters
\right..\] (57) _Equation (_57_) holds as long as the SNR is high, as detailed in Appendix B._  
The derivation of the approximation (55) in Assumption 5 can be found in Appendix B. Using Assumption 5 and after some manipulations, similarly to what was done for (44), it is possible to derive the expression of the cross-EMSE from (54):  
\[\xi_{\infty}^{(12)}=\lim_{n\to\infty}\mathrm{E}\left\{\left|e_{\mathrm{a}}^{(1 )}\left[n\right]e_{\mathrm{a}}^{(2)}\left[n\right]\right|\right\}=\frac{\mu_{ 1}\mu_{2}\sigma_{\mathrm{v}}^{2}\mathrm{Tr}\left(\mathrm{E}\left\{\mathbf{A }_{n}\right\}\right)}{\left(\mu_{1}+\mu_{2}\right)\mathrm{Tr}\left(\mathbf{S} ^{(12)}\mathrm{E}\left\{\mathbf{B}_{n}\right\}\right)-\mu_{1}\mu_{2}\mathrm{ Tr}\left(\mathbf{S}^{(12)}\mathrm{E}\left\{\mathbf{A}_{n}\right\}\right)}. \tag{58}\]  
As done in the previous section for \(\xi_{\infty}^{(j)}\), we may further simplify also (58) when the regularization factor \(\delta_{j}\) is small enough to be neglected, so that \(\mathbf{A}_{n}\approx\mathbf{B}_{n}\), which implies:  
\[\xi_{\infty}^{(12)}=\frac{\mu_{12}\sigma_{\mathrm{v}}^{2}\mathrm{Tr}\left( \mathrm{E}\left\{\mathbf{A}_{n}\right\}\right)}{\left(1-\mu_{12}\right)\mathrm{ Tr}\left(\mathbf{S}^{(12)}\mathrm{E}\left\{\mathbf{A}_{n}\right\}\right)} \tag{59}\]  
where \(\mu_{12}=\mu_{1}\mu_{2}/\left(\mu_{1}+\mu_{2}\right)\). Equation (59) can be further reduced according to the values of the step sizes. In particular, similarly to what was done for \(\xi_{\infty}^{(j)}\) in Section 4.3, we can consider the following approximations. * Small values for both \(\mu_{j}\). This case is typical when we want to differentiate the combined scheme according to the projection order and we choose the same small value for both step sizes \(\mu_{1}=\mu_{2}=\mu\). Based on Assumption 5, we have that \(\mathbf{S}^{(12)}=\mathbf{I}\); hence, (59) becomes: \[\xi_{\infty}^{(12)}=\frac{\mu_{12}\sigma_{\mathrm{v}}^{2}}{\left(1-\mu_{12} \right)}.\] (60)
* Large value for at least one \(\mu_{j}\). We can also consider the case for which \(\mathbf{S}^{(j)}=\mathbf{F}\), according to Assumption 5. This may occur when we want to characterize the combined scheme according to the projection order and we choose the same large value for both step sizes \(\mu_{1}=\mu_{2}=\mu\), or also when we choose the same projection order, but one step-size value small and the other one large (close to one). In both of these situations, after some approximations similarly to (50), we have that (59) becomes:  
\[\xi_{\infty}^{(12)}=\frac{\mu_{12}\sigma_{v}^{2}}{(1-\mu_{12})}\mathrm{Tr}\left( \mathbf{R}_{n}\right)\mathrm{E}\left\{\frac{K_{\mathrm{max}}}{\left\|\mathbf{x }_{n}\right\|^{2}}\right\} \tag{61}\]  
which depends proportionally on the value of the maximum projection order \(K_{\mathrm{max}}\) between the two MISO filters. We can note that, similarly to the case of the individual EMSEs in (44), even the steady-state cross-EMSE (58) provides a unique value for all the projections, although it considers the information related to them. This also implies a unique value at steady state for the optimum mixing parameter for all the projections. Therefore, considering (44) and (58), and related approximations, the expression of the theoretical EMSE of the combined MISO scheme (24) can be written at steady state as:  
\[\xi_{\infty} =\xi_{\infty}^{(1)}\left[n\right]-\left(1-\lambda_{\infty}\right) \Delta_{\infty}^{\lambda(1)} \tag{62}\] \[\equiv\xi_{\infty}^{(2)}-\lambda_{\infty}\Delta_{\xi\infty}^{ \lambda(2)}\]  
according to which the four different cases described in Section 3 can be easily derived.