We study the secrecy of wireless channels in the presence of an eavesdropper, where the channels are random and the transmitter only has knowledge of the channel statistics. We investigate the optimal input distribution with respect to several secrecy metrics: the Secrecy Outage Probability (SOP), defined as the probability that the coding rate $r$ exceeds the instantaneous secrecy rate; the Ergodic Secrecy Rate (ESR), defined as the expected secrecy rate over channel realizations; and the Ergodic Positive Secrecy Rate (EPSR), defined as the expected value of the positive part of the secrecy rate. We introduce two partial orderings for random channels: the uniformly less noisy order and the less noisy on average order. We show that when the main channel is uniformly less noisy than the eavesdropper channel, the optimal input distribution is a non-precoded Gaussian input for both the SOP and the EPSR. Furthermore, we show that the same input distribution is optimal for the ESR when the less noisy on average order holds. In addition, similar optimality results for the SOP and the EPSR are obtained for single-transmit-antenna channels without requiring any channel ordering assumptions. Closed-form expressions of the secrecy metrics are derived for special cases of Rayleigh fading channels.

翻译：我们研究了存在窃听者时无线信道的保密性问题，其中信道具有随机性且发射端仅知晓信道统计特性。我们针对若干保密度量指标优化了输入分布：保密中断概率（SOP），定义为编码速率$r$超过瞬时保密率的概率；遍历保密率（ESR），定义为信道实现上的期望保密率；以及遍历正保密率（EPSR），定义为保密率正部期望值。我们引入两种随机信道的偏序关系：一致较弱噪声序和平均较弱噪声序。研究表明，当主信道相对于窃听信道满足一致较弱噪声序时，对于SOP和EPSR的最优输入分布均为非预编码高斯输入。此外，我们证明当平均较弱噪声序成立时，相同输入分布对于ESR也是最优的。在无信道序假设条件下，我们针对单发射天线信道获得了SOP和EPSR的类似最优性结论。针对瑞利衰落信道的特例，推导出了各保密度量指标的闭式表达式。