## 1 Introduction
In [28], the authors investigated the dual-hop AF cooperative systems in the presence of I/Q imbalance. In [29; 30], the authors analyzed the impact of I/Q imbalance in full-duplex relay networks. In [31], the authors concluded all the factors and proposed a general model which has been cited by most of the related papers. In [32], the authors studied the impact of hardware impairments (HIs) on the considered two-way relay networks by deriving the analytical expression of the OP. To its regret, they only paid attention to the HIs at relay and neglected the HIs at sources. In [33], the two-way multiple antenna multiple relay networks with HIs were studied and all nodes were considered suffering HIs, and the closed-form expression of the OP for the considered network was also derived. In [34], the authors analyzed the impact of HIs on the multiple relays network with partial relay selection scheme. In [35], the impact of HIs was analyzed based on the cognitive network in the presence of interference. In [36], HIs has been considered as a key problem that needs to be solved in 5G communication networks. In [37], the authors have given the new HIs model, which is the extension of [31]; it is the practical and commonly applied approach in the communication network. The HIs of the transmitted node not only have an effect on the terrestrial network but they also have an impact on satellite networks. HIs of satellite could reduce the HSTCN performance; it is essential to research the effect of HIs on the Satcom. Until now there is some literature studying the effect of HIs on the Satcom networks, and even HSTCN systems. In [38], the authors first considered the HIs in two-hop Satcom systems and analyzed the effect of HIs on the system performance. The analytical expression of OP for the considered system was derived in shadowed-Rician channel. In [39], the authors derived the analytical expression of the OP for the satellite relay network in the presence of interference and HIs. In [40], the authors studied the OP for the considered networks by means of relay selection scheme without direct link in HIs environment. In [41], the OP was studied for the multiple terrestrial HSTCN with a switch-and-stay combining scheme in the presence of HIs. However, two-way relay can bring better system performance by increasing the utilization of the spectrum. In two-way networks, the source and destination can receive the signal at the same time with the help of two-way relays. The research for HIs on two-way satellite-terrestrial network is quite limited. We know that only the authors of [42] derived the closed-form expressions of the OP and the throughput for the considered satellite two-way terrestrial networks. Unfortunately, the authors of [42] just considered one terrestrial relay for AF protocol and ignored the performance improvement by the diversity of multiple terrestrial relays. Motivated by the above discussions, we take the HIs into account and investigate the performance of the two-way hybrid satellite multi-terrestrial cooperative network, where the two-way terrestrial relays are equipped with multiple antennas. Particularly, our main contributions can be summarized as follows:  
* Firstly, taking the HIs into account, we propose a framework of two-way hybrid satellite multi-terrestrial cooperative network, where the two-way terrestrial relays are used to assist the signal transmission. Besides, the HIs system model used in this paper is established according to the literatures [37] which is the extension of [31] and the practical and commonly applied approach in the relay communication network. * Secondly, based on the principle of opportunistic relay selection scheme [17; 18], the closed-form expressions for the OP and the throughput of the considered network are derived with AF and decode-and-forward (DF) protocols, which give clear views on the difference between the AF and DF protocol.