On the Capacity of Low-Rank Fading Channels with Full CSI at Low SNR

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). Two more wireless communication scenarios can be cited as relevant examples: low-rate communication on the control channels in the existing wireless local area networks (WLANs), as well as the low-rank Reconfigurable Intelligent Surface (RIS) assisted millimeter-wave (mmWave) communication link operating at a low transmit power budget. 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 distribution of deep fades increases significantly with fading severity resulting in poor performance. Our analysis of the double-fading pinhole channel at low-SNR shows a very counter-intuitive result that-higher fading severity enables higher spectral efficiency at sufficiently low SNRs. 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 SNRs to exploit channel peaks 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 varies inversely to the product of fading (severity) parameters of the two independent Nakagami-m fadings involved.