  In this paper, we consider cognitive beamforming for multiple secondary data
streams subject to individual signal-to-noise ratio (SNR) requirements for each
secondary data stream. In such a cognitive radio system, the secondary user is
permitted to use the spectrum allocated to the primary user as long as the
caused interference at the primary receiver is tolerable. With both secondary
SNR constraint and primary interference power constraint, we aim to minimize
the secondary transmit power consumption. By exploiting the individual SNR
requirements, we formulate this cognitive beamforming problem as an
optimization problem on the Stiefel manifold. Both zero forcing beamforming
(ZFB) and nonzero forcing beamforming (NFB) are considered. For the ZFB case,
we derive a closed form beamforming solution. For the NFB case, we prove that
the strong duality holds for the nonconvex primal problem and thus the optimal
solution can be easily obtained by solving the dual problem. Finally, numerical
results are presented to illustrate the performance of the proposed cognitive
beamforming solutions.
