## 3 Scheme Design
### Low-Complexity Diagonalizing Design
#### 3.2.1 Channel Parallelization
Substituting Equations (19)-(21) into each of the constraints in Equation (11), the original problem can be expressed as  
\[\min_{\boldsymbol{\Lambda}_{r},\boldsymbol{\Lambda}_{i}} \mathcal{J}_{1}^{*}+\mathcal{J}_{2}^{*}\] (23a) s.t. \[\text{Tr}\left(\boldsymbol{\Lambda}_{i}\boldsymbol{\Lambda}_{i}^{H}\right)\leqslant p _{i} \tag{23b}\] \[(1-\beta_{i}) Tr\left[\boldsymbol{\Gamma}_{G}+\sigma_{S_{i}}^{2}\mathbf{I}_{M_{R}} \right]\geqslant\overline{e}_{i},\quad\quad\forall i\] (23c) \[\text{Tr}\left[\boldsymbol{\Lambda}_{r}\left(\boldsymbol{\Gamma}+ \mathbf{B}_{h}\sigma_{r}^{2}\mathbf{I}_{N_{R}}\right)\boldsymbol{\Lambda}_{r}^ {H}\right]\leqslant p_{r}, \tag{23d}\]  
where \(\boldsymbol{\Gamma}_{G}=\mathbf{B}_{i}^{-1}\boldsymbol{\Lambda}_{dl}\boldsymbol {\Lambda}_{r}\left(\boldsymbol{\Gamma}+\mathbf{B}_{h}\sigma_{r}^{2}\mathbf{I}_{ N_{R}}\right)\boldsymbol{\Lambda}_{r}^{H}\boldsymbol{\Lambda}_{dl}^{H}\) and \(\boldsymbol{\Gamma}=\boldsymbol{\Sigma}_{h_{1}}\boldsymbol{\Lambda}_{1} \boldsymbol{\Lambda}_{1}^{H}\boldsymbol{\Sigma}_{h_{1}}^{H}+\boldsymbol{\Sigma}_ {h_{2}}\boldsymbol{\Lambda}_{2}\boldsymbol{\Lambda}_{2}^{H}\boldsymbol{\Sigma}_ {h_{2}}^{H}\). To solve the nonconvexity caused by Equation (23c), we propose Theorem 2 as follows. The left side of the energy-harvesting constraint in Equation (23c) can be replaced by its lower-bound \((1-\beta_{i})\text{Tr}\left[\boldsymbol{\Lambda}_{\Gamma_{G}}+\sigma_{S_{i}}^{2} \mathbf{I}_{M_{R}}\right]\). Proof. : We take \(N>M\) as an example to illustrate the proof procedure and the optimization problem. Expanding the left side of Equation (23c), defining \(\mathbf{A}=\mathbf{R}_{dli}\boldsymbol{\Lambda}_{dl}\boldsymbol{\Lambda}_{r}\), and ignoring the constant matrix \(\sigma_{r}^{2}\mathbf{I}_{N_{R}}\), the part \(Tr\left[\mathbf{B}_{i}^{-1}\boldsymbol{\Lambda}_{dl}\boldsymbol{\Lambda}_{r} \mathbf{B}_{h}\boldsymbol{\Lambda}_{r}^{H}\boldsymbol{\Lambda}_{dl}^{H}\right]\) becomes  
\[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]0  
Define the \((i,j)\) entry of \(\mathbf{C}\mathbf{B}_{h}\) as \((\mathbf{C}\mathbf{B}_{h})_{ij}=\sum_{k=1}^{M}c_{ik}b_{h_{ij}}\); then, we have \(Tr(\mathbf{C}\mathbf{B}_{h})\geqslant Tr(\mathbf{C}\boldsymbol{\Lambda}_{B_{h}})\) based on the relationship \(\sum_{i=1}^{M}\sum_{k=1}^{M}c_{ik}b_{h_{ki}}\geqslant\sum_{i=1}^{M}c_{ii}b_{h_{ ii}}\). Following a similar procedure, the lower-bound of \(Tr\left[\mathbf{B}_{i}^{-1}\boldsymbol{\Lambda}_{dl}\boldsymbol{\Lambda}_{r} \boldsymbol{\Gamma}\boldsymbol{\Lambda}_{r}^{H}\boldsymbol{\Lambda}_{dl}^{H}\right]\) can be expressed as \(Tr\left[\boldsymbol{\Lambda}_{B_{i}}^{-1}\boldsymbol{\Lambda}_{dl}\boldsymbol{ \Lambda}_{r}\boldsymbol{\Gamma}\boldsymbol{\Lambda}_{r}^{H}\boldsymbol{\Lambda}_{dl}^{H}\right]\). Then, Theorem 2 is proved. By using Theorem 2, Equation (23) can be reformulated as  
\[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]1 (25a) s.t. \[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]2 \[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]3 (25c) \[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]4  
where \(\mathbf{\Lambda}_{\Gamma_{G}}=\mathbf{\Lambda}_{B_{i}}^{-1}\mathbf{\Lambda}_{dl}\mathbf{\Lambda }_{r}\left(\mathbf{\Gamma}+\mathbf{\Lambda}_{B_{n}}\sigma_{T}^{2}\text{I}_{N_{R}} \right)\mathbf{\Lambda}_{r}^{H}\mathbf{\Lambda}_{dl}^{H}\).