## 3 Model of the Digital Beamforming Architecture in System-Level CAD
### Radiation Patterns Analysis  
The first step is focused on evaluating the radiation pattern of the antenna to assess the influence of non-ideal feeding amplitudes and phases provided by the transmitting and receiving chains and the inter-element mutual couplings. The antenna lattice considered is the one reported in Figure 2b. All radiating elements share the same pattern, computed by full-wave e.m. analysis. Slight pattern modifications, which do occur from element to element in the array, are not considered since their effect is negligible. To assess the effect of inter-element coupling, we compare the antenna analyses carried out by considering or neglecting said couplings. Figure 9 presents the broadside radiation pattern results. It is evident that mutual coupling reduces the maximum gain by approximately 1.5 dB and causes a slight increase in side-lobe levels, particularly around \(\pm 37^{\circ}\). The figure also includes a scenario with severely degraded performance due to an artificially intensified coupling effect. This extreme case is created by increasing by 50% the inter-element coupling from its nominal value, numerically computed. This is intended to represent a highly unlikely degraded case condition. Despite this increase in coupling, the radiation pattern metrics exhibit only about a 1 dB reduction in peak gain and a 1.5 dB increase in side-lobe level, which remains a satisfactory outcome under such severe impairment.  
Table 6 provides a comprehensive numerical comparison of the analyses discussed above. The results indicate that, by increasing the beam steering angle, the SLL tends to rise, and this effect is stronger if mutual coupling effects are taken into account. Conversely, the array gain shows relatively less degradation under these conditions. These findings highlight the significant impact of element coupling on radiation pattern characteristics, particularly in off-broadside directions, underscoring the importance of considering coupling in accurate system-level performance evaluations.  
\begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline \multicolumn{2}{c}{**PA**} & \multicolumn{2}{c}{**LO**} & \multicolumn{2}{c}{**ADC/DAC**} & \multicolumn{2}{c}{**IQ Mod.**} & \multicolumn{2}{c}{**LNA**} \\ \hline
**Param.** & **Val.** & **Param.** & **Val.** & **Param.** & **Val.** & **Param.** & **Val.** & **Param.** & **Val.** \\ \hline Gain & 12 & PN @ 10 kHz & \(-70\) & ENOB & 6 & I/ Q PH & 1.7\({}^{\circ}\) & Gain & 29 \\ OP1 dB & 27 & PN @ 100 kHz & \(-96\) & & & I/ Q AM & 0.07 & NF & 1.4 \\ OIP3 & 40 & PN @ 1 MHz & \(-120\) & & & & & OP1 dB & 6 \\  & & & & & & & OIP3 & 11 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Nominal parameters adopted in simulations. Abbreviations: Param. = Parameter, Val. = Value, PN = Phase Noise, ENOB = Effective Number of Bits, I/ Q AM and PH = I/ Q Amplitude and Phase Imbalance. Units: Gains, noise figures, and imbalances are in dB; output powers (OP1 dB, OIP3) are in dBm; PN values are in dB\(\mathrm{/Hz}\). Frequency is in \(\mathrm{Hz}\) (as specified).  
Figure 9: Effect of coupling between array elements in the array radiation.  
Next, we report the results of the simulations in multibeam operation. For this purpose, 13 simultaneous beams are considered, with pointing directions as in Table 7. Beams are chosen at two values for \(\phi\) angle to allow just two cuts to represent all of them. The choice does not impair generality.  
Figure 10 shows the radiation pattern analysis in the multibeam case, both as a full gain map and two \(\phi\) cuts. In the latter it is also possible to appreciate the scan loss, while the slight asymmetry, more apparent on the 45\({}^{\circ}\) cut, is due to the corresponding asymmetry in the radiation pattern of the radiating elements, which, in turn, is due to the asymmetry in the feed pins (Figure 5d).