## 2 System Model
Among these relays, we focused on the \(M\) nodes which are nearer to \(SS_{1}\) than \(SS_{2}\)._  
Because all the A-B channels are Rayleigh fading, the probability density function (PDF) and cumulative distribution function (CDF) of \(g_{\rm AB}\) can be written, respectively, as  
\[f_{g_{\rm AB}}(x)=\Omega_{\rm AB}\exp(-\Omega_{\rm AB}x),F_{g_{\rm AB}}(x)=1- \exp(-\Omega_{\rm AB}x), \tag{1}\]  
where \(\Omega_{\rm AB}=(d_{\rm AB})^{\mu}\) and \(\mu\) (\(2\leq\mu\leq 8\)) denotes a path-loss exponential. For the ease of presentation and analysis, we assume that the random variables (RVs) \(g_{\rm PT_{u}B}\) (\(g_{\rm APR_{r}}\), \(g_{\rm ASR_{m}}\) and \(g_{\rm SR_{m}B}\)) are independent and identical. Therefore, we can use the following notations:  
\[\Omega_{\rm PT_{u}B}\stackrel{{\Delta}}{{=}}\Omega_{\rm PTB}, \ \Omega_{\rm APR_{r}}\stackrel{{\Delta}}{{=}}\Omega_{\rm APR},\ \Omega_{\rm ASR_{m}}\stackrel{{\Delta}}{{=}}\Omega_{\rm ASR},\ \Omega_{\rm SR_{m}B}\stackrel{{\Delta}}{{=}}\Omega_{\rm SRB},\ \forall u,v,m. \tag{2}\]  
**Remark 2**. : _This paper considers the block Rayleigh fading channel, where \(g_{AB}\) remains unchanged during one data exchange cycle, but changes independently after each cycle. As a result, we have \(g_{AB}\equiv g_{BA}\) for all the \(A\) and \(B\) nodes._  
Since \(d_{\rm SS_{1}SR}\leq d_{\rm SS_{2}SR}\), we hence apply PRS for the second hop between \(\rm SS_{2}\) and \(\rm SR_{m}\) as (see [31])  
\[\rm SR_{b}:g_{\rm SS_{2}SR_{b}}=\max_{m=1,2,\ldots,M}\bigl{(}g_{\rm SS_{2}SR_{ m}}\bigr{)}. \tag{3}\]  
Because \(\rm SS_{2}\) and \(\rm SR_{m}\) are in radio range of one another, we can assume that the CSI of the \(\rm SR_{m}-SS_{2}\) links are available at \(\rm SS_{2}\). Therefore, \(\rm SS_{2}\) can select the best relay \(\rm SR_{m}\) as presented in (3). From (3), and using CDF in (1), we obtain CDF of \(g_{\rm SS_{2}SR_{b}}\) as  
\[F_{g_{\rm SS_{2}SR_{b}}}(x) =\Pr\Bigl{(}\max_{m=1,2,\ldots,M}\bigl{(}g_{\rm SS_{2}SR_{m}} \bigr{)}<x\Bigr{)}=\Bigl{[}F_{g_{\rm SS_{2}SR_{m}}}(x)\Bigr{]}^{M}\] \[=1+\sum_{p=1}^{M}\,(-1)^{p}C_{M}^{p}\exp\Bigl{(}-p\Omega_{g_{\rm SS _{2}SR}}x\Bigr{)}, \tag{4}\]  
where \(C_{M}^{p}\) denotes a binomial coefficient, i.e., \(C_{M}^{p}=\frac{M!}{p!(M-p)!}\). Then, the corresponding PDF of \(g_{\rm SS_{2}SR_{b}}\) can be expressed as  
\[f_{g_{\rm SS_{2}SR_{b}}}(x) =M\Omega_{g_{\rm SS_{2}SR}}\exp\Bigl{(}-\Omega_{g_{\rm SS_{2}SR}} x\Bigr{)}\Bigl{[}1-\exp\Bigl{(}-\Omega_{g_{\rm SS_{2}SR}}x\Bigr{)}\Bigr{]}^{M-1}\] \[=\sum_{p=0}^{M-1}\,(-1)^{p}C_{M-1}^{p}M\Omega_{g_{\rm SS_{2}SR}} \exp\Bigl{(}-(p+1)\Omega_{g_{\rm SS_{2}SR}}x\Bigr{)}. \tag{5}\]  
We now describe the operation principle of the proposed scheme. At the first time slot, \(\rm SS_{1}\) and \(\rm SS_{2}\), respectively, send \(x_{1}\) and \(x_{2}\) to \(\rm SR_{b}\), and PT uses the \(t\)-th antenna to send its data \((x_{\rm P})\) to PR. The received signals at the \(r\)-th antenna of PR and at \(\rm SR_{b}\), under the impact of the cross interference, can be given, respectively, as  
\[y_{\rm PR_{r}}=\sqrt{P_{\rm PT}}h_{\rm PT_{r}PR_{r}}x_{\rm P}+\sqrt{P_{\rm SS _{1}}}h_{\rm SS_{1}PR_{r}}x_{1}+\sqrt{P_{\rm SS_{2}}}h_{\rm SS_{2}PR_{r}}x_{2} +n_{\rm PR_{r}}, \tag{6}\]\[y_{\text{SR}_{b}}=\sqrt{P_{\text{SS}_{1}}}h_{\text{SS}_{1}\text{SR}_{b}}x_{1}+ \sqrt{P_{\text{SS}_{2}}}h_{\text{SS}_{2}\text{SR}_{b}}x_{2}+\sqrt{P_{\text{PT}}}h _{\text{PT}_{i}\text{SR}_{b}}x_{\text{P}}+n_{\text{SR}_{b}}. \tag{7}\]  
In (6) and (7), \(P_{\text{A}}\) is the transmit power of transmitter A, where \(\text{A}\in\{\text{PT},\text{SS}_{1},\text{SS}_{2}\}\), \(n_{\text{PR}}\), and \(n_{\text{SR}_{b}}\) denote Gaussian noises at PR, and SR\({}_{b}\), respectively. For the ease of presentation and analysis, we can assume that the Gaussian noises at all receivers B have zero mean and unit variance. In addition, we transmit the power \(P_{\text{SS}_{1}}\) and \(P_{\text{SS}_{2}}\), which will be derived by closed-form expressions in the next section.