## 2 Triangular vs. Square Lattice Planar Arrays
Furthermore, \(\eta_{Gain}\) grows almost linearly up to \(\gamma=30^{\circ}\), then gradually decreases with the increase of the \(\gamma\) value up to around \(\gamma=45^{\circ}\), where a reversal of the trend occurs. Subsequently, the angular average gain reaches the second peak at \(\gamma=60^{\circ}\), after which it quickly drops again. Therefore, by considering the minimum element spacing (Figure 4) and the angular average gain (Figure 6), it is possible to infer that with a circular sector cell there are two optimal \(\gamma\) values in cases of triangular grids, namely, \(\gamma=30^{\circ}\) and \(\gamma=60^{\circ}\). In fact, these values ensure the largest minimum antenna element distance and,  
Figure 5: RP in the case of an antenna array with 8 \(\times\) 8 isotropic elements arranged in a triangular lattice in the event of (**a**) \(\gamma=30^{\circ}\), and (**b**) \(\gamma=60^{\circ}\). The red circle represents the visible region, whereas the black circle highlights the circular sector cell. hence, the lowest performance degradation due to the elements' mutual coupling [41; 42], in addition to having the highest angular average gain as a function of the frequency. In view of highlighting the impact of the antenna array lattice, two \(8\times 8\) planar arrays were considered--the former with elements arranged on a square lattice, and the latter on an equilateral triangular (\(\gamma=60^{\circ}\)) grid. It is worth noting that, as stated before, in a triangular lattice placement of the elements there are two most suitable values, namely, \(\gamma=30^{\circ}\) and \(\gamma=60^{\circ}\). However, in the following performance comparison, it was decided to use an equilateral triangular lattice due to its more uniform footprint in the \(u\)-\(v\) plane, as shown in Figure 5b. By considering the Equations (3)-(5), with \(R=1.1\), for a square lattice \(d_{0}\) turns out to be \(5.17\) mm (\(0.508\)\(\lambda_{\text{HF}}\)), whereas the equilateral triangular grid exhibits a minimum element distance of \(5.97\) mm (\(0.587\)\(\lambda_{\text{HF}}\)), hence offering a \(13.5\%\) improvement. It is worth observing that values of \(R\) greater than \(1\) assure the absence of grating lobes during the beam steering inside the circular sector [43]. The same \(R\) value for both lattices provides the same distance in the \(u\)-\(v\) plane between the closest grating lobes and the center of the visible region (\(u=0\), \(v=0\)), and hence, the same PSLL value as a function of the beam steering inside the circular sector cell. Furthermore, PSLL is around -13.3 dB at the broadside, whereas with the increase in the \(\theta\) steering angle, PSLL degrades up to around -9 dB for \(\theta=60^{\circ}\) due to the cosine-shaped elements' radiation pattern. The larger minimum distance obtained via a triangular lattice arrangement allows us to achieve a lower level of mutual coupling among antenna elements by ensuring greater robustness of the active impedance of the array elements along the beam steering, a superior linearity of the employed power amplifiers (PAs), and a general improvement of the massive MIMO performance [41]. The wider element distance offered by the triangular lattice can be also considered advantageous for the thermal aspect, by helping the cooling system of the transceiver, which is critical in large mm-wave phased arrays [31]. The overlapped array geometry of triangular and square lattices is illustrated in Figure 7.  
Figure 6: Angular average gain (\(\eta_{Gain}\)) as a function of the \(\gamma\) value within the working frequencies (24.25 –29.5 GHz) in the case of a triangular lattice planar array of \(8\times 8\) elements. The larger array area of the triangular element arrangement also guarantees a better angular resolution and, in a massive MIMO scenario, a reduced angular interval within which users cannot be spatially resolved [2].