## Appendix A  
Proof. : The processing procedure for digital precoder design in (36). The power constraint in Equation (36) can be transformed into  
\[\mathrm{Tr}\bigg{\{}\bigg{[}\bar{\mathbf{G}}_{I}\Big{(}\mathbf{F}_{I}^{\mathbf{A} }\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{(}\mathbf{F}_{I}^{\mathbf{A}}\mathbf{F }_{I}^{\mathbf{D}}\Big{)}^{H}\bar{\mathbf{G}}_{I}^{H}\bigg{]}+[\mathbf{D}_{I} ]\bigg{\}}=\mathrm{Tr}\Big{\{}\Big{[}\bar{\mathbf{G}}_{I}\mathbf{F}_{I}^{ \mathbf{F}}\bar{\mathbf{G}}_{I}^{H}\Big{]}+[\mathbf{D}_{I}]\Big{\}}. \tag{11}\]  
Expanding the distortion section, we can obtain  
\[ \begin{split}\Big{\|}\bar{\mathbf{G}}_{I}\mathbf{F}_{I}^{ \mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{\|}_{F}^{2}&=\big{\|} \bar{\mathbf{G}}_{I}\mathbf{F}_{I}\big{\|}_{F}^{2}+2|\beta_{3}|^{2}\mathrm{Tr} \bigg{\{}\mathbf{F}_{I}\mathbf{F}_{I}^{H}\odot\Big{|}\mathbf{F}_{I}\mathbf{F }_{I}^{H}\big{|}^{2}\bigg{\}}\\ &\hskip 14.226378 pt-2|\beta_{3}|^{2}\mathrm{Tr}\Bigg{\{}\Big{(} \mathbf{F}_{I}^{\mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{(}\mathbf{F }_{I}^{\mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}^{H}\odot\Big{|}\Big{(} \mathbf{F}_{I}^{\mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{(}\mathbf{ F}_{I}^{\mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}^{H}\bigg{|}^{2}\Bigg{\}}. \end{split}  \tag{12}\]  
For convenience in processing, the results of the last iteration are used in each iteration for both the analog precoder and digital precoder. Then, we need to deal with the paradigm expression part in Equation (12), where the matrices are not easy to handle. Hence, we columnize the matrices first. \[\Big{\|}\mathbf{F}_{I}-\mathbf{F}_{I}^{\mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}} \Big{\|}_{F}^{2}=\Big{\|}\mathrm{vec}\Big{(}\mathbf{F}_{I}-\mathbf{F}_{I}^{ \mathbf{A}}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{\|}_{2}^{2}=\Big{\|} \mathrm{vec}(\mathbf{F}_{I})-\Big{(}\mathbf{I}_{K}\otimes\mathbf{F}_{I}^{ \mathbf{A}}\Big{)}\mathrm{vec}\Big{(}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{\|} _{2}^{2}. \tag{13}\]  
In the following, we let \(\mathbf{f}=\mathrm{vec}(\mathbf{F}_{I})\), \(\mathbf{b}=\mathrm{vec}(\mathbf{F}_{I}^{\mathbf{D}})\), \(\mathbf{E}=\mathbf{I}_{K}\otimes\mathbf{F}_{I}^{\mathbf{A}}\). By adding an auxiliary variable \(t\), the objective function can be written as  
\[\|t\mathbf{f}-\mathbf{E}\mathbf{b}\|_{F}^{2}=\Big{[}\mathbf{b}^{H}\quad t \Big{]}\Bigg{[} \begin{array}{cc}\mathrm{E}^{H}\mathrm{E}&-\mathrm{E}^{H} \mathbf{f}\\ -\mathbf{f}^{H}\mathrm{E}&\mathbf{f}^{H}\mathbf{f}\end{array} \Bigg{]}\Bigg{[} \begin{array}{c}\mathbf{b}\\ t\end{array} \Bigg{]}, \tag{14}\]  
where \(\mathbf{y}=\Bigg{[} \begin{array}{c}\mathbf{b}\\ t\end{array} \Bigg{]}\), \(\mathbf{C}=\Bigg{[} \begin{array}{cc}\mathrm{E}^{H}\mathrm{E}&-\mathrm{E}^{H} \mathbf{f}\\ -\mathbf{f}^{H}\mathrm{E}&\mathbf{f}^{H}\mathbf{f}\end{array} \Bigg{]}\). Thus, Equation (14) is equal to \(\mathbf{y}^{H}\mathbf{C}\mathbf{y}\), where \(\mathbf{C}=[\mathrm{E}-\mathbf{f}]^{H}\left[\mathrm{E}-\mathbf{f}\right]\). After dealing with the objective function, we deal with the power constraints in Equation (36) next. \[\Big{\|}\mathbf{\Sigma}_{I}\mathbf{F}_{I}^{\mathbf{D}}\Big{\|}_{F}^{2}=\Big{\|} \mathrm{vec}\Big{(}\mathbf{\Sigma}_{I}\mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{\|} _{2}^{2}=\Big{\|}(\mathbf{I}_{K}\otimes\mathbf{\Sigma}_{I})\mathrm{vec}\Big{(} \mathbf{F}_{I}^{\mathbf{D}}\Big{)}\Big{\|}_{2}^{2}. \tag{15}\]  
Define \(\mathbf{\widetilde{\Sigma}}_{I}=\mathbf{I}_{K}\otimes\mathbf{\Sigma}_{I}\). Thus, Equation (15) can be expressed as  
\[\Big{\|}\mathbf{\widetilde{\Sigma}}_{I}\mathbf{b}\Big{\|}_{2}^{2}=\Big{[} \mathbf{b}^{H}\quad t\Big{]}\Bigg{[} \begin{array}{cc}\mathbf{\widetilde{ \Sigma}}_{I}^{H}\mathbf{\widetilde{\Sigma}}_{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{array} \Bigg{]}\Bigg{[} \begin{array}{c}\mathbf{b}\\ t\end{array} \Bigg{]}=\mathrm{Tr}\Big{\{}\mathbf{b}^{H}\mathbf{\widetilde{ \Sigma}}_{I}^{H}\mathbf{\widetilde{\Sigma}}_{I}\mathbf{b}\Big{\}}, \tag{16}\]  
\[\text{where }\mathbf{A}_{1}=\Bigg{[}