On the Second Order Asymptotics of Covert Communications over AWGN Channels

This work tackles the asymptotics of the maximal throughput of covert communications over AWGN channels when the covert metric is Kullback-Leibler divergence (KL divergence). It is shown that the first and second order asymptotics of the maximal throughput are
$\sqrt{n\delta\log e}$
and (2)
${ }^{\frac{1}{2}}(n \delta)^{\frac{1}{4}}(\log e)^{\frac{3}{4}} \cdot Q^{-1}(\epsilon)$
, respectively by
$n$
channel uses, where
$\delta$
and
$\epsilon$
are constraints imposed on covertness and channel decoding error probabilities, 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 second order moment constraint is extended from dimension 1 to dimension
$n$
, which further leads to the direct converse bound in terms of covert metric.