## 2 Jamming Classification and Mathematical Modeling
### Spoofing  
Based on the number of transmitters, spoofing can be categorized into single-source and multi-source types; according to the signal generation method, it can be divided into replay-based and generative spoofing, and it can be classified into simple, moderate, and advanced levels of interference complexity.  
1. Simple spoofing adopts a two-step strategy of \"jamming first, then spoofing\". When a navigation terminal is already locked onto authentic satellite signals, it will maintain tracking as long as the code delay and Doppler shift of the spoofed signal fall outside the pull-in range of the tracking loop. Therefore, the attacker must first transmit high-power jamming signals to cause the receiver to lose lock. After maintaining the jamming for a certain period, the transmitter switches to broadcasting the spoofed signal. Once the terminal loses lock, it attempts reacquisition based on historical parameters and due to the power advantage of the spoofed signal, it locks onto it and includes it in the PVT solution. Thus, in single-interference-source scenarios, it can be assumed that suppression and spoofing interference, as well as different categories of suppression jamming signals, do not occur simultaneously.
2. Moderate-level spoofing achieves takeover without loss of lock by aligning the spoofed signal's code phase and Doppler frequency with those of the authentic signal. The spoofing device first receives genuine satellite signals and estimates the relative position and velocity of the target receiver. It then generates spoofed signals with identical parameters and transmits them in an overlapping manner. As the spoofed signal gradually increases in power and introduces a slow code phase shift, the receiver's tracking loop becomes dominated by the stronger spoofed signal, ultimately leading to takeover of the positioning solution. During this process, the receiver remains locked, making detection difficult and the spoofing highly covert.
3. Advanced-level spoofing extends moderate spoofing by introducing multiple transmitters. Each antenna independently simulates the signal of a specific satellite, collectively forming a spatial signal field that closely mimics real satellite geometry, thereby enhancing the resistance to multi-antenna spoofing detection techniques. However, due to the complexity of transmission synchronization, phase control, and environmental dynamics, practical implementation remains highly challenging, and engineering feasibility is currently low. As a result, this form of spoofing does not yet pose a significant real-world threat.  
Considering the combined effects of channel fading, Doppler shift, and phase offset, this section extends the diagnostic targets to include high-power delay/frequency-shift spoofing with observable features, as well as complex scenarios involving changes in the number of GNSS signals. Spoofed and authentic GNSS signals typically use the same PRN pseudorandom code sequence but embed falsified navigation message bits, expressed as  
\[z_{sp}[n]=\sum_{k=1}^{K}G_{k}q_{k}[n]\otimes d_{k}(nT-\delta_{k})c_{k}(nT- \delta_{k})\text{cos}\Big{(}2\pi f_{kp}nT+\psi_{k}\Big{)} \tag{6}\]  
In this expression, \(G_{k}\) is the gain factor determined by the spoofing signal-to-noise power ratio, \(q_{k}[n]\) is the channel coefficient vector of the spoofing link corresponding to the \(k\)-th PRN code, and \(\hat{d}_{k}\) denotes the falsified navigation message information being transmitted. \(\delta_{k}\), \(f_{kp}\), and \(\psi_{k}\) represent the time delay, Doppler frequency offset, and phase deviation of the \(k\)-th spoofed signal, respectively.  
In the absence of interference, the received signal consists only of authentic navigation signals and additive white Gaussian noise. Since the power of the GNSS signals is significantly lower than the noise floor during the correlation preprocessing stage, the power spectral density of the composite signal is dominated by characteristics of white Gaussian noise.