Hybrid analog and digital beamforming (HBF) has recently emerged as an attractive technique for millimeter-wave (mmWave) communication systems. It well balances the demand for sufficient beamforming gains to overcome the propagation loss and the desire to reduce the hardware cost and power consumption. In this paper, the mean square error (MSE) is chosen as the performance metric to characterize the transmission reliability. Using the minimum sum-MSE criterion, we investigate the HBF design for broadband mmWave transmissions. To overcome the difficulty of solving the multi-variable design problem, the alternating minimization method is adopted to optimize the hybrid transmit and receive beamformers alternatively. Specifically, a manifold optimization based HBF algorithm is firstly proposed, which directly handles the constant modulus constraint of the analog component. Its convergence is then proved. To reduce the computational complexity, we then propose a low-complexity general eigenvalue decomposition based HBF algorithm in the narrowband scenario and three algorithms via the eigenvalue decomposition and orthogonal matching pursuit methods in the broadband scenario. A particular innovation in our proposed alternating minimization algorithms is a carefully designed initialization method, which leads to faster convergence. Furthermore, we extend the sum-MSE based design to that with weighted sum-MSE, which is then connected to the spectral efficiency based design. Simulation results show that the proposed HBF algorithms achieve significant performance improvement over existing ones, and perform close to full-digital beamforming.Index Terms-Millimeter-wave (mmWave) communications, Minimum mean square error (MMSE), Hybrid analog and digital beamforming (HBF), Alternating optimization, Manifold optimization (MO)

This letter investigates the hybrid analog and digital beamforming (HBF) design for multiuser millimeter wave (mmWave) communication systems based on the minimum mean square error (MMSE) criterion. Using the alternating minimization method, the hybrid precoder of the base station (BS) and the hybrid combiners of the users are alternatively optimized. It is shown that both the optimized digital precoder of the BS and the digital combiners of the users have closed-form expressions, and their corresponding analog ones can be efficiently obtained via generalized eigen-decomposition. Simulation results show that the proposed MMSE HBF scheme has fast convergence and performs close to the fully digital beamforming.

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Hybrid analog and digital beamforming (HBF) has been recognized as an attractive technique offering a tradeoff between hardware implementation limitation and system performance for future broadband millimeter wave (mmWave) communications. In contrast to most current works focusing on the HBF design for orthogonal frequency division multiplexing based mmWave systems, this paper investigates the HBF design for single carrier (SC) systems due to the advantage of low peak-to-average power ratio in transmissions. By applying the alternating minimization method, we propose an efficient HBF scheme based on the minimum mean square error criterion. Simulation results show that the proposed scheme outperforms the conventional HBF scheme for SC systems.Index Terms-Millimeter-wave communications, single carrier, hybrid analog and digital beamforming