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.

中文翻译：

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峰值功率约束下隐蔽通信的最佳信令

先前工作中研究的隐蔽通信通常仅考虑发射信号的平均功率约束。在本文中，考虑到发射机的实际限制，我们探索了峰值功率约束下隐蔽通信的最佳信令。我们的主要结果是，在隐蔽性约束下，速率最优的发射信号分布形成有限的超球面，每个超球面上的点均匀分布。为了证明这一点，首先推导一个引理来探索最大化可实现速率和最小化由 Kullback-Leibler 散度测量的隐蔽性之间的等价性。然后利用等价性质将隐蔽通信表征为特定的二用户广播信道，这简化了棘手的隐蔽性约束。然后，利用高斯噪声的球对称性和全纯函数的恒等定理，导出了主要结果。此外，按照 Shamai 的方法研究了低信噪比 (SNR) 区域中最优信号传输的分析表示。对于高信噪比区域，开发了非线性动态规划算法来生成最佳信号分布。为了便于实际实现，离散星座基于顺序二次规划算法进行优化。仿真结果验证了所提出的超球面信令的最优性，并证明了优化星座相对于典型调制星座的性能增益。