# Localization of GNSS Spoofing Interference Source Based on a Moving Array Antenna
## 3 Proposed Methods
### Extraction of Spoofing Interference Based on the Double-Differenced Carrier Phase  
After successfully receiving both authentic and spoofing GNSS signals, this method effectively extracts and confirms the deceptive signals among the previously received signals, thereby enabling subsequent localization of the spoofing source. The proposed method leverages the inherent characteristics of same-origin signals and the spread spectrum properties of spoofing interference to effectively extract the deceptive interference by observing and statistically analyzing the double-differenced carrier phase feature. According to the content referenced in [31], we present the observation model for the carrier phase measurement value obtained in Section 2 as follows:  
\[\phi=\frac{2\pi}{\lambda}(\rho+\delta\rho+c(\delta tr-\delta ts)+\delta\rho c+T- I+\delta\rho m)-2\pi N+\varepsilon \tag{1}\]  
in which \(\phi\) is the carrier phase observation value of the navigation signal, \(\lambda\) is the carrier wavelength of the signal, \(\rho\) is the distance from the satellite to the receiver antenna, \(\delta\rho\) is the orbit error of the satellite, \(c\) is the speed of light, \(\delta tr\) is the receiver clock difference, \(\delta ts\) is the satellite clock difference, \(\delta\rho c\) is the error caused by the receiver cable, \(T\) is the error caused by the troposphere, \(I\) is the error caused by the ionosphere, \(\delta\rho m\) is the error caused by the multi-path effect, \(N\) is the number of complete cycles of carrier phase, and \(\varepsilon\) is the measurement noise. The units of \(\phi\) and \(\varepsilon\) are rad, the units of \(\rho\), \(\delta\rho\), \(\delta\rho c\), \(T\), \(I\), and \(\delta\rho m\) are m, the units of \(\delta tr\) and \(\delta ts\) are s, the unit of \(c\) is m/s, and the unit of \(N\) is 1. According to Equation (1), the carrier phase single-difference between array element \(i\) and array element 0 can be calculated, i.e., \(\varphi_{i}=\phi_{i}-\phi_{0}\). Taking satellite signal \(h\) as an example, its single-difference calculation equation is as follows. \[ \begin{split}\varphi_{i}^{h}&=\phi_{i}^{h}-\phi_{0}^{h}= \frac{2\pi}{\lambda}\left(\rho_{i}^{h}+\delta\rho_{i}^{h}+c(\delta tr_{i}^{h}- \delta ts_{i}^{h})+\delta\rho c_{i}^{h}+T_{i}^{h}-I_{i}^{h}+\delta\rho m_{i}^{h} \right)-2\pi N_{i}^{h}+\\ &\varepsilon_{i}^{h}-\frac{2\pi}{\lambda}\left(\rho_{0}^{h}+ \delta\rho_{0}^{h}+c(\delta tr_{0}^{h}-\delta ts_{0}^{h})+\delta\rho c_{0}^{h }+T_{0}^{h}-I_{0}^{h}+\delta\rho m_{0}^{h}\right)+2\pi N_{0}^{h}-\varepsilon_{ 0}^{h}\end{split}  \tag{2}\]  
According to reference [32], for the same signal source, due to the very close distance between the array elements, it can be assumed that the parameters \(\delta\rho\), \(T\), \(I\), and \(\delta ts\) between the array elements are the same. By implementing an equal-length design for the antenna element feed lines, an equal-length layout design for the sampling circuit, and digital delay correction for each channel, it becomes feasible to maintain consistent \(\delta\rho c\) values across all array elements. By employing the same highly stable crystal oscillator as the time-frequency reference for the sampling circuit, the RF circuit, and the navigation signal processing circuit associated with each array element, the parameter \(\delta tr\) between each array element can be consistent. Moreover, since the distances between antenna array elements are designed without integer ambiguities, the formulation of the carrier phase single-difference measurement \(\varphi_{i}^{h}\) can be further simplified as follows. \[\varphi_{i}^{h}=\frac{2\pi}{\lambda}(\rho_{i}^{h}-\rho_{0}^{h})+\varepsilon_{i }^{h}-\varepsilon_{0}^{h} \tag{3}\]  
The conditions described here are overly idealized, but they can be achieved in practice. Reference [32] adopts a similar approach to this article to achieve the above conditions.