# Localization of Radio Sources Using High Altitude Platform Station (HAPS)
## 2 Materials and Methods  
### Antenna Configuration  
The antenna specifications used in the localization simulation are shown in Table 1.  
The reception antenna used in the simulation was modeled as a cylindrical antenna, as employed when utilizing a HAPS as a base station [16]. The basic structure of the cylindrical antenna is shown in Figure 1. In reality, the antenna consists of 31 elements arranged in a circular configuration and 6 elements stacked vertically, but it is simplified for representation.  
For the simulation, an antenna with elements arranged as shown in Figure 2 was used. The specifications of the cylindrical antenna are listed in Table 2. Each element is a patch antenna with directivity, as illustrated in Figure 3. In Table 2, \(\lambda\) represents the signal wavelength.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Tx} & Center Frequency [GHz] & 2 \\  & Frequency Bandwidth [MHz] & 10 \\  & Transmission Power & Scenario dependent \\  & Antenna Type & Isotropic \\  & Antenna Height [m] & 1 \\ \hline \multirow{2}{*}{Rx} & Antenna Type & cylindrical antenna \\  & Antenna Height [m] & 20,000 \\ \hline \multicolumn{2}{c}{Number of Reflections} & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Antenna specifications.  
Figure 1: Schematic diagram of the cylindrical antenna.  
The reception pattern of the patch antenna is given by the following theoretical formula [17]. \(\epsilon_{r}\) represents the permittivity.  
\[E_{\theta}=-jKf(\theta,\phi)\cos\phi\frac{\epsilon_{r}-\sin^{2}\theta}{\epsilon_ {r}-(\sin\theta\cos\phi)^{2}} \tag{1}\]  
\[E_{\phi}=-jKf(\theta,\phi)\cos\theta\sin\phi\frac{\epsilon_{r}}{\epsilon_{r}-( \sin\theta\cos\phi)^{2}} \tag{2}\]  
\(f(\theta,\phi)\) and the constant \(K\) are given by the following equations. Additionally, \(a\) and \(b\) represent the length and width of the antenna element, respectively, as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**Direction/Type** & **Item** & **Value** \\ \hline Horizontal & Number of Elements & 31 \\  & Element spacing & 0.6\(\lambda\) \\ \hline Vertical & Number of Elements & 6 \\  & Element spacing & 0.5\(\lambda\) \\ \hline Bottom & Number of Elements & 10 \\  & Element spacing & 0.3\(\lambda\) \\ \hline \end{tabular}
\end{table}
Table 2: Cylindrical antenna specifications.  
Figure 3: Patch antenna.  
Figure 2: Cylindrical antenna used in the simulation.  
\(V_{0}\) is a value determined by the feeding voltage, \(R\) is the distance from the antenna to the observation point, and \(k_{0}\) is the wavenumber of a plane wave in free space.  
\[f(\theta,\phi)=\left(\frac{\sin u}{u}\right)\cos v \tag{3}\]  
\[u=\left(\frac{k_{0}b}{2}\right)\sin\theta\sin\phi \tag{4}\]  
\[v=\left(\frac{k_{0}a}{2}\right)\sin\theta\cos\phi \tag{5}\]  
\[K=\left(\frac{V_{0}k_{0}b}{\pi}\right)\left(\frac{\exp(-jk_{0}R)}{R}\right) \tag{6}\]  
In this simulation, the relative permittivity was set to \(\epsilon_{r}=2.6\), with the element size in the \(x\)-direction as \(a=0.5\lambda\) and in the \(y\)-direction as \(b=0.6\lambda\). Using these parameters, the directivity gain can be expressed as follows.  
\[G_{d}(\theta,\phi)=\left(\frac{4\pi|E(\theta,\phi)|^{2}}{\int_{0}^{2\pi}d\phi \int_{0}^{\pi/2}|E(\theta,\phi)|^{2}\sin\theta\,d\theta}\right) \tag{7}\]  
The total electric field magnitude is given by \(|E(\theta,\phi)|^{2}=|E_{\theta}(\theta,\phi)|^{2}+|E_{\phi}(\theta,\phi)|^{2}\). The normalized reception pattern obtained from these equations is shown in Figure 4. The maximum gain was adjusted to 8 dBi to ensure consistency in the simulation results.  
Additionally, to ensure that the antenna can direct its beam toward transmission sources located below the HAPS, a modified antenna configuration was also used in the simulation. As shown in Figure 5, this was achieved by tilting the element panel with attached patch antennas, thereby directing the beam downward.  
Figure 4: Antenna pattern.