Covert Communication in Continuous-Time Systems in the Presence of a Jammer

Abstract

Covert communication considers the ability of transmitter Alice to communicate reliably to receiver Bob without being detected by warden Willie. Previous work has generally considered a discrete-time model, and, in the standard Alice-Bob-Willie scenario, it has been shown that a discrete-time model captures the salient aspects of the underlying continuous-time covert communications system. However, in the presence of a jammer assisting Alice, where it has been shown in previous work that Alice can achieve a positive covert rate on a discrete-time model, we demonstrate here that a straightforward extension of the discrete-time construction to the continuous-time system does not guarantee a positive covert rate, hence indicating that the discrete-time model does not capture the salient aspects of the continuous-time system in such a scenario. This is because Willie is able to exploit excess bandwidth for co-channel interference suppression, as we demonstrate with an interference cancellation receiver. Hence, for the case when Alice is assisted by an uninformed jammer, we consider the continuous-time channel directly and study whether efficient covert communication can still be achieved against any possible receiver that Willie might employ. By presenting and characterizing an approach much different than that suggested by previous work for discrete-time systems in such a scenario, we establish that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(WT)$ </tex-math></inline-formula> information bits can be transmitted covertly and reliably on a continuous-time channel of asymptotic bandwidth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> seconds, regardless of the receiver that Willie employs. This is done for two separate scenarios: 1) when there is perfect frame synchronization between Alice and the jammer’s signals; 2) when there is no frame synchronization between Alice and the jammer’s signals. Numerical results are provided to validate the theory.