Fundamental Limits of Covert Communication over MIMO AWGN Channel
Fundamental limits of covert communication have been studied in literature for different models of scalar channels. It was shown that, over $n$ independent channel uses, $\mathcal{O}(\sqrt{n})$ bits can transmitted reliably over a public channel while achieving an arbitrarily low probability of detection (LPD) by other stations. This result is well known as square-root law and even to achieve this diminishing rate of covert communication, some form of shared secret is needed between the transmitter and the receiver. In this paper, we establish the limits of LPD communication over the MIMO AWGN channel. We define the notion of $\epsilon$-probability of detection ($\epsilon$-PD) and provide a formulation to evaluate the maximum achievable rate under the $\epsilon$-PD constraint. We first show that the capacity-achieving input distribution is the zero-mean Gaussian distribution. Then, assuming channel state information (CSI) on only the main channel at the transmitter, we derive the optimal input covariance matrix, hence, establishing the $\epsilon$-PD capacity. We evaluate $\epsilon$-PD rates in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO). We show that, in the asymptotic block-length regime, while the SRL still holds for the MIMO AWGN, the number of bits that can be transmitted covertly scales exponentially with the number of transmitting antennas. Further, we derive the $\epsilon$-PD capacity \textit{with no shared secret}. For that scenario, in the massive MIMO limit, higher covert rate up to the non LPD constrained capacity still can be achieved, yet, with much slower scaling compared to the scenario with shared secret. The practical implication of our result is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay.