## 4 Numerical Results and Discussions
### Performance Analysis with Comparative Schemes  
The superiority of the proposed scheme with benchmark schemes is investigated according to the following figures. Figure 5 compares the max-min achievable rate (\(R_{\min}\)) obtained by the proposed JPCRA scheme with the end-to-end rate(we use \(R_{i},i=1,2\) to denote the transmission rate of source (\(S_{i}\)) obtained by the maximization of achievable sum-rate (MASR) scheme [28]. In the simulation, parameters are set as \(d_{1}=\{2,\,3\}\) m, \(d_{ru}=\{2,\,3\}\) m, \((P_{1,\max},P_{2,\max})=(15,\,15)\) dBm and \(\sigma_{u}^{2}=\{10^{-6},10^{-4}\}\) W. This figure clearly shows that \(R_{2}\) increases as \(d_{1}\) decreases while \(R_{1}\) remains the same, and \(R_{\min}\) also increases, because the system with MASR scheme will enhance the transmission ability of the high channel quality to maximize the achievable sum-rate. Thus, the MASR scheme cannot enhance the transmission ability of the poor channel, which may mean that the transmission links with a poor channel quality do not meet their transmission needs. The proposed scheme considers the transmission fairness, effectively alleviating the effects caused by channel asymmetry. Therefore, when the system values the enhancement of the worst performance of a user most in this system, it is better to choose the proposed JPCRA scheme, whereas when the system values the enhancement of the system achievable sum rate most, it is better to choose the MASR scheme. Figure 6 depicts the max-min achievable rate of the proposed JPCRA scheme versus the interference threshold of the primary network, which is compared with three traditional transmission schemes. The parameters are set as \((d_{1},d_{2})=(3,\,7)\) m, \(P_{1,\max}=20\) dBm, \(P_{2,\max}=15\) dBm, \(\sigma_{u}^{2}=10^{-6}\) W. It is observed that as the interference threshold increases, the \(R_{\min}\) of each scheme increases and saturates at a stable value. This is because the transmit power of the cognitive user is mainly constrained by power control when the interference threshold is small; as the interference threshold increases to a certain value, the transmit power of the cognitive user is mainly constrained by the peak power limits. In this figure, the peak power is a fixed parameter; thus, the performance saturates as the interference threshold increases to a certain value. The proposed JPCRA scheme outperforms the other three schemes in the whole range of \(Q\). When the interference threshold is a high SNR value, it is possible to use the FPA-JoTPS scheme to replace JPCRA, and when the interference threshold is a low SNR value, it is possible to use the FT-JoPSPA scheme to replace JPCRA. Figure 7 shows the max-min achievable rate versus peak transmit power limit \(P_{1,\max}\). The proposed JPCRA scheme is compared with three traditional transmission schemes. The parameters are set as \((d_{1},d_{2})=(3,\,7)\) m, \(Q=\)5 dBm, \(P_{2,\max}\)= 15 dBm and \(\sigma_{u}^{2}=10^{-6}\) W. The proposed JPCRA scheme clearly outperforms the three other schemes. It is also observed that as \(P_{1,\max}\) increases, the \(R_{\min}\) of each schemes increases and saturates to a stable value. This is because the transmit power is curved by the PU threshold, which curves the transmission rates. In this figure, it can be seen that when the peak transmit power limit \(P_{1,\max}\) is sufficient, the proposed JPCRA scheme is definitely a good choice, whereas when \(P_{1,\max}\) is in the low-SNR range, it is possible to use the three other traditional schemes to replace JPCRA. Figure 8 illustrates the max-min achievable rate versus the interference power \(\sigma_{u}^{2}\) caused by the primary user to cognitive user. The parameters are set as \((d_{1},d_{2})=(3,\,7)\) m, \(Q=5\) dBm, \(P_{1,\max}=20\) dBm and \(P_{2,\max}=15\) dBm. As \(\sigma_{u}^{2}\) increases, \(R_{\min}\) decreases and approaches 0.