# Sparse Multiuser Receiver Design in Large Scale Array System
## 1 Introduction  
The demand for broadband wireless communication has seen explosive growth, accelerating the development of physical layer technology in wireless communication [1]. To improve the communication capacity of wireless networks, mmWave and THz massive multiple-input multiple-output (MIMO) technologies are expected to be promising solutions [2; 3; 4]. This method utilizes the short wavelength property of mmWave and THz, thus allowing more antennas to be placed in a fixed size. In particular, the array element is tens of times larger than traditional microwave arrays, which significantly improves the gain of antenna arrays, and enables multiple beams to serve multiple users simultaneously. Hundreds or even thousands of antennas can be arranged in some scenarios with less strict constraints on physical size. However, despite the apparent advantages of mmWave and THz, there are many challenges to overcome in practice. The massive signal bandwidth in mmWave and THz communication puts more constraints on the sampling rate of the analog-to-digital converter (ADC), digital-to-analog converter (DAC), and other devices. Moreover, high-speed ADC and DAC, along with fewer quantization bits, make it challenging to meet the requirement of high-order modulation, such as orthogonal frequency division multiplexing (OFDM). Instead, simple modulation, such as binary phase shift keying (BPSK) or on-off keying (OOK), is adopted. In addition, in the case of a considerable number of antennas inmassive MIMO, the hardware cost and complexity of all-digital structures will be unacceptably high. Therefore, the hybrid structure is usually applied to simplify the system design. Meanwhile, the number of RF chains can be reduced to a great extent by using shifters and combiners [5]. Furthermore, we tend to adopt the DFT matrix to construct the shifter matrix in a hybrid structure. The massive antenna array can allocate each user an independent RF chain to form a beam pointing to the user. Benefiting from the vast size array as well as narrow beams, we reap good interuser orthogonal performance. However, narrow beams are susceptible to angle changes. Additionally, a tiny fluctuation of angle will cause a sharp deterioration in performance, resulting in an inevitable requirement for high-accuracy channel estimations. While the traditional DFT matrix is applied to building the analog matrix, the number of users will be limited by the number of RF chains. It is necessary to redesign the beam to take advantage of the limited RF chains to service more users. The idea of multi-user beamforming in uplink is to design the received beam pattern without sacrificing any users' signal-noise ratio (SNR). Multicast beamforming in a single group of the network is discussed in [6], where all users request public information from transmitters. Its application is expanded to scenarios with multiple groups of networks in [7; 8], where different user groups request a variety of information from transmitters. There are two different formulated problems in the design of traditional multi-user beams: minimizing the transmission power while meeting the minimum required SNR (QoS problem), and maximizing the minimum SNR that meets the total transmission power limit (MMF problem). Both issues have been proved NP-hard in [6], leading to a flurry of research activities in approximation techniques suitable for different scenarios. In this paper, we reconsider the design problem of multi-user beamforming at the receiver. The basic idea is to maximize the minimum SNR of the received signal to satisfy the shifter's constant modulus constraint. In [6], the two formulated problems are approximated as semi-definite programming (SDP) issues, which can be addressed by the modern interior point method. The authors of [9] developed a successive linear approximation (SLA) algorithm for QoS problems, which is based on second-order cone programming (SOCP).