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## ABSTRACT

Hardware impairments, such as phase noise, quantization errors, non-linearities, and noise amplification, have baneful effects on wireless communications. In this paper, we investigate the effect of hardware impairments on multipair massive multiple-input multiple-output (MIMO) two-way full-duplex relay systems with amplify-and-forward scheme. More specifically, novel closed-form approximate expressions for the spectral efficiency are derived to obtain some important insights into the practical design of the considered system. When the number of relay antennas $N$ increases without bound, we propose a hardware scaling law, which reveals that the level of hardware impairments that can be tolerated is roughly proportional to $\sqrt{N}$. This new result inspires us to design low-cost and practical multipair massive MIMO two-way full-duplex relay systems. Moreover, the optimal number of relay antennas is derived to maximize the energy efficiency. Finally, Motor-Carlo simulation results are provided to validate our analytical results.