## 3 Preliminaries and System Model
### Analytical Modeling  
In order to meet the power constraint requirements, the transmissions from the secondary source and the relay network must satisfy the following condition:  
\[|I_{(SS\to PR)}|^{2}P_{SS}+|I_{TRj\to PR}|^{2}P_{TRj}\leq Q_{th}, \tag{2}\]  
where \(Q_{th}\) is the maximum allowable interference power at the primary receiver. Similarly, the acceptable interference at the receiving relay of interest is:  
\[|g_{P\to R_{i}}|^{2}P_{p}+|g_{R\to D}|^{2}P_{TRj}\leq\gamma_{ir}, \tag{3}\]  
where \(|g_{S\to Ri}|^{2}\) is the channel gain for the source-to-relay link and \(|g_{R\to Dj}|^{2}\) is the channel gain for the relay-to-destination link. \(P_{ss}\) is the secondary source transmit power, and \(P_{TRj}\) is the transmitting relay transmit power, with the constraint imposed on these power levels. Furthermore, we assume that the source and the relay have rate-adaptation capabilities and can exploit the CSI to their advantage. Similarly, the interference channel gains from the secondary source and the \(j^{\text{th}}\) transmitting relay to the primary receiver are \(|I_{SS\to PR}|^{2}\) and \(|I_{TRj\to PR}|^{2}\), respectively; while \(|I_{TRj\to SR}|^{2}\) is the interference caused by the transmitting relay to the receiving relay. The primary transmitter will also cause interference to the secondary network; this interference is represented by channel gains \(|I_{P\to R_{i}}|^{2}\) and \(|I_{P\to D}|^{2}\) (secondary network receiving relay and secondary network destination, respectively). As we are considering virtual duplex relaying, therefore, in the secondary system, \((S\to R)\) and \((R\to D)\) transmissions are performed simultaneously by using the best pair of receiving and transmitting relays. For a given selected relay pair \((i,j)\), the received signal at the receiving relay \(i\) and at the destination (\(D\)) is expressed in Equations (4) and (5), respectively. \[y_{R}=|g_{S\to R_{i}}|^{2}P_{ss}+|I_{P\to D}|^{2}P_{PT}+|I_{TR_{j}\to SR}|^{2} P_{TR_{j}}+n, \tag{4}\]  
Figure 8: System model. \[y_{D}=|g_{R\to D}|^{2}P_{TR_{j}}+|I_{PD}|^{2}P_{PT}+n, \tag{5}\]  
where \(n\) is the additive white Gaussian Noise (AWGN) with zero mean and unit variance and \(P_{SS}\), \(P_{TRj}\), and \(P_{PT}\) are the transmitter powers of the secondary source, the transmitting relay and the primary source, respectively. As we assume that direct communication between the secondary source and secondary destination is not possible, so in Equation (4), the interference by secondary transmission is not considered. For error-free decoding at the receiving relay, the \(|g_{S\to R_{i}}|^{2}P_{ss}\) will be greater than \(|I_{P\to D}|^{2}P_{PT}+|I_{TRj\to SR}|^{2}P_{TRj}+n\). The ratio of these two is termed as SINR and is represented by \(\gamma_{RR}\), and it should be greater than or equal to \(\gamma_{th}\), which is the SINR required at the receiving relay for error-free decoding at the required data rate \(R\). Similarly, at the secondary destination, \(|g_{R\to D}|^{2}P_{TRj}\) should be greater than \(|I_{P\to D}|^{2}P_{PT}\), and we again define the ratio as \(\gamma_{SD}\), which should be greater than \(\gamma_{th}\). Thus, \(\gamma_{th}\) is the minimum SINR required at the receiving relay, as well as the secondary destination to decode the message at the required data rate. The secondary users cannot control the primary transmitter power, but must curtail its own transmit power to ensure that it remains below the threshold set by the primary network. This is ensured by applying \(|I_{S\to R}|^{2}P_{SS}+|I_{R\to D}|^{2}P_{TRj}\leq Q_{th}\). Here, we need to find the maximum power permitted, both at the secondary source and the relay, as well as the SINR required for the error-free decoding of information at the prescribed data rate for the secondary network.