Gaussian Signalling for Covert Communications

In this paper, we examine the optimality of Gaussian signalling for covert communications with an upper bound on <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{1}}||p_{_{0}})$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{0}}||p_{_{1}})$ </tex-math></inline-formula> as the covertness constraint, where <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{1}}||p_{_{0}})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{0}}||p_{_{1}})$ </tex-math></inline-formula> are different due to the asymmetry of Kullback–Leibler divergence, <inline-formula> <tex-math notation="LaTeX">$p_{_{0}}(y)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$p_{_{1}}(y)$ </tex-math></inline-formula> are the likelihood functions of the observation <inline-formula> <tex-math notation="LaTeX">${y}$ </tex-math></inline-formula> at the warden under the null hypothesis (no covert transmission) and alternative hypothesis (a covert transmission occurs), respectively. Considering additive white Gaussian noise at both the receiver and the warden, we prove that the Gaussian signalling is optimal in terms of maximizing the mutual information of transmitted and received signals for covert communications with an upper bound on <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{1}}||p_{_{0}})$ </tex-math></inline-formula> as the constraint. More interestingly, we also prove that the Gaussian signalling is not optimal for covert communications with an upper bound on <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{0}}||p_{_{1}})$ </tex-math></inline-formula> as the constraint, for which as we explicitly show skew-normal signalling can outperform the Gaussian signalling in terms of achieving higher mutual information. Finally, we prove that, for Gaussian signalling, an upper bound on <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{1}}||p_{_{0}})$ </tex-math></inline-formula> is a tighter covertness constraint in that it leads to lower mutual information than the same upper bound on <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{0}}||p_{_{1}})$ </tex-math></inline-formula>, by proving <inline-formula> <tex-math notation="LaTeX">$\mathcal {D}(p_{_{0}}||p_{_{1}}) \leq \mathcal {D}(p_{_{1}}||p_{_{0}})$ </tex-math></inline-formula>.

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