## 3 Proposed Max-SNR-FP Scheme
### Optimize \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)
Let us define

\[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]8 \[\mathbf{h}_{b}=\left[ \begin{array}{c}\sqrt{\rho_{srb}P}\text{diag} \{\mathbf{h}_{rb}^{H}\}\mathbf{H}_{sr}\mathbf{v}\\ \sqrt{\rho_{sb}P}\mathbf{h}_{sb}^{H}\mathbf{v}\end{array} \right]_{(M+1)\times 1},\] (45) \[\mathbf{H}_{b}=\left[ \begin{array}{c}\sqrt{\rho_{rb}}\text{diag} \{\mathbf{h}_{rb}^{H}\mathbf{E}_{M_{a}}\}\\ \mathbf{0}_{1\times M}\end{array} \right]_{(M+1)\times M}. \tag{46}\]

Then, the SNR in (10) can be reformulated as follows:

\[\text{SNR} =\frac{P|(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{T}\mathbf{H }_{sr}+\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Phi}\mathbf{H}_{sr}+\sqrt{ \rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}|^{2}}{\sigma_{r}^{2}|\sqrt{\rho_{rb} }\mathbf{h}_{rb}^{H}\mathbf{v}|^{2}+\sigma_{b}^{2}}\] \[=\frac{P|(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Theta} \mathbf{H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}|^{2}}{\sigma_{ r}^{2}|\sqrt{\rho_{rb}}\mathbf{h}_{rb}^{H}\mathbf{E}_{M_{a}}\mathbf{\Theta}|^{2}+ \sigma_{b}^{2}}\] \[=\frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^ {2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}. \tag{47}\]

Accordingly, the power constraint in (25) can be reformulated as follows:

\[P_{r} =\text{Tr}\Big{(}\mathbf{\Psi}\Big{(}\rho_{sr}\text{Pr}_{sr}\mathbf{v }\mathbf{v}^{H}\mathbf{H}_{sr}^{H}+\sigma_{r}^{2}\mathbf{I}_{M}\Big{)}\mathbf{ \Psi}^{H}\Big{)}\] \[=\text{Tr}\Big{(}\mathbf{E}_{M_{a}}\mathbf{\Theta}\Big{(}\rho_{sr}P \text{Pr}_{sr}\mathbf{v}\mathbf{v}^{H}\mathbf{H}_{sr}^{H}+\sigma_{r}^{2} \mathbf{I}_{M}\Big{)}\mathbf{E}_{M_{a}}\mathbf{\Theta}^{H}\Big{)}\] \[=\mathbf{\theta}^{T}(\rho_{sr}P\text{E}_{M_{a}}\text{diag}\{\mathbf{v }^{H}\mathbf{H}_{sr}^{H}\}\text{diag}\{\mathbf{H}_{sr}\mathbf{v}\}\mathbf{E}_{ M_{a}}+\sigma_{r}^{2}\mathbf{E}_{M_{a}}\mathbf{E}_{M_{a}})\mathbf{\theta}^{*}\] \[=\widehat{\mathbf{\theta}}^{T}\text{blkdiag}\{\rho_{sr}P\text{E}_{M_{a }}\text{diag}\{\mathbf{v}^{H}\mathbf{H}_{sr}^{H}\}\text{diag}\{\mathbf{H}_{sr} \mathbf{v}\}\mathbf{E}_{M_{a}}+\sigma_{r}^{2}\mathbf{E}_{M_{a}}\mathbf{E}_{M_{ a}};0\}\widehat{\mathbf{\theta}}^{*}\] \[\leq P_{r}^{\text{max}}. \tag{48}\]

Then, Optimization Problem (12) regarding \(\mathbf{\Theta}\) can be recast as follows:

\[\underset{\widehat{\mathbf{\theta}}}{\text{max}} \frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^{ 2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}\] (49a) s.t. \[|\widehat{\mathbf{\theta}}(m)|=1,\text{if}\,m\
ot\in\Omega,\] (49b) \[=\frac{P|(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Theta} \mathbf{H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}|^{2}}{\sigma_{ r}^{2}|\sqrt{\rho_{rb}}\mathbf{h}_{rb}^{H}\mathbf{E}_{M_{a}}\mathbf{\Theta}|^{2}+ \sigma_{b}^{2}}\]0 (49c) \[=\frac{P|(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Theta} \mathbf{H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}|^{2}}{\sigma_{ r}^{2}|\sqrt{\rho_{rb}}\mathbf{h}_{rb}^{H}\mathbf{E}_{M_{a}}\mathbf{\Theta}|^{2}+ \sigma_{b}^{2}}\]1

Then, Problem (49) can be reformulated as follows:

\[=\frac{P|(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Theta} \mathbf{H}_{sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}|^{2}}{\sigma_{ r}^{2}|\sqrt{\rho_{rb}}\mathbf{h}_{rb}^{H}\mathbf{E}_{M_{a}}\mathbf{\Theta}|^{2}+ \sigma_{b}^{2}}\]2 (53a) s.t.