# Recurrent Deep Learning for Beam Pattern Synthesis in Optimized Antenna Arrays
## 4 Discussion  
Conducting case study analyses in terms of the array factor has the advantage of generalizability, allowing the obtained findings to be efficiently and straightforwardly translated to various applications in different areas without the need to be tied to a specific antenna type or application.  
Deep hidden layers were used in ANNs to provide antenna design parameters for arbitrary beam patterns, which previously may not have been possible. Due to this characteristic in antenna radiation patterns, the proposed network could be used to solve practical antenna problems, such as those related to beamforming and coupling, among others. It is worth noting, furthermore, that the development of DNNs that are capable of generating adaptive beamforming, such as the proposed RNN in this work, makes them suitable for use in dynamic applications, where quasi-optimal solutions may become the better option.  
Improving the beam pattern beforehand through a metaheuristic algorithm helped us enhance the antenna parameters, specifically, the SLL and D. This ultimately translates into a DNN, which is capable of solving interference mitigation and pointing problems. Furthermore, the combined use of EAs with DL allowed us to significantly reduce the computational burden by eliminating the need to simulate a specific result for each possible beam position.  
In the same vein, data pre-conditioning based on the isolation level and directivity allowed us to utilize quality metrics that are even more meaningful for antenna designers. However, the network was not directly trained with these metrics, mainly because they are not differentiable with respect to the input parameters, and this issue has been previously mentioned by some authors [23]. However, it is worth noting, as this behavior is also observed in our analysis, where we used the MSE as the loss function, thus aiding the RNN in generating similar beam patterns to those it was trained with.  
Another notable aspect is that, unlike MLPs, CNNs and LSTMs are neural networks that have been widely employed for time-series analysis. CNNs excel at capturing the relationships among various data points within each time series, allowing them to provide accurate predictions. On the other hand, LSTMs are particularly effective at capturing temporal and sequential relationships in the input data due to their ability to retain past inputs. This capability is crucial for analyzing radiation patterns, as it involves understanding the evolution of patterns over time and modeling long-term dependencies among the input data. Therefore, LSTMs are a favorable choice for beam pattern synthesis, as they can effectively learn and adequately represent the changing complexity of radiation patterns in the antenna array.  
Figure 7: Beam pattern synthesis obtained from input patterns with mutual coupling: (**a**) Test pattern at 65\({}^{\circ}\) and (**b**) test pattern at 115\({}^{\circ}\).  
Finally, before achieving the results shown in Table 1, we trained the proposed architectures with fewer training patterns (e.g., 2264 training patterns), and error levels of magnitude ten times higher were obtained. To solve this problem, we added more training patterns and performed tuning in the neural architectures until we finished with the presented results. This occurred as deep learning essentially involves curve fitting and, for a model to achieve high performance, it must be trained on a comprehensive and dense representation of its input space. In this context, \"dense sampling\" means that the training data should thoroughly encompass the entire manifold of the input data, particularly around the decision boundaries. When the sampling is sufficiently dense, the model can interpret new inputs by interpolating between previously seen training examples [31]. Otherwise, although deep learning approaches have achieved outstanding results in almost all areas of knowledge, deep algorithms still require a large amount of data to achieve good performance [32; 33; 34].