## 3 Multiple RFI Sources Geolocation Method
### RFI DOA Estimation
\begin{array}{l}\mathbf{S}_{Y1}=\mathbf{A}_{Y1}\mathbf{J}+\mathbf{G }_{Y1}\\ \mathbf{S}_{Y2}=\mathbf{A}_{Y1}\mathbf{\Phi}_{Y}\mathbf{J}+\mathbf{G}_{Y2}\\ \mathbf{S}_{Z1}=\mathbf{A}_{Z1}\mathbf{J}+\mathbf{G}_{Z1}\\ \mathbf{S}_{Z2}=\mathbf{A}_{Z1}\mathbf{\Phi}_{Z}\mathbf{J}+\mathbf{G}_{Z2} \end{array} \right.. \tag{26}\]  
It can be easily found that \(\mathbf{\Phi}_{Y}\) and \(\mathbf{\Phi}_{Z}\) are diagonal matrices consisting of angle information, and DOA can be estimated by solving them. Then, we perform pairwise covariance operations on the signals in Equation (26) to construct the covariance matrix of these signal matrices:  
\[\left\{ \begin{array}{l}\mathbf{W}_{1}=E\Big{[}\mathbf{S}_{Y1}\mathbf{S}_{Z1 }^{H}\Big{]}=\mathbf{A}_{Y1}E\big{[}\mathbf{J}^{H}\big{]}\mathbf{A}_{Z1}^{H}+ E\Big{[}\mathbf{G}_{Y1}\mathbf{G}_{Z1}^{H}\Big{]}=\mathbf{A}_{Y1}\mathbf{R}_{ J}\mathbf{A}_{Z1}^{H}+\mathbf{R}_{g1}\\ \mathbf{W}_{2}=E\Big{[}\mathbf{S}_{Y2}\mathbf{S}_{Z1}^{H}\Big{]}=\mathbf{A}_{Y1 }\mathbf{\Phi}_{Y}E\big{[}\mathbf{J}^{H}\big{]}\mathbf{A}_{Z1}^{H}+E\Big{[} \mathbf{G}_{Y2}\mathbf{G}_{Z1}^{H}\Big{]}=\mathbf{A}_{Y1}\mathbf{\Phi}_{Y} \mathbf{R}_{J}\mathbf{A}_{Z1}^{H}+\mathbf{R}_{g2}\\ \mathbf{W}_{3}=E\Big{[}\mathbf{S}_{Y1}\mathbf{S}_{Z2}^{H}\Big{]}=\mathbf{A}_{Y1 }E\big{[}\mathbf{J}^{H}\big{]}\mathbf{\Phi}_{Z}^{H}\mathbf{A}_{Z1}^{H}+E\Big{[} \mathbf{G}_{Y1}\mathbf{G}_{Z2}^{H}\Big{]}=\mathbf{A}_{Y1}\mathbf{R}_{J} \mathbf{\Phi}_{Z}^{H}\mathbf{A}_{Z1}^{H}+\mathbf{R}_{g3}\\ \mathbf{W}_{4}=E\Big{[}\mathbf{S}_{Y2}\mathbf{S}_{Z2}^{H}\Big{]}=\mathbf{A}_{Y1 }\mathbf{\Phi}_{Y}E\big{[}\mathbf{J}^{H}\big{]}\mathbf{\Phi}_{Z}^{H}\mathbf{A}_ {Z1}^{H}+E\Big{[}\mathbf{G}_{Y2}\mathbf{G}_{Z2}^{H}\Big{]}=\mathbf{A}_{Y1} \mathbf{\Phi}_{Y}\mathbf{R}_{J}\mathbf{\Phi}_{Z}^{H}\mathbf{A}_{Z1}^{H}+ \mathbf{R}_{g4}\end{array} \right. \tag{27}\]  
where \(\mathbf{R}_{J}=E\big{[}\mathbf{J}^{H}\big{]}\) is the autocorrelation matrix of RFI signal vector, \(\mathbf{R}_{g1}\), \(\mathbf{R}_{g2}\), \(\mathbf{R}_{g3}\), and \(\mathbf{R}_{g4}\) are the noise power matrices. The formulas in Equation (27) are combined into a new matrix \(\mathbf{W}\):  
\[\mathbf{W}=\big{[} \begin{array}{ccc}\mathbf{W}_{1}&\mathbf{W}_{2}&\mathbf{W} _{3}&\mathbf{W}_{4}\end{array} \big{]}^{T}. \tag{28}\]  
When it is considered that the signal energy of RFI is much greater than the energy of echo and noise, \(\mathbf{R}_{g1}\), \(\mathbf{R}_{g2}\), \(\mathbf{R}_{g3}\), and \(\mathbf{R}_{g4}\) in \(\mathbf{W}\) can be considered to be approximately zero. On this basis, the singular value decomposition of \(\mathbf{W}\) can get its signal subspace \(\mathbf{D}\)[14; 25]:  
\[\mathbf{D}=\left[ \begin{array}{c}\mathbf{D}_{1}\\ \mathbf{D}_{2}\\ \mathbf{D}_{3}\\ \mathbf{D}_{4}\end{array} \right]=\left[ \begin{array}{c}\mathbf{A}_{Y1} \mathbf{T}\\ \mathbf{A}_{Y1}\mathbf{\Phi}_{Y}\mathbf{T}\\ \mathbf{A}_{Y1}\mathbf{\Phi}_{Y}^{H}\mathbf{T}\\ \mathbf{A}_{Y1}\mathbf{\Phi}_{Y}\mathbf{\Phi}_{Z}^{H}\mathbf{T}\end{array} \right], \tag{29}\]where \(\mathbf{D}_{1}\), \(\mathbf{D}_{2}\), \(\mathbf{D}_{3}\), and \(\mathbf{D}_{4}\) are the singular value decomposition results of \(\mathbf{W}_{1}\), \(\mathbf{W}_{2}\), \(\mathbf{W}_{3}\), and \(\mathbf{W}_{4}\), respectively, and \(\mathbf{T}\) is an invertible eigenvector matrix. The relationship between them can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{D}_{3}=\mathbf{A}_{Y1}\mathbf{\Phi}_{Z}^{H} \mathbf{T}=(\mathbf{A}_{Y1}\mathbf{T})\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Z}^{H} \mathbf{T}=\mathbf{D}_{1}\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{T}= \mathbf{D}_{1}\mathbf{\Theta}\\ \mathbf{D}_{4}=\mathbf{A}_{Y1}\mathbf{\[PERSON]}_{Y}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{T }=(\mathbf{A}_{Y1}\mathbf{\[PERSON]}_{Y}\mathbf{T})\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Z }^{H}\mathbf{T}=\mathbf{D}_{2}\mathbf{T}^{-1}\mathbf{\[PERSON]}_{Z}^{H}\mathbf{T}= \mathbf{D}_{2}\mathbf{\[PERSON]}\end{array}