## 2 System Model and Problem Formulation  
As provided in Figure 1, this paper studies the two-way hybrid satellite multi-terrestrial communication network, where the satellite and mobile user communicate with each other instead of having the help of multi-two-way terrestrial relays. Due to the heavy fading or huge obstructions, we assume that there is no direct link between the satellite and mobile user (For example, the mobile user is indoors or in the cave, where there is no direct link between the satellite and the mobile user. This assumption has been used in many previous papers [2; 3; 8; 10; 11].). In the system, a satellite source node (\(S_{1}\)), \(N\) terrestrial nodes (\(R\)) and a mobile source node (\(S_{2}\)) are considered. \(S_{1}\) and \(S_{2}\) are equipped with a single antenna, respectively. \(R\) is equipped with \(M\) antennas. The opportunistic relay selection scheme is used to get better system performance. Opportunistic relaying selection scheme is expressed as follows: the system selects the relay link which has the largest signal-to-noise-and-distortion ratio (SNDR) among all the relay links.  
It takes two time slots for the communication. In the first time slot, \(S_{1}\) and \(S_{2}\) transmit the information signals \(x_{1}\left(t\right)\) and \(x_{2}\left(t\right)\) to the \(i\)-th \(R\), respectively. Hence the received signal at the \(i\)-th \(R\) in the first time slot is presented as  
\[y_{r}\left(t\right)=\mathbf{w}_{i}^{H}\mathbf{h}_{1i}\left[\sqrt{k_{1}P  
where \(\mathbf{w}_{i}^{H}\) is the BF vector at the \(i\)-th R with \(E\left[\left|\left|\mathbf{w}_{i}\right|\right|^{2}\right]=1\), \(\mathbf{h}_{1i}\) the channel coefficient vector for \(S_{1}\)-the \(i\)-th R link satisfying shadowed-Rician fading. \(k_{1}\) denotes the impairments level at \(S_{1}\) satisfying \(0\leq k_{1}\leq 1\), where \(k_{1}=1\) denotes the ideal hardware. \(P_{1}\) is the transmitted power at \(S_{1}\), \(x_{1}\left(t\right)\) the transmitted signal from \(S_{1}\) with \(E\left[\left|x_{1}\left(t\right)\right|^{2}\right]=1\), \(\eta_{1}\left(t\right)\) the distortion noise due to HIs satisfying \(\eta_{1}\left(t\right)\sim\mathcal{CN}\left(0,\left(1-k_{1}\right)P_{1}\right)\). \(\mathbf{h}_{2i}\) is the channel coefficient vector for \(S_{2}\)-the \(i\)-th R obeying Rayleigh fading. \(k_{2}\) presents the impairments level at \(S_{2}\) with \(0\leq k_{2}\leq 1\), \(P_{2}\) the transmitted power from \(S_{2}\), \(x_{2}\left(t\right)\) the signal transmitted from \(S_{2}\) obeying \(E\left[\left|x_{2}\left(t\right)\right|^{2}\right]=1\), \(\eta_{2}\left(t\right)\) the distortion noise due to HIs satisfying \(\eta_{2}\left(t\right)\sim\mathcal{CN}\left(0,\left(1-k_{2}\right)P_{2}\right)\), \(n_{i}\left(t\right)\) the additive white Gaussian noise (AWGN) at the \(i\)-th R which is represented as \(n_{i}\left(t\right)\sim\mathcal{CN}\left(0,\delta_{i}^{2}\right)\).  
In the second time slot, the \(i\)-th \(R\) simultaneously forwards the received signal to \(S_{j}\) (\(\mathrm{j}=1,2\)). Particularly, we use both AF and DF protocols in this paper. In what follows, without loss of generality, the received signals of \(S_{j}\) from the \(i\)-th \(R\) for these two protocols are derived, respectively.