Low SNR Capacity of Keyhole MIMO Channel in Nakagami-m Fading With Full CSI

In this paper, we derive asymptotic expressions for the ergodic capacity of the multiple-input multiple-output (MIMO) keyhole channel at low SNR in independent and identically distributed (i.i.d.) Nakagami-$m$ fading conditions with perfect channel state information available at both the transmitter (CSI-T) and the receiver (CSI-R). We show that the low-SNR capacity of this keyhole channel scales proportionally as $\frac{\mathrm{SNR}}{4} \log^2 \left(1/{\mathrm{SNR}}\right)$. With this asymptotic low-SNR capacity formula, we find a very surprising result that contrary to popular belief, the capacity of the MIMO fading channel at low SNR increases in the presence of keyhole degenerate condition. Additionally, we show that a simple one-bit CSI-T based On-Off power scheme achieves this low-SNR capacity; surprisingly, it is robust against both moderate and severe fading conditions for a wide range of low SNR values. These results also extend to the Rayleigh keyhole MIMO channel as a special case.