  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 achieves ergodic
secrecy capacity subject to a power constraint where only statistical
information about the eavesdropper channel is available at the transmitter.
This is a non-convex optimization problem that is in general difficult to
solve. Existing results address the case in which the eavesdropper or/and
legitimate channels have independent and identically distributed Gaussian
entries with zero-mean and unit-variance, i.e., the channels have trivial
covariances. This paper addresses the general case where eavesdropper and
legitimate channels have nontrivial covariances. A set of equations describing
the optimal input covariance matrix are proposed along with an algorithm to
obtain the solution. Based on this framework, we show that when full
information on the legitimate channel is available to the transmitter, the
optimal input covariance has always rank one. We also show that when only
statistical information on the legitimate channel is available to the
transmitter, the legitimate channel has some general non-trivial covariance,
and the eavesdropper channel has trivial covariance, the optimal input
covariance has the same eigenvectors as the legitimate channel covariance.
Numerical results are presented to illustrate the algorithm.
