  In recent years there has been growing interest in study of multi-antenna
transmit designs for providing secure communication over the physical layer.
This paper considers the scenario of an intended multi-input single-output
channel overheard by multiple multi-antenna eavesdroppers. Specifically, we
address the transmit covariance optimization for secrecy-rate maximization
(SRM) of that scenario. The challenge of this problem is that it is a nonconvex
optimization problem. This paper shows that the SRM problem can actually be
solved in a convex and tractable fashion, by recasting the SRM problem as a
semidefinite program (SDP). The SRM problem we solve is under the premise of
perfect channel state information (CSI). This paper also deals with the
imperfect CSI case. We consider a worst-case robust SRM formulation under
spherical CSI uncertainties, and we develop an optimal solution to it, again
via SDP. Moreover, our analysis reveals that transmit beamforming is generally
the optimal transmit strategy for SRM of the considered scenario, for both the
perfect and imperfect CSI cases. Simulation results are provided to illustrate
the secrecy-rate performance gains of the proposed SDP solutions compared to
some suboptimal transmit designs.
