# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 1 Introduction
### Previous Works on Hybrid Architectures  
In recent years, some works have been proposed for narrowband hybrid analog-digital systems [15; 16; 17; 18; 19; 20; 21; 22]. In [15], the spatially scattering structure of mmWave channels was explored to formulate the hybrid RF/baseband beamforming scheme as a sparse reconstruction signal recovery problem. Then, the principle of orthogonal matching pursuit (OMP) algorithm is applied to develop the algorithms that accurately approximate optimal unconstrained precoders and combiners as a linear combination of beam steering vectors. To handle the inter-user interference, a hybrid MU algorithm was proposed for the uplink in [16]. This algorithm is based on the Gram-Schmidt method to compute the analog vectors, and then the minimum mean square error (MMSE) beamforming is performed over the effective channel. The authors in [17] proposed an efficient iterative algorithm for a single antenna user terminals (UTs) system to overcome the multiple access interference obtained in the scatterer-share circumstance, based on the joint hybrid analog-digital precoder and combiner design for mmWave by exploiting the duality of the uplink and downlink MU-MIMO channels. The OMP algorithm was employed for selecting the analog precoder-combiner pair to increase the channel gain, while the MMSE approach was used to get the digital combiner. A hybrid precoder for the downlink MU-MIMO system based on angles of departure (AoDs) in frequency division duplex (FDD) systems was proposed in [18]. First, the beam steering and zero-interference conditions are assumed, and then a decomposition is performed in the analog and digital parts without losses, as these parts are jointly designed. The authors of [19] proposed beamforming methods, where the phase shifters are replaced by switches. In the first method a sub-connected phase shifter network is combined with a full connected switch architecture to reduce the number of the phase shifters. Then, the full connected switches are replaced by a sub-connected switch network with the aim of simplifying the switch network. A hybrid beamforming for optimizing beam control vectors based on a low complexity codebook and signal to interference and noise ratio (SINR) maximization was proposed in [20]. In the analog part, an optimization algorithm is used to alternately optimize the vectors of the receiver and transmitter. Then, the digital part is computed using the equivalent channel and singular value decomposition (SVD). The authors of [21] proposed a low-complexity hybrid beamforming scheme, considering an alternating MMSE algorithm on the basis of the traditional MMSE algorithm to minimize the MSE of the transmitted and received signals, using the orthogonality of the digital matrix and the idea of an iterative update. In order to maximize the energy efficiency of narrowband mmWave MIMO interference channels involving internet of things (IoT) devices, the authors in [22] designed a low-complexity two-stage hybrid transceiver, considering both perfect and imperfect channel state information (CSI). Although the previous solutions focused on hybrid beamforming approaches for narrowband systems, wideband communications systems are expected for future systems (beyond 5G or 6G). Therefore, the solutions for hybrid mmWave mMIMO wideband systems are of paramount importance. Some works were addressed for hybrid full connected architectures in [23; 24; 25]. In [23], a precoding algorithm is proposed and efficient hybrid codebooks are developed in order to maximize the mutual information that can be obtained for any particular RF codebook. For each subcarrier, different digital precoding is performed in the frequency domain, while the analog precoder remains constant over the subcarriers. In [24], hybrid precoders for downlink space division multi access (SDMA) and orthogonal frequency division multi access (OFDMA) systems were addressed.