## 3 Proposed Max-SNR-FP Scheme
### Optimize \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)  
First, we transform the power constraint in (12b) into a convex constraint with respect to \(\mathbf{v}\) as follows:  
\[P_{r}=\mathbf{v}^{H}\left(\rho_{sr}P\mathbf{H}_{sr}^{H}\boldsymbol{\
abla}^{H }\mathbf{\Psi H}_{sr}\right)\mathbf{v}+\text{Tr}\left(\sigma_{r}^{2}\mathbf{ \Psi}\mathbf{\Psi}^{H}\right)\leq P_{r}^{\text{max}}. \tag{13}\]  
Then, given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\), the optimal beamforming vector \(\mathbf{v}\) is found by addressing the following problem:  
\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\] \[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]  
where  
\[\mathbf{A}= (\sqrt{\rho_{sr}}\mathbf{h}_{rb}^{H}\mathbf{\Phi H}_{sr}+\sqrt{ \rho_{sr}}\mathbf{h}_{rb}^{H}\mathbf{\Psi H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_{ sb}^{H})^{H}\] \[(\sqrt{\rho_{sr}}\mathbf{h}_{rb}^{H}\mathbf{\Phi H}_{sr}+\sqrt{ \rho_{sb}}\mathbf{h}_{rb}^{H}\mathbf{\Psi H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_ {sb}^{H}). \tag{15}\]  
\[P_{r} =\text{Tr}\Big{(}\mathbf{\Psi}\Big{(}\rho_{sr}\text{Pr}\mathbf{\mathrm{H}}_{sr }\mathbf{\mathrm{v}}\mathbf{\mathrm{v}}^{H}\mathbf{\mathrm{H}}_{sr}^{H}+\sigma_{r}^{2}\mathbf{ \mathrm{I}}_{M}\Big{)}\mathbf{\Psi}^{H}\Big{)}\] \[=\mathbf{\psi}^{T}(\rho_{sr}P\text{diag}\{\mathbf{\mathrm{v}}^{H}\mathbf{ \mathrm{H}}_{sr}^{H}\}\text{diag}\{\mathbf{\mathrm{H}}_{sr}\mathbf{\mathrm{v}}\}+ \sigma_{r}^{2}\mathbf{\mathrm{I}}_{M})\mathbf{\psi}^{*}\] \[\leq P_{r}^{\max}. \tag{25}\]  
By neglecting the constant terms, the subproblem with respect to \(\mathbf{\Psi}\) is given by  
\[\max_{\mathbf{\Psi}} \frac{|(\sqrt{\rho_{sr}}\mathbf{\mathrm{h}}_{rb}^{H}\mathbf{\mathrm{v}} \mathbf{\mathrm{H}}_{sr}+\sqrt{\rho_{sr}}\mathbf{\mathrm{h}}_{rb}^{H}\mathbf{\mathrm{v}} \mathbf{\mathrm{H}}_{sr}+\sqrt{\rho_{sb}}\mathbf{\mathrm{h}}_{sb}^{H})\mathbf{\mathrm{v}} |^{2}}{\sigma_{r}^{2}|\sqrt{\rho_{rb}}\mathbf{\mathrm{h}}_{rb}^{H}\mathbf{\mathrm{v}} |^{2}+\sigma_{b}^{2}} \tag{26a}\] \[\text{s.t.}\quad\eqref{eq:P_r},\eqref{eq:P_r},\eqref{eq:P_r}. \tag{26b}\]  
Let us define  
\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]0  
Then, the objective function in (26) can be converted into  
\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]1  
Optimization Problem (26) became a nonlinear fractional optimization problem. On the basis of the FP strategy in [43], we introduce parameter \(\tau\) and transform Objective Function (28) as follows:  
\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]2 \[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]3  
The optimal solution can be achieved if and only if \(\mathbf{\psi}^{H}\mathbf{\mathrm{C}}\mathbf{\psi}+2\Re\{\mathbf{\psi}^{H}\sqrt{\rho_{sr}}\text {diag}\{\mathbf{\mathrm{h}}_{rb}^{H}\}\mathbf{\mathrm{H}}_{sr}\mathbf{\mathrm{v}}\mathbf{ \mathrm{D}}^{*}\}+|D|^{2}-\tau(\sigma_{r}^{2}\rho_{rb}|\mathbf{\psi}^{H}\text{diag} \{\mathbf{\mathrm{h}}_{rb}^{H}\}|^{2}+\sigma_{b}^{2})=0\). We linearize the \(\mathbf{\psi}^{H}\mathbf{\mathrm{C}}\mathbf{\psi}\) by employing the Taylor series expansion at a given vector \(\mathbf{\psi}\); the subproblem with respect to \(\mathbf{\mathrm{\
abla}}\) can be recast as follows:  
\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]4 \[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]5 \[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]6  
Problem (30) is convex and can be addressed effectively with the convex optimization tool. The whole procedure of the Max-SNR-FP scheme is described in Algorithm 1. ```
1:Initialize feasible solutions \(\mathbf{\mathrm{v}}^{(0)}\), \(\mathbf{\mathrm{\Phi}}^{(0)}\), and \(\mathbf{\mathrm{\
abla}}^{(0)}\), and calculate achievable rate \(R_{b}^{(0)}\) on the basis of (9). 2:Set iteration number \(k=0\), accuracy value \(\epsilon\).