## Appendix D  
**Proof. ** Now, defining \(\sqrt{\widehat{B^{(1)}_{1}}}=\frac{\eta}{N_{1}}\rho_{S_{1}D_{2}}d^{-\alpha/2}_ {S_{1}}d^{-\alpha/2}_{I_{1}D_{2}}\sum\limits_{n=1}^{N_{1}}\hat{h}_{S_{1},n} \hat{h}_{I_{1}D_{2},n}\) and \(\sqrt{\widehat{B^{(1)}_{2}}}=\frac{\eta}{N_{2}}\rho_{S_{1}D_{2}}\),  
\[d^{-\alpha/2}_{S_{2}}d^{-\alpha/2}_{I_{2}D_{2}}\sum\limits_{n=1}^{N_{2}}\hat {h}_{S_{2},n}\hat{h}_{I_{2}D_{2},n}\] (A17)  
\[\rho_{S_{1}D_{2}}=\sqrt{\frac{K_{S_{1}}K_{1}D_{2}}{(K_{S_{1}}+1)(K_{I_{2}}D_{ 2}+1)}}.\] (A18)  
Moreover, \(\widehat{B^{(1)}_{1}}\) and \(\widehat{B^{(1)}_{2}}\) have constant mean values as [36]  
\[\mathrm{E}\left[\widehat{B^{(1)}_{1}}\right]=\frac{\eta^{2}\rho_{S_{1}D_{2}}^ {2}}{d^{\alpha}_{S_{1}}d^{\alpha}_{I_{2}D_{2}}}.\] (A19)  
The expected value of \(\gamma^{(1)}_{x_{1},D_{2}}\) can be derived as  
\[\mathrm{E}\left[\widehat{\gamma^{(1)}_{x_{1},D_{2}}}\right]= \mathrm{E}\left[\frac{\gamma\chi_{1}\left(N_{1}\sqrt{\widehat{B^{(1)}_{1}}}+ N_{2}\sqrt{\widehat{B^{(1)}_{2}}}\right)^{2}}{\gamma\chi_{2}\left(N_{1}\sqrt{ \widehat{B^{(1)}_{1}}}+N_{2}\sqrt{\widehat{B^{(1)}_{2}}}\right)^{2}+\left(Y^{ 2}_{5}+Y^{2}_{D_{2}}\right)\gamma\left(N_{1}\sqrt{\widehat{B^{(1)}_{1}}}+N_{ 2}\sqrt{\widehat{B^{(1)}_{2}}}\right)^{2}+1}\right]\] (A20) \[=\frac{\gamma\chi_{1}\left(N_{1}^{2}\mathrm{E}\left[\widehat{B^{ (1)}_{1}}\right]+2N_{1}N_{2}\mathrm{E}\left[\sqrt{\widehat{B^{(1)}_{1}}}\sqrt {\widehat{B^{(1)}_{2}}}\right]+N^{2}_{2}\mathrm{E}\left[\widehat{B^{(1)}_{2} }\right]\right)}{\gamma\chi_{2}\left(N_{1}^{2}\mathrm{E}\left[\widehat{B^{(1) }_{1}}\right]+2N_{1}N_{2}\mathrm{E}\left[\sqrt{\widehat{B^{(1)}_{1}}}\sqrt {\widehat{B^{(1)}_{2}}}\right]+N^{2}_{2}\mathrm{E}\left[\widehat{B^{(1)}_{2} }\right]\right)+\left(Y^{2}_{5}+Y^{2}_{D_{2}}\right)\gamma\left(N_{1}^{2} \mathrm{E}\left[\widehat{B^{(1)}_{1}}\right]+2N_{1}N_{2}\mathrm{E}\left[ \sqrt{\widehat{B^{(1)}_{1}}}\sqrt{\widehat{B^{(1)}_{2}}}\right]+N^{2}_{2} \mathrm{E}\left[\widehat{B^{(1)}_{2}}\right]\right)+1}.\]  
We let \(\Xi_{4}=N_{1}^{2}\mathrm{E}\left[\widehat{B^{(1)}_{1}}\right]+2N_{1}N_{2} \mathrm{E}\left[\sqrt{\widehat{B^{(1)}_{1}}}\sqrt{\widehat{B^{(1)}_{1}}}\right] +N_{2}^{2}\mathrm{E}\left[\widehat{B^{(1)}_{2}}\right]\). From (A17) and (A18), \(\Xi_{4}\) can be calculated as  
\[\Xi_{4}=N_{1}^{2}\frac{\eta^{2}\rho_{S_{1}D_{2}}^{2}}{\frac{d^{ \alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}}}+2N_{1}N_{2}\sqrt{\frac{ \eta^{2}\rho_{S_{1}D_{2}}^{2}}{d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{\alpha}_ {S_{2}}}}\sqrt{\frac{\eta^{2}\rho_{S_{1}D_{2}}^{2}}{d^{\alpha}_{S_{1}}d^{\alpha }_{S_{2}}}}+N_{2}^{2}\frac{\eta^{2}\rho_{S_{1}D_{2}}^{2}}{\frac{d^{\alpha}_{S_{ 1}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}}}\] (A21) \[=\frac{\eta^{2}N_{1}^{2}\rho_{S_{1}D_{2}}^{2}}{d^{\alpha}_{S_{1}} d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{\alpha}_{S_{ 2}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}}+\frac{\eta^{2}N_{1} ^{2}\rho_{S_{1}D_{2}}^{2}}{d^{\alpha}_{S_{1}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{ 2}}}\] \[=\omega_{1}+2\omega_{2}+\omega_{3},\]  
where \(\omega_{1}=\frac{\eta^{2}N_{1}^{2}\rho_{S_{1}D_{2}}^{2}}{\frac{d^{\alpha}_{S_{1}} d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}D_{2}}{d^{\alpha}_{S_{1}}d^{\alpha}_{S_{1}}d^{ \alpha}_{S_{2}}}}\), \(\omega_{2}=\frac{\eta\eta N_{1}N_{2}\rho_{S_{1}D_{2}}\rho_{S_{1}D_{2}}}{d^{\alpha}_{S_{1} }d^{\alpha}_{S_{1}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}}\), \(\omega_{3}=\frac{\eta^{2}N_{1}^{2}\rho_{S_{1}D_{2}}^{2}}{d^{\alpha}_{S_{1}}d^{ \alpha}_{S_{2}}d^{\alpha}_{S_{2}}d^{\alpha}_{S_{2}}}\). From (A20) and (A20), \(\mathrm{E}\left[\widehat{\gamma^{(1)}_{x_{1},D_{2}}}\right]\) is formulated by  
\[\mathrm{E}\left[\widehat{\gamma^{(1)}_{x_{1},D_{2}}}\right]\leq\frac{\gamma\chi_{1} (\omega_{1}+2\omega_{2}+\omega_{3})}{\gamma\chi_{2}(\omega_{1}+2\omega_{2}+ \omega_{3})+\left(Y^{2}_{5}+Y^{2}_{D_{2}}\right)\tau(\omega_{1}+2\omega_{2}+ \omega_{3})+1}.\] (A22)Similar to (A19), the expected value of \(\gamma^{(1)}_{\text{T}_{\text{2}}\text{D}_{2}}\) can be derived as  
\[