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In this article, we consider an overlay cognitive hybrid satellite-terrestrial network consisting of a primary satellite source-receiver pair and a secondary transmitter-receiver pair on the ground while taking into account the practical hardware impairments (HIs) at the user devices. Herein, a secondary terrestrial user acts as a cooperative relay to assist the transmission of primary satellite user. In return for this kind of favor, the secondary user gets access to the primary's spectrum for its own communication. Further, we consider the distortion noises induced by all the hardware-impaired terrestrial nodes and demonstrate their detrimental impact on the system performance. We manifest analytically that, for the higher data rates, the HIs invoke ceiling effects which reprehensibly cap the fundamental capacity of the system. To mitigate the effect of HIs, we propose an adaptive relaying (AR) protocol for both amplify-and …