Ergodic Capacity of Dyadic Fading Channels in Ultra Low-SNR Regime

In a mobile wireless channel, the small-scale multipath fading induces temporal channel fluctuations in the form of peaks and deep fades. The channel capacity degradation with fading severity in the high signal-to-noise ratio (SNR) regime is well known in the wireless communication literature: the probability of deep fades increases significantly with higher fading severity resulting in poor performance. In this paper, we focus on double-fading pinhole channels under perfect CSIT to show a very counter-intuitive result that - higher fading severity enables higher ergodic capacity at sufficiently low SNR. The underlying reason is that at low SNRs, ergodic capacity depends crucially on the probability distribution of channel peaks (simply tail distribution); for the pinhole channel, the tail distribution improves with increased fading severity. This allows a transmitter operating at low SNR to exploit channel peaks more efficiently resulting in a net improvement in achievable spectral efficiency. We derive a new key result quantifying the above dependence for the double-Nakagami-$m$ fading pinhole channel - that is, the ergodic capacity ${C} \propto (m_T m_R)^{-1}$ at low SNR, where $m_T m_R$ is the product of fading (severity) parameters of the two independent Nakagami-$m$ fadings involved.