## Appendix A
Thus, the optimal \(\mathbf{A}_{\mathrm{D},k}\) can be directly obtained through the first-order conditions  
\[\
abla_{a_{\mathrm{D},k}}(g_{\mathrm{D},k})=0\Rightarrow a_{\mathrm{D},k}^{*}= \epsilon_{\mathrm{D},k}^{-1}. \tag{11}\]  
Similarly, the optimal \(u_{\mathrm{D},k}\) can be derived as  
\[\
abla_{u_{\mathrm{D},k}^{*}}(g_{\mathrm{D},k}) =a_{\mathrm{D},k}\frac{\partial\Bigl{(}u_{\mathrm{D},k}^{*}u_{ \mathrm{D},k}\hat{\mathbf{h}}_{\mathrm{D},k}^{H}\mathbf{F}\mathbf{F}^{H}\hat {\mathbf{h}}_{\mathrm{D},k}-2\operatorname{Re}\Bigl{\{}u_{\mathrm{D},k}^{*} \hat{\mathbf{h}}_{\mathrm{D},k}^{H}\hat{k}_{\mathrm{D},k}\Bigr{\}}+\sigma_{ \mathrm{D},k}^{2}u_{\mathrm{D},k}^{*}u_{\mathrm{D},k}+1\Bigr{)}}{\partial u_{ \mathrm{D},k}^{*}} \tag{12}\] \[=a_{\mathrm{D},k}\Bigl{(}u_{\mathrm{D},k}\hat{\mathbf{h}}_{ \mathrm{D},k}^{H}\mathbf{F}\mathbf{F}^{H}\hat{\mathbf{h}}_{\mathrm{D},k}-\hat {\mathbf{h}}_{\mathrm{D},k}^{H}\mathbf{f}_{k}+\sigma_{\mathrm{D},k}^{2}u_{ \mathrm{D},k}\Bigr{)},\]  
\[\
abla_{u_{\mathrm{D},k}^{*}}(g_{\mathrm{D},k})=0\Rightarrow u_{\mathrm{D},k} ^{*}=\Bigl{(}\hat{\mathbf{h}}_{\mathrm{D},k}^{H}\mathbf{F}\mathbf{F}^{H}\hat {\mathbf{h}}_{\mathrm{D},k}+\sigma_{\mathrm{D},k}^{2}\Bigr{)}^{-1}\hat{ \mathbf{h}}_{\mathrm{D},k}^{H}\mathbf{f}_{k}. \tag{13}\]  
By substituting \(u_{\mathrm{D},k}^{*}\) into \(\epsilon_{\mathrm{D},k}\), the minimized MSE can be derived as follows:  
\[\begin{split}\epsilon_{\mathrm{D},k}^{\mathrm{MMSE}}& =1-\hat{\[\
abla_{\mathbf{u}_{\mathrm{U},l}}(\mathrm{U}_{\mathrm{U},l}) =0 \tag{12}\] \[\Rightarrow\mathbf{u}_{\mathrm{U},l}{}^{\star} =\sqrt{P_{\mathrm{U}}}\Big{(}P_{\mathrm{U}}\hat{\mathbf{H}}_{ \mathrm{U}}\hat{\mathbf{H}}_{\mathrm{U}}^{H}+\mathbf{H}_{\mathrm{SI}}\mathbf{F} \mathbf{F}^{H}\mathbf{H}_{\mathrm{SI}}^{H}+\sigma_{\mathrm{U}}^{2}\mathbf{I} \Big{)}^{-1}\hat{\mathbf{h}}_{\mathrm{U},l},\]  
where  
\[\arg\max_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}R_{\mathrm{U},j} =\arg\min_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}\big{(}a_{\mathrm{ U},j}\epsilon_{\mathrm{U},j}-\log_{2}\big{|}a_{\mathrm{U},j}\big{|}\big{)}, \quad\forall l.\]0 \[\arg\max_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}R_{\mathrm{U},j} =\arg\min_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}\big{(}a_{\mathrm{ U},j}\epsilon_{\mathrm{U},j}-\log_{2}\big{|}a_{\mathrm{U},j}\big{|}\big{)}, \quad\forall l.\]1 \[\arg\max_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}R_{\mathrm{U},j} =\arg\min_{\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{F}}\big{(}a_{\mathrm{ U},j}\epsilon_{\mathrm{U},j}-\log_{2}\big{|}a_{\mathrm{U},j}\big{|}\big{)}, \quad\forall l.\]2  
By substituting \(\mathbf{u}_{\mathrm{U},l}{}^{\star}\) into (16), we have  
\[ \begin{split}\varepsilon_{\mathrm{U},l}^{\mathrm{MMSE}}& =\mathbf{u}_{\mathrm{U},l}^{H}\Big{(}P_{\mathrm{U}}\hat{\mathbf{H}}_{ \mathrm{U}}\hat{\mathbf{H}}_{\mathrm{U}}^{H}+\mathbf{H}_{\mathrm{SI}} \mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\mathrm{SI}}^{H}+\sigma_{\mathrm{U}}^{2} \mathbf{I}\Big{)}\mathbf{u}_{\mathrm{U},l}-2\sqrt{P_{\mathrm{U}}}\operatorname {Re}\Big{\{}\mathbf{u}_{\mathrm{U},l}^{H}\hat{\mathbf{h}}_{\mathrm{U},l} \Big{\}}+1\\ &=1-P_{\mathrm{U}}\hat{\mathbf{h}}_{\mathrm{U},l}^{H}\Big{(}P_{ \mathrm{U}}\hat{\mathbf{H}}_{\mathrm{U}}\hat{\mathbf{H}}_{\mathrm{U}}^{H}+ \mathbf{H}_{\mathrm{SI}}\mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\mathrm{SI}}^{H}+ \sigma_{\mathrm{U}}^{2}\mathbf{I}\Big{)}^{-1}\hat{\mathbf{h}}_{\mathrm{U},l}. \end{split}  \tag{14}\]  
Thus, according to (11), \(a_{\mathrm{U},l}^{\star}\) can be rewritten as  
\[ \begin{split} a_{\mathrm{U},l}{}^{\star}&=\varepsilon_ {\mathrm{U},l}^{\mathrm{MMSE}-1}\\ &\overset{(b)}{=}1+P_{\mathrm{U}}\hat{\mathbf{h}}_{\mathrm{U},l} \Bigg{(}P_{\mathrm{U}}\sum_{i=1,i\
eq l}^{K_{\mathrm{U}}}\hat{\mathbf{h}}_{ \mathrm{U},l}\hat{\mathbf{h}}_{\mathrm{U},i}^{H}+\mathbf{H}_{\mathrm{SI}} \mathbf{F}\mathbf{F}^{H}\mathbf{H}_{\mathrm{SI}}^{H}+\sigma_{\mathrm{U}}^{2} \mathbf{I}\Bigg{)}^{-1}\hat{\mathbf{h}}_{\mathrm{U},l}^{H},\end{split}