## 4 Massive MIMO Performance Evaluation  
The comparison of triangular and square lattices was carried out in terms of array gain, ASLL, active \(S_{ii}\) parameters, and matching efficiency, by considering a circular sector in which to serve users. However, one of the key factors that will allow us to drastically improve the data throughput of the upcoming 5G wireless technology will be the use of massive MIMO technology capable of serving different users within the same time-frequency resource through multibeam radiation patterns. In general, various beamforming methods are available for the achievement of a multibeam system [10; 45; 46]. Let us consider a HAPS equipped with an array of \(M=64\) (\(8\times 8\)) antenna elements that serves \(K\) concurrent ground users, equipped with a single isotropic antenna, located inside the above-mentioned circular angular sector (Figure 2), through a multibeam radiation pattern in a line-of-sight (LOS) scenario. The assumption of a LOS channel is consistent since, at mm-wave, the LOS ray represents the predominant mode of propagation between the base station (BS) and the users, due to large path loss as well as the use of high-gain antennas [47; 48]. Furthermore, the LOS channel condition turns out to be further emphasized in the case of HAPS wireless communication [49]. The received signal-to-interference-plus-noise ratio (SINR) for the \(n^{th}\) user in case of a multi-user scenario can be written as [50; 51; 40]:  
\[SINR_{n}=\frac{\delta\eta_{\rm{T}}(\theta_{n},\phi_{n})G(\theta_{tr},\phi_{n}) }{\delta\sum\limits_{i=1}^{K}G(\theta_{i},\phi_{i})\left|h_{i}*W_{n}\right|^{2} +1} \tag{9}\]  
Figure 15: Simulated realized gain for a planar array composed of 64 elements at a broadside direction: (**a**) 24.25 GHz \(\phi=0^{\circ}\); (**b**) 24.25 GHz \(\phi=90^{\circ}\); (**c**) 29.5 GHz \(\phi=0^{\circ}\); and (**d**) 29.5 GHz \(\phi=90^{\circ}\). where \(G(\theta_{n}\phi_{n})\) represents the HAPS array's gain toward the \(n^{th}\) user located at (\(\theta_{n}\),\(\phi_{n}\)); \(\delta\) represents the ratio between the transmitted power (\(P_{t}\)) and the noise power (\(\sigma_{0}\)); \(h_{n}\in 1\times M\) (\(n\) = 1,2,...,\(K\)) corresponds to the channel propagation between the \(n^{th}\) user and the HAPS antenna array; and \(w_{n}\in M\times 1\) consists of the precoding vector that, in general, depends on the selected beamforming method. It is worth noting that, differently from Reference [37], the simulated matching efficiency (\(\eta_{T}\)) was included in (9) in order to achieve a more accurate SINR estimate. To evaluate the system quality of service, a maximum ratio (MR) precoding technique [2] with a perfect channel state information (CSI) was assumed in the following analysis. However, it is worth noting that more efficient precoding and combining algorithms able to reduce interference could be exploited [52] but, in general, they require more complexity and turn out to be more sensitive to channel estimation error. Additionally, since array elements' mutual coupling (MC) undermines the MIMO performance [53] due to the unwanted correlation among elements, the channel matrix \(H\) is modelled as [54]:  
\[H=H_{0}\left(\mathrm{I}_{mxm}-SS^{H}\right) \tag{10}\]  
where \(H_{0}\in K\times M\) denotes the complex channel gain among the users and the HAPS antenna elements in the absence of antenna mutual coupling; \(I_{mxm}\) is an identity matrix; and \(S\in M\times M\) corresponds to the scattering parameters matrix of the HAPS planar array. Unlike [37], the channel matrix was evaluated by taking into account in (10) the full-wave simulation of the planar array, and not just the antenna element distance. Once the SINR is known, the maximum achievable bitrate over 1 Hz of bandwidth for the \(n^{th}\) user, which is the spectral efficiency (SE) per user, is:  
\[SE_{n}=\log_{2}(1+SINR_{n}) \tag{11}\]  
For the SE assessment both the simulated array gain and the \(S\)-matrix of the planar arrays shown in Figure 11 were used.