Optimal Spectrum Sharing in MIMO Cognitive Radio Networks via Semidefinite Programming

In cognitive radio (CR) networks with multiple-input multiple-output (MIMO) links, secondary users (SUs) can exploit "spectrum holes" in the space domain to access the spectrum allocated to a primary system. However, they need to suppress the interference caused to primary users (PUs), as the secondary system should be transparent to the primary system. In this paper, we study the optimal secondary-link beamforming pattern that balances between the SU's throughput and the interference it causes to PUs. In particular, we aim to maximize the throughput of the SU, while keeping the interference temperature at the primary receivers below a certain threshold. Unlike traditional MIMO systems, SUs may not have the luxury of knowing the channel state information (CSI) on the links to PUs. This presents a key challenge for a secondary transmitter to steer interference away from primary receivers. In this paper, we consider three scenarios, namely when the secondary transmitter has complete, partial, or no knowledge about the channels to the primary receivers. In particular, when complete CSI is not available, the interference-temperature constraints are to be satisfied with high probability, thus resulting in chance constraints that are typically hard to deal with. Our contribution is fourfold. First, by analyzing the distributional characteristics of MIMO channels, we propose a unified homogeneous quadratically constrained quadratic program (QCQP) formulation that can be applied to all three scenarios, in which different levels of CSI knowledge give rise to either deterministic or probabilistic interference-temperature constraints. The homogeneous QCQP formulation, though non-convex, is amenable to semidefinite programming (SDP) relaxation methods. Secondly, we show that the SDP relaxation admits no gap when the number of primary links is no larger than two. A polynomial-time algorithm is presented to compute the optimal solution to the QCQP problem efficiently. Thirdly, we propose a randomized polynomial-time algorithm for constructing a near-optimal solution to the QCQP problem when there are more than two primary links. Finally, we show that when the secondary transmitter has no CSI on the links to primary receivers, the optimal solution to the QCQP problem can be found by a simple matrix eigenvalue-eigenvector computation, which can be done much more efficiently than solving the QCQP directly.