# 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 MUSIC Method  
This section describes in detail the MUSIC method, which uses eigenvalues and eigenvectors to estimate the direction of arrival at a higher angle resolution than the Beamformer method [12]. For the sake of simplicity, let assume a K-element linear array antenna is used, as shown in Figure 6. In this case, the input vector \(\mathbf{X}(t)\) of each antenna is the same as in Equation (12). \[\mathbf{X}(t)=\mathbf{A}\mathbf{F}(t)+\mathbf{N}(t) \tag{19}\]  
The correlation matrix of the input vectors is then as in Equation (15). \[\mathbf{R}_{xx}\triangleq E\Big{[}\mathbf{X}(t)\mathbf{X}^{H}(t)\Big{]}=\mathbf{ASA}^{H}+ \sigma^{2}\mathbf{I} \tag{20}\]  
where the signal correlation matrix \(\mathbf{S}\) can be expressed as follows. \[\mathbf{S}= \begin{bmatrix}E\Big{[}\big{|}F_{1}(t)\big{|}^{2}\Big{]}&E\big{[}F_{1} (t)F_{2}^{*}(t)\big{]}&\cdots&E\big{[}F_{1}(t)F_{L}^{*}(t)\big{]}\\ E\big{[}F_{2}(t)F_{1}^{*}(t)\big{]}&E\Big{[}\big{|}F_{2}(t)\big{|}^{2}\Big{]}& \cdots&E\big{[}F_{2}(t)F_{L}^{*}(t)\big{]}\\ \vdots&\vdots&\ddots&\vdots\\ E\big{[}F_{L}(t)F_{1}^{*}(t)\big{]}&E\big{[}F_{L}(t)F_{2}^{*}(t)\big{]}&\cdots&E \Big{[}\big{|}F_{L}(t)\big{|}^{2}\Big{]}\end{bmatrix}  \tag{21}\]  
In the absence of thermal noise, if the arriving waves are uncorrelated, \(\mathbf{S}\) becomes a diagonal matrix and its rank is \(L\). If the direction matrix \(\mathbf{A}\) also differs from the direction of arrival, its column vectors become independent, and its rank becomes \(L\). Therefore, in this case, the correlation matrix \(\mathbf{R}_{xx}\) is a non-negative definite Hermitian matrix of rank \(L\). Denoting the eigenvalues of this matrix by \(\mu_{i}(i=1,2,\ldots,K)\) and the corresponding eigenvectors by \(\mathbf{e}_{i}(1,2,\ldots,K)\)[12],  
\[\mathbf{ASA}^{H}\mathbf{e}_{i}=\mu_{i}\mathbf{e}_{i}(i=1,2,\ldots K) \tag{22}\]  
and each eigenvalue can be expressed as a real number [12]. \[\mu_{1}\geq\mu_{2}\geq\cdots\mu_{L}>\mu_{L}+1=\cdots=\mu_{K}=0 \tag{23}\]  
The eigenvectors are orthogonal to each other, as follows [12]. \[\mathbf{e}_{i}^{H}\mathbf{e}_{k}=\delta_{ik}\quad(i,k=1,2,\ldots,K) \tag{24}\]  
In the presence of thermal noise, the thermal noise power is added to the eigenvalues and the eigenvectors remain unchanged [12]. \[\mathbf{R}_{xx}\mathbf{e}_{i}=\Big{(}\mathbf{ASA}^{H}+\sigma^{2}\mathbf{I}\Big{)}\mathbf{e}_{i}\]  
\[=\mathbf{ASA}^{H}\mathbf{e}_{i}+\sigma^{2}\mathbf{e}_{i}\]  
\[=\mu_{i}\mathbf{e}_{i}+\sigma^{2}\mathbf{e}_{i}\]  
\[=\Big{(}\mu_{i}+\sigma^{2}\Big{)}\mathbf{e}_{i}\quad(i=1,2,\ldots K) \tag{25}\]  
Then we have:  
\[\lambda_{i}\triangleq\mu_{i}+\sigma^{2}\qquad(i=1,2,\ldots,K) \tag{26}\]  
Denoting the eigenvalues of the correlation matrix by:  
\[\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{L}>\lambda_{L+1}=\cdots= \lambda_{K}=\sigma^{2} \tag{27}\]the number of arriving waves L can be obtained by finding the number of eigenvalues of the correlation matrix that have values larger than the noise power. Also, if we focus on the eigenvectors corresponding to the eigenvalues equal to the thermal noise power:  
\[R_{xx}\mathbf{e}_{i}=\Big{(}ASA^{H}+\sigma^{2}\mathbf{I}\Big{)}\mathbf{e}_{i}=\lambda_{i} \mathbf{e}_{i}=\sigma^{2}\mathbf{e}_{i}\ \ \ \ \ (i=L+1,\ldots,K) \tag{28}\]  
based on Equation (20), we can derive the following property [12]. \[\mu_{1}\geq\mu_{2}\geq\cdots\mu_{L}>\mu_{L}+1=\cdots=\mu_{K}=0 \tag{23}\]0  
Also, from the fact that both \(\mathbf{A}\) and \(\mathbf{S}\) are full ranks. \[\mu_{1}\geq\mu_{2}\geq\cdots\mu_{L}>\mu_{L}+1=\cdots=\mu_{K}=0 \tag{23}\]1  
In other words:  
\[\mu_{1}\geq\mu_{2}\geq\cdots\mu_{L}>\mu_{L}+1=\cdots=\mu_{K}=0 \tag{23}\]2  
From this equation, it can be seen that the eigenvectors corresponding to eigenvalues equal to the thermal noise power are orthogonal to the direction vectors of all arriving waves. Using this property, the evaluation function of the MUSIC method can be expressed as follows [12] with normalization.