The First and Second-Order Asymptotics of Covert Communication over AWGN Channels

TL;DR

The paper analyzes finite-blocklength covert communication over AWGN channels under a KL-divergence constraint, deriving the first- and second-order throughput terms $\sqrt{n\delta\log e}$ and $\sqrt{2}\,(n\delta)^{1/4}(\log e)^{3/4}Q^{-1}(\epsilon)$. It develops an achievability framework using truncated Gaussian codebooks and a quasi-$\varepsilon$-neighborhood in information geometry, and a matching converse based on Gaussian extremality under a second-moment constraint, establishing tight bounds. A local information-geometric interpretation justifies the optimality of truncated Gaussian generating distributions, and the analysis reveals a square-root power law $\Psi(n)=O(n^{-1/2})$ that balances covert constraints with main-channel throughput. The results highlight a fundamental AWGN–DMC divergence: even with favorable adversary noise, the key size scales as $O(\sqrt{n})$, and the derived asymptotics coherently explain the role of power, noise, and truncation in practical covert communication designs.

Abstract

This paper investigates the asymptotics of the maximal throughput of communication over AWGN channels by $n$ channel uses under a covert constraint in terms of an upper bound $δ$ of Kullback-Leibler divergence (KL divergence). It is shown that the first and second order asymptotics of the maximal throughput are $\sqrt{nδ\log e}$ and $(2)^{1/2}(nδ)^{1/4}(\log e)^{3/4}\cdot Q^{-1}(ε)$, respectively. The technique we use in the achievability is quasi-$\varepsilon$-neighborhood notion from information geometry. For finite blocklength $n$, the generating distributions are chosen to be a family of truncated Gaussian distributions with decreasing variances. The law of decreasing is carefully designed so that it maximizes the throughput at the main channel in the asymptotic sense under the condition that the output distributions satisfy the covert constraint. For the converse, the optimality of Gaussian distribution for minimizing KL divergence under the second order moment constraint is extended from dimension $1$ to dimension $n$. Based on that, we establish an upper bound on the average power of the code to satisfy the covert constraint, which further leads to the direct converse bound in terms of covert metric.

Figures (4)

• Figure 1: The channel model of Gaussian LPD communication in Section \ref{['model']}
• Figure 2: $\Delta$ tends to $1$ with proper chosen $\mu$.
• Figure 3: Numerical behavior of $\Delta$ with proper chosen $\eta$.
• Figure 4: The metric of weak convergence between $\bar{P}_n^{(n)}$ and $\delta_0^n$. On one hand, the distributions $\bar{P}_l^{(m)}$ in the $l$-th row converge to the unique Dirac measure $\delta_0^l$ as $m \rightarrow \infty$ due to the decreasing variance $P(m)$; on the other hand, the distances between $\bar{P}_n^{(n)}$ and $\delta_0^n$ converge to $0$ as $n \rightarrow \infty$.

Theorems & Definitions (59)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Lemma 2
• Corollary 1
• proof
• Definition 4
• Theorem 1
• Lemma 3