## 3 Scheme Design
### Low-Complexity Diagonalizing Design
#### 3.2.1 Channel Parallelization  
Substituting Equation (12) into Equation (10) and employing  
\[\mathbf{E}^{-1}-\left(\mathbf{F}^{-1}\mathbf{E}+\mathbf{I}\right)^{-1} \mathbf{E}^{-1}=\left(\mathbf{E}+\mathbf{F}\right)^{-1}, \tag{17}\]  
where \(\mathbf{E}\) and \(\mathbf{F}\) are arbitrary matrices and \(\mathbf{I}\) is the identity matrix, the function \(\mathcal{J}_{i}\) can be simplified as  
\[\mathcal{J}_{i}=Tr\left[\left(\mathbf{I}_{M_{T}}+\mathbf{J}_{i_{R}}^{H} \mathbf{Z}_{1i}^{-1}\mathbf{J}_{i_{R}}\right)^{-1}\right], \tag{18}\]  
where \(\mathbf{Z}_{1i}=\sigma_{\mathbf{r}}^{2}\mathbf{J}_{i_{A}}\mathbf{J}_{i_{A}}^ {H}+\beta_{i}\sigma_{\mathbf{S}_{i}}^{2}\mathbf{I}_{M_{R}}\). Applying GSVD on the uplink channel matrix pair \(\left\{\mathbf{H}_{\mathbf{S}_{1}R}^{H},\mathbf{H}_{\mathbf{S}_{2}R}^{H}\right\}\) and SVD on the downlink channels \(\mathbf{H}_{dl}=\left[\mathbf{H}_{RS_{1}}^{T},\mathbf{H}_{RS_{2}}^{T}\right]^{T}\), we can obtain  
\[\mathbf{H}_{\mathbf{S}_{1}R}=\mathbf{R}_{h}\mathbf{\Sigma}_{h_{1}}\mathbf{U}_{ h_{1}}^{H},\quad\mathbf{H}_{\mathbf{S}_{2}R}=\mathbf{R}_{h}\mathbf{\Sigma}_{h_{2}} \mathbf{U}_{h_{2}}^{H}, \tag{19}\]  
\[\mathbf{H}_{RS_{1}}=\mathbf{R}_{dl1}\mathbf{\Sigma}_{dl}\mathbf{U}_{dl}^{H}, \quad\mathbf{H}_{RS_{2}}=\mathbf{R}_{dl2}\mathbf{\Sigma}_{dl}\mathbf{U}_{dl}^{ H}, \tag{20}\]where \(\mathbf{R}_{h}\in\mathbb{C}^{N\times N}\), \(\mathbf{U}_{h_{i}}\in\mathbb{C}^{M\times M}\), \(\mathbf{R}_{dl}\in\mathbb{C}^{2M\times 2M}\), and \(\mathbf{U}_{dl}\in\mathbb{C}^{N\times N}\) are four unitary matrices; \(\boldsymbol{\Sigma}_{h_{1}}=\left(\mathbf{0}_{(N-M)\times M^{\prime}}^{T} \boldsymbol{\Lambda}_{h_{1}}^{T}\right)^{T}\); \(\boldsymbol{\Sigma}_{h_{2}}=\left(\boldsymbol{\Lambda}_{h_{2}}^{T},\mathbf{0 }_{(N-M)\times M}^{T}\right)^{T}\); \(\boldsymbol{\Sigma}_{dl}=\left(\boldsymbol{\Lambda}_{dl}^{T},\mathbf{0}_{(2M- N)\times N}^{T}\right)^{T}\); \(\boldsymbol{\Lambda}_{h_{i}}\) and \(\boldsymbol{\Lambda}_{dl}\) are \(M\times M\) and \(N\times N\) nonnegative diagonal matrices; \(\mathbf{R}_{dl1}=\mathbf{R}_{dl}\)\((1:M,1:N)\); \(\mathbf{R}_{dl2}=\mathbf{R}_{dl}\)\((M+1:N,1:N)\) for \(N>M\); and \(\mathbf{R}_{dl1}=\mathbf{\tilde{R}}_{dl1}=\mathbf{R}_{dl}(1:M,1:M)\), \(\mathbf{R}_{dl2}=\mathbf{\tilde{R}}_{dl2}=\mathbf{R}_{dl}(M+1:2M,1:M)\) for \(N=M\). In order to parallelize the channels in Equations (19) and (20), the relay and source beamformers \(\mathbf{F}_{r}\) and \(\mathbf{F}_{i}\) can be proposed as  
\[\mathbf{F}_{r}=\mathbf{U}_{dl}\boldsymbol{\Lambda}_{r}\mathbf{R}_{h}^{-1}, \quad\mathbf{F}_{i}=\mathbf{U}_{h_{i}}\boldsymbol{\Lambda}_{i}\mathbf{V}_{ \mathbf{F}_{i}}^{H}, \tag{21}\]  
where \(\boldsymbol{\Lambda}_{r}\) and \(\boldsymbol{\Lambda}_{i}\) are \(N\times N\) and \(M\times M\) nonnegative diagonal matrices, respectively. Substituting Equations (19)-(21) into Equation (18), the resultant objective function \(\mathcal{J}_{i}^{*}\) becomes  
\[\mathcal{J}_{i}^{*}= Tr\left[\left(\mathbf{I}_{M_{T}}+\beta_{i}\boldsymbol{ \Delta}_{i}^{H}\boldsymbol{\Sigma}_{h_{l}}^{H}\boldsymbol{\Lambda}_{r}^{H} \boldsymbol{\Lambda}_{dl}^{H}\left(\sigma_{r}^{2}\beta_{i}\boldsymbol{\Lambda }_{dl}\boldsymbol{\Lambda}_{r}\boldsymbol{\Lambda}_{B_{h}}\boldsymbol{\Lambda}_ {r}^{H}\boldsymbol{\Lambda}_{dl}^{H}+\boldsymbol{\Lambda}_{B_{i}}\beta_{i} \sigma_{S_{i}}^{2}\mathbf{I}_{M_{R}}\right)^{-1}\boldsymbol{\Lambda}_{dl} \boldsymbol{\Lambda}_{r}\boldsymbol{\Sigma}_{h_{l}}\boldsymbol{\Lambda}_{l} \right)^{-1}\right], \tag{22}\]  
where \(\boldsymbol{\Lambda}_{B_{h}}\) and \(\boldsymbol{\Lambda}_{B_{i}}\) are two diagonal matrices containing the \((k,k)\)th entries of \(\mathbf{B}_{h}\) and \(\mathbf{B}_{i}\), \(\mathbf{B}_{h}=\left(\mathbf{R}_{h}\mathbf{R}_{h}^{H}\right)^{-1}\), and \(\mathbf{B}_{i}=\left(\mathbf{R}_{dili}^{H}\mathbf{R}_{dli}\right)^{-1}\) for \(N>M\), \(\mathbf{B}_{i}=\left(\mathbf{R}_{dili}^{H}\mathbf{R}_{dli}\right)^{-1}\) for \(N=M\).