## 1 Introduction  
In recent years, the massive multiple input multiple output (MIMO) technology has emerged as a highly promising solution for modern wireless communication. With the growing demand for high-speed data transfer and low latency, the implementation of massive MIMO has become increasingly important, especially in millimeter wave (mmWave) communication, which is a crucial aspect for the future of 5G networks. The central idea behind massive MIMO is to equip base stations with a large number of antennas, which allows multiple users to be served at the same time in the same frequency band. This results in a significant increase in both capacity and spectral efficiency compared to traditional MIMO systems. The high number of antennas in a massive MIMO system enables it to provide much higher data rates than traditional MIMO systems [1; 2; 3; 4; 5; 6]. As a result, the system is able to better utilize the available bandwidth and effectively mitigate the effects of fading and interference. In mmWave communication, massive MIMO systems address the problem of severe propagation attenuation and make efficient use of the signal bandwidth [7; 8; 9]. Additionally, massive MIMO is becoming increasingly popular in radar sensing due to its ability to enhance target detection and tracking accuracy, reduce false alarms, increase capacity, and improve coverage [10; 11]. Despite the numerous benefits offered by massive MIMO systems, their practical implementation is a challenging endeavor. In a typical MIMO system, each antenna is equipped with its own radio frequency (RF) chain, composed of components, such as a band-pass filter, a low-noise amplifier, a mixer, a low-pass filter, and a high-resolution analog-to-digital converter (ADC). With the implementation of massive MIMO systems, the number of antennas and RF chains required at each base station is significantly increased, thereby leading to an increase in cost, complexity, and power consumption. To make the implementation of a massive MIMO system practical, one approach is to adopt the hybrid beamforming technique. Hybrid beamforming addresses this limitation by reducing the number of RF chains required in a massive MIMO system. It accomplishes this by splitting the beamforming process into two parts: a digital part and an analog part. In the analog part, the signals from multiple antennas are combined before they are passed through a reduced number of the RF chain. To achieve this effectively, a compressive measurement matrix (CMM), which projects the high-dimensional array received signal onto a low-dimensional signal considering the sparsity nature of the signals. Numerous studies have investigated the design of beamformer and precoder matrices for MIMO systems [12; 13; 14]. The approach in [12] involves alternating optimization to optimize the transmit and receive beamformers using a minimum mean square error (MMSE) criterion between the received signal and the transmitted symbol vectors. Reference [13] considers the optimization of the precoder matrix based on the singular vectors of the channel matrix. In [14], a beamformer is optimized for MIMO-integrated sensing and communication (ISAC) scenarios, where the beamforming matrix is designed to achieve the desired radar beampattern, while maintaining a signal-to-interference-plus-noise ratio constraint for communication users. However, the aforementioned method requires knowledge of the signal directions of arrival (DOAs), which may not be available in many scenarios and is a parameter that needs to be estimated in our problem. Several papers have also explored compressive sampling-based DOA estimation techniques, such as [15; 16]. In [15], a sparse localization framework for the MIMO radar is proposed by randomly placing transmitting and receiving antennas, and a random measurement matrix is used for target localization. Similarly, Ref. [16] develops a compressive sampling framework for 2D DOA and polarization estimation in mmWave polarized massive MIMO systems using a Gaussian measurement matrix.