# DNN Beamforming for LEO Satellite Communication at Sub-THz Bands
## 3 Numerical Results  
The parameters for performing the numerical simulation are mentioned in Table 1. In order to calculate the atmospheric losses due to gases at an angle of arrival \(\theta_{0}\), we have used the ITU recommendation by considering a multi-layered homogeneous atmosphere with total of \(L=922\) layers and a total height of 100 km [42]. The recommendation P676-13 [42] provides the method to compute the absorption loss due to the gaseous atmosphere for the frequency range upto 1000 GHz. The major contributor is the oxygen and water vapor. The reference standard atmosphere as given in the ITU recommendation P.835-6 [45] provide the atmospheric models to be used in the computation of gaseous absorption loss along the Earth-space paths. It provides the calculation of atmospheric parameters i.e., pressure, temperature and water vapor density as a function of height. In order to calculate the pressure, temperature and water vapor profile of the standard atmosphere model, we assume the mean annual global reference atmosphere as given in ITU recommendation [45], which is the most commonly used model [24; 46]. We consider the bandwidth of the signal to be 2 GHz at a center frequency of 100 GHz and the signal occupying the frequency range of 99-101 GHz. The value of absorption loss in the atmospheric channel will vary from 0.98 dB at the elevation angle of 90\({}^{\circ}\) to 5.58 dB at the elevation angle of 10\({}^{\circ}\) for the signal frequency of 100 GHz. The transmitter antenna is considered as the parabolic reflector [47], whose gain can be found as \((\pi D/\lambda)^{2}\). The link-budget calculation is shown in the Appendix A for the given parameters. The data for training and testing of the DNN beamformer are generated by varying the angle \(\theta_{0}\) from 0\({}^{\circ}\) to 80\({}^{\circ}\) in the steps of 0.1\({}^{\circ}\). For each of these AoA, we vary the frequency of the received signal in the range of 99-101 GHz with a step size of 0.1 GHz and generate the corresponding DNN beamformer output voltage \(V_{out}^{DNN}\). This creates 16,821 data samples for the training, testing and validation of the DNN beamformer. For each of these angles \(\theta_{0}\) and frequency of the received signal (99-101 GHz), we also obtain the output voltage of a TTD beamformer \(V_{out}^{TDD}\). The network is then trained to minimize the MSE loss between these two output voltages over all the training samples. Each of the antenna elements has been assumed as an isotropic radiator with unity gain. The entire dataset is divided into a ratio of 70%, 15%, and 15% for training, validation and test respectively. The performance metric for the neural net is the mean square error between the output voltage of a TTD beamformer and the output of the DNN beamformer as shown in (10). \begin{table}
\begin{tabular}{l l l} \hline
**Definition** & **Symbol** & **Value** \\ \hline Transmitter power & \(P_{T}\) & 2 W \\ Transmitter antenna diameter & \(G_{T}\) & 3 m \\ Satellite altitude & \(h_{s}\) & 1000 km \\ Atmospheric height & & 100 km \\ Number of array elements & N & 1000 \\ Antenna efficiency & \(\eta_{T},\eta_{R}\) & 0.9 \\ Center frequency & \(f_{0}\) & 100 GHz \\ Antenna spacing at \(f_{0}\) & \(d\) & 0.5\(\lambda_{0}\) \\ Bandwidth & \(B\) & 2 GHz \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters. A neural network having 10 hidden layers with each layer consisting of 15 neurons is used for the design of DNN beamformer. The value of \(M\) i.e., the last layer of DNN is 15 and the value of \(K\) i.e., the number of training examples in our case corresponds to 11,775. The activation function for each of the neuron is tan-sigmoid function. The training algorithm uses Fletcher-Powell Conjugate Gradient method [48; 49] to minimize the loss function over the training samples as defined in (10). In Figure 3, we show the out-of-sample performance of the proposed DNN beamformer with the output voltage (true) by a TTD beamformer and the output voltage predicted by the DNN beamformer.