# Localization of Radio Sources Using High Altitude Platform Station (HAPS)
## 2 Materials and Methods
### Signal Processing System  
The received signal at the antenna is generated based on the propagation characteristics. For simplicity, we considered a scenario where a single transmission source exists, and \(L\) incoming waves arrive at the receiving antenna. Using the propagation distance and the number of reflections obtained from the ray-tracing simulation, the propagation loss \(PL\)\([\mathrm{dB}]\) for each incoming wave \((l=1,2,\ldots,L)\) was calculated using Equation (9). The input power of each incoming wave was then determined using the following equation.  
\[P_{l}=P_{\mathrm{source}}\times 10^{-\left(\frac{P_{l}}{10}\right)}\ [\mathrm{W}]\quad(l=1,2,\ldots,L) \tag{10}\]  
\(P_{\mathrm{source}}\) represents the transmission power of the source. Furthermore, using the phase shift \(\Delta\phi\)\((l=1,2,\ldots,L)\) obtained from the ray-tracing simulation, the complex signal of the \(l\)-\(th\) wave was calculated using the following equation.  
\[F_{l}(t)=\sqrt{P_{l}}\exp(2\pi ft+\Delta\phi_{l})\quad(l=1,2,\ldots,L) \tag{11}\]  
The steering matrix \(A\) for each element of the cylindrical antenna was calculated using the steervec function in MATLAB R2023a's Phased Array System Toolbox. The thermal noise power \(P_{N}\) generated at the receiving antenna was calculated using the following equation.  
\[P_{N}=10\log_{10}(\mathrm{k}TB)+NF\ [\mathrm{dBW}] \tag{12}\]  
\(\mathrm{k},T,B,\) and \(NF\) represent [PERSON]'s constant, the absolute temperature of the noise source, the bandwidth, and the noise figure, respectively. In this study, the absolute temperature at an altitude of 20 km was set to 216.5 \(K\), and the bandwidth was assumed to be 10 MHz for the simulation. The noise figure was set to 3 dB, based on previous research on HAPS systems [20].  
Using the signal of each incoming wave \(F_{l}\), the steering matrix \(A\), and the thermal noise power \(P_{N}\), the input vector \(\mathbf{X}(t)\) at each antenna was calculated using the following equation.  
\[\mathbf{X}(t)=\mathbf{A}\mathbf{F}(t)+\mathbf{N}(t) \tag{13}\]  
\[\mathbf{F}(t)=[\mathbf{F}_{1}(t),\mathbf{F}_{2}(t),\ldots,\mathbf{F}_{L}(t)]^{T} \tag{14}\]  
\(\mathbf{N}(t)\) represents the thermal noise vector, where each component is an independent complex Gaussian process with a mean of 0 and a variance of \(P_{N}\). By substituting random values for \(t\) in Equation (13), the generated signals were used as the complex input signals for each array element. Here, the number of substitutions corresponds to the number of samples.  
The height of the spherical surface, representing the estimated region where the radio source exists, can be derived based on the estimated elevation and azimuth angles. Ideally, this height could be obtained from average terrain elevation data. However, in this simulation, for simplicity, a fixed value was used for calculation. When performing localization using HAPSs, compared to ground-based antenna DOA estimation, the elevation angle is often close to vertical relative to the ground. This characteristic helps minimize horizontal positioning errors even if there is some uncertainty in the source height. Therefore, this approach was adopted in the simulation. Additionally, if neither the direct wave nor the reflected wave (and in some cases, the diffracted wave) reaches the HAPS, or if the received signal power is too low to determine a reliable location, the estimation is considered unsuccessful.  
Furthermore, as shown in Sections 2.3 and 2.4, our method merely uses fundamental linear algebra, that requires no specific additional computation resource. Such computation can be performed onboard at the HAPS and the estimation results then can be sent to the ground-based control center via the communication link (feeder link) between the HAPS and the ground station.