In this paper, we obtain asymptotic expressions for the ergodic capacity of the keyhole multiple-input multiple-output (MIMO) channel at low signal-to-noise ratio (SNR) in independent and identically distributed Nakagami-$m$ fading conditions with perfect channel state information at the transmitter and receiver. We show that the low-SNR capacity of this keyhole MIMO channel scales proportionally as $\frac{\textrm{SNR}}{4} \log^2 \left(1/{\textrm{SNR}}\right)$. Our main contribution is to identify a surprising result that the low-SNR capacity of the MIMO fading channel increases in the presence of keyhole degenerate condition, which is in direct contrast to the well-known MIMO capacity degradation at high SNR under keyhole conditions. To explain why rank-deficient keyhole fading channel outperforms the full-rank MIMO fading channel at sufficiently low-SNR, we remark that the rank of the MIMO channel matrix has no impact in the low-SNR regime and that the double-faded (or double-scattering) nature of the keyhole MIMO channel creates more opportunistic communications at low-SNR when compared with pure MIMO fading channel which leads to increased capacity. Finally, we also show that a simple one-bit channel information based on-off power control achieves this low-SNR capacity; surprisingly, this power adaptation is robust against both moderate and severe fading for a wide range of low SNR values. These results also hold for the keyhole MIMO Rayleigh channel as a special case.
翻译：在本文中,我们在独立和同样分布的Nakagami-$$淡化条件下,在发射机和接收机的完美频道状态信息中,我们获得对关键孔多投入多输出输出(MIMO)的低信号-噪音比率(SNR)频道在独立和同样分布的Nakagami-$m$淡化条件中,在发射机和接收机接收机的完美频道状态信息中,对关键孔多输出(MIMO)的低信号-脉冲能力在独立和同样分布的Nakagami-m$的低信号-噪音率(SNRR)中,对关键孔-mmmmmml流频道的低信号-负能量(1/textrm{SNRQrrright)(我们的主要贡献是确定一个令人惊讶的结果:MIMO的低信号-RR(S)频道的低信号-NRR(S)的低信号流淡化能力增加,而S-ral-ral-ral的双声流系统又显示双频-ral-ro-ro-ro-ral-ral-ral-ral-la-la-la-l-l-lax-l-l-l-l-l-l-l-la-l-l-l-l-l-l-l-l-l-l-l-l-l-l-lation-l-l-l-l-l-lation-lation-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-
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