## 1 Introduction  
The application of the Global Navigation Satellite System (GNSS) in military and civilian fields is gradually becoming widespread, and society's dependence on PNT (positioning, navigation, and timing) services provided by GNSS is rapidly increasing. At this time, the safety and stability of the GNSS have also received great attention. Due its weak signal strength, open signal modulation methods, and predictable partial navigation data, the GNSS is highly susceptible to spoofing [1, 2], which is fatal to navigation systems. Therefore, the detection of spoofing is very important. Currently, many spoofing methods have been proposed. In the early stages of research on spoofing-detection methods, scholars conducted spoofing detection by searching for signal features that changed after a single antenna receiver was spoofed. Due to the inability of spoofing to perfectly simulate the power of real signals, the addition of spoofing can cause changes in the signal power [3], the carrier-to-noise ratio (CNR) [4], etc. In the acquisition and tracking stage, under the covert spoofing strategy, the spoofing needs to enter the tracking loop, slowly increase the power, and change the phase, which can cause misalignment between the real code and the spoofing code phases and the appearance of multiple correlation peaks [5]. Based on the changed signal features mentioned above, spoofing detection can be performed, which has good detection performance for simple spoofingand is currently the most widely used spoofing-detection algorithm. The combination of inertial navigation and GNSS can achieve spoofing detection by conducting consistency checks on features such as trajectory, acceleration, and position. [PERSON] et al. [6] compared the trajectory estimated by a GNSS receiver with the trajectory obtained by the INS. [PERSON] et al. [7] compared the accelerations calculated using accelerometers and GNSS receivers. Currently, many scholars use encryption and authentication techniques in cryptography to encrypt civilian navigation signals, making it difficult to predict the navigation signals. Moreover, receivers can judge the integrity of the received signals, thereby better resisting spoofing attacks. According to the different encryption-authentication methods and objects, this method can be divided into navigation message authentication (NMA) [8; 9] and spreading code authentication (SCA) [10]. The spoofing-detection method based on signal encryption authentication requires modifying the navigation signal system, which is difficult to achieve in the short term. In addition, many scholars detect spoofing based on the direction differences between spoofed signals and real signals and other variables caused by different directions. [PERSON] et al. [11] proposed a method for the spoofing detection and suppression of pre-spread signals using array antennas. This method utilizes the original baseband signal to establish a cyclic correlation matrix, perform singular value decomposition, and perform a cyclic correlation eigenvalue test (CCET) to determine whether there is spoofing. If there is spoofing, the feature vector is used to establish a shadow space to eliminate spoofing. [PERSON] et al. [12] proposed a spoofing-detection technology based on dual antennas. The essence of this method is to accurately estimate the frequency difference of arrival between a pair of fixed antennas based on carrier phase observation and navigation information. When there is no spoofing, the observations should be consistent with the predictions. Otherwise, due to the geometric and kinematic differences between the GNSS satellite and the spoofer, the spoofing will disrupt consistency, so multi-agent spoofing can be detected. [PERSON] et al. [13] used the position information of multiple antennas to derive the intersection angle between two directions of arrival (IA-DOA) of different signals.