## 1 Introduction  
MIMO systems employ multiple antennas at both the transmitter and receiver ends to enhance communication performance. MIMO technology leverages spatial diversity and multiplexing to increase data throughput and link reliability without requiring additional bandwidth or transmit power, making it fundamental in modern wireless communication systems, including 4G and 5G networks. A key component of MIMO systems is beamforming, a signal processing technique used in antenna arrays for directional signal transmission or reception. By combining elements in an antenna array so that signals at particular angles experience constructive interference, beamforming improves signal quality and reduces interference. This spatial filtering capability enhances the communication range and capacity. Building on the MIMO concept, massive MIMO employs hundreds or even thousands of antennas at the base station. This large-scale antenna system significantly boosts spectral and energy efficiency by spatially multiplexing a large number of users. Massive MIMO is crucial for 5G and future networks, enabling high data rates and improved reliability. Massive MIMO and hybrid beamforming are interdependent in modern wireless communication systems, especially at millimeter-wave (mmWave) frequencies. While massive MIMO boosts data rates and spectral efficiency, its practical implementation faces challenges such as high hardware complexity and power consumption due to the need for dedicated radio frequency (RF) chains for each antenna. Hybrid beamforming addresses these challenges by combining analog and digital beamforming techniques, reducing the number of RF chains needed while still achieving high array gains and spatial multiplexing benefits. This makes massive MIMO systems more cost-effective and energy-efficient, facilitating widespread deployment in 5G and beyond. Moreover, hybrid beamforming in mmWave massive MIMO systems effectively addresses practical issues like severe path loss and high power consumption. Recent studies show that hybrid beamforming can achieve a performance close to that of full digital beamforming with significantly lower complexity, enhancing the feasibility of implementing massive MIMO in real-world scenarios [1]. Hybrid beamforming merges analog and digital techniques for efficient signal transmission using compact, cost-effective quantized phase shifters. These phase shifters adjust the signal phase at the antenna level, improving signal quality. Analog weights form beams toward user groups, and digital weights handle MIMO tasks like interference cancellation within the groups. Hybrid setups with fully connected subarray antennas can produce multiple simultaneous beams, with phase-shifting states controlled by digital circuits for real-time adaptation. This integration offers practical, near-optimal performance in modern wireless systems. Advanced MIMO antenna designs integrating phase shifters highlight the potential for enhanced beamforming performance in complex wireless environments [2]. Efficient performance in multi-user mmWave systems involves serving multiple pieces of user equipment (UE) from each base station (BS) simultaneously. Precoding multi-plexes different data streams to different users. However, fully digital baseband beamforming is impractical for multi-stream mmWave systems due to high costs and power consumption [3]. In mmWave systems, the large number of antennas and very low signal-to-noise ratio (SNR) before beamforming make it impractical to obtain full CSI for conventional closed-loop precoding matrix calculations [4]. Therefore, alternative beamforming and precoding techniques are necessary to achieve efficient performance while managing cost, power, and CSI availability. This requirement drives research interest in the field of hybrid beamforming in massive MIMO. Hybrid precoding enables multiplexing multiple data streams by dividing processing between analog and digital domains [5; 6].