Achieving Positive Rate of Covert Communications Covered by Randomly Activated Overt Users
Bichen Kang, Neng Ye, Jianping An
Abstract
This paper studies the fundamental limits of covert communications covered by randomly activated overt users in both single-frame and multi-frame transmission scenarios. While traditional covert communications mainly consider concealing signal power characteristics, the existence of overt users provides opportunities such that covert communications can be achieved through the confusion between the users. This benefit is first revealed in single-frame transmission scenario. The major obstacle in analyzing performance limits is that the conventional Kullback-Leibler divergence based covertness measurement becomes infinite. To overcome the intractability, a tighter upper bound of the total variation distance (TVD) is then developed using a novel recursive-iterative approximation. On this basis, the collapse effect of the TVD is derived, which shows that the TVD is strictly less than 1 if the covert user sets the transmit power to be an integer multiple of that of the overt users. Then, we find that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(N)$ </tex-math></inline-formula>-bit information can be transmitted over N channel uses under the above setting, which breaks the well-known square root law. If the above setting is violated, the TVD instantly approaches 1 as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\rightarrow \infty $ </tex-math></inline-formula>, and only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(\sqrt {N})$ </tex-math></inline-formula>-bit information can be covertly transmitted. To prove this, the detection method of the warden is modified to cope with the random activation of overt users. These conclusions also hold for the transmission with uncertain powers or in fading channels, which resembles realistic wireless transmissions. In multi-frame transmission scenario, however, the access characteristics of overt users can be exposed from a statistical perspective, such that the rate gain disappears and the covert transmission rate drops to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(\sqrt {N})$ </tex-math></inline-formula> bits per frame. To obtain a positive covert transmission rate, we propose a rate-splitting based covert transmission scheme that introduces an opportunistic access branch to bring randomness, through which the covert user can transmit up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(NL)$ </tex-math></inline-formula>-bit information over L frames.