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

Kamal Singh

TL;DR

This paper analyzes the capacity of low-rank, dyadic (pinhole) fading channels in the low-$SNR$ regime under perfect CSI. It uses a double-Nakagami-$m$ model with h = h_T h_R, deriving the end-to-end gain distribution and an asymptotic low-$SNR$ capacity bound: $C \approx \left(\frac{\Omega_T \Omega_R}{m_T m_R}\right) \frac{\mathrm{SNR}}{4} \log^2\left(\frac{1}{\mathrm{SNR}}\right)$. A key finding is that capacity increases as the fading severity grows (i.e., smaller $m_T m_R$), due to improvements in the tail distribution of the channel gain and the ability to exploit rare high-gain events at very low $SNR$. The results are validated numerically and extend to multi-antenna pinhole channels by a multiplicative gain factor, underscoring a counter-intuitive yet practically relevant advantage of higher fading severity for low-power, short-range communications.

Abstract

We characterize the capacity of a low-rank wireless channel with varying fading severity at low signal-to-noise ratios (SNRs). The channel rank deficiency is achieved by incorporating pinhole condition. The 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 dyadic 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 (tail distribution); for the pinhole channel, the tail distribution improves with fading severity. This allows a transmitter operating at low SNR to exploit channel peaks `more efficiently' and hence improves spectral efficiency. We derive a new key result quantifying the above dependence for the double-Nakagami-$m$ fading pinhole channel - the capacity ${C} \propto (m_T m_R)^{-1}$ at low SNR, where $m_T m_R$ is the severity parameters (product) of the fadings involved.

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

TL;DR

This paper analyzes the capacity of low-rank, dyadic (pinhole) fading channels in the low- regime under perfect CSI. It uses a double-Nakagami- model with h = h_T h_R, deriving the end-to-end gain distribution and an asymptotic low- capacity bound: . A key finding is that capacity increases as the fading severity grows (i.e., smaller ), due to improvements in the tail distribution of the channel gain and the ability to exploit rare high-gain events at very low . The results are validated numerically and extend to multi-antenna pinhole channels by a multiplicative gain factor, underscoring a counter-intuitive yet practically relevant advantage of higher fading severity for low-power, short-range communications.

Abstract

We characterize the capacity of a low-rank wireless channel with varying fading severity at low signal-to-noise ratios (SNRs). The channel rank deficiency is achieved by incorporating pinhole condition. The 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 dyadic 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 (tail distribution); for the pinhole channel, the tail distribution improves with fading severity. This allows a transmitter operating at low SNR to exploit channel peaks `more efficiently' and hence improves spectral efficiency. We derive a new key result quantifying the above dependence for the double-Nakagami- fading pinhole channel - the capacity at low SNR, where is the severity parameters (product) of the fadings involved.
Paper Structure (5 sections, 1 theorem, 24 equations, 5 figures)

This paper contains 5 sections, 1 theorem, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Dyadic/Double-fading (Pinhole) Channel
  3. Full CSI Dyadic (Pinhole) Channel Capacity at Low SNR
  4. Numerical Results & Discussion
  5. Concluding Remarks

Key Result

Theorem 3

Under full CSI, the asymptotic (low-SNR) capacity of the double-fading pinhole channel subjected to independent Nakagami-$m$ fadings between the source-to-pinhole and pinhole-to-destination side respectively [described by eq:sys_model$/$eq:pinholeH] is given by

Figures (5)

  • Figure 1: Illustration of signal propagation through two local rich-scattering wireless environments connected via a pinhole.
  • Figure 2: Cumulative distribution function $F_{g}(\cdot)$ of single & dyadic Nakagami-$m$ channel gains. The channel gain $g := r^2$ is the squared fading envelope, i.e., $r = |h|$ for dyadic and $r = |h_i|$ for single-hop channels (see \ref{['eq:pinholeH22']}). The mean channel gain is normalized to unity (i.e., $\mathbb{E} \,[g] = 1$ or $0$ dB) in all cases.
  • Figure 3: Comparison of exact & asymptotic capacity of the double-Nakagami-$m$ pinhole channel at low SNRs. For simplicity, $m_R = m_T$ (say $m$) and $\Omega_R = \Omega_T = 1$.
  • Figure 4: Comparison of exact capacity performance of the Nakagami-$m$ fading (single-hop) and double-Nakagami-$m$ fading (double-hop) channels at low SNRs. We assume $m_R = m_T = m$ and $\Omega_R = \Omega_T = 1$ for the dyadic channel. Mean channel gain is also normalized to unity for the single-hop channel.
  • Figure 5: PDF of the double-Nakagami-$m$ pinhole channel gain (see \ref{['eq:cap:csit0']}). For simplicity, we keep $m_R = m_T = m$ (say) and $\Omega_R = \Omega_T = 1$.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Remark 4