# Advanced Noncoherent Detection in Massive MIMO Systems via Digital Beamspace Preprocessing
## 2 System Model
### Channel Model  
The massive MIMO propagation channel is modeled using a geometry-based stochastic approach, more precisely using an adapted version of the COST 2100 channel model [33, 34]. Here,  
Figure 1: Digital beamspace preprocessing in noncoherent massive multiple-input/multiple-output (MIMO) systems: (**a**) Full-array and (**b**) sub-array architecture.  
randomly placed clusters in the topological simulation environment emulate physical scattering objects, where each cluster consists of groups of multi-path components (MPC). Its parameters, such as complex amplitude and angular direction, are computed on the basis of the simulation area. Generally, the clusters can be assigned to the following four categories: (i) local clusters (scatterers surrounding the users and/or the BS), (ii) single clusters (scattering on one object), (iii) twin clusters (scattering on multiple objects), and (iv) common clusters (propagation through the same set of clusters). In addition to multi-path propagation, a line-of-sight (LOS) between the users and the BS can be considered as well. In order to reduce the complexity of the channel and to ease the interpretation of the subsequent numerical results, the propagation via LOS and local clusters around the users are taken into account.  
Considering omni-directional single-antenna users located in the far-field and a BS employing a uniform linear array of directional antennas with pattern \(C_{\text{rx}}(\phi)\), the complex channel vector \(h_{u}=[h_{1,u},\dots,h_{N_{\text{rx}},u}]^{\mathsf{T}}\) of the \(u\)-th user in case of a flat-fading channel is given by  
\[\mathbf{h}_{u} =C_{\text{rx}}(\phi_{u}^{\text{LOS}})\ a_{u}^{\text{LOS}}\ \mathbf{a}_{N_{\text{rx}}}(\phi_{u}^{\text{LOS}})\] \[\quad+\ \sum_{p=1}^{N_{p}}C_{\text{rx}}(\phi_{u,p}^{\text{MPC}})\ a _{u,p}^{\text{MPC}}\ \mathbf{a}_{N_{\text{rx}}}(\phi_{u,p}^{\text{MPC}})\, \tag{1}\]  
where \(a_{u}^{\text{LOS}}\) is the complex amplitude of the LOS component and \(\phi_{u}^{\text{LOS}}\) its corresponding direction of arrival (DOA). The \(p\)-th MPC originating from \(N_{p}\) local clusters is denoted by \(a_{u,p}^{\text{MPC}}\) with DOA \(\phi_{u,p}^{\text{MPC}}\). \(\mathbf{a}_{N_{\text{rx}}}(\phi)\) represents the steering vector of a uniform linear array with \(N_{\text{rx}}\) elements of distance \(d_{\alpha}\), i.e.,  
\[\mathbf{a}_{N_{\text{rx}}}(\phi)=\frac{1}{\sqrt{N_{\text{rx}}}}\left[\begin{array} []{c}1\\ \mathrm{e}^{-\mathrm{j}kd_{\alpha}\sin(\phi)}\\ \vdots\\ \mathrm{e}^{-\mathrm{j}k(N_{\text{rx}}-1)d_{\alpha}\sin(\phi)}\end{array} \right]\, \tag{2}\]  
where \(\phi\) denotes the DOA, \(k=2\pi/\lambda\) is the wavenumber, and \(\lambda\) is the wavelength.  
Since the local clusters are uniformly distributed in the azimuthal plane around the users, \(a_{u,p}^{\text{MPC}}\) and its DOA \(\phi_{u,p}^{\text{MPC}}\) becomes random for different channel realizations, where \(\phi_{u,p}^{\text{MPC}}\) can be generally characterized by an angular spread with standard deviation \(\sigma_{\text{s}}\). On the contrary, \(\phi_{u}^{\text{LOS}}\) is deterministic and the power of the LOS component is adjusted by a LOS-to-MPC ratio (LMR), which is defined as  
\[\text{LMR}=10\log_{10}\frac{|a_{u}^{\text{LOS}}|^{2}}{\sum_{p=1}^{N_{p}}|a_{u, p}^{\text{MPC}}|^{2}}\ \text{dB}\,. \tag{3}\]