Optimizing beamforming in quaternion signal processing using projected gradient descent algorithm

Recent advances in quaternion signal processing have drawn attention to the Quaternion Beamforming Problem (QBP). By leveraging appropriate relaxation techniques, QBP can be transformed into a constrained quaternion matrix optimization problem, aiming to develop a simple and effective solution. To this end, this paper first establishes a comprehensive theory of convex optimization for quaternion matrices based on the GHR calculus, covering quadratic upper bounds and projection theorems. In particular, we propose a quaternion projected gradient descent (QPGD) for constrained quaternion matrix optimization problems and prove the convergence of the QPGD algorithms, showing the monotonic decrease of the objective function. The numerical experiments verify the applicability and effectiveness of the QPGD algorithm in solving constrained quaternion matrices least squares problems in Frobenius norm and the quaternion beamforming problem.