# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 1 Introduction  
The number of users in telecommunication systems has grown rapidly in recent years, which led to work for the next generations of wireless networks in order to achieve the desired requirements in terms of reaching to higher data rates, reliable services, robustness, and spectral efficiency. For that reason, the predict key enabling technologies for the evolution of mobile technologies towards beyond 5G and 6G include the following: (1) the use of millimeter-wave (mmWave) frequency bands, as the conventional sub-6 GHz frequency spectrum, used by cellular communications networks, is currently saturated; and (2) the use of a massive number of antennas, called massive multiple input multiple output (MIMO) (mMIMO) or large-scale MIMO [1]. The implementation of mMIMO is very desirable for the smaller wavelength of the mmWave frequencies as it can improve the link reliability [2] andmitigate the antenna array size restrictions to deal with the possibility of arranging more antennas into the confined space [3; 4].  
It is of paramount importance to look forward to the features of the beamforming techniques introduced by mmWave mMIMO, which has an important and attractive role in combating the limitation of high path loss (PL) [5; 6]. Although the use of mmWave mMIMO is very attractive, the high cost and power consumption of mixed-signal components can be considered the main drawbacks to achieve full digital systems, that is, systems where each antenna is connected with a fully dedicated radio frequency (RF) chain [7; 8]. A simple approach, so-called full analog beamforming with only phase shifters, has been proposed to overcome the large number of RF chains issue [9]. However, the full analog signal processing is just used for single-stream transmission, and its performance is limited by constraints on the amplitude and quantized phase shifters [10]. Then, a new architecture was addressed in [11], so-called hybrid analog-digital architecture, where the number of RF chains is much lower than the number of antennas in order to cope with the hardware constraints. In this approach, part of the signal processing is implemented in the analog domain, while a lower complexity processing is left to the digital domain.  
Concerning wireless channel access and sharing, digital communication technologies have become more sophisticated and reliable, as they can operate with higher spectral efficiency. One of the most relevant specifications in the physical layer of the wireless system is the technology used to access and share the wireless channel. For this reason, the communications systems integrate multi carrier transmission technologies in order to accomplish their target, and thus orthogonal frequency division multiplexing (OFDM) is implemented in 4G (LTE downlink) systems owing to its ability to achieve high data rates transmission [12]. Although the use of OFDM is suitable for transmitting signals via a multipath propagation scenario, where frequency selective fading is verified at the channel response, the large peak signal that formed by the random sum of the phase subcarriers results in a high peak to average power ratio (PAPR). Therefore, a large power back-off is required for the power amplifiers, or then non-linear distortions will degrade the system performance [12]. Moreover, OFDM presents relatively high levels of out-of-band (OOB) emissions induced because of using rectangular pulse shaping filter in the transmitter, which results in strong interference into neighboring frequency bands [13]. That is why other waveform candidates were evaluated and proposed as an alternative of OFDM to overcome such problems, such as constant envelope OFDM (CE-OFDM), filter bank multi carrier (FBMC), universal filtered multi carrier (UFMC), and generalized frequency division multiplexing (GFDM). These modulation schemes are suitable for future systems (beyond 5G/6G), and GFDM can be considered the most flexible non-orthogonal multi carrier transmission scheme, among others [14].