# Iterative Analog-Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems
## 2 System Model Characterization  
In this manuscript, we consider an uplink mmWave system with \(U\) users sharing the same radio resources and equipped with a single RF chain and \(N_{tx}\) transmit antennas, where each user transmits a single data stream per subcarrier. The base station is equipped with \(N_{rx}^{RF}\) RF chains and \(N_{rx}\) receive antennas, with \(U\leq N_{rx}^{RF}\leq N_{rx}\)[14]. The considered uplink system uses SC-FDMA as the access technique, with \(N_{c}\) available subcarriers. We consider a clustered channel discussed in [23], where the delay--\(d\) MIMO channel matrix of the \(u\)th user, \(\mathbf{H}_{u,d}\), represents the sum of the contribution of \(N_{cl}\) clusters, each of which contribute with \(N_{ray}\) propagation paths, and may be expressed as  
\[\mathbf{H}_{u,d}=\sqrt{\frac{N_{rx}N_{tx}}{\rho_{PL}}}\sum_{q=1}^{N_{cl}}\sum _{l=1}^{N_{ray}}\biggl{(}\alpha_{q,l}^{u}p_{rc}(dT_{s}-\tau_{q}^{u}-\tau_{q,l} ^{u})\ \mathbf{a}_{rx}\biggl{(}\phi_{q,l}^{rx,u},\phi_{q,l}^{rx,u}\biggr{)} \mathbf{a}_{tx}\biggl{(}\phi_{q,l}^{rx,u},\phi_{q,l}^{rx,u}\biggr{)}^{H} \biggr{)}. \tag{1}\]  
where \(\alpha_{q,l}^{u}\) is the complex path gain of the \(l\)th ray in the \(q\)th scattering cluster, and a raised-cosine filter is adopted for the pulse shaping function \(p_{rc}(. )\) for \(T_{S}\)-spaced signaling as in [23]. The \(q\)th cluster has a time delay \(\tau_{q}^{u}\) while each ray \(l\) from \(q\)th cluster has a relative time delay \(\tau_{q,l}^{u}\). The angles \(\phi_{q,l}^{rx,u}\), \(\theta_{q,l}^{rx,u}\), \(\phi_{q,l}^{tx,u}\), and \(\theta_{q,l}^{tx,u}\) are the azimuth and elevation angles of arrival and departure, respectively. For instance, \(\phi_{q,l}^{rx,u}\) has a Laplacian distribution, with mean \(\phi_{q}^{rx,u}\) uniformly distributed in \([0,2\pi]\) and variance \(\sigma_{\phi_{q}^{rx,u}}^{2}\). The remaining angles have similar distributions. The paths delay is uniformly distributed in \([0,DT_{S}]\), and the angles follow the random distribution mentioned in [23], such that \(\mathbb{E}[\|\mathbf{H}_{u,d}\|_{F}^{2}]=N_{rx}N_{tx}\). Finally, the vectors \(\mathbf{a}_{rx,u}\) and \(\mathbf{a}_{tx,u}\) denote the receive and transmit array vectors, respectively. For a uniform planar array (UPA) in the \(yz\)-plane with \(N_{y}\) and \(N_{z}\) elements on the \(y\) and \(z\) axes respectively, the array response vector is given by  
\[\mathbf{a}_{UPA}(\phi,\theta)=\frac{1}{\sqrt{N_{y}N_{z}}}\biggl{[}1,\ldots,e^{2 \pi\frac{\
u}{4}d(\sin{(\phi)}\sin{(\theta)}+\alpha\cos{(\theta)})},\ldots,e^{2 \pi\frac{\
u}{4}d(|N_{y}-1)\sin{(\phi)}\sin{(\theta)}+(N_{z}-1)\cos{(\theta)} )}\biggr{]}^{T}, \tag{2}\]where \(\lambda\) is the wavelength, \(\gamma\) is the inter-element spacing, \(0\leq m<N_{y}\) and \(0\leq n<N_{z}\). The uniform linear array (ULA) is obtained, making \(N_{y}=1\) or \(N_{z}=1\). \(\rho_{PL}\) denotes the path-loss between the transmitter and the receiver separated by a distance \(d\), such that [32]  
\[\rho_{PL,dB}=\rho_{PL0,dB}+10n_{p}\log_{10}\left(\frac{d}{d_{0}} \right)+S_{\sigma_{s}}, \tag{3}\]  
where \(\rho_{PL,dB}=10\log_{10}(\rho_{PL})\) and \(d_{0}\) is the reference distance. \(\rho_{PL0,dB}\) represents the free-space loss at distance \(d_{0}\), \(n_{p}\) is the path-loss exponent and \(S_{\sigma_{s}}\) is the log-normal shadowing with a standard deviation of \(\sigma_{s}(dB)\). The frequency domain channel \(\mathbf{H}_{u,k}\in\mathbb{C}^{N_{rx}\times N_{tx}}\) of the \(u\)th user at the subcarrier \(k\) can be given by,  
\[\mathbf{H}_{u,k}=\sum_{d=0}^{D-1}\mathbf{H}_{u,d}e^{-j\frac{2\pi _{k}}{N_{c}}d}, \tag{4}\]  
Which can also be expressed as  
\[\mathbf{H}_{u,k}=\mathbf{A}_{rx,u}\mathbf{\Delta}_{u,k}\mathbf{A} _{tx,u}^{H}, \tag{5}\]  
where \(\mathbf{\Delta}_{u,k}\) is a diagonal matrix, with entries \((q,l)\) that correspond to the paths gain of the \(l\)th ray in the \(q\)th scattering cluster.