## 2 Triangular vs. Square Lattice Planar Arrays  
The radiation pattern (RP) generated by a planar array with \(N\times M\) elements in the event of uniform amplitude excitation, and when neglecting the mutual coupling, is equal to [39]:  
\[ \begin{split}&\text{RP}(\theta,\phi)=\sum\limits_{n=0}^{N-1}\sum \limits_{m=0}^{M-1}e^{j\varphi_{nm}}E_{nm}(\theta,\phi)\,e^{j\beta F}\\ & F=x_{nm}\sin(\theta)\cos(\phi)+y_{nm}\sin(\theta)\sin(\phi) \end{split}  \tag{1}\]  
where \(\beta=2\pi/\lambda_{0}\) is the phase constant; \(E_{nm}(\theta,\phi)\) represents the elements' radiation pattern; and \(\varphi_{nm}\) represents the phase associated with the (\(n\),\(m\))-_th_ element necessary to steer the main beam toward the desired direction (\(u_{0}\), \(v_{0}\)), which depends on the element'sposition (\(x_{mn}\), \(y_{mn}\)) within the employed lattice. The relation between the phase \(\varphi_{mn}\) and the element's position (\(x_{mn}\), \(y_{mn}\)) is given by the following equation:  
\[ \begin{split}\varphi_{nm}&=\beta(x_{mn}u_{0}+y_{mn}v _{0})\\ u_{0}&=\sin(\theta_{0})\cos(\phi_{0})\\ v_{0}&=\sin(\theta_{0})\sin(\phi_{0})\end{split}  \tag{2}\]  
The minimum distance between elements (\(d_{0}\)) is a fundamental parameter for array radiation performance. Furthermore, \(d_{0}\) should be selected in order to avoid the appearance of grating lobes within the visible region during the beam steering inside the angular sector selected to serve the users. In fact, high lateral lobes do not only provide a considerable reduction in the maximum array gain, but also produce interference for all other users served within the same time-frequency resource. In general, the maximum value of the minimum distance between the antenna elements depends on the maximum steering angle necessary to cover the area of interest. The HAPS scenario differs from that of a 5G ground BS--whose coverage spans 30\({}^{\circ}\) in elevation and 120\({}^{\circ}\) in azimuth [37; 40]--since a circular scan area turns out to be more appropriate [18]. Therefore, it is assumed to cover an angular sector with a maximum steering angle of 60\({}^{\circ}\) off broadside (-60 \(\leq\theta_{0}\leq\) 60\({}^{\circ}\)) for all \(\phi\) angles (0 \(\leq\phi_{0}\leq\) 180\({}^{\circ}\)), as shown in Figure 2 in the \(u\)-\(v\) plane. As stated before, most of the designed phased arrays with uniform lattices rely upon a square arrangement of the antenna elements. In this case, the condition on \(d_{0}\) for avoiding the appearance of grating lobes inside the visible region is given by:  
\[d_{0}\leq\frac{\lambda_{HF}}{R+\sin(\theta_{\max})} \tag{3}\]  
where \(\lambda_{\rm HF}\) represents the wavelength evaluated at the highest frequency (i.e., 29.5 GHz); \(\theta_{\max}\) is the maximum antenna array steering angle, which is 60\({}^{\circ}\) for the considered circular angular sector shown in Figure 2; and \(R\) is a real number that represents the distance in the \(u\)-\(v\) plane between the closest grating lobes and the center of the visible region (\(u=0\), \(v=0\)). Therefore, in order to guarantee the absence of undesirable high lobes inside the visible region, \(R\) has to be set to greater than 1. In the case of an antenna array with uniform spacing but with a triangular grid, the spacing \(d_{x}\) and \(d_{y}\)--namely, the height and the base of the triangle (Figure 3)--are related to the \(\gamma\) angle by the following equations:  
\[ \begin{split}& d_{x}=D\sin(\gamma)\\ & d_{y}=2D\cos(\gamma)\end{split}  \tag{4}\]  
Figure 2: Circular sector cell in the \(u\)–\(v\) plane within which the HAPS antenna array has to steer the main beam to serve the 5G users. where \(D\) is the distance between the (\(n\),\(m\))-\(th\) element and the (\(n\)+1,\(m\))-\(th\), as depicted in Figure 3.  
Differently from a square lattice, there are different choices of grid spacing (\(d_{x}\), \(d_{y}\)) in an array with a triangular grid, which are able to guarantee the absence of grating lobes.