## Appendix A Outage probability can be calculated as  
\[P_{out}=\Pr(SNR<z)=\Pr\!\left(\frac{\eta\rho(1-\rho)\psi|h_{sr}|^{2}|h_{rd}|^{ 2}}{\eta\rho|h_{rd}|^{2}d_{sr}^{m}+(1-\rho)d_{i}^{m}}<z\right) \tag{1}\]  
where \(z=2^{2R}-1\) is a threshold, and R is the source rate. Equation (1) can be rewritten as  
\[ \begin{array}{rl}P_{out}&=\Pr\!\left[\eta\rho(1-\rho)\psi|h_{sr}|^{2}|h_{rd} |^{2}<z\eta\rho|h_{rd}|^{2}d_{sr}^{m}+z(1-\rho)d_{i}^{m}\right]\\ &=\Pr\!\left[|h_{sr}|^{2}<\frac{\gamma_{0}|h_{rd}|^{2}d_{sr}^{m}+z(1-\rho)d_{ i}^{m}}{\eta\rho(1-\rho)\psi|h_{rd}|^{2}}\right]\\ &=\Pr\!\left[|h_{sr}|^{2}<\frac{\gamma_{0}^{m}}{\eta\rho\psi|h_{rd}|^{2}}+ \frac{\gamma_{0}^{m}}{(1-\rho)\psi}\right]\\ \end{array}  \tag{2}\]  
\[P_{out}=\int\limits_{0}^{\infty}F_{|h_{rd}|^{2}}\Bigg{\{}\frac{zd_{i}^{m}}{ \eta\rho\psi|h_{rd}|^{2}}+\frac{zd_{sr}^{m}}{(1-\rho)\psi}\!\left|\left|h_{rd }\right|^{2}\right\}\!f_{|h_{rd}|^{2}}(|h_{rd}|^{2})d\!\left(|h_{rd}|^{2}\right). \tag{3}\]  
Figure 7: The outage probability (**a**) and ergodic capacity (**b**) versus the numbers of users. Since \(\rho^{*}=\frac{1}{1-|h_{rd}|}\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}\) results in a value of \(\rho^{*}>1\) or \(\rho^{*}<0\), we choose \(\rho^{*}=\frac{1}{1+|h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}}\) as the solution. If Equation (A3) is placed into Equation (A1), \(SNR_{max}\) can be obtained as  
\[SNR_{\max}=\frac{\eta\psi|h_{sr}|^{2}|h_{rd}|^{2}}{\left(1+|h_{rd}|\sqrt{\frac{ \eta\rho_{rd}^{m}}{d_{i}^{m}}}\right)^{2}d_{i}^{m}}.\] (A10)  
EC analysis can be demonstrated as follows:  
\[C_{AF}=\int\limits_{0}^{\infty}f_{SNR_{\max}}(z)\log_{z}(1+z)dz=\frac{1}{\ln 2 }\int\limits_{0}^{\infty}\frac{1-F_{SNR_{\max}}(z)}{1+z}dz\] (A11)  
\[F_{SNR_{\max}}(z)=\Pr(SNR_{\max}<z)=\Pr\left[\frac{\eta\psi|h_{rd}|^{2}|h_{rd }|^{2}}{\left(1+|h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}\right)^{2} d_{i}^{m}}<z\right]\] (A12)  
\[F_{SNR_{\max}}(z)=\int\limits_{0}^{\infty}F_{|h_{rd}|^{2}}\left[\frac{\left(1+ |h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}\right)^{2}zd_{i}^{m}}{\eta \psi|h_{rd}|^{2}}\right]^{2}|h_{rd}|^{2}\right]^{2}|f_{|h_{rd}|^{2}}\left(|h_{rd }|^{2}\right)d\left(|h_{rd}|^{2}\right).\] (A13)  
As in the previous section, we have  
\[F_{SNR_{\max}}(z)=1-\frac{1}{\lambda_{rd}}\int\limits_{0}^{\infty}\exp[-\frac {\left(1+|h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}\right)^{2}zd_{i}^{ m}}{\Omega_{|h_{rd}|^{2}}\eta\psi|h_{rd}|^{2}}]^{m_{|h_{rd}|^{2}}}\sum\limits_{t=0}^{ \infty}\frac{1}{t^{\left(\Omega_{|h_{rd}|^{2}}\right)^{t}}}\times\left[\frac {\left(1+|h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i}^{m}}}\right)^{2}zd_{i}^{ m}}{\eta\psi|h_{rd}|^{2}}\right]^{t}\] (A14)  
In Equation (A14), we consider  
\[A =\exp[-\frac{\left(1+|h_{rd}|\sqrt{\frac{\eta\rho_{rd}^{m}}{d_{i} ^{m}}}\right)^{2}zd_{i}^{m}}{\Omega_{|h_{rd}|^{2}}\eta\psi|h_{rd}|^{2}}]=\exp \left[-\frac{zd_{i}^{m}+2z|h_{rd}|\sqrt{\eta\rho_{rd}^{m}}d_{i}^{m}+2|h_{rd}|^ {2}\eta d_{nr}^{m}}{\Omega_{|h_{rd}|^{2}}\eta\psi|h_{rd}|^{2}}\right]\] (A15) \[=\exp[-\frac{zd_{i}^{m}}{\Omega_{|h_{rd}|^{2}}\eta\psi|h_{rd}|^{2 }}]\exp[-\frac{2z\sqrt{d_{i}^{m}d_{i}^{m}}}{\Omega_{|h_{rd}|^{2}}\psi|h_{rd}| \sqrt{\eta}}]\exp[-\frac{zd_{nr}^{m}}{\Omega_{|h_{rd}|^{2}}\psi}]\]  
We then apply a Taylor series as follows:  
\[\exp[-\frac{2z\sqrt{d_{sr}^{m}d_{i}^{m}}}{\Omega_{|h_{rd}|^{2}}\psi|h_{rd}| \sqrt{\eta}}]=\sum\limits_{q=0}^{\infty}\frac{\left(-\frac{2z\sqrt{d_{sr}^{m}d _{i}^{m}}}{\Omega_{|h_{rd}|^{2}}\psi|h_{rd}|\sqrt{\eta}}\right)^{q}}{q!}=\sum \limits_{q=0}^{\infty}\frac{(-1)^{q}\left(\frac{2\sqrt{d_{sr}^{m}d_{i}^{m}}}{ \Omega_{|h_{rd}|^{2}}\psi\sqrt{\eta}}\right)^{q}}{q!}|h_{rd}|^{-q}.\]Equation (A15) can then be rewritten as  
Figure 7:0 (A16)  
If Equation (A14) is placed into Equation (A16) and the expression \((x+y)^{m}=\sum\limits_{n=0}^{m}\bigg{(} \begin{array}{c}m\\ n\end{array} \bigg{)}x^{m-n}y^{n}\) is applied, we have  
\[