## 2 Triangular vs. Square Lattice Planar Arrays
The array gain as a function of the main beam direction (\(\theta_{0}\), \(\phi_{0}\)) for both arrays of Figure 7 is illustrated in Figure 8 by considering an element pattern equal to \(E(\theta\),\(\phi)=cos(\theta)\). It can be noted that, although the employed array lattices present similar trends during beam steering, the triangular lattice outperforms the square one due to a higher array gain. Furthermore, the gain value presents the highest value along the broadside (\(\theta_{0}=0^{\circ}\)), then decreases during the main beam steering due to beam widening and the approaching of the grating lobes inside the visible region [39]. For a more comprehensive analysis, the impact of the employed lattice on the array gain was also examined from a statistical point of view, by calculating the mean value (\(\eta_{Gain}\)), the variance (\(\sigma^{2}_{Gain}\)), the minimum value (\(min_{Gain}\)), and the maximum value (\(max_{Gain}\)). The results reported in Table 1 emphasize that the triangular lattice enables the attainment of an average linear array gain improvement of around 13 %, when compared to a square lattice. Moreover, as visible from the color maps of Figure 8, the use of a triangular lattice provides both a superior minimum value (\(min_{Gain}\)) and maximum value (\(max_{Gain}\)) to those of a square grid. Figure 8: _Cont._  
Figure 7: Array geometry of a planar array of 64 elements (\(8\times 8\)) in a triangular lattice (\(\gamma=60^{\circ}\)) and a square lattice. Since the user's interference plays a key role in massive MIMO systems, the impact of the employed array lattice on the average sidelobe level (ASLL) was evaluated from a statistical point of view. For the ASLL evaluation, the addressed region was selected for each main beam pointing at a user by the following ellipse equation:  
\[ \begin{array}{l}\frac{(u-u_{0})}{r_{u}^{2}}+\frac{(v-v_{0})}{r_{v}^{2}}>1\\ r_{u}=1.2\frac{\lambda}{L_{x}}\,,\,r_{v}=1.2\frac{\lambda}{L_{y}}\end{array}  \tag{6}\]  
where \(L_{x}\) and \(L_{y}\) represent the array side length along the \(x\) and \(y\) directions; \(\lambda\) is the wavelength; \((u_{0}.v_{0})\) the desired direction in the \(u\)-\(v\) plane in which to steer the main beam; and \(r_{u}\) and \(r_{v}\) identify the main beam in the \(u\) and \(v\) planes, respectively. Once the sidelobe region has been selected for a desired (\(u_{0}\),\(v_{0}\)) direction, the ASLL is calculated by averaging all of the array's radiation pattern values inside the sidelobe region. Moreover, since the array can steer the main beam inside a circular area (Figure 2), we decided to evaluate the ASLL mean (\(\eta_{ASLL}\)), minimum (\(min_{ASLL}\)), and maximum (\(max_{ASLL}\)) values by considering different scan angles (\(N_{\theta}\) = 15, \(N_{\phi}\) = 91) uniformly distributed inside the circular sector through the following equations:  
\[ \begin{array}{l}\eta_{ASLL}=\frac{1}{N_{\theta}N_{\phi}}\sum\limits_{i=1}^{ N_{\theta}}\sum\limits_{j=1}^{N_{\theta}}ASLL(i,j)\\ \min_{ASLL}=\min\{ASLL(i,j)\}\,,\,i=1,2,3,\ldots,N_{\theta},j=1,2,3,\ldots,N_{ \phi}\\ \max_{ASLL}=\max\{ASLL(i,j)\}\,,\,i=1,2,3,\ldots,N_{\theta},j=1,2,3,\ldots,N_{ \phi}\end{array}  \tag{7}\]  
where ASLL(\(i\),\(j\)) represents the ASLL when the array main beam direction is toward the direction identified by (\(i\),\(j\)). The calculated ASLL mean (\(\eta_{ASLL}\)), minimum (\(min_{ASLL}\)), and maximum (\(max_{ASLL}\)) values are illustrated in Figures 9 and 10. Specifically, Figure 9 highlights the ASLL statistical comparison in the event that the sidelobe region is represented by the intersection between Equation (6) and the \(u\)-\(v\) points inside the unit radio's circle, whereas the ASLL assessment of Figure 10 takes into account a sidelobe region represented