On The Capacity of Low-Rank Dyadic Fading Channels in the Low-SNR Regime

We analyze low-power short-range wireless communications through a low-rank fading channel - a bonafide use case in many communication scenarios requiring simple wireless connectivity with much relaxed constraints on throughput and data latency. This is certainly true, for instance, in low-complexity wireless channels in the low-rate wireless personal area networks (LR-WPANs). Low-rate communication on control channels in wireless networks is another relevant example. Specifically, we characterize the capacity of a low-rank wireless channel with varying fading severity at low signal-to-noise ratios (SNRs). The rank deficiency is incorporated by introducing pinhole condition in the channel. The channel capacity degradation with fading severity at high SNRs is well known: the probability of deep fades increases significantly with higher fading severity resulting in poor performance. Our analysis of the double-fading pinhole channel at low-SNR shows a very counter-intuitive result that - \emph{higher fading severity enables higher 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 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.