## 3 Phased Array Comparison of Triangular and Square Lattices  
In this section, a planar array composed of 64 elements (8 \(\times\) 8) is analyzed to better emphasize the advantages of using a triangular lattice. The full-wave electromagnetic simulations were carried out using Ansys HFSS [44]. Each single element consists of a square patch antenna printed on a 0.7 mm thick grounded dielectric layer (RO5880), fed through a coaxial cable. As mentioned in the previous section, the minimum antenna element spacing  
Figure 10: ASLL statistical comparison of square and equilateral triangular lattices by considering the circular sector cell. Figure 9: ASLL statistical comparison of square and equilateral triangular lattices by considering the whole visible region. for a square lattice is 5.17 mm (0.508 \(\lambda_{\rm HF}\)), whereas the equilateral triangular grid provides a minimum element distance of 5.97 mm (0.587 \(\lambda_{\rm HF}\)). Since antenna array design is beyond the scope of this paper, the lattice comparison as a function of frequency was pursued by tuning the antenna array elements to be matched in one narrowband frequency band at time. The finite array with 64 elements, with the antenna elements tuned to 29.5 GHz, is illustrated in Figure 11. Specifically, the \(x\)-\(z\) plane represents the E-plane of the array, whereas \(y\)-\(z\) represents the H-plane. At the beginning, the mutual coupling (\(S_{ij}\)) among antenna elements was addressed. The simulated mutual coupling average value (\(\eta_{Sij}\)) among antenna elements, and the maximum mutual coupling as a function of the frequency, are illustrated in Table 2. It can be seen that a triangular lattice enables us to considerably reduce both the mutual coupling average value and the maximum mutual coupling. For instance, the peak mutual coupling achieved at the highest frequency (29.5 GHz) with a square lattice (\(-14.4\) dB) turns out to be worse than the peak mutual coupling achieved at the lowest frequency (24.25 GHz) for the triangular lattice planar array (\(-14.91\) dB). In general, the mutual coupling difference between triangular and square lattices is more pronounced at the lowest frequencies, and then the coupling difference tends to reduce with the increase in frequency. One of the detrimental effects of mutual coupling is the variation of the antenna elements' input impedance during beam steering. The impact of the employed array lattice on active \(S_{ii}\) parameters (\(i\) = 1,..., 64) was evaluated from a statistical point of view through the cumulative distribution function (CDF) of the active \(S_{ii}\) of all of the antenna array elements during the coverage. From Figure 12, a better statistical behavior of active \(S_{ii}\) parameters in triangular lattice than in square one is clearly recognizable. Indeed, the triangular lattice, thanks to a lower mutual coupling (Table 2), provides an average value of the active \(S_{ii}\) of around 13.8 dB within the investigated frequencies, whereas the square one is characterized by a mean value between \(-11.8\) dB and \(-12.5\) dB. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**24.25 GHz**} & \multicolumn{2}{c}{**26.875 GHz**} & \multicolumn{2}{c}{**29.5 GHz**} \\ \hline  & Tri & Squ & Tri & Squ & Tri & Squ \\ \(\eta_{Sij}\) & \(-48.6\) & \(-42.52\) & \(-49.35\) & \(-44.8\) & \(-50.74\) & \(-46.2\) \\
**Max(\(S_{ij}\))** & \(-14.91\) & \(-11.84\) & \(-15.62\) & \(-12.55\) & \(-16.9\) & \(-14.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulated mutual coupling average values (\(\eta_{Sij}\)) among antenna elements and the maximum value comparison in dB between triangular and square lattice planar arrays composed of 64 elements (\(8\times 8\)). Figure 11: Planar array with 64 elements arranged with (**a**) an equilateral triangular lattice; and (**b**) a square lattice.