## 3 Multiple RFI Sources Geolocation Method
### RFI DOA Estimation
\right., \tag{30}\]  
where the relationship between \(\mathbf{\[PERSON]}\) and \(\mathbf{\[PERSON]}_{Z}^{H}\) is:  
\[\mathbf{\[PERSON]}=\mathbf{T}^{-1}\mathbf{\Phi}_{Z}^{H}\mathbf{T}. \tag{31}\]  
Further, we can solve Equation (30) and get:  
\[\mathbf{\Theta}=\left[ \begin{array}{c}\mathbf{D}_{1}\\ \mathbf{D}_{2}\end{array} \right]^{+}\left[ \begin{array}{c}\mathbf{D}_{3}\\ \mathbf{D}_{4}\end{array} \right], \tag{32}\]  
where the symbol \"+\" represents the Moore-Penrose pseudo-inverse of the matrix. It is obvious that the diagonal channels of \(\mathbf{\Phi}_{Z}^{H}\) are identical to the eigenvalues of \(\mathbf{\Theta}\), which can be expressed as [14]:  
\[\mathbf{\Phi}_{Z}^{H}=\mathbf{T}\mathbf{\Theta}\mathbf{T}^{-1}. \tag{33}\]  
According to Equation (31), with the assistance of performing a singular value decomposition on \(\mathbf{\Theta}\), its eigenvalue matrix \(\mathbf{C}_{Z}=diag\big{(} \begin{array}{ccc}C_{1}&\cdots&C_{K}\end{array}  \big{)}\) is the estimation of \(\mathbf{\Phi}_{Z}^{H}\). In accordance with Equation (25), the elevation angle \(\theta^{k}\) of the \(k\)-th RFI source can be estimated:  
\[\hat{\theta}^{k}=\arccos\bigg{(}\frac{angle(C_{k})\cdot\lambda}{2\pi d}\bigg{)}. \tag{34}\]  
Correspondingly, according to Equations (29) and (30), we can also get the relationship between \(\mathbf{D}_{1}\) and \(\mathbf{D}_{2}\), \(\mathbf{D}_{3}\) and \(\mathbf{D}_{4}\):  
\[\left\{ \begin{array}{l}\mathbf{D}_{2}=\mathbf{A}_{Y1}\mathbf{\Phi}_{Y} \mathbf{T}=(\mathbf{A}_{Y1}\mathbf{T})\mathbf{T}^{-1}\mathbf{\Phi}_{Y} \mathbf{T}=\mathbf{D}_{1}\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Y}\mathbf{T}=\mathbf{D }_{1}\mathbf{\[PERSON]}\\ \mathbf{D}_{4}=\mathbf{A}_{Y1}\mathbf{\[PERSON]}_{Y}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{T }=\mathbf{A}_{Y1}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{\[PERSON]}_{Y}\mathbf{T}=\big{(} \mathbf{A}_{Y1}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{T}\big{)}\mathbf{T}^{-1}\mathbf{ \[PERSON]}_{Y}\mathbf{T}=\mathbf{D}_{3}\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Y}\mathbf{T}= \mathbf{D}_{3}\mathbf{\Psi}\end{array} \right., \tag{35}\]  
where \(\mathbf{\Phi}_{Y}\mathbf{\Phi}_{Z}^{H}=\mathbf{\Phi}_{Z}^{H}\mathbf{\Phi}_{Y}\) is used because both \(\mathbf{\Phi}_{Y}\) and \(\mathbf{\Phi}_{Z}\) are diagonal matrices. The solution of Equation (35) is:  
\[\mathbf{\Psi}=\left[ \begin{array}{c}\mathbf{D}_{1}\\ \mathbf{D}_{3}\end{array} \right]^{+}\left[ \begin{array}{c}\mathbf{D}_{2}\\ \mathbf{D}_{4}\end{array} \right]. \tag{36}\]  
Then, the estimation of \(\mathbf{\Phi}_{Y}\) is derived from the eigenvalue matrix \(\mathbf{E}_{Y}=diag(E_{1}\cdots E_{K})\) of the singular value decomposition of \(\mathbf{\Psi}\):  
\[\mathbf{\Phi}_{Y}=\mathbf{T}\mathbf{\Psi}\mathbf{T}^{-1}. \tag{37}\]  
The azimuth angle \(\varphi^{k}\) can be estimated by combining Equations (34) and (37):  
\[\hat{\varphi}^{k}=\arccos\bigg{(}\frac{angle(E_{k})\cdot\lambda}{2\pi d\cdot \sin\hat{\theta}^{k}}\bigg{)}. \tag{38}\]  
It is worth mentioning that Equations (33) and (37) independently perform singular value decomposition of \(\mathbf{\Theta}\) and \(\mathbf{\Psi}\), and their eigenvalues are the estimated \(\mathbf{\Phi}_{Z}^{H}\) and \(\mathbf{\Phi}_{Y}\). However, the order of the eigenvalues and eigenvectors obtained by the two singular value decompositions may not be correspond. It means that after getting the estimation of \(\mathbf{\Phi}_{Z}^{H}\) and \(\mathbf{\Phi}_{Y}\), we need to perform the pair-matching procedure before we can continue to use Equations (34) and (38) to obtain the DOA estimation. For this reason, the eigenvector matrices in Equations (33) and (37) may not be the same, they should be rewritten as:  
\[ \begin{array}{l}\mathbf{\Phi}_{Z}^{H}=\mathbf{T}_{1}\mathbf{\Theta}\mathbf{T}_ {1}{}^{-1}\\ \mathbf{\Phi}_{Y}=\mathbf{T}_{2}\mathbf{\Psi}\mathbf{T}_{2}{}^{-1}\end{array}  \tag{39}\]  
where \(\mathbf{T}_{1}\) and \(\mathbf{T}_{2}\) are the eigenvectors obtained by performing singular value decomposition on \(\mathbf{\Theta}\) and \(\mathbf{\Psi}\), respectively.