## 2 System Model and Problem Formulation
### Information and Energy Transfer  
Let the total time of the whole transmission phase be normalized to be 1; if the MA phase time period is \(t\), then the BC phase time period is \(1-t\). In the MA phase, the transmit power of each source node is \(P_{i}\). Due to the peak power constraint, \(P_{i}\) should satisfied the following equation:  
\[P_{i}\leq P_{i,\max},i=1,2 \tag{1}\]  
where \(P_{i,\max}\) is the peak power of source node \(S_{i}\). Since the cognitive network utilizes underlay spectrum sharing, the received interference power for the primary user should be less than an interference threshold \(Q\) to satisfy the primary performance, i.e.,  
\[\sum_{i=1}^{2}|l_{i}|^{2}P_{i}\leq Q,i=1,2 \tag{2}\]  
where \(l_{i}\) is the CSI from source \(S_{i}\) to PU, and \(|l_{i}|^{2}P_{i},i=1,2\) is the interference caused by spectrum sharing from source node \(S_{i}\) to PU. Let \(\alpha(0<\alpha<1)\) be the interference constraint ratio of two cognitive sources to the primary network. The restrictions of \(P_{1}\) and \(P_{2}\) can be reformulated as follows. \[P_{1} \leq\min\biggl{(}P_{1,\max}\frac{\alpha Q}{|l_{1}|^{2}}\biggr{)} \tag{3}\] \[P_{2} \leq\min\biggl{(}P_{2,\max}\frac{(1-\alpha)Q}{|l_{2}|^{2}}\biggr{)} \tag{4}\]  
The signal received at the cognitive relay is  
\[y_{R}=h_{1}\sqrt{P_{1}}\x_{1}+h_{2}\sqrt{P_{2}}\x_{2}+\tau_{u,r}+n_{r,a} \tag{5}\]  
where \(\tau_{u,r}\sim\mathcal{CN}(0,\sigma_{u,r}^{2})\) is the interference introduced by the primary network, \(h_{i}\) is the CSI between source node \(S_{i}\) and the relay \(R\), and \(n_{r,a}\sim\mathcal{CN}(0,\sigma_{a}^{2})\) is the white noise at the receiver. The cognitive relay then splits the received signal \(y_{R}\) into two parts: \(\sqrt{p}y_{R}\) for energy harvesting and \(\sqrt{1-\rho}y_{R}\) for information decoding. With linear energy harvesting, the harvested energy can be expressed as  
\[E=\eta\rho(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+\sigma_{u,r}^{2}+\sigma_{a}^{2})\cdot t \tag{6}\]  
where \(0<\eta<1\) is energy conversion efficiency. Because for practical cases the noise power is far less than the signal power, we neglect white noise for simplicity of analysis. Thus, Equation (6) is written as  
\[E=\eta\rho(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}+\sigma_{u,r}^{2})\cdot t \tag{7}\]  
The signal used to decode information is written as  
\[y_{ID}=\sqrt{1-\rho}(h_{1}\sqrt{P_{1}}\x_{1}+h_{2}\sqrt{P_{2}}\x_{2}+\tau_{u, r}+n_{r,a})+n_{r,b} \tag{8}\]  
where \(n_{r,b}\sim\mathcal{CN}(0,\sigma_{b}^{2})\) is the noise generated by signal conversion from band-pass to baseband [27]. Since \(\sigma_{a}^{2}\ll\sigma_{b}^{2}\), in practice, for simplicity, we neglect \(n_{r,a}\) in the following analysis. According to Equation (8) and [10], the rate region of the MA phase is obtained as  
\[C_{MA}=\left\{(R_{1},R_{2}): \begin{Bmatrix}R_{1}\leq t\cdot C(Y_{1r})\\ R_{2}\leq t\cdot C(Y_{2r})\\ R_{1}+R_{2}\leq t\cdot C(Y_{MA})\end{Bmatrix} \right\} \tag{9}\]  
where \(C(x)=\log_{2}(1+x)\), \(Y_{ir},i=1,2\) is the signal to interference plus noise power ratio (SINR) from source \(S_{i}\) to relay \(R\), \(Y_{MA}\) is the SINR of multiple access transmission, and  
\[Y_{1r}=\frac{(1-\rho)\gamma_{1}}{(1-\rho)\sigma_{u,r}^{2}+\sigma _{b}^{2}} \tag{10}\] \[Y_{2r}=\frac{(1-\rho)\gamma_{2}}{(1-\rho)\sigma_{u,r}^{2}+\sigma _{b}^{2}}\] (11) \[Y_{MA}=\frac{(1-\rho)\gamma_{\Sigma}}{(1-\rho)\sigma_{u,r}^{2}+ \sigma_{b}^{2}} \tag{12}\]  
where \(\gamma_{1}=|h_{1}|^{2}P_{1}\), \(\gamma_{2}=|h_{2}|^{2}P_{2}\) and \(\gamma_{\Sigma}=|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2}\). In the broadcast phase, the cognitive relay performs information decoding utilizing the harvested energy. Assuming perfect CSI, the relay can utilize physical layer network coding to encode the received signal \(y_{ID}\) into \(x_{R}=x_{1}\oplus x_{2}\).