# A Study on a Radio Source Location Estimation System Using High Altitude Platform Stations (HAPS)
## 3 Numerical Analysis and Results
### Location Estimation Simulation
#### 3.2.3 Signal Processing System  
This section describes the signal processing performed in the location estimation simulation. First, the analytical signal input to the cylinder antenna is obtained by generating a single radio source in a 3D model and considering the case where \(L\) radio waves are incident from the source. The propagation loss \(PL_{l}[dB]\) (\(l=1,2,\ldots,L\)) is obtained from Equation (37) using the propagation distance and other data obtained from the ray tracing simulation, and the input power is calculated by following formula.  
\[P_{l}=10\times 10^{-(\frac{PL_{l}}{10})}\ \ [W]\ (l=1,2,\ldots,L) \tag{37}\]  
Using this value, the propagation time delay against the direct wave \(\tau_{l}(l=1,2,\ldots,L)\), \((\tau_{1}=0)\) and the times of reflection \(m_{l}(l=1,2,\ldots,L)\), the complex waveform of the \(l\)-th wave can be written by the following formula.  
\[F_{l}(t)=\sqrt{P_{l}}exp\{j2\pi f(t-\tau_{l})\}\cdot(-1)^{m_{l}}\ \ (l=1,2,\ldots L) \tag{38}\]  
The array matrix for the cylinder antennas is then expressed by the following equation [12].  
\[\mathbf{A}=[\mathbf{a}(\theta_{1},\phi_{1},f),\ldots,\mathbf{a}(\theta_{L},\phi_{L},f)] \tag{39}\]  
\[\mathbf{a}(\theta_{l},\phi_{l},f)=\bigg{[}\exp\biggl{\{}j2\pi\frac{f}{c}\mathbf{r}_{1 }^{T}\mathbf{L}(\theta_{l},\phi_{l})\biggr{\}}\,\ldots,\exp\biggl{\{}j2\pi\frac{f}{c}\mathbf{r}_{N}^{T}\mathbf{L}(\theta_{l},\phi_{l })\biggr{\}}\ \ \ \bigg{]}^{T}(l=1,2,\ldots,L) \tag{40}\]  
\[\mathbf{L}(\theta,\phi)\triangleq[\text{sin}\theta\text{cos}\phi,\text{sin}\theta \text{sin}\phi,\text{cos}\theta] \tag{41}\]  
where \(\mathbf{r}_{k}(k=1,2,\ldots,N)\) is the position vector of each element, and \(\theta\) and \(\phi\) are the elevation and azimuth angles to the reference point, respectively.  
The thermal noise power \(P_{N}\) generated at the receiving antenna is given by following equation [18].  
\[P_{N}=10 log_{10}(kTB)+NF\ \ \ [dBW] \tag{42}\]  
where \(k,\ T,\ B,\ NF\) are [PERSON]'s constant, absolute temperature of the noise source, bandwidth, and noise figure, respectively. The absolute temperature in the HAPS environment was assumed to be 216.5 [K] and the bandwidth was 10 [MHz]. The noise figure was calculated as 3 dB [19]. As a result, the thermal noise power \(P_{N}\) was about \(-102\) [dBm] in the HAPS environment while in the UAV environment, the thermal noise power was slightly higher since the absolute temperature at the UAV altitude is about 288 [K]. Using Equations (38), (39) and (42), the input vector \(\mathbf{X}(t)\) to each antenna is expressed by the following equation.  
\[\mathbf{X}(t)=\mathbf{A}\mathbf{F}(t)+\mathbf{N}(t) \tag{43}\]\[\mathbf{F}(t)=\left[F_{1}(t),F_{2}(t),\ldots,F_{L}(t)\right]^{T} \tag{44}\]  
In the above equation, \(\mathbf{N}(t)\) is the thermal noise vector, whose components are independent complex Gaussian processes with mean zero and variance (power) of \(P_{N}\). The input vector, obtained by substituting appropriate values for \(t\) in Equation (43) for different time samples, was used as the input complex signal for the cylinder array. The number of samples was set to 100 in this paper. By applying the aforementioned Beamformer and MUSIC methods to the obtained input vector \(\mathbf{X}(t)\), the direction of arrival from the radio source was estimated and used for the location estimation method described in the next section. The direction of arrival with the highest power obtained by the Beamformer and MUSIC methods was used as the estimated direction of arrival.