The constant modulus constraint is widely used in analog beamforming, hybrid beamforming, intelligent reflecting surface design, and radar waveform design. The quadratically constrained quadratic programming (QCQP) problem is also widely used in signal processing. However, the QCQP with extra constant modulus constraints was not systematically studied in mathematic programming and signal processing. For example, the multiple quality of service (QoS) constrained analog beamforming is rare, while the QoS constrained digital beamforming methods are abundant. We propose to tackle the QCQP with extra constant modulus constraints problem by solving a series of subproblems with linear programming (LP) under extra constant modulus constraints. Under mild condition, the strong duality between the LP with extra constant modulus constraints and its dual problem is established. Then, by using the optimal solutions from the subproblems, the QCQP with extra constant modulus constraints problem is solved with a monotonically converged algorithm, and all converged solutions are K.K.T. points. As an application of the positive semidefinite quadratic form, the mean squared error (MSE) constrained hybrid beamforming is firstly proposed and solved in the past decades. As an application of the indefinite quadratic form, the signal-to-interference-plus-noise-ratio (SINR) constrained hybrid beamforming is solved. Simulation results show that the transmit power of the proposed method is similar to that of the semidefinite relaxation (SDR) method, while the computational time of proposed method is much faster than the SDR method.

## Citation

Xin He was with the College of Electronics and Information Engineering, Shenzhen University, China. 2021.8.5