# Iterative Analog-Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems
### Related Works  
In this section, we briefly review the state-of-the-art on hybrid analog-digital architectures. Namely, beamforming approaches designed for narrowband single-user mmWave communications systems have been proposed in [11; 12; 13]. Particularly, the authors of [11] designed a bird swarm algorithm based on a matrix-inversion bypass precoder algorithm to overcome the large complexity of orthogonal matching pursuit method, namely the implicit matrix inversion. In [12], a general solution for single user narrowband systems was proposed to convert any existing precoder/combiner designed for the full digital structure into an analog-digital precoder/combiner for the hybrid structure. The nonconvex problem is decoupled into a series of convex subproblems and then solved by a singular value decomposition-based technique to obtain an initial solution. Next, the phase increment of each entry in the RF precoder/combiner from iteration to iteration is restricted. The approach discussed above assumed a fully connected hybrid architecture, where each RF chain is connected to all antennas. However, this architecture may require a large number of connections; thus, one possible solution is the use of the subconnected hybrid architectures presented in [13], where each RF chain is connected only to a subset of antennas. Transmit and receive beamforming approaches designed for narrowband multi-user mmWave communications systems have also been proposed in [14; 15; 16; 17; 18; 19; 20; 21]. The authors of [14] designed an uplink receive beamforming with single antenna user terminals (UTs) that consider the multi-user interference at two stages: analog and digital. In [15], it was proposed four different digital precoding techniques with a hybrid beamformer for multicell systems, to reduce inter-beam interference. An iterative matrix decomposition based on the subspace projection of angles of departure (AoD) of the channel paths to perform the block diagonalization was proposed for downlink mmWave narrowband channel was proposed in [16] to obtain interference free channels for the UTs. The authors of [17] designed an iterative hybrid precoder and combiner algorithm by exploiting the duality of the uplink and downlink multi-user mmWave MIMO channels. A hybrid weighted minimum mean square error (MMSE) precoding and combining scheme was proposed in [18]. In this algorithm, the hybrid precoderand combiners are optimized and updated in an iterative manner to minimize a weighted sum-MSE cost function. The authors of [19] proposed a linear detector for uplink systems, where in order to avoid the direct matrix inverse of the MMSE, which is computationally expensive, they proposed an improved Jacobi iterative algorithm. This algorithm works by accelerating the convergence rate of the signal detection process, and where a whole-correction method is applied to update the iterative process. In [20], it was proposed a precoder for a downlink multiuser system, with partially connected hybrid beamforming architecture, and where the analog part of precoder is composed by both variable phase shifters and constant phase shifters. Additionally, the sum rate was considered as a metric and a greedy algorithm was employed to reduce the complexity of the algorithm, since the exact solution of the combinatorial problem is mathematically intractable. A hybrid iterative block space-time equalizer, based on IB-DFE principles [22], was proposed [21]. The previous works mainly focused on hybrid beamforming approaches for either single or multi-user narrowband systems. However, the design of a hybrid solution for mmWave mMIMO wideband communications is of paramount importance. Hybrid approaches specifically designed for single user can be found in [23; 24; 25]. Hybrid precoding solutions and codebooks for limited feedback wideband mmWave systems were discussed in [23]. It was assumed that the digital precoding is performed in the frequency domain and can be different for each subcarrier, and the analog precoder is constant over the subcarriers.