Covert Communications Under Environmental Uncertainty on a Continuous-Time Channel.

Covert communication is achieved when a transmitter Al-ice can successfully transmit a message to a receiver Bob without being detected by an attentive and capable adversary Willie. Early results demonstrated the difficulty of the covert communications problem: with AWGN discrete-time channels between all parties, only $O(\sqrt{n})$ bits can be sent in $n$ channel uses. But it was soon recognized that uncertainty about the environment at Willie, for example, uncertainty in his own noise statistics, could allow for a positive rate: $O(n)$ bits can be sent covertly in $n$ channel uses. However, most covert communication results, including this promising positive rate result, have been obtained for a discrete-time communications channel. Here, we demonstrate that the assumption of a discrete-time channel is problematic when trying to exploit Willie's noise uncertainty. In particular, we demonstrate that if Alice transmits $\omega(\sqrt{T})$ bits in a length $T$ interval to Bob on a continuous-time channel, then there exists a detector at Willie that can detect her transmission, as the probability of false alarm and missed detection $P_{MD}+P_{FA}\rightarrow 0$ as $T\rightarrow\infty$ . In other words, the communication is not covert, unlike the case of a discrete-time channel.