## 3 Multiple RFI Sources Geolocation Method
### RFI DOA Estimation  
RFI DOA estimation result can be obtained by applying the two-dimensional ESPRIT method to the array signal [24; 25]. The ESPRIT algorithm obtains the incident angle information of the signal through the rotation invariance of the signal subspace of the array [24]. The L-shaped array, as shown in Figure 2, has two orthogonal uniform linear arrays. Each linear array can be regarded as composed of two sub-arrays, the channels on the \(Y\)-axis are divided into two sub-arrays: \(Y_{1}\) and \(Y_{2}\). Similarly, the channels on the \(Z\)-axis are divided into \(Z_{1}\) and \(Z_{2}\) as shown in Figure 4. The special arrangement of all the channels in the L-shaped array makes the signal received by the sub-array contain the DOA information of the RFI, which can be estimated by the two-dimensional ESPRIT method. The following is the DOA estimation process. The Equations (10) and (13) give the signal form received by the channels on the \(Y\)-axis and the \(Z\)-axis in the L-shaped array. Further, the signals received by their sub-arrays \(Y_{1}\), \(Y_{2}\),\(Z_{1}\), and \(Z_{2}\) can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{S}_{Y1}=\left[\begin{array}{ccc}s_{y1}(t_{ r},t_{a})&\ldots&s_{y(M-1)}(t_{r},t_{a})\end{array}\right]^{T}=\mathbf{A}_{Y1} \mathbf{J}+\mathbf{G}_{Y1}\\ \mathbf{S}_{Y2}=\left[\begin{array}{ccc}s_{y2}(t_{r},t_{a})&\ldots&s_{yM}(t_ {r},t_{a})\end{array}\right]^{T}=\mathbf{A}_{Y2}\mathbf{J}+\mathbf{G}_{Y2}\\ \mathbf{S}_{Z1}=\left[\begin{array}{ccc}s_{z1}(t_{r},t_{a})&\ldots&s_{z(M-1) }(t_{r},t_{a})\end{array}\right]^{T}=\mathbf{A}_{Z1}\mathbf{J}+\mathbf{G}_{Z1 }\\ \mathbf{S}_{Z2}=\left[\begin{array}{ccc}s_{z2}(t_{r},t_{a})&\ldots&s_{zM}(t_ {r},t_{a})\end{array}\right]^{T}=\mathbf{A}_{Z2}\mathbf{J}+\mathbf{G}_{Z2} \end{array} \right. \tag{21}\]  
Figure 4: Sub-arrays of the L-shaped array. Each linear array can be regarded as composed of two sub-arrays. The two sub-arrays on the \(Y\)-axis are \(Y_{1}\) and \(Y_{2}\), the two sub-arrays on the \(Z\)-axis are \(Z_{1}\) and \(Z_{2}\). where \(\mathbf{A}_{Y1}\) and \(\mathbf{A}_{Y2}\) are the sub-matrices of \(\mathbf{A}_{Y}\), \(\mathbf{A}_{Z1}\), and \(\mathbf{A}_{Z2}\) are the sub-matrices of \(\mathbf{A}_{Z}\), \(\mathbf{G}_{Y1}\), \(\mathbf{G}_{Y2}\), \(\mathbf{G}_{Y2}\), and \(\mathbf{G}_{Z2}\) are the corresponding noise vectors.\(\mathbf{A}_{Y1}\) and \(\mathbf{A}_{Y2}\) have the following forms:  
\[\mathbf{A}_{Y1}=\left[ \begin{array}{ccc}e^{-i2\pi\frac{d}{\lambda}\sin\theta ^{1}\cos\varphi^{1}}&\dots&e^{-i2\pi\frac{d}{\lambda}\sin\theta^{K}\cos\varphi ^{K}}\\ \vdots&\ddots&\vdots\\ e^{-i2\pi\frac{(M-1)d}{\lambda}\sin\theta^{1}\cos\varphi^{1}}&\dots&e^{-i2\pi \frac{(M-2)d}{\lambda}\sin\theta^{K}\cos\varphi^{K}}\end{array} \right], \tag{22}\]  
\[\mathbf{A}_{Y2}=\left[ \begin{array}{ccc}e^{-i2\pi\frac{d}{\lambda}\sin\theta ^{1}\cos\varphi^{1}}&\dots&e^{-i2\pi\frac{d}{\lambda}\sin\theta^{K}\cos\varphi ^{K}}\\ \vdots&\ddots&\vdots\\ e^{-i2\pi\frac{(M-1)d}{\lambda}\sin\theta^{1}\cos\varphi^{1}}&\dots&e^{-i2\pi \frac{(M-1)d}{\lambda}\sin\theta^{K}\cos\varphi^{K}}\end{array} \right]. \tag{23}\]  
The relationship between Equations (22) and (23) can be written as:  
\[\left\{ \begin{array}{l}\mathbf{A}_{Y2}=\mathbf{A}_{Y1}\mathbf{\Phi}_{Y}\\ \mathbf{\Phi}_{Y}=diag\Big{(}\begin{array}{ccc}e^{-i2\pi\frac{d}{\lambda} \sin\theta^{1}\cos\varphi^{1}}&\dots&e^{-i2\pi\frac{d}{\lambda}\sin\theta^{K} \cos\varphi^{K}}\end{array}\Big{)}\end{array} \right.. \tag{24}\]  
Similarly, the relationship between \(\mathbf{A}_{Z1}\) and \(\mathbf{A}_{Z2}\) can be presented as:  
\[\left\{ \begin{array}{l}\mathbf{A}_{Z2}=\mathbf{A}_{Z1}\mathbf{\Phi}_{Z}\\ \mathbf{\Phi}_{Z}=diag\Big{(}\begin{array}{ccc}e^{-i2\pi\frac{d}{\lambda} \cos\theta^{1}}&\dots&e^{-i2\pi\frac{d}{\lambda}\cos\theta^{K}}\end{array} \Big{)}\end{array} \right.. \tag{25}\]  
So, according to Equations (24) and (25), Equation (21) is further derived as:  
\[\left\{