# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 2 System Model  
In this manuscript, we consider an uplink mmWave mMIMO hybrid analog-digital system that uses GFDM as the access technique. The GFDM block has \(K\) subcarriers, \(M\) timeslots, and \(N\) samples per symbol (\(N=KL\)), where \(L\) is the oversampling factor. This system is designed for \(U\) UTs sharing the same radio resources. Each user transmits one data stream per subcarrier, and has \(N_{tx}\) transmitting antennas and a single RF chain. The base station is equipped with a number of RF chains (\(N_{rx}^{RF}\)) lower or equal than the number of receiving antennas (\(N_{rx}\)), that is, \(U\leq N_{rx}^{RF}\leq N_{rx}\).  
The frequency domain clustered channel matrix of the \(u\)th user is \(\mathbf{H}_{u,l}\in\mathbb{C}^{N_{rx}\times N_{rx}},l=0,\ldots,NM-1\), where \(N_{cl}\) denotes the number of clusters and \(N_{ray}\) denotes the number of paths per cluster, as discussed in [25] and given by  
\[\mathbf{H}_{u,l}=\sum_{d=0}^{D-1}\mathbf{H}_{u,d}e^{-j\frac{2\pi il}{NM}d}, \tag{1}\]  
where \(\mathbf{H}_{u,d}\) is the channel in time domain and can be expressed as  
\[\mathbf{H}_{u,d}=\sqrt{\frac{N_{rx}N_{tx}}{\rho_{PL}}}\sum_{q}^{N_{dq}}\sum_ {r}^{N_{mr}}\Bigl{(}\alpha_{q,r}^{u}p_{rc}(dT_{s}-\tau_{q}^{u}-\tau_{q,r}^{u}) \ \mathbf{a}_{tx,u}(\theta_{q}^{u}-\delta_{q,r}^{u})\mathbf{a}_{rx,u}^{H}( \phi_{q}^{u}-\varphi_{q,r}^{u})\Bigr{)}, \tag{2}\]  
noting that \(\mathbb{E}\Bigl{[}\|\mathbf{H}_{u,d}\|_{F}^{2}\Bigr{]}=N_{rx}N_{tx}\) and the remaining variables are as follows:  
* \(\rho_{PL}\) is the path loss (PL) between the UTs and the BSs;
* \(\alpha_{q,r}^{u}\) is the complex path gain at the \(r\)th ray of the \(q\)th scattering cluster;
* \(p_{rc}(.)\) is the pulse shaping filter function, where \(T_{S,}\tau_{q}^{u}\), and \(\tau_{q,r}^{u}\) are the sampling interval, the time delay of \(q\)th scattering cluster, and the relative time delay, respectively;
* \(\mathbf{a}_{tx,u}(\theta_{q}^{u}-\delta_{q,r}^{u})\) denotes the normalized transmitting array response vector with the AoD \(\theta_{q}^{u}\) and the relative angle of departure \(\delta_{q,r}^{u}\) at the \(r\)th ray of the \(q\)th scattering cluster;
* \(\mathbf{a}_{rx,u}(\phi_{q}^{u}-\varphi_{q,r}^{u})\) denotes the normalized receiving array response vector with the angle of arrival (AoA) \(\phi_{q}^{u}\) and the relative angle of arrival \(\varphi_{q,r}^{u}\) at the \(r\)th ray of the \(q\)th scattering cluster.  
For a uniform linear array (ULA), the normalized array response vector is [15]  
\[\mathbf{a}_{ULA}(\phi)=\frac{1}{\sqrt{N_{mt}}}\Bigl{[}1,e^{i\frac{2\pi}{ \lambda}d\sin(\phi)},e^{i2\frac{2\pi}{\lambda}d\sin(\phi)},\ldots,e^{i(N_{mt} -1)\frac{2\pi}{\lambda}d\sin(\phi)}\Bigr{]}^{T}, \tag{3}\]where \(\lambda\) is the wavelength, \(d\) is the inter element spacing, and \(N_{ant}\) is the number of array antennas elements. The path delays have a uniform distribution in \([0,DT_{s}]\), where \(DT_{s}\) denotes the maximum channel ray delay with significant power and, for the angles, the random distribution presented in [23] is assumed. The power \(\alpha_{q,t}^{u}\), has a decay equal to \(\beta_{q_{1},q_{2}}\) from the \(q_{1}\)th cluster to the \(q_{2}\)th cluster, and a decay equal to \(\beta_{q_{1},r_{1},r_{2}}\) from the \(r_{1}\)th ray to \(r_{2}\)th ray of the \(q\)th cluster.