## 3 Problem Formulations
### Sensing-Based Spectrum Sharing under Imperfect Sensing Conditions  
In Section 3.1, the perfect sensing scenario is presented, and the optimal capacity is the upper bound of Equation (19). Herein, the imperfect sensing condition, where the sensing time \(\tau\) is not zero, is considered. To protect the primary user in a conventional sensing-based spectrum cognitive radio model, the detection probability \(P_{d}\) is usually larger than detection threshold \(P_{d}\), and \(P_{d}\) is usually close to 1 but less than 1. For instance, in the IEEE 802.22 WRAN, \(P_{d}\) is 0.9 as the SNR is -20 dB [37]; therefore, \(P_{d}\) is usually chosen to be larger than 0.9, and \(P_{f}\) is controlled to be very small. Under the assumptions, we can introduce the fact that \(P_{d}\) is larger than \(\bar{P}_{d}\), close to 1, and \(P_{f}\) is closed to zero. Therefore, Equation (19) can be simplified as follows:  
\[C=\max_{\left\{P_{s}^{(i)},P_{s}^{(i)},\tau\right\}}E\left\{\frac{ T-\tau}{T}\left[\mathcal{P}\left(\mathcal{H}_{i}\right)\left(1-P_{f}\right)C_{ii}+ \mathcal{P}\left(\mathcal{H}_{b}\right)P_{d}C_{bb}\right]\right\},\quad\text{( e0)}\] \[s.t.\left\{ \begin{array}{ll}E\left\{P_{s}^{(i)}\right\} \mathcal{P}\left(\mathcal{H}_{i}\right)\left(1-P_{f}\right)+E\left\{P_{s}^{( b)}\right\}\mathcal{P}\left(\mathcal{H}_{b}\right)P_{d}\leq P_{av},&\quad\text{(e1)}\\ E\left\{P_{s}^{(b)}G_{t}\left(\theta^{\prime}\right)G_{BS}L_{sp}h_{sp}\right\}P _{d}\leq l_{av},&\quad\text{(e2)}\\ 0\leq\tau\leq T.&\quad\text{(e3)}\end{array} \right. \tag{28}\]  
Obviously, (e0) is a convex optimization problem with respect to \(P_{s}^{(i)}\) and \(P_{s}^{(b)}\). In the following, we prove that it is concave in \(\tau\), when \(\tau\  
Proposition 1 shows that, when \(\tau\in(0,T)\), if \(P_{f}\left(\tau\right)\leq 0.5\), \(P_{f}\left(\tau\right)\) is decreasing, \(P_{f}^{\prime}\left(\tau\right)<0\) and \(P_{f}^{\prime}\left(\tau\right)\) is monotonically increasing; similarly, if \(P_{d}(\tau)\geq 0.5\), \(P_{d}(\tau)\) is increasing, \(P_{d}^{\prime}(\tau)>0\) and \(P_{d}^{\prime}(\tau)\) is monotonically decreasing. Therefore, \(R^{\prime}(\tau)\) is decreasing when \(\tau\in(0,T)\), which further indicates that \(R(\tau)\) is concave in \(\tau\). Since the concavity is not affected by the expectation, (e0) is a concavity function with respect to \(\tau\). It can be verified that the constraints Equations (e1) and (e2) are concave with \(\tau\) by the same method. Therefore, Equation (28) is a convex optimization problem with \(P_{s}^{\left(i\right)}\), \(P_{s}^{\left(b\right)}\), and \(\tau\). Without loss of generality, \(\mathcal{P}(\mathcal{H}_{b})  
the proposed approach leads to higher values of ergodic capacities compared with the opportunistic spectrum access approach in all the considered scenarios. In Figure 4, it can be observed that the satellite network capacities increase with the increase of \(\mathcal{P}\left(\mathcal{H}_{i}\right)\). This is due to the fact that the more idle the terrestrial user is, the larger the ergodic capacity that the satellite user can obtain. Furthermore, under the same \(\mathcal{P}(\mathcal{H}_{i})\), the ergodic capacities of the proposed approach are significantly higher than that of the opportunistic spectrum access approach. However, the capacity gap between the two approaches is smaller when the \(\mathcal{P}(\mathcal{H}_{i})\) is higher; this is because the opportunistic spectrum access approach has excellent performance if the terrestrial user is always idle. Moreover, we can see that the ergodic capacities of the proposed method become saturated when the \(I_{av}\) is large, since the transmit power \(P_{av}\) becomes the main constraint. In Figure 5, we study the performance of the ergodic capacity of the satellite user versus \(I_{av}\) for different \(m_{sp}\) with \(\mathcal{P}\left(H_{i}\right)=0.6\) and \(P_{av}=15\,\mathrm{dB}\).