Optimal Signaling for Covert Communications Under Peak Power Constraint

Covert communications studied in prior works typically consider only the average power constraint on the transmit signal. In this paper, we explore the optimal signaling for covert communication under the peak power constraint, in view of the realistic limitation at the transmitter. Our main result is that the rate-optimal transmit signal distribution under the covertness constraint forms finite hyperspheres, on each of which the points distribute uniformly. To prove this, a lemma is first deduced to explore the equivalence between maximizing the achievable rate and minimizing the covertness measured by the Kullback-Leibler divergence. The equivalence property is then exploited to characterize the covert communication as a specific two-user broadcast channel, which simplifies the intractable covertness constraint. After that, by exploiting the spherical symmetry property of Gaussian noise and the identity theorem of holomorphic functions, the main result is derived. Furthermore, the analytical representation of the optimal signaling in low signal-to-noise ratio (SNR) region is studied following Shamai's approach. For high SNR region, a nonlinear dynamic programming algorithm is developed to generate the optimal signal distribution. For ease of practical implementation, the discrete constellations are optimized based on the sequential quadratic programming algorithm. Simulation results verify the optimality of the proposed hyper-sphere signaling and demonstrate the performance gain of the optimized constellations over typical modulation constellations.