Riemannian Conjugate Gradient Method Based on Exact Penalty for Phase-Only Beamforming Synthesis

The phase-only null beamforming synthesis plays an important role in suppressing interference within radar systems. This problem poses a challenge as it involves nonconvex constraints, including constant modulus and inequality constraints. Existing methods are mainly divided into two types. The first type is the relaxation method, which inevitably brings relaxation errors. The second type is nonrelaxed methods, which are very sensitive to parameter selection. We noticed that complex circle manifold naturally satisfies the constant module constraint, and further noticed that in actual optimization, the inequality constraint are not necessarily strictly satisfied, and the penalty function method can punish the parts that do not satisfy the constraint. Based on the above considerations, a Riemannian conjugate gradient method based on exact penalty is proposed. Simulation results demonstrate the following advantages of the proposed method: its performance is insensitive to parameter selection; null depth is 10.16 dB deeper than those in Zhong et al. (2022) and An et al. (2023), 12 dB deeper than those in Cao et al. (2017) and Zhang et al. (2021), and 16.81 dB deeper than those in Zhuang et al. (2021) and Lin (2017). Moreover, the advantages of the proposed method become more pronounced as the number of array elements increases.