# Multiuser Equalizer for Hybrid Massive MIMO mmWave CE-OFDM Systems
### Previous Work on Hybrid Architectures  
In the recent years, some works based on hybrid architectures have been proposed for narrowband single-user systems [10; 11]. In [10], it is proposed a solution based on the concept of orthogonal matching pursuit (OMP), where the sparse-scattering structure of mmWave channels is exploited to derive both the precoder and combiner. Basically, a number of dominant propagation paths are selected and then they are digitally combined at baseband. To avoid the complexity of the OMP, the authors of [11] proposed a precoder, where the singular value decomposition (SVD) is used to obtain the phase information of the channel.  
Some designs for narrowband multi-user systems were addressed in [12; 13]. In [12], an iterative hybrid equalizer was designed to efficiently remove the multiuser interference. The analog and digital parts of the hybrid equalizer are jointly optimized using the average bit-error-rate as metric. To compute the analog part, a set of vectors are selected from a dictionary based on the array response vectors of the channel. The authors of [13], proposed a precoder only using switches and inverters, which is an alternative to the traditional phase shifters. First, several precoder candidates are randomly generated, based on the initial probability distribution. Then, they are weighted based on their achievable relative sum-rate, and optimized by minimizing the cross entropy.  
Because mmWave channels are expected to be wideband, solutions for frequency-selective channels have also been proposed [14; 15; 16; 17]. In [14], it is proposed a precoder for a single-user system, where first the optimal hybrid precoding design for a given RF codebook is derived, then the hybrid codebooks are designed, and finally, a near-optimal greedy frequency selective hybrid precoding algorithm is proposed, based on Gram-Schmidt orthogonalization. In [15], hybrid precoder and equalizer designs are proposed for single-user and multiuser OFDM systems. The analog beamformer, based on the average of the covariance matrices of the frequency domain channels, is fixed for the entire band, while the digital beamformers are computed on per subcarrier basis. The authors of [16] proposed a hybrid precoder for downlink space-division multi-access and OFDM systems. The total transmit power of the BS is minimized, considering the coverage constraint of signaling and data rate requirements of users. In [17], the beamformers are computed only with the knowledge of the statistics of the channel. Since the dominant eigenvector of each covariance multi-cluster channel matrix is approximately a linear combination of the dominant eigenvectors of the single-cluster subchannels, the multi-cluster channel was decomposed into multiple single-cluster subchannels. Then, these dominant eigenvectors are used to design the beamformers, forming beams in the statistically more important directions.  
All approaches referred to above are related to hybrid systems, where each RF chain is connected to all antennas, named hybrid full-connected architectures. This is the approach adopted in this paper. However, a hybrid subconnected architecture, where each RF chain is connected only to a subset of antennas, is addressed in [18; 39], where two variants are presented: In the first one, each RF chain is connected to a fixed subset of antennas, while in the second one, these connections can change dynamically. In the hybrid subconnected architectures, a lower number of phase shifter and connections are required, however, a better performance can be achieved with the hybrid full-connected architectures.