Second-Order Asymptotics for Covert Communication over Quasi-Static Multiple-Antenna Fading Channels

Abstract

We study the second-order asymptotics of optimal codes for covert communication over quasi-static multi-antenna fading channels, under the covertness metric of Kullback--Leibler (KL) divergence. In particular, we study all four cases regarding the availability of channel state information (CSI) for the legitimate transmitter and receiver, assume that the warden knows perfect CSI for the channel from the legitimate transmitter to itself, whereas the legitimate transmitter knows only that the warden's channel matrix belongs to a bounded deterministic uncertainty set. Specifically, we show that, when the blocklength is $n$, the first-order covert rate satisfies the square root law, scaling as $\Theta (n^{-\frac{1}{2}})$ with the coefficient determined by the traces of the channel matrices of the legitimate users and the warden, and the second-order rate vanishes. We also show that the availability of CSI at the transmitter enables optimal covert power allocation across the spatial sub-channels, which can increase the covert rate, relatively to equal power allocation, without changing the square root law. Furthermore, we reveal the significant spatial diversity gain provided by multiple-antenna systems for covert communication and demonstrate the critical role of the number of antennas to achieve high throughput covert communication. For the covertness analysis, we extend the quasi-$\eta$-neighborhood framework to quasi-static fading channels. For the reliability analysis, due to the vanishing power imposed by the covertness constraint, we refine the non-covert analysis by Yang et al. (TIT, 2014), by carefully controlling higher-order terms and exploiting the properties of covert outage probability.