# Multiuser Equalizer for Hybrid Massive MIMO mmWave CE-OFDM Systems
## 2 System Characterization
### Channel Model  
A clustered channel with \(N_{cl}\) clusters, and \(N_{ny}\) rays per cluster is considered [14]. For the \(u\)th UT, at the \(k\)th subcarrier, the channel matrix \(\mathbf{H}_{u,k}\in\mathbb{C}^{N_{rx}\times N_{tx}},k=1,\ldots,N_{c}\) is given in the frequency domain by:  
\[\mathbf{H}_{u,k}=\sum_{d=0}^{D-1}\mathbf{H}_{u,d}e^{-j\frac{2\pi k}{N_{c}}d}, \tag{3}\]  
Figure 1: Block diagram of the \(u\)th user terminal. IFFT: inverse fast Fourier transform; CP: cyclic prefix; RF: radio frequency.  
With  
\[\mathbf{H}_{u,d}=\gamma\sum_{q=1}^{N_{cl}}\sum_{l=1}^{N_{ray}}\alpha_{q,l}p_{rc}(dT _{s}-\tau_{q}-\tau_{q,l})\mathbf{a}_{rx,u}\bigg{(}\big{[}\bigotimes_{q,l^{\prime \prime}}^{rx}\theta_{q,l}^{rx}\big{]}\mathbf{a}_{tx,u}\bigg{(}\big{[}\bigotimes_ {q,l^{\prime}}^{tx}\theta_{q,l}^{tx}\big{)}^{H}, \tag{4}\]  
where \(\gamma=\sqrt{N_{rx}N_{tx}/\big{[}\overline{\rho}_{l}\big{]}_{E}}\) is the normalization factor, which ensures that \(\mathbb{E}\Big{[}\|\mathbf{H}_{u,d}\|_{F}^{2}\Big{]}=N_{rx}N_{tx}\). The path-loss between the \(u\)th UT and the BS is \(\overline{\rho}_{l}\). The function \(p_{rc}(.)\), is the pulse shaping function, being adopted the raised-cosine filter. For the \(l\)th ray in the \(q\)th scattering cluster: \(\alpha_{q,l}\sim\mathcal{CN}\bigg{(}0,\sigma_{q,l}^{2}\bigg{)}\) is the complex path gain, \(\tau_{q}\) is the cluster time delay, while \(\tau_{q,l}\) is the relative time delay, and finally the angles \(\big{[}\bigotimes_{q,l^{\prime}}^{rx}\theta_{q,l^{\prime}}^{rx}\bigotimes_{q, l^{\prime}}^{tx}\big{]}\theta_{q,l}^{tx}\) and \(\theta_{q,l^{\prime}}^{tx}\) are azimuth and elevation angles of arrival and departure, respectively.  
The value \(\sigma_{q,l}^{2}\) is the power of the \(l\)th ray in the \(q\)th scattering cluster, with 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 \(\beta_{q_{1},l_{2}}\) from the \(l_{1}\)th ray to the \(l_{2}\)th cluster of \(q\)th cluster. The path delays are uniformly distributed in \([0,DT_{s}]\), where \(T_{S}\) is the sampling interval and \(D\) is the cyclic prefix length. The angles \(\big{[}\bigotimes_{q,l^{\prime}}^{rx}\theta_{q,l^{\prime}}^{rx}\bigotimes_{q, l^{\prime}}^{tx}\bigotimes_{q,l^{\prime}}^{tx}\bigotimes_{q,l^{\prime}}^{tx}\bigotimes _{q,l^{\prime}}^{tx}\bigotimes_{q,l^{\prime}}^{tx}\bigotimes_{q,l^{\prime}}^{tx}\) have a Laplacian distribution, where for instance, \(\big{[}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes _{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{ rx}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{ \prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}\) uniformly distributed in \([0,2\pi]\) and variance \(\sigma_{\big{[}\bigotimes_{q,l^{\prime}}^{rx}\bigotimes_{q,l^{\prime}}^{rx}}^{ \sigma^{2}}\).  
Finally, the vectors \(\mathbf{a}_{rx}\) and \(\mathbf{a}_{tx}\) are the normalized receive and transmit array response vectors, respectively. For a uniform linear array (ULA), the normalized array response vector is:  
\[\mathbf{a}_{ULA}\bigg{(}\big{[}\bigotimes]\bigg{)}=\frac{1}{\sqrt{N}}\bigg{[}1,e^{j1\frac{2\pi}{\lambda}d\sin\big{(}\bigotimes_{q,l^{\prime}}^{rx}\big{)}}, e^{j2\frac{2\pi}{\lambda}d\sin\big{(}\bigotimes_{q,l^{\prime}}^{rx}\big{)}}, \ldots,e^{j(N-1)\frac{2\pi}{\lambda}d\sin\big{(}\bigotimes_{q,l^{\prime}}^{rx }\big{)}}\bigg{]}^{T}, \tag{5}\]  
where \(N\) is the number of elements of array antennas, \(\lambda\) is the wavelength, and finally \(d\) is the inter-element spacing.