  This paper considers a spectrum sharing based cognitive radio (CR)
communication system, which consists of a secondary user (SU) having multiple
transmit antennas and a single receive antenna and a primary user (PU) having a
single receive antenna. The channel state information (CSI) on the link of the
SU is assumed to be perfectly known at the SU transmitter (SU-Tx). However, due
to loose cooperation between the SU and the PU, only partial CSI of the link
between the SU-Tx and the PU is available at the SU-Tx. With the partial CSI
and a prescribed transmit power constraint, our design objective is to
determine the transmit signal covariance matrix that maximizes the rate of the
SU while keeping the interference power to the PU below a threshold for all the
possible channel realization within an uncertainty set. This problem, termed
the robust cognitive beamforming problem, can be naturally formulated as a
semi-infinite programming (SIP) problem with infinitely many constraints. This
problem is first transformed into the second order cone programming (SOCP)
problem and then solved via a standard interior point algorithm. Then, an
analytical solution with much reduced complexity is developed from a geometric
perspective. It is shown that both algorithms obtain the same optimal solution.
Simulation examples are presented to validate the effectiveness of the proposed
algorithms.
