RMOCG: A Riemannian Manifold Optimization-Based Conjugate Gradient Method for Phase-Only Beamforming Synthesis

The phase-only null beamforming synthesis is widely used in communication, radar, and sonar. Due to the phase-only constraint (constant modulus constraint), the problem is nonconvex and NP-hard. The existing methods mainly solve the problem in an indirect way, by relaxing the problem to a more tractable form. Nevertheless, these methods either need huge computational cost or degrade the performance. To break the tradeoffs between the performance and computational complexity, a direct method is proposed without relaxation. In the proposed method, the problem is first transformed into an unconstrained problem on a complex circle manifold. Then, by deriving the gradient descent direction and the step size, a Riemannian manifold optimization-based conjugate gradient algorithm is derived to obtain the direct solution for the transformed problem. Compared with the existing methods, the proposed method can achieve 4 dB null gain reduction and 1 magnitude complexity decreasing.