## 2 System Model and Problem Formulation  
As illustrated in Figure 1, in this paper, the CSTRN, which has one source (S), one secondary destination (D), a terrestrial relay (R) and \(M\) terrestrial primary users (PUs), is considered. All of them are equipped with a single antenna. Furthermore, all nodes in the CSTRN are assumed suffering from HIs. S and D can communicate with each other only via R due to fog, rain, haze or heavy shadowing [25]. One R with the half-duplex decode-and-forward (DF) protocol is applied in the network, and the whole transmission occurs in two time slots. Through the first time slot, the signal \(s\left(t\right)\) with \(E\left[\left|s\left(t\right)\right|^{2}\right]=1\) will be transmitted from S to R; hence, the signal received at R is given by:  
\[y_{R}\left(t\right)=h_{SR}\left[\sqrt{P_{S}}^{S}\left(t\right)+\eta_{1}\left( t\right)\right]+n_{R}\left(t\right), \tag{1}\]  
where \(h_{SR}\) represents the complex channel coefficient between S and R whose absolute value follows the shadowed Rician (SR) fading. \(P_{S}\) denotes the transmit power from S; \(\eta_{1}\left(t\right)\) presents the distortion  
Figure 1: Illustration of the system model. noise owing to HIs, which can be shown as \(\eta_{1}\left(t\right)\sim\mathcal{CN}\left(0,k_{1}^{2}P_{S}\right)\); \(k_{1}\) is the error vector magnitude (EVM), which indicates the extent of the non-ideality of the hardware [26]. \(n_{R}\left(t\right)\) is the additive white Gaussian noise (AWGN) at R, which can be distributed as \(\mathcal{CN}\left(0,\delta_{R}^{2}\right)\). During the second time slot, the signal received will be transmitted to D by R via the DF protocol. The signal received at D is presented as:  
\[y_{D}\left(t\right)=h_{RD}\left[\sqrt{P_{R}}^{\sim}\left(t\right)+\eta_{2} \left(t\right)\right]+n_{D}\left(t\right), \tag{2}\]  
where \(h_{RD}\) denotes the complex channel coefficient between R and D whose absolute value follows Rayleigh fading, \(P_{R}\) is the power of R, \(\eta_{2}\left(t\right)\) is the distortion noise caused by HIs, which has the form as \(\eta_{2}\left(t\right)\sim\mathcal{CN}\left(0,k_{2}^{2}P_{R}\right)\) and \(k_{2}\) quantifies the level of impairments and is measured experimentally as EVM. \(n_{D}\left(t\right)\) is the AWGN at D presented as \(n_{D}\sim\mathcal{CN}\left(0,\delta_{D}^{2}\right)\). Now, as pointed out earlier, only non-ideal CSI of both satellite and terrestrial links is known; thus, non-ideal channel gains need to be found before the performance evaluations. Hence, the channel can be modeled as [23; 24] (we should note that this model has been proven and extensively used in existing works such as [23; 24]),  
\[h_{X}=\widetilde{h}_{X}+e_{h_{X}},X\in\left\{SR,RP,SP,RP\right\}, \tag{3}\]  
where \(h_{X}\) and \(\widetilde{h}_{X}\) represent the practical and estimated fading coefficients, respectively, \(h_{X}\) and \(\widetilde{h}_{X}\) are assumed to have combined ergodicity and \(e_{h_{X}}\) denotes the estimation error, which is orthogonal to the channel estimate coefficient \(\widetilde{h}_{X}\) with modeling as a zero mean complex Gaussian distribution [23] with its variance:  
\[\overline{\Upsilon}_{h_{X}}=E\left\{\left|h_{X}\right|^{2}\right\}-E\left\{ \left|\widetilde{h}_{X}\right|^{2}\right\}=\frac{1}{T_{X}\overline{\Upsilon} _{X}+1}, \tag{4}\]  
where \(T_{X}\) is the length of training symbols and \(\overline{\Upsilon}_{X}=E\left\{\left\{\Upsilon_{X}\right\}\right\}=\frac{P_ {X}E\left\{\left|h_{X}\right|^{2}\right\}}{N_{X}}\) is the mean SNR of the training symbol for the transmitting link with HIs. Besides \(P_{X}=\left(1-\sigma\right)\overline{P}_{total}\) with the scale coefficient \(\sigma\in\left(0,1\right)\), \(P_{X}\) presents the power of the pilot symbols, \(P_{total}\) the total transmission power and \(N_{X}\) the AWGN variance of the transmitting links.