# A Study on a Radio Source Location Estimation System Using High Altitude Platform Stations (HAPS)
## 2 Methods for Estimating the Location of Radio Transmission Sources
### Estimation Method Using the Beamformer Method  
The Beamformer method is a basic method for estimating the direction of arrival. It estimates the direction of a radio source by scanning the main lobe of a directional antenna with high gain in all directions, as shown in Figure 5, outputting the output voltage change of the array at each angle, and obtaining the maximum output value. Figure 4: Triangulation method. Figure 5: An example of an antenna pattern. For simplicity, the estimation of the direction of arrival using the Beamformer method is described in detail using a K-element linear array antenna, as shown in Figure 6[12]. In this case, the output of the array is \(y(t)\) can be expressed as follows [12]. \[y(t)=\mathbf{W}^{H}\mathbf{X}(t) \tag{5}\]  
\[\mathbf{X}(t)=\left[x_{1}(t),x_{2}(t),\ldots,x_{K}(t)\right]^{T} \tag{6}\]  
\[\mathbf{W}=\left[w_{1},w_{2},\ldots,w_{K}\right]^{T} \tag{7}\]  
\((\cdot)^{H}\) denotes the transpose operation, where the rows and columns of a matrix are swapped. \((\cdot)^{T}\) denotes the Hermitian transpose operation, where the matrix is first transposed and then each element is replaced by its complex conjugate. The output voltage \(P_{out}\) is provided as follows [12]. \[P_{out}=\frac{1}{2}E\Big{[}\big{|}y(t)\big{|}^{2}\Big{]}=\frac{1}{2}\mathbf{W}^{H} R_{xx}\mathbf{W} \tag{8}\]  
\[\mathbf{R}_{xx}=E\Big{[}\mathbf{X}(t)\mathbf{X}^{H}(t)\Big{]} \tag{9}\]  
Assuming that \(L\) waves are incident on the incoming wave and that the respective signal waveforms and angles of arrival are \(F_{l}(t)\) and \(\theta_{l}(l=1,2,\ldots,L)\), the respective direction vectors \(\mathbf{V}_{l}\) can be expressed as follows [12]. \[\mathbf{V}_{l}=\left[\exp\left(-j\frac{2\pi}{\lambda}d_{1}\sin\theta_{l}\right), \ldots,\exp\left(-j\frac{2\pi}{\lambda}d_{K}\sin\theta_{l}\right)\right]^{T} \equiv\mathbf{a}(\theta_{l}) \tag{10}\]  
Also, \(d_{k}(k=1,2,\ldots,K)\) is the distance from the reference array to the position of the \(k\)-th element. At this time, the input vector is expressed as follows [12]. \[\mathbf{X}(t)=\sum_{l=1}^{L}F_{l}(t)\mathbf{a}(\theta_{l})+\mathbf{N}(t) \tag{11}\]  
\[=\mathbf{A}\mathbf{F}(t)+\mathbf{N}(t) \tag{12}\]  
Figure 5:0  
Figure 5:1 K-element linear array. Figure 5:2  
In the above equation, \(\mathbf{N}(t)\) is the thermal noise vector, whose components are independent complex Gaussian processes with zero mean and variance (power) \(\sigma^{2}(=P_{n})\). Also, the correlation matrix \(\mathbf{R}_{xx}\) is expressed as follows [12]. Figure 5:3  
Figure 5:4  
Figure 5:5  
Figure 5:6  
The matrix \(\mathbf{S}\) in Equation (16), which represents the correlation between arriving waves, is called the signal correlation matrix and is a diagonal matrix with \(P_{l}(l=1,2,\dots,L)\) (each input power) in the diagonal components, if all arriving waves are uncorrelated with each other. In order to direct the main lobe of the array antenna to angle \(\theta\) using the above equation and others, the weight \(w_{k}\) expressed in Equation (17) [12] can be multiplied to each input signal based on the common-phase condition (the condition to align the phases to be in-phase). Figure 5:7  
The angle \(\theta\) is varied from \(-\pi/2\) to \(\pi/2\) to find the peak of the output voltage of the array. The output voltage of the array is expressed by Equation (18), and the angular distribution by the Beamformer method is obtained by normalizing Equation (18) [12]. Figure 5:8  
The Beamformer method estimates the direction of arrival and the input power of the arriving wave from the position of the peak of the evaluation function, as shown in Equation (18) using the input correlation matrix \(\mathbf{R}_{xx}\) and the weight expressed in Equation (17).