## 7 Conclusions
\tag{10}\]  
Assuming a small value for the regularization parameter \(\delta_{j}\), the relation between the a priori and the a posteriori estimation error signals in (27) becomes:  
\[\mathbf{e}_{\mathrm{p}_{n}}^{(j)}=\mathbf{e}_{\mathrm{a}_{n}}^{(j)}-\mu_{j} \mathbf{e}_{n}^{(j)}=\left(1-\mu_{j}\right)\mathbf{e}_{\mathrm{a}_{n}}^{(j)}- \mu_{j}\mathbf{v}_{n}^{(j)}. \tag{11}\]  
Taking the expectation of the square of both sides of (11) and considering the error signals to be statistically independent of the noise, we have:  
\[\mathrm{E}\left\{\mathbf{e}_{\mathrm{p}_{n}}^{(j)}\mathbf{e}_{\mathrm{p}_{n}} ^{(j),T}\right\}=\left(1-\mu_{j}\right)^{2}\mathrm{E}\left\{\mathbf{e}_{ \mathrm{a}_{n}}^{(j)}\mathbf{e}_{\mathrm{a}_{n}}^{(j),T}\right\}+\mu_{j}^{2} \sigma_{v}^{2}. \tag{12}\]  
At steady state, we know that the coefficients of the weight vectors no longer change [41]. We also know that, due to the construction of the input data matrix, in the APA, the \(k-1^{\mathrm{th}}\) projection of the input signal at the \(n^{\mathrm{th}}\) time instant is equivalent to the \(k^{\mathrm{th}}\) projection at the time instant \(n-1\)[41; 42]. Therefore, at steady state, the \(k-1\) projection of the a posteriori error is equivalent to the \(k^{\mathrm{th}}\) projection of the a priori error signal. For this reason, the expectation in (12) can be rewritten, for \(k=1,\ldots,K_{j}-1\), as:  
\[\mathrm{E}\left\{\left|e_{\mathrm{a}_{k}}^{(j)}\left[n\right]\right|^{2} \right\}=\left(1-\mu_{j}\right)^{2}\mathrm{E}\left\{\left|e_{\mathrm{a}_{k-1} }^{(j)}\left[n\right]\right|^{2}\right\}+\mu_{j}^{2}\sigma_{v}^{2} \tag{13}\]  
while for \(k=0\), \(\mathrm{E}\left\{\left|e_{\mathrm{a}_{0}}^{(j)}\left[n\right]\right|^{2}\right\}\) is initialized to its own value, which we write omitting the index \(k=0\). Generalizing (13) for all the projections, we have:  
\[\mathrm{E}\left\{\mathbf{e}_{\mathrm{a}_{n}}^{(j)}\mathbf{e}_{\mathrm{a}_{n} }^{(j),T}\right\}=\mathrm{E}\left\{\left|e_{\mathrm{a}}^{(j)}\left[n\right] \right|^{2}\right\}\cdot\mathbf{S}^{(j)}+\mu_{j}^{2}\sigma_{v}^{2}\cdot \mathbf{S}_{v}^{(j)} \tag{14}\]  
where \(\mathbf{S}^{(j)}\) and \(\mathbf{S}_{v}^{(j)}\) are diagonal matrices \(\in\mathbb{R}^{K_{j}\times K_{j}}\), respectively given by:  
\[\mathbf{S}^{(j)} =\mathrm{diag}\left\{ \begin{array}{cccc}1&\zeta&\zeta^{2}& \ldots&\zeta^{K_{j}-1}\end{array} \right\} \tag{15}\] \[\mathbf{S}_{v}^{(j)} =\mathrm{diag}\left\{ \begin{array}{cccc}0&1&(\zeta+1)&\left( \zeta^{2}+\zeta+1\right)&\ldots&\left(\zeta^{K_{j}-2}+\zeta^{K_{j}-3}+\ldots+ 1\right)\end{array} \right\} \tag{16}\]  
where \(\zeta=\left(1-\mu_{j}\right)^{2}\). It is worth noting that, when \(\mu_{j}\) is small, \(\mathbf{S}^{(j)}\approx\mathbf{I}\). On the other hand, when \(\mu_{j}\) is close to one, \(\mathbf{S}^{(j)}\approx\mathbf{F}\), where \(\mathbf{F}=\mathrm{diag}\left\{ \begin{array}{cccc}1&0&\ldots&0\end{array} \right\}\). Regarding the second term of the right-hand side of (14), \(\mu_{j}^{2}\sigma_{v}^{2}\mathbf{S}_{v}^{(j)}\approx\mathbf{0}\), when \(\mu_{j}\) is small enough or when the SNR (signal-to-noise ratio) is high enough (i.e., \(\sigma_{v}^{2}\) very small). In addition, considering that the choice of large values of \(\mu_{j}\) might not lead to satisfactory convergence performance when the SNR is low, we can always neglect the term \(\mu_{j}^{2}\sigma_{v}^{2}\mathbf{S}_{v}^{(j)}\) in (14). Therefore, steady-state EMSE for the \(j^{\mathrm{th}}\) MISO filter evaluated at all the projections can be finally written as:  
\[\mathrm{E}\left\{\mathbf{e}_{\mathrm{a}_{n}}^{(j)}\mathbf{e}_{\mathrm{a}_{n}}^{ (j),T}\right\}=\mathrm{E}\left\{\left|e_{\mathrm{a}_{n}}^{(j)}\left[n\right] \right|^{2}\right\}\cdot\mathbf{S}^{(j)}=\xi_{\infty}^{(j)}\mathbf{S}^{(j)}. \tag{17}\]