Weighted Unequal Error Protection over a Rayleigh Fading Channel

TL;DR

This work analyzes the asymptotic and finite blocklength performance of two achievability schemes, one based on power-domain superposition (PDS) and another based on orthogonal resource allocation (ORA), also known as time-sharing.

Abstract

We study a variant of unequal error protection in channel coding, where the message bit string is divided into a finite number of blocks and the maximization objective is a weighted sum of per-block decoding success probabilities. The channel model is quasi-static Rayleigh fading with channel state information available to the receiver but unavailable to the transmitter. We analyze the asymptotic and finite blocklength performance of two achievability schemes, one based on power-domain superposition (PDS) and another based on orthogonal resource allocation (ORA), also known as time-sharing. Upper bounds on the optimal number of blocks to transmit are derived. Algorithms to compute the optimal power and time splits for the two schemes are given. Simplified algorithms to compute locally optimal power and time splits are also given, and their outputs match those of the previous algorithms across all tested parameters. Our results show that PDS outperforms ORA, but the performance differential is less than 2% in both the asymptotic and finite blocklength regimes (Figures 4 - 6). For both PDS and ORA, numerical results also upper bound the gap between the asymptotic and finite blocklength performance by approximately 10% for n = 1000 and 3% for n = 5000 (Figures 7 - 10).

Figures (12)

• Figure 1: The error exponent bound is tighter than the normal approximation bound in the low-rate regime, but the opposite is true in the high-rate/near-capacity regime. The horizontal axis is the coding rate represented as a fraction of the capacity $C(\rho) = \log(1 + \rho)$.
• Figure 2: For fixed $R = 0.1$ and $\overrightarrow d = \frac{1}{14}(5,4,3,2)$, the objective value $G(\overrightarrow x)$ is plotted against $\theta$, when $\overrightarrow x$ is taken as the output from Algorithm \ref{['algorithmlocalmax']}. For a fixed $R$, $\theta \propto \frac{1}{P \sigma^2}$ so the $x$-axis should be interpreted as inverse average SNR up to some scaling.
• Figure 3: For fixed $R = 0.1$ and $\overrightarrow d = \frac{1}{14}(5,4,3,2)$, the power fractions $\overrightarrow \alpha = (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ are plotted versus $\theta$. The plots are based on the output of Algorithm \ref{['algorithmlocalmax']}; using the output of Algorithm \ref{['algorithmglobalmax']} yields identical plots.
• Figure 4: For $R = 0.1$, importance vector $\overrightarrow d = \frac{1}{440}[100, 85, 70, 60, 50, 40, 25, 10]$, the percentage difference $\frac{N_4}{N_2} \times 100$ is plotted against $\theta$, where $N_1$ is computed using Algorithm \ref{['algorithmlocalmax']} and $N_4$ is computed using Algorithm \ref{['algorithmlocalmaxORA']}. The asymptotic performance of the ORA scheme is only slightly less than that of the PDS scheme. The troughs represent points where the PDS and ORA drop packets as the channel gets worse (see, e.g., Figure \ref{['power_plots']}).
• Figure 5: For $n=1000$, $R = 0.1$, $P = 1$, importance vector $\overrightarrow d = \frac{1}{440}[100, 85, 70, 60, 50, 40, 25, 10]$, the percentage difference $\frac{N_6}{N_5} \times 100$ is plotted against $\theta$. Similar to Figure \ref{['asymptotic_PDS_vs_ORA']}, the ORA performance even at finite blocklength is within $2\%$ of the PDS scheme.

Theorems & Definitions (38)

• Lemma 1
• Lemma 2: Error Exponent
• Lemma 3: Normal Approximation
• Theorem 1
• Theorem 2
• Definition 1
• Theorem 3: Structural Properties of an optimal solution
• Corollary 1
• Theorem 4
• Remark 1