## 5 Mean Squared Performance of the Combination of MISO Filters  
Now, we want to derive the expression of the cross-EMSE \(\tilde{\varsigma}_{k}^{(12)}\left[n\right]\) by carrying out similar procedural steps of the previous section, but considering the cross-information, as explained in detail in the following. In order to derive the energy conservation relation for the combined scheme, similarly to [6], we consider (29) with \(\mathbf{X}_{n}\in\mathbb{R}^{K_{\max}\times 1}\) and multiply the transpose of (29) for \(j=1\) by (29) itself for \(j=2\), and get:  
\[\widetilde{\mathbf{w}}_{n}^{(1),T}\widetilde{\mathbf{w}}_{n}^{(2)}+\mathbf{e} _{a}^{(1),T}\left(\mathbf{X}_{n}\mathbf{X}_{n}^{T}\right)^{-1}\mathbf{e}_{a}^ {(2)}=\widetilde{\mathbf{w}}_{n-1}^{(1),T}\widetilde{\mathbf{w}}_{n-1}^{(2)} +\mathbf{e}_{p_{n}}^{(1),T}\left(\mathbf{X}_{n}\mathbf{X}_{n}^{T}\right)^{-1} \mathbf{e}_{p_{n}}^{(2)}. \tag{51}\]  
Taking the expectations of both sides of (51) and considering that at steady state \(\operatorname{E}\left\{\widetilde{\mathbf{w}}_{n}^{(1),T}\widetilde{\mathbf{ w}}_{n}^{(2)}\right\}=\operatorname{E}\left\{\widetilde{\mathbf{w}}_{n-1}^{(1),T} \widetilde{\mathbf{w}}_{n-1}^{(2)}\right\}\), we obtain:  
\[\operatorname{E}\left\{\mathbf{e}_{a}^{(1),T}\left(\mathbf{X}_{n}\mathbf{X}_{ n}^{T}\right)^{-1}\mathbf{e}_{a}^{(2)}\right\}=\operatorname{E}\left\{\mathbf{e}_{p _{n}}^{(1),T}\left(\mathbf{X}_{n}\mathbf{X}_{n}^{T}\right)^{-1}\mathbf{e}_{p _{n}}^{(2)}\right\}. \tag{52}\]  
Replacing (27) (with \(\mathbf{X}_{n}\in\mathbb{R}^{K_{\max}\times 1}\)) in (52) and after some simplifications, similarly to what was done for (36), it is possible to achieve the variance relation for the combined scheme at steady state:  
\[\mu_{1}\mu_{2}\operatorname{E}\left\{\mathbf{e}_{n}^{(1),T}\mathbf{A}_{n} \mathbf{e}_{n}^{(2)}\right\}=\mu_{1}\operatorname{E}\left\{\mathbf{e}_{n}^{( 1),T}\mathbf{B}_{n}\mathbf{e}_{a}^{(2)}\right\}+\mu_{2}\operatorname{E}\left\{ \mathbf{e}_{a}^{(1),T}\mathbf{B}_{n}\mathbf{e}_{n}^{(2)}\right\} \tag{53}\]  
where \(\mathbf{A}_{n}\) and \(\mathbf{B}_{n}\) are defined similarly to (34) and (35), respectively, but over \(K_{\max}\). Considering (14), we can rewrite (53), similarly to (37), as:  
\[\mu_{1}\mu_{2}\operatorname{E}\left\{\mathbf{e}_{a}^{(1),T}\mathbf{A}_{n} \mathbf{e}_{a}^{(2)}\right\}+\mu_{1}\mu_{2}\operatorname{E}\left\{\mathbf{v}_{ n}^{T}\mathbf{A}_{n}\mathbf{v}_{n}\right\}=\left(\mu_{1}+\mu_{2}\right) \operatorname{E}\left\{\mathbf{e}_{a}^{(1),T}\mathbf{B}_{n}\mathbf{e}_{a}^{(2 )}\right\}. \tag{54}\]Now, similarly to Assumption 4, but with extension to the case of combined MISO filters, we can define the following assumption. **Assumption 5**. : _At steady state, the input signal \(\mathbf{x}_{n}\) is statistically independent of the a priori estimation error signals \(\mathbf{e}_{\mathrm{a}t}^{(j)}\), \(j=1,2\), and moreover,_  
\[\mathrm{E}\left\{\mathbf{e}_{\mathrm{a}t}^{(1)}\mathbf{e}_{\mathrm{a}t}^{(2),T }\right\}=\mathrm{E}\left\{\left|e_{\mathrm{a}}^{(1)}\left[n\right]e_{ \mathrm{a}}^{(2)}\left[n\right]\right|\right\}\cdot\mathbf{S}^{(12)} \tag{55}\]  
_where the value of \(\mathbf{S}^{(12)}\in\mathbb{R}^{K_{\mathrm{max}}\times K_{\mathrm{max}}}\) is defined according to the following cases._  
* _If we characterize the combination scheme according to the step-size values, generally_ \(\mu_{1}\) _small and_ \(\mu_{2}\) _large, e.g., to find an optimal selection of filter parameters, we have that:_ \[\mathbf{S}^{(12)}\approx\mathbf{F}\] (56) _where_ \(\mathbf{F}\) _is defined similarly to (_39_)._
* _On the other hand, if we want to provide diversity to the combined scheme and choose different projection orders, but the same step-size value, i.e.,_ \(\mu_{1}=\mu_{2}=\mu\)_, we have that:_ \[\mathbf{S}^{(12)}\approx\left\{ \begin{array}{ll}\mathbf{I},&\text{for small values of }\mu\\ \mathbf{F},&\text{for large values of }\mu\end{array}