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In this paper, we investigate the optimal probabilistic constellation shapingdesign for covert communication systems from a practical view. Different fromconventional covert communications with equiprobable constellations modulation,we propose nonequiprobable constellations modulation schemes to further enhancethe covert rate. Specifically, we derive covert rate expressions for practicaldiscrete constellation inputs for the first time. Then, we study the covertrate maximization problem by jointly optimizing the constellation distributionand power allocation. In particular, an approximate gradient descent method isproposed for obtaining the optimal probabilistic constellation shaping. Tostrike a balance between the computational complexity and the transmissionperformance, we further develop a framework that maximizes a lower bound on theachievable rate where the optimal probabilistic constellation shaping problemcan be solved efficiently using the Frank-Wolfe method. Extensive numericalresults show that the optimized probabilistic constellation shaping strategiesprovide significant gains in the achievable covert rate over thestate-of-the-art schemes.

Optimal Probabilistic Constellation Shaping for Covert Communications