# DNN Beamforming for LEO Satellite Communication at Sub-THz Bands
## 2 DNN Beamforming
The free-space path loss as a function of AoA is given by [24]  
\[L_{FS}(\theta_{0})=(\frac{4\pi}{\lambda})^{2}\big{[}-R_{0}\cos\theta_{0}+ \sqrt{(R_{0}+h_{s})^{2}-R_{0}^{2}\sin^{2}\theta_{0}}\big{]}^{2} \tag{3}\]  
where \(R_{0}=6371\) km (the radius of the earth) and \(h_{s}\) is the altitude of LEO satellite. The absorption losses over the atmospheric channel depends on many factors including the frequency, gaseous constituents, pressure, temperature, humidity and the slant path length through the atmospheric media. The atmospheric loss can be calculated by considering the atmospheric media as several homogeneous layers of gases and calculating the specific attenuation and path length through each layer. The atmospheric absorption loss of the signal can be expressed as [42]  
\[L_{A}(\theta_{0})=\sum_{i=1}^{L}\gamma_{i}d_{i}(\theta_{0}) \tag{4}\]  
where \(L\) is the number of homogeneous layers of atmospheric media, \(\gamma_{i}\) is the specific attenuation of each of such layer, and \(d_{i}(\theta_{0})\) is the slant path length through each layer for a given AoA \(\theta_{0}\). The proposed DNN beamformer has to be trained in order to mimic the performance of a TTD beamformer so that we need to compute the output voltage of a TTD beamformer to train the network in a supervised manner. The output voltage of the TTD beamformer can be expressed as  
\[V_{out}^{TTD}=\sum_{n=1}^{N}V_{n}e^{-i[\frac{2 nd}{\lambda}(n-1)\cos\theta_{0} -2\pi f\tau_{n}]} \tag{5}\]  
where \(\tau_{n}\) is the time delay provided at the \(n\)th antenna element. These delay elements have to provide a time delay of \((n-1)(d/c)\cos\theta_{0}\) at the \(n\)th antenna element for a constructive interference at the output with no beam squinting problem as occurs with phase shift beamforming. For the phase shift beamformer (PSB), the signal voltage at the output of phase shifter network of antenna array can be expressed as  
\[V_{out}^{PSB}=\sum_{n=1}^{N}V_{n}e^{-i[\frac{2 nd}{\lambda}(n-1)\cos\theta_{0} -\frac{2 nd}{\lambda_{0}}(n-1)\cos\theta_{0}]} \tag{6}\]  
where \(\lambda\) is the wavelength of the signal arriving at the array elements in the direction of \(\theta_{0}\) and \(\lambda_{0}\) is the center frequency at which the phase shifter is designed to provide a phase shift for constructive interference of the signals. We note that when the arriving signal have a frequency of \(c/\lambda_{0}\), we get unity gain in the direction of \(\theta_{0}\) while as the signal frequency deviates, there will be degradation in the gain of beamforming array. For the DNN beamformer, we provide the input to the network as the output voltage amplitude of each of the array elements through an ADC and the output of DNN is the combined voltage signal that is trained to mimic the output of a TTD beamformer. The schematic of DNN beamformer is presented in Figure 2 showing the array elements voltages through ADC fed into the network that is combined to produce a regression output voltage. The input to the each of the neurons in the first layer will be expressed as [43]  
\[f(X)=\sum_{i=1}^{n}W_{i}^{1}V_{i}+b_{1} \tag{7}\]  
where \(W_{i}^{1}\) is the weight matrix for a neuron in the first hidden layer, \(V_{i}\) is the column vector of the voltages obtained from individual antenna array elements, \(n\) is the total number of array elements employed, and \(b_{1}\) is the bias for the first hidden layer. The input received to each neuron is then passed through an activation function such that the output of the neuron can be expressed as  
\[g(X)=\sigma(f(X)) \tag{8}\]  
where \(\sigma\) is the sigmoid function. The output of this first layer of neurons are then fed as the input to the second layer of neurons and so on. The calculation can be repeated for any given number of layers and the output of beamformer \(V_{out}^{DNN}\) can be evaluated.