# Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems
## 1 Introduction  
Compared with microwave communication systems, millimeter wave (mmWave) communication systems have larger bandwidth and higher data rate [1], receiving great attention from both academic and industry. However, high propagation loss is a major characteristic of mmWave signals [2], which should be compensated for by deploying large antenna arrays to guarantee the communication quality. Fortunately, due to the shorter wavelength of mmWave, more antennas can be integrated into the same space, enabling Massive MIMO technology in mmWave communication systems [3]. Nevertheless, the drastic increase in the number of antennas leads to a high cost, based on which researchers proposed the hybrid beamforming (HBF) technology. mmWave communication is mostly used in line-of-sight (LOS) dominant scenarios owing to their high vulnerability to blockages. However, the source-destination link is probably non-line-of-sight (NLOS). Practically, relay nodes are commonly deployed to solve the problem [3, 4]. Combined with the relay technology, hybrid transceivers cancombat severe attenuation of mmWave signals and extend network convergence. In addition, an mmWave relay system operating in full-duplex (FD) mode can achieve higher spectral efficiency (SE) than the one based on half-duplex (HD) by enabling simultaneous reception and transmission of relay nodes [5]. However, the self-interference (SI) imposed by simultaneous transmission should be mitigated to ensure the communication quality [6]. With effective suppression of SI, the FD technology can reduce end-to-end communication delay and improve spectral efficiency significantly. Unfortunately, the HBF design problem in mmWave relay communication systems has not been extensively investigated. In [7], the authors exploited a transceiver HBF design on the basis of orthogonal matching pursuit (OMP), whose complexity is relatively low while the performance significantly relates to the orthogonality of the pre-defined analog beamforming (ABF) matrices. In [8], a wideband HBF algorithm was adopted in the multiple-relays scenario. The HBF design techniques of a relay node has been explored in [9; 10]. In particular, the alternating direction method of multipliers (ADMM) algorithm was proposed to reduce the minimum square error (MSE) between received signals and transmit signals in [9]. To overcome high complexity, the optimization problem was decomposed into three sub-problems in [10], which is solved by an iterative successive approximation (ISA) algorithm. However, the above two algorithms only optimize the relay node. The authors in [11] developed an HBF algorithm to jointly design all nodes in the mmWave amplify-and-forward (AF) relay system, including the source node, the relay node and the destination node. In [12], the HBF design in mmWave multi-user MIMO relay system was investigated. The ABF of the base station, relay and users could be iteratively designed by solving the weighted minimum MSE (WMMSE) problem. It is noteworthy that all of these works were under accurate channel state information (CSI). Due to channel estimation errors and user mobility, CSI is not available perfectly in practice. The works of [13; 14] studied the robust HBF design for the mmWave FD relay system with the premise of imperfect CSI. To be specific, the work in [13] introduced a robust OMP HBF scheme for mmWave channels with Gaussian-distributed errors to maximize the average receive signal-to-noise ratio (SNR). To reduce complexity, the authors in [14] designed the radio frequency (RF) and baseband separately and showed a robust WMMSE-based algorithm with respect to the sum of rate for the fully connected AF relay networks. On the other hand, the robust design for mmWave HD relay systems were investigated in [15; 16]. Given the impact of SI, a minimum MSE (MMSE) optimization problem for the relay node was deduced in [15]. In [16], the SI power constraint was taken into account in the worst-case sum rate maximization problem.