## 2 System Model  
Figure 1 presents the system model of the proposed CR WSNs. In the secondary network, a source S communicates with a destination D in the dual-hop fashion. In addition, there are \(M\) secondary relays (denoted by \(\mathrm{R}_{1}\), \(\mathrm{R}_{2},...,\mathrm{R}_{M}\)), and one of them is selected to serve the source-destination communication. In the primary network, there are \(N\) licensed users (or primary users), denoted as \(\mathrm{P}_{1}\), \(\mathrm{P}_{2},...,\mathrm{P}_{N}\). To support dynamic spectrum access in a strict manner, the secondary transmitters must adjust their transmit power so that the interferences generated by their operations are not harmful to the quality of service (QoS) of the primary users. It is assumed that the source and relays are single-antenna and energy-constrained devices that have to harvest energy from a \(K\)-antenna power beacon (B) deployed in the secondary network. Due to deep shadow fading or far distance, the direct link between S and D does not exist, and the data transmission is realized by two orthogonal time slots via the selected relay. Denote \(\gamma_{\mathrm{SR}_{m}}=\left|h_{\mathrm{SR}_{m}}\right|^{2}\) and \(\gamma_{\mathrm{R}_{m}\mathrm{D}}=\left|h_{\mathrm{R}_{m}\mathrm{D}}\right|^{2}\) as the channel gains of the \(\mathrm{S}\rightarrow\mathrm{R}_{m}\) and \(\mathrm{R}_{m}\rightarrow\mathrm{D}\) links, respectively, where \(m=1\), 2,..., \(M\). For the interference links, \(\gamma_{\mathrm{SR}_{m}}\) and \(\gamma_{\mathrm{R}_{m}\mathrm{P}_{n}}\) denote the channel  
Figure 1: System model of PB-assisted relaying protocols in underlay CR with relay selection methods. gains of the \(\mathrm{S}\rightarrow\mathrm{P}_{n}\) and \(\mathrm{R}_{m}\rightarrow\mathrm{P}_{n}\) links, where \(n=1,2,\,...,\,N\). Next, the channel gains between the \(k\)-th antenna of the beacon and the source and relay \(\mathrm{R}_{m}\) are \(\gamma_{\mathrm{B}_{\mathrm{B}}\mathrm{S}}\) and \(\gamma_{\mathrm{B}_{\mathrm{B}}\mathrm{R}_{m}}\), respectively, where \(k=1,\,2,\,...,\,K\). Assume that all of the channels experience Rayleigh fading, and hence the channel gains have exponential distributions. Denote \(\lambda_{\mathrm{XY}}\) as a parameter of the random variable (RV) \(\gamma_{\mathrm{XY}}\), which is given as \(\lambda_{\mathrm{XY}}=1/E\left\{\gamma_{\mathrm{XY}}\right\}\), where \(\left(\mathrm{X},\mathrm{Y}\right)\in\left\{\mathrm{S},\mathrm{R}_{m},\mathrm{ D},\mathrm{B}_{k},\mathrm{P}_{n}\right\}\), and \(E\left\{Z\right\}\) is the expected value of a RV \(Z\). Therefore, the cumulative distribution function (CDF) and probability density function (PDF) of the RV \(\gamma_{\mathrm{XY}}\) can be expressed, respectively, as  
\[F_{\gamma_{\mathrm{XY}}}\left(x\right)=1-\exp\left(-\lambda_{\mathrm{XY}}x \right),\,\,f_{\gamma_{\mathrm{XY}}}\left(x\right)=\lambda_{\mathrm{XY}}\exp \left(-\lambda_{\mathrm{XY}}x\right). \tag{1}\]  
To take path-loss into account, we can model these parameters as in [30]:  
\[\lambda_{\mathrm{XY}}=d_{\mathrm{XY}}^{\beta}, \tag{2}\]  
where \(\beta\) is the path-loss exponent, and \(d_{\mathrm{XY}}\) is the link distance between the nodes \(\mathrm{X}\) and \(\mathrm{Y}\). Assume that the relays (and primary uses) are close together and form a cluster. Hence, \(d_{\mathrm{SR}_{m}}=d_{\mathrm{SR}}\), \(d_{\mathrm{SP}_{n}}=d_{\mathrm{SP}}\) and \(d_{\mathrm{R}_{m}\mathrm{P}_{n}}=d_{\mathrm{RP}}\) can be assumed for all \(m\) and \(n\). Hence, \(\gamma_{\mathrm{SR}_{m}}\) (and \(\gamma_{\mathrm{R}_{m}\mathrm{D}}\), \(\gamma_{\mathrm{SP}_{n}}\), \(\gamma_{\mathrm{R}_{m}\mathrm{P}_{n}}\)) are i.i.d. RVs can be assumed, where \(\lambda_{\mathrm{SR}_{m}}=\lambda_{\mathrm{SR}}\), \(\lambda_{\mathrm{R}_{m}\mathrm{D}}=\lambda_{\mathrm{RD}}\), \(\lambda_{\mathrm{SP}_{n}}=\lambda_{\mathrm{SP}}\) and \(\lambda_{\mathrm{R}_{m}\mathrm{P}_{n}}=\lambda_{\mathrm{RP}}\) for all \(m\) and \(n\). Similarly, \(\gamma_{\mathrm{B}_{\mathrm{B}}\mathrm{S}}\) and \(\gamma_{\mathrm{B}_{\mathrm{R}_{m}}}\) are also assumed to be i.i.d.