Ergodic secrecy rate for Gaussian MISO wiretap channels with Rician fading

A Gaussian multiple-input single-output (MISO) wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with a single antenna. We study the problem of finding the optimal input covariance that maximizes the ergodic secrecy rate subject to a power constraint, where the full information on the legitimate channel is known to the transmitter, but only statistical information on the eavesdropper channel is available at the transmitter. Existing results address the case in which the eavesdropper channel has independent and identically distributed Gaussian entries with zero-mean, i.e., the channel has trivial covariance. This paper addresses the general case where eavesdropper channel has non-zero mean and nontrivial covariance. We show that the optimal input covariance has always rank one. Based on this, for the non-zero mean but trivial covariance, we reduce the original problem to an one variable optimization problem and propose Newton's method to solve the resulting optimization problem. Numerical results are presented to illustrate the algorithm.