On the Optimum Secrecy Outage Probability and Ergodic Secrecy Rate over Wireless Channels
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.