# Recurrent Deep Learning for Beam Pattern Synthesis in Optimized Antenna Arrays
## 3 Numerical Experiments
### Deep Neural Network Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of** & **Triangular Pattern** & **Square Pattern** & **Test Pattern 1** & **Test Pattern 2** \\
**Antenna** & **(Figure 5a)** & **(Figure 5b)** & **(Figure 6a)** & **(Figure 6b)** \\ \hline
1: & 7.55\(\angle\)99.55\({}^{\circ}\) & 8.94\(\angle\)82.95\({}^{\circ}\) & 7.99\(\angle\)74.92\({}^{\circ}\) & 8.49\(\angle\)104.92\({}^{\circ}\) \\
2: & 9.04\(\angle\)62.87\({}^{\circ}\) & 8.56\(\angle\)128.55\({}^{\circ}\) & 5.79\(\angle\)91.29\({}^{\circ}\) & 8.80\(\angle\)73.23\({}^{\circ}\) \\
3: & 11.06\(\angle\)75.49\({}^{\circ}\) & 8.02\(\angle\)108.07\({}^{\circ}\) & 8.41\(\angle\)59.00\({}^{\circ}\) & 7.88\(\angle\)91.96\({}^{\circ}\) \\
4: & 6.52\(\angle\)69.78\({}^{\circ}\) & 7.19\(\angle\)90.36\({}^{\circ}\) & 6.27\(\angle\)91.80\({}^{\circ}\) & 6.59\(\angle\)80.65\({}^{\circ}\) \\
5: & 8.70\(\angle\)93.20\({}^{\circ}\) & 10.73\(\angle\)101.71\({}^{\circ}\) & 8.19\(\angle\)105.79\({}^{\circ}\) & 8.44\(\angle\)96.91\({}^{\circ}\) \\
6: & 5.29\(\angle\)91.71\({}^{\circ}\) & 8.29\(\angle\)70.66\({}^{\circ}\) & 8.54\(\angle\)109.25\({}^{\circ}\) & 8.99\(\angle\)82.86\({}^{\circ}\) \\
7: & 6.31\(\angle\)133.32\({}^{\circ}\) & 4.39\(\angle\)107.45\({}^{\circ}\) & 9.34\(\angle\)90.69\({}^{\circ}\) & 8.97\(\angle\)72.30\({}^{\circ}\) \\
8: & 7.23\(\angle\)84.77\({}^{\circ}\) & 9.36\(\angle\)95.39\({}^{\circ}\) & 7.77\(\angle\)120.59\({}^{\circ}\) & 6.85\(\angle\)96.77\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Amplitudes and phases (output data) of the LSTM-RNN. Figure 6: Radiation pattern synthesis resulting from test input patterns: (**a**) Test pattern 1 and (**b**) test pattern 2. To analyze the capabilities of the proposed approach (LMST-RNN), a scenario involving strong interactions between array antenna elements was considered. The analysis, based on the array factor, modeled the geometric and phase interaction between the array elements without considering the specific electromagnetic details of each radiator. This approach provides an initial approximation of how the array elements affect the radiation pattern due to their separation and relative phase. The effects of recurrent deep learning were analyzed with the antenna elements separated by only \(0.3\lambda\) (where the typical distance is \(0.5\lambda\)). By applying the same methodology used previously but changing the distance between elements to \(0.3\lambda\), the results represented in Figure 7 were obtained. From Figure 7, we can clearly observe the effects of the interaction between array elements on the beam pattern in both cases, directly impacting the visible characteristics of the radiation pattern, such as the beam width, through reducing directivity and causing issues related to beam steering despite the pre-conditioning phase. On the other hand, in general, we can observe a significant improvement in the output of the network in both cases, making the beam more directive and adjusting the beam steering. Specifically, in Figure 7a, a beam pattern at \(65^{\circ}\) with geometric and phase coupling interference is shown as input to the network, with an isolation level of \(-12.3\) dB and a directivity of \(6.25\) dB. At the network's output, a better performance was achieved, with a directivity of \(8.33\) dB and an SLL of \(-12.3\) dB. In the case of Figure 7b, there was a similar behavior at \(115^{\circ}\) with an SLL value of \(-12.9\) dB and only \(5.9\) dB in directivity. At the output, a sidelobe level of \(-11.23\) dB and a directivity of \(8.07\) dB were achieved in the beam pattern. It is worth highlighting that geometric and phase coupling is a significant concern in applications where antennas are arranged in close proximity, such as in beamforming arrays, MIMO systems, or satellite communications. In such cases, minimizing this phenomenon is essential to prevent signal quality degradation and to maintain optimal system performance.