# Multi-User Linear Equalizer and Precoder Scheme for Hybrid Sub-Connected Wideband Systems
## 2 System Model
### Channel Model Description  
We assume a channel given by the sum of the contribution of \(N_{cl}\) clusters; each one contributes with \(N_{ray}\) propagation paths. The considered delay-\(d\) MIMO channel matrix of the \(u\)th user can be written as  
\[\mathbf{H}_{u,d}=\sqrt{\frac{N_{tx}N_{rx}}{\rho_{PL}}}\sum_{q}^{N_{cl}}\sum_{ l}^{N_{ray}}\left(\alpha_{j,l}^{u}p_{r}\!\left(dT_{S}-\tau_{q}^{u}-\tau_{q,l}^{u} \right)\mathbf{a}_{tx,u}\!\left(\theta_{q}^{u}-\mathfrak{g}_{q,l}^{u}\right) \mathbf{a}_{rx,u}\!\left(\phi_{q}^{u}-\varphi_{q,l}^{u}\right)\right)\!, \tag{2}\]  
Figure 1: Schematic of the \(u\)th user terminal transmitter. CP is cycle prefix, RF is radio frequency, FFT and IFFT are the fast Fourier transform and inverse fast Fourier transform.  
and the corresponding frequency domain channel matrix \(\mathbf{H}_{u,k}\in\mathbb{C}^{N_{rx}\times N_{rx}}\) of the \(u\)th user at the \(k\)th subcarrier is given by  
\[\mathbf{H}_{u,k}=\sum_{d=0}^{D-1}\mathbf{H}_{u,d}e^{-j\frac{2\pi k}{N_{rx}}d}, \tag{3}\]  
where \(N_{rx}\) represents the number of receive antennas, \(\rho_{PL}\) represents the path-loss between the transmitter and the receiver, \(\alpha_{q,l}^{u}\) 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 [22]. The \(q\)th cluster has a time delay \(\tau_{q}^{u}\), angles of departure \(\theta_{q}^{u}\), and arrival \(\phi_{q}^{u}\). Each ray \(l\) from \(q\)th cluster has a relative time delay \(\tau_{q,l}^{u}\), relative angles of departure \(\delta_{q,l}^{u}\), and arrival \(\varphi_{q,l}^{u}\). The paths delay is uniformly distributed in \([0,DT_{s}]\) where \(D\) denotes the length of the CP, and the angles follow the random distribution mentioned in [22], such that \(\mathbb{E}[\|\mathbf{H}_{u,d}\|_{F}^{2}]=N_{rx}N_{tx}\). Finally, the vectors \(\mathbf{a}_{rx,u}\) and \(\mathbf{a}_{tx,u}\) represent the normalized receive and transmit array response vectors, respectively. For an \(N\)-element uniform linear array (ULA), the array response vector can be given by  
\[\mathbf{a}_{ULA}(\theta)=\frac{1}{\sqrt{N}}\big{[}1,e^{j\Re p\sin(\theta)}, \ldots,e^{j(N-1)\Re p\sin(\theta)}\big{]}^{T}, \tag{4}\]  
where \(k=2\pi/\lambda\), \(\lambda\) is the wavelength, and \(p\) is the inter-element spacing. The channel matrix of the \(u\)th user 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 entry \((q,l)\) that corresponds to the path gain of the \(l\)th ray in the \(q\)th scattering cluster. \(\mathbf{A}_{tx,u}=[\mathbf{a}_{tx,u}(\theta_{1}^{u}-\delta_{1,1}^{u}),\ldots, \mathbf{a}_{tx,u}(\theta_{N_{xl}}^{u}-\delta_{N_{xl},N_{ny}}^{u}))]\) and \(\mathbf{A}_{rx,u}=[\mathbf{a}_{rx,u}(\phi_{1}^{u}-\varphi_{1,1}^{u}),\ldots, \mathbf{a}_{rx,u}(\phi_{N_{xl}}^{u}-\varphi_{N_{xl},N_{ny}}^{u}))]\) hold the transmit and receive array response vectors of the \(u\)th user, respectively.