# Localization of GNSS Spoofing Interference Source Based on a Moving Array Antenna
## 3 Proposed Methods
### Spoofing Interference Direct Location Based on Carrier Phase Single-Difference  
The method described in Section 3.1 enables the extraction and determination of spoofing interference signals from the received GNSS signals by the array antenna. Expanding on this, in this section, we directly obtain the position of the spoofing interference source by fusing raw carrier phase observations using a moving antenna array. Due to the dynamic observing scenario in this paper, there are some carrier phase observation errors and array antenna positioning errors. We use carrier phase single-difference measurements to construct the observation equations for the spoofing source and directly determine the location of the spoofing source by moving the antenna array and combining the observation equations. Compared with traditional two-stage positioning methods, the direct location method avoids spatial correlation of signal estimation parameters and enhances the robustness of spoofing localization in dynamic scenarios [23]. We set up the scenario model for the spoofing interference source location in this paper, illustrated in Figure 6, employing the same Cartesian coordinate system approach as depicted in Figure 2. In this model, we designate the initial position \(\mathbf{p}_{1}\) of antenna element 0 as the origin. \(\mathbf{p}_{t}\) represents the vector of coordinates corresponding to the location of the spoofing source. \(\mathbf{p}_{j}\) denotes the \(j\)-th signal acquisition position of element 0, while \(M\) signifies the total number of observations. Figure 6: Scenario model of spoofing interference direct localization. The observation equation can be established at observation position \(\mathbf{p_{j}}\) by utilizing the carrier phase single-difference measurement \(\varphi_{j,i}\) of array element \(i\). The formulated expression is as follows:  
\[\left\|\mathbf{p_{t}}-\mathbf{p_{j,i}}\right\|-\left\|\mathbf{p_{t}}-\mathbf{p_{j}}\right\|= \frac{\lambda\varphi_{j,i}}{2\pi} \tag{6}\]  
in which \(\mathbf{p_{j,i}}\) is the coordinate of array element \(i\) at position \(\mathbf{p_{j}}=\left[x_{j},y_{j},z_{j}\right]^{T}\), and \(\mathbf{p_{j,i}}=\mathbf{p_{j}}+\left[Rcos\gamma_{i},Rsin\gamma_{i},0\right]^{T}\). The coordinate \(\mathbf{p_{t}}\), which represents the interference source in Equation (6), is the unknown variable to be determined. By substituting \(\mathbf{p_{t}}=\left[x,y,z\right]^{T}\), Equation (6) can be reformulated as follows. \[\sqrt{\left(x-x_{j,i}\right)^{2}+\left(y-y_{j,i}\right)^{2}+\left(z-z_{j,i} \right)^{2}}-\sqrt{\left(x-x_{j}\right)^{2}+\left(y-y_{j}\right)^{2}+\left(z- z_{j}\right)^{2}}=\frac{\lambda\varphi_{j,i}}{2\pi} \tag{7}\]  
We take the equations of position \(\mathbf{p_{j}}\) as an example, and we can obtain six nonlinear equations at the position for array elements 1 to 6. In Figure 6, there are \(M\) observation positions in the scene, each of which gives the above six nonlinear equations; thus, the number of equations we can obtain is \(6M\), where the number of unknowns in the equations is \(3\) (\(x\), \(y\), and \(z\)), and the rest variables are known observations. We express the nonlinear Equation (7) as follows:  
\[g(x,y,z,\mathbf{v_{j,i}})=u_{j,i} \tag{8}\]  
In the equation, \(g(\cdot)\) represents the nonlinear relationship in Equation (7), while \(\mathbf{v_{j,i}}\) and \(u_{j,i}\), respectively, denote known variables associated with the position of the receiving antenna and carrier phase observation values, where \(\mathbf{v_{j,i}}=\left(\mathbf{p_{jt}}^{T},\mathbf{p_{j}}^{T}\right)^{T}\), \(u_{j,i}=\lambda\varphi_{j,i}/2\pi\). By combining Equation (8) from various positions and array elements, the following system of equations is formulated. \[\left\{ \begin{aligned} g(x,y,z,\mathbf{v_{1,1}})&=u_{1,1} \\ g(x,y,z,\mathbf{v_{1,2}})&=u_{1,2}\\ \vdots\\ g(x,y,z,\mathbf{v_{j,i}})&=u_{j,i}\\ \vdots\\ g(x,y,z,\mathbf{v_{M,6}})&=u_{M,6}\end{aligned} \right. \tag{9}\]  
We linearize Equation (8) by Taylor expansion and obtain the following result:  
\[