## 1 Introduction  
The target of next generation wireless communication networks contains many connected devices, data rates in the range of Gbps, increased reliability, lower latencies, environmentally-friendly and improved coverage, energy-efficiency and low-cost operation [1]. Satellite mobile communication has drawn extensive interest in radio technology research, which aims to offer enough overlap with ordinary industrial facilities as a result of the performance limits in the existing cellular spectrum and the growing interest in the exploration of supplementary resource [2]. In this regard, researchers have recently envisaged the integration of satellite and terrestrial systems to form a satellite-terrestrial relay network (STRN) architecture [3, 4], which basically implants terrestrial relay cooperation into satellite mobile communications. Such STRN can provide broadcast/multicast services and uninterrupted coverage to portable and mobile users. Among the existing works, the authors of [3] derived the asymptotic results for the STRN in the presence of the amplify and forward relaying protocol with fixed gain. In [4], the outage probability (OP) of STRN was studied in the presence of multiple users with opportunistic user scheduling. In [5], the authors examined the problem of amplify-and-forward (AF)-based relaying in a hybrid satellite-terrestrial link and derived the novel expressions for the symbol error rate (SER) of the considered system. In [6], the authors investigated the performance of integrated wireless sensor and multi-beam satellite networks under terrestrial interference and derived the closed-form approximations of capacity per beam. In [7], the authors proposed the energy-efficient optimal power allocation schemes in the cognitive satellite-terrestrial networks for non-real-time and real-time applications and maximized the energy efficiency of the cognitive satellite user. In [8], the OP was studied for the multiple terrestrial STRN with the switch-and-stay combining scheme in the presence of HIs. In [9], the authors studied the OP of the STRN in the presence of multi-antenna multiple users. Besides, the outdated channel state information (CSI) and co-channel interference were both considered in the system. Cognitive radio is considered as an important technology that is regarded as a method of spectrum management in the usual wireless communication system due to the reason that primary and secondary networks coexist by utilizing identical resources [10; 11; 12]. The fusion of the cognitive satellite-terrestrial relay network (CSTRN) has been extensively investigated in both academic and industry areas. In [10], the authors minimized the transmit power by optimizing the artificial noise and cooperative beamforming in the CSTRN. In [11], the authors investigated the impact of multiple primary users and fading on the spectrum sensing of a classical energy detector. In [12], to minimize the transmit power and satisfy the outage in the CSTRN, a beamforming method was proposed. The authors in [13] explored the possibility of maximizing spectrum efficiency for satellite-terrestrial uplink transmissions. Moreover, in [14], the main aspects of the CSTRN were studied, and possible practical scenarios for the CSTRN were presented. In [15], a mathematic approach was provided to achieve higher efficiency of the OP for the CSTRN. In [16], the authors investigated the OP of the CSTRN in the presence of multiple secondary networks. To its detriment, the hardware of the aforementioned works is assumed to be perfect, which cannot be realized in practical systems. In fact, the transceiver node in wireless networks often suffers several types of hardware impairments (HIs) such as phase noise (PN), high power amplifier (HPA) non-linearities and in-phase quadrature-phase imbalance (IQI) in the oscillator [17]. Due to these reasons, the wanted transmit signal and practical emitted signal will be mismatched as a result of distortions for the received signal during the transmission course [18].