## 4 Mean Squared Performance of Individual MISO APA Filters
### Steady-State Performance for MISO Filters
 \tag{40}\]  
Similarly, we can write the remaining terms of (37) respectively as follows:  
\[\mu_{j}\mathrm{E}\left\{\mathbf{v}_{n}^{(j),T}\mathbf{A}_{n}^{(j) }\mathbf{v}_{n}^{(j)}\right\} =\mu_{j}\sigma_{v}^{2}\mathrm{Tr}\left(\mathrm{E}\left\{\mathbf{ A}_{n}^{(j)}\right\}\right) \tag{41}\] \[2\mathrm{E}\left\{\mathbf{e}_{a}^{(j),T}\mathbf{B}_{n}^{(j)} \mathbf{e}_{a}^{(j)}\right\} =2\mathrm{E}\left\{\left|e_{a}^{(j)}\left[n\right]\right|^{2} \right\}\mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{B}_{n}^{(j) }\right\}\right). \tag{42}\]  
Therefore, (37) becomes:  
\[\left(2\mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{B}_{n}^{(j)} \right\}\right)-\mu_{j}\mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{ \mathbf{A}_{n}^{(j)}\right\}\right)\right)\mathrm{E}\left\{\left|e_{a}^{(j)} \left[n\right]\right|^{2}\right\}=\mu_{j}\sigma_{v}^{2}\mathrm{Tr}\left( \mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\}\right) \tag{43}\]  
from which we can derive the expression of the steady-state EMSE of the individual MISO filters as:  
\[\xi_{\infty}^{(j)}=\lim_{n\to\infty}\mathrm{E}\left\{\left|e_{a}^{(j)}\left[n \right]\right|^{2}\right\}=\frac{\mu_{j}\sigma_{v}^{2}\mathrm{Tr}\left( \mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\}\right)}{\left(2\mathrm{Tr}\left( \mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{B}_{n}^{(j)}\right\}\right)-\mu_{j} \mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\} \right)\right)}. \tag{44}\]  
It is worth noting that Equation (44) takes into account the information related to each projection and contained in the matrices \(\mathbf{A}_{n}^{(j)}\), \(\mathbf{B}_{n}^{(j)}\) and \(\mathbf{S}^{(j)}\), but at the same time, it provides a unique steady-state EMSE value for all the projections; therefore, the dependence of \(\xi_{\infty}^{(j)}\) on index \(k\) disappears. The expression above can be simplified when the regularization factor \(\delta_{j}\) is small enough, so that \(\mathbf{A}_{n}^{(j)}\approx\mathbf{B}_{n}^{(j)}\). This implies \(\mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\} \right)\approx\mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{B}_{n}^ {(j)}\right\}\right)\); hence, the EMSE (44) reduces to:  
\[\xi_{\infty}^{(j)}=\frac{\mu_{j}\sigma_{v}^{2}}{\left(2-\mu_{j}\right)}\frac{ \mathrm{Tr}\left(\mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\}\right)}{ \mathrm{Tr}\left(\mathbf{S}^{(j)}\mathrm{E}\left\{\mathbf{A}_{n}^{(j)}\right\} \right)}. \tag{45}\]  
Now, we can distinguish two further approximations. 1. [label=()]
2. Small value of \(\mu_{j}\). If we assume a small value of the step size \(\mu_{j}\), we have for Assumption 4 that \(\mathbf{S}^{(j)}=\mathbf{I}\); hence, (45) becomes: \[\xi_{\infty}^{(j)}=\frac{\mu_{j}\sigma_{v}^{2}}{\left(2-\mu_{j}\right)}.\] (46)
2. Large value of \(\mu_{j}\). If we assume a large value of the step size \(\mu_{j}\) (i.e., close to one), we have for Assumption 4 that \(\mathbf{S}^{(j)}=\mathbf{F}\); hence, (45) becomes: \[\xi_{\infty}^{(j)}=\frac{\mu_{j}\sigma_{v}^{2}}{\left(2-\mu_{j}\right)}\frac{ \operatorname{Tr}\left(\operatorname{E}\left\{\mathbf{A}_{n}^{(j)}\right\} \right)}{\operatorname{E}\left\{a_{11}^{(j)}\left[n\right]\right\}}\] (47) where \(a_{11}^{(j)}\left[n\right]\) is the element on the first row and first column of \(\mathbf{A}_{n}^{(j)}\).