## 2 System Model and Problem Formulation
Since the detection and estimation of training signals are also impacted by HIs, \(\overline{\Upsilon}_{X}\) can be rewritten with the help of the training symbols' SNR for the system that has ideal hardware \(\overline{\Upsilon}_{idX}\) as:  
\[\overline{\Upsilon}_{X}=\frac{\overline{\Upsilon}_{idX}}{\overline{\Upsilon }_{idX}k_{X}^{2}+1}, \tag{5}\]  
where \(k_{X}\) presents the EVM parameters, which influence the training symbols of the transmitting links. Then, substituting Equation (5) into Equation (4), we can obtain:  
\[\overline{\Upsilon}_{h_{X}}=\frac{\overline{\Upsilon}_{idX}k_{X}^{2}+1}{ \left(T_{X}+k_{X}^{2}\right)\overline{\Upsilon}_{idX}+1}, \tag{6}\]  
where \(\overline{\Upsilon}_{h_{X}}\) reports the accuracy of channel estimation and is presented as the minimum mean square error (MMSE). Further, in the CSTRN with CEEs and HIs, to limit the interference power at PUs below a pre-determined threshold \(Q\), the instantaneous powers at S and R should be constrained as:\[E\left\{\sum\limits_{i=1}^{M}\left|\left(\widetilde{h}_{SP_{i}}+e_{ h_{SP_{i}}}\right)\left(s\left(t\right)+\eta_{SP_{i}}\left(t\right)\right) \right|^{2}\right\}\leq Q,\] \[E\left\{\sum\limits_{i=1}^{M}\left|\left(\widetilde{h}_{RP_{i}}+ e_{h_{RP_{i}}}\right)\left(s\left(t\right)+\eta_{RP_{i}}\left(t\right)\right) \right|^{2}\right\}\leq Q,\]  
where \(\eta_{SP_{i}}\left(t\right)\sim\mathcal{CN}\left(0,k_{SP_{i}}^{2}P_{S}\right)\) and \(\eta_{RP_{i}}\left(t\right)\sim\mathcal{CN}\left(0,k_{RP_{i}}^{2}P_{R}\right)\) are the distortion noise components caused by HIs in transmitting processing at S and R and \(k_{SP_{i}}\) and \(k_{RP_{i}}\) represent the HI level at PUs. Consequently, we have:  
\[P_{S}=\frac{Q}{\sum\limits_{i=1}^{M}\left|\left(\left|\widetilde{h}_{SP_{i}} \right|^{2}+\overline{\Upsilon}_{h_{SP_{i}}}\right)\left(1+k_{SP_{i}}^{2} \right)\right|^{2}}, \tag{7}\]  
Figure 1:0  
Now, it has assumed that at S and R, the maximum transmit power is large enough and can be neglected to meet the interference constraint [27]. Then, from Equations (1) and (7), the final SNDER of \(y_{R}\left(t\right)\) is given by:  
Figure 1:1  
where \(\gamma_{SR}=\frac{\sigma Q}{\delta_{R}^{2}}\left|\widetilde{h}_{SR}\right|^{2}\), \(\gamma_{SP}=\frac{\sigma Q}{\delta_{R}^{2}}\sum\limits_{i=1}^{M}\left|\widetilde {h}_{SP_{i}}\right|^{2}\), \(A=\frac{\sigma Q^{2}}{\delta_{R}^{2}}\overline{\Upsilon}_{h_{SR}}\left(1+k_{ 1}^{2}\right)+Q\sum\limits_{i=1}^{M}\overline{\Upsilon}_{h_{SP_{i}}}\left(1+k _{SP}^{2}\right)\) with the assumption \(k_{SP_{1}}=\cdots=k_{SP_{i}}=\cdots=k_{SP_{M}}=k_{SP}\). In the same manner, the final SNDER at D is given by:  
Figure 1:2  
where \(\gamma_{RD}=\frac{\sigma Q}{\delta_{D}^{2}}\left|\widetilde{h}_{RD}\right|^{2}\), \(\gamma_{RP}=\frac{\sigma Q}{\delta_{D}^{2}}\sum\limits_{i=1}^{M}\left| \widetilde{h}_{RP_{i}}\right|^{2}\), \(B=\frac{Q^{2}\sigma}{\delta_{D}^{2}}\overline{\Upsilon}_{h_{RD}}\left(1+k_{2}^ {2}\right)+Q\sum\limits_{i=1}^{M}\overline{\Upsilon}_{h_{RP_{i}}}\left(1+k_{ RP}^{2}\right)\) and \(k_{RP_{1}}=k_{RP_{2}}=\cdots k_{RP_{M}}=k_{RP}\). As discussed before, R forwards the signal received with the DF protocol, and the received SNDER of the system is given by:  
Figure 1:3  
**Remark 1**: _It should be mentioned that we establish a more general framework of the CSTRN by taking multiple PUs, CEEs and HIs into consideration, where the propagation factors of path loss, channel shadowing conditions and satellite beam pattern are also included. Specifically, our paper consists of the system model in [4] as a special case, where only a single PU, perfect hardware and accurate CSI are assumed. Furthermore, our work extends the works in [22; 23] as a special case where only HIs and CEEs were considered, respectively._