Globally optimal precoder design with finite-alphabet inputs for cognitive radio networks

This paper investigates the linear precoder design for spectrum sharing in multi-antenna cognitive radio networks with finite-alphabet inputs. It formulates the precoding problem by maximizing the constellation-constrained mutual information between the secondary-user transmitter and secondary-user receiver while controlling the interference power to primary-user receivers. This formulation leads to a nonlinear and nonconvex problem, presenting a major barrier to obtain optimal solutions. This work proposes a global optimization algorithm, namely Branch-and-bound Aided Mutual Information Optimization (BAMIO), that solves the precoding problem with arbitrary prescribed tolerance. The BAMIO algorithm is designed based on two key observations: First, the precoding problem for spectrum sharing can be reformulated to a problem minimizing a function with bilinear terms over the intersection of multiple co-centered ellipsoids. Second, these bilinear terms can be relaxed by its convex and concave envelopes. In this way, a sequence of relaxed problems is solved over a shrinking feasible region until the tolerance is achieved. The BAMIO algorithm calculates the optimal precoder and the theoretical limit of the transmission rate for spectrum sharing scenarios. By tuning the prescribed tolerance of the solution, it provides a trade-off between desirable performance and computational complexity. Numerical examples show that the BAMIO algorithm offers near global optimal solution with only several iterations. They also verify that the large performance gain in mutual information achieved by the BAMIO algorithm also represents the large gain in the coded bit-error rate.