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

TL;DR

This paper derives the asymptotic low-SNR capacity of keyhole MIMO channels under Nakagami-m fading, revealing that keyhole conditions can enhance capacity at low SNR, contrary to high SNR behavior.

Contribution

It provides the first asymptotic expressions for low-SNR capacity in keyhole MIMO channels with full CSI, highlighting the capacity increase due to keyhole effects at low SNR.

Findings

Low-SNR capacity scales as (SNR/4) * log^2(1/SNR).

Keyhole channels outperform full-rank MIMO at low SNR.

One-bit on-off power control achieves this capacity effectively.

Abstract

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.