## 3 Scheme Design
### Low-Complexity Diagonalizing Design
#### 3.2.2 Alternating Optimization of \(\mathbf{F}_{r}\) and \(\mathbf{F}_{i}\)  
In this section, an iterative approach is utilized to convert the multivariate non-convex problem in Equation (25) into three convex subproblems. We first study how to optimize \(\mathbf{\Lambda}_{r}\) with a fixed \(\mathbf{\Lambda}_{i}\), and then, the alternating optimization of \(\mathbf{\Lambda}_{1}\) and \(\mathbf{\Lambda}_{2}\) is performed with a given \(\mathbf{\Lambda}_{r}\). 1. Optimization of \(\mathbf{\Lambda}_{r}\)  
Using Equation (17), \(\mathcal{J}_{i}^{*}\) can be simplified and rewritten as  
\[\mathcal{J}_{i}^{*}=M-\text{Tr}\left\{\left[\beta_{i}\mathbf{\Lambda}_{i}^{H}\mathbf{ \Sigma}_{h_{i}^{H}}^{H}\mathbf{\mathrm{D}}_{i}^{-1}\mathbf{\Sigma}_{h_{i}^{H}}\mathbf{ \Lambda}_{i}-\mathbf{MSE}_{i}^{*}\right]\right\}, \tag{26}\]  
where \(\mathbf{MSE}_{i}^{*}=\beta_{i}\left(\mathbf{\Lambda}_{i}^{H}\mathbf{\Sigma}_{h_{i}^{H} }^{H}\mathbf{\mathrm{D}}_{i}^{-1}\mathbf{\Lambda}_{i}\mathbf{\Sigma}_{h_{i}^{H}}\right) \left[\frac{1}{\beta_{i}\sigma_{S_{i}}^{2}}\mathbf{\Lambda}_{r}^{H}\mathbf{\Lambda}_{ dl}^{H}\mathbf{\Lambda}_{dl}\ \ \mathbf{\Lambda}_{r}\mathbf{\Lambda}_{B_{i}}^{-1}\mathbf{\mathrm{D}}_{i}+\mathbf{\mathrm{I}}_{M} \right]^{-1}\) and \(\mathbf{\mathrm{D}}_{i}=\beta_{i}\mathbf{\Sigma}_{h_{i}^{*}}\mathbf{\Lambda}_{i}^{H}\mathbf{ \Sigma}_{h_{i}^{*}}^{H}\mathbf{\Sigma}_{h_{i}^{*}}^{H}\)\(+\sigma_{r}^{2}\beta_{i}\mathbf{\Lambda}_{B_{n}}\). Since \(\mathbf{\Lambda}_{r}\) exists in \(\mathbf{MSE}_{i}^{*}\) only, the problem of minimizing \(\mathcal{J}_{1}^{*}+\mathcal{J}_{2}^{*}\) is equivalent to that of minimizing \(\mathbf{MSE}_{1}^{*}+\mathbf{MSE}_{2}^{*}\). Defining \(a_{in}\), \(a_{h_{i}n}\), \(a_{rn}\), \(a_{dn}\), \(\lambda_{B_{i}n}\), and \(\lambda_{B_{n}n}\) as the \(n\)th diagonal element of \(\mathbf{\Lambda}_{i}\), \(\mathbf{\Lambda}_{h_{i}}\), \(\mathbf{\Lambda}_{r}\), \(\mathbf{\Lambda}_{dl}\), \(\mathbf{\Lambda}_{B_{i}}\), and \(\mathbf{\Lambda}_{B_{n}}\), respectively, \(\mathbf{MSE}_{i}^{*}\) can be given by  
\[\mathbf{MSE}_{1}^{*}=\sum_{n=1}^{M}\frac{\beta_{1}^{2}\sigma_{S_{1}}^{2}a_{2n} ^{2}a_{h_{2}n}^{2}\lambda_{B_{1}n}}{\left(a_{rn}^{2}a_{dn}^{2}\lambda_{1n}+ \lambda_{B_{1}n}\beta_{1}\sigma_{S_{1}}^{2}\right)\lambda_{1n}}, \tag{27}\]  
\[\mathbf{MSE}_{2}^{*}=\sum_{n=1}^{N}\frac{\beta_{2}^{2}\sigma_{S_{2}}^{2}a_{1n} ^{2}a_{h_{1}n}^{2}\lambda_{B_{2}n}}{\left(a_{rn}^{2}a_{dn}^{2}\lambda_{2n}+ \lambda_{B_{2}n}\beta_{2}\sigma_{S_{2}}^{2}\right)\lambda_{2n}}, \tag{28}\]  
where \(\varsigma=N-M+1\), \(\lambda_{in}=\beta_{i}a_{h_{in}}^{2}a_{in}^{2}+\sigma_{r}^{2}\beta_{i}\lambda_{ B_{n}n}\). Moreover, we define \(\phi_{in}=a_{rn}^{2}a_{h_{in}}^{2}\), \(a_{rn}^{2}=a_{rn}^{2}\lambda_{B_{n}n}\), \(\Phi_{in}=\phi_{in}a_{dn}^{2}\lambda_{B_{i}n}^{-1}\), and \(\Theta_{rn}=\theta_{rn}\lambda_{B_{i}n}^{-1}a_{dn}^{2}\). Accordingly, the problem related to \(\mathbf{\Lambda}_{r}\) can be described as  
\[\min_{a_{rn}^{2}} \mathbf{MSE}_{1}^{*}+\mathbf{MSE}_{2}^{*}\] (29a) s.t. \[(1-\beta_{i})[\mathbf{\Lambda}_{r_{c}}]\geqslant\overline{e}_{i} \tag{29b}\] \[\sum_{n=1}^{M}\phi_{1n}+\sum_{n=1}^{N}\phi_{2n}+\sum_{n=1}^{N} \sigma_{r}^{2}\theta_{rn}\leqslant p_{r}\] (29c) \[a_{rn}\geqslant 0, \tag{29d}\]  
where \(\mathbf{\Lambda}_{r_{c}}=\sum_{n=1}^{M}\Phi_{1n}+\sigma_{S_{i}}^{2}\mathbf{\mathrm{I}}_ {M}+\sum_{n=\varsigma}^{N}\Phi_{2n}+\sum_{n=1}^{N}\sigma_{r}^{2}\Theta_{m}\). 2. Optimization of \(\mathbf{\Lambda}_{i}\)Similarly, the solution for \(a_{in}\) can be described in the following scalar form  
\[\underset{a_{in}^{2}}{\text{min}} \mathcal{J}_{1}^{\square}+\mathcal{J}_{2}^{\square}\] (30a) s.t.