Beamforming in underlay cognitive radio: Null-shaping design for efficient Nash equilibrium

We consider a multiuser cognitive radio setting where multiple secondary systems, consisting of transmitter-receiver pairs, coexist with multiple primary systems. Each secondary 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 restricted amount of interference at the primary users. We consider two settings. In the first setting, the primary systems are assumed to tolerate an amount of interference originating from secondary systems. This amount of interference is controlled by a pricing mechanism that penalizes the secondary systems in proportion to the interference they produce on the primary users. In the second setting, the primary systems are assumed not to tolerate any interference from secondary systems. Such constraints on the secondary systems are referred to as null-shaping constraints. We characterize for both settings transmission strategies that correspond to Pareto optimal operation points for the secondary systems. The interesting result is that all points on the Pareto boundary can be achieved as the outcome of a non-cooperative game by imposing certain null shaping constraints.