Beamforming in underlay cognitive radio: Null-shaping constraints and greedy user selection

In a multiuser cognitive radio setting, multiple secondary systems coexist with multiple primary systems. We assume the secondary systems consist of transmitter-receiver pairs, and each transmitter is equipped with multiple antennas while all receivers use a single antenna. In this setting, the secondary transmitters are to operate under the constraints of producing no interference at the primary users. Such constraints on the secondary systems are referred to as null-shaping constraints. Three results are derived in this work. First, the Pareto optimal operation points for the secondary systems under null-shaping constraints are characterized by real-valued parametrization of the transmission strategies. Second, we show that all points on the Pareto boundary of the secondary systems achievable rate region without interference constraints can be achieved as the outcome of a noncooperative game by imposing certain virtual null-shaping constraints. Third, motivated by the surprising result that all Pareto efficient points can be achieved as a Nash equilibrium, we consider the problem of selecting a subset of the noncooperative secondary users for operation under the objective of maximizing their achievable sum rate. A low complexity suboptimal greedy secondary user selection algorithm is proposed and its performance is illustrated by simulations.