## 2 Jamming Classification and Mathematical Modeling
### Suppressive Jamming  
(1) Amplitude modulation (AM) jamming: As a subclass of continuous wave (CW) interference, AM jamming can be categorized into single-tone and multi-tone modulation types. Its defining characteristic lies in the carrier amplitude being modulated by a low-frequency signal. The interference at the \(n\)-th sampling point can be expressed as [20]  
\[z_{am}[n]=Pp[n]\otimes\sum_{k=1}^{M_{d}}R_{P_{k}}\text{exp}\big{(}j\big{(}2\pi f _{P_{k}}nT+\theta_{P_{k}}\big{)}\big{)} \tag{1}\]  
In this expression, \(M_{d}\) denotes the number of amplitude-modulated tones. \(P\) is the interference gain factor derived from the jamming signal ratio (JSR), representing the power ratio between the interference and the GNSS signal. \(p[n]\) denotes the channel coefficient vector of the suppressive interference link, and \(\otimes\) represents the convolution operator, accounting for multipath fading effects within the interference  
Figure 1: Time–frequency spectrograms of six types of typical jamming. channel. \(R_{p_{k}}\), \(f_{p_{k}}\), and \(\theta_{p_{k}}\) represent the amplitude, frequency, and phase of the \(k\)-th AM interference component, respectively. 2. Frequency modulation (FM) jamming: This type of interference features a time-varying carrier frequency, characterized by a large instantaneous bandwidth and low power spectral density. Such properties enable it to effectively evade adaptive filtering and frequency-domain notch filtering strategies at the receiver. It is defined as [20] \[z_{fm}[n]=Pp[n]\otimes\sum_{k=1}^{M_{f}}a\cdot\exp\bigl{(}j\bigl{(}2\pi f_{m_{ k}}nT+\beta_{k}\text{sin}\bigl{(}2\pi f_{m_{k}}nT\bigr{)}\bigr{)}\bigr{)}\] (2) \(M_{f}\) denotes the number of multi-tone carriers, \(f_{m_{k}}\) is the frequency of the \(k\)-th jamming component, and \(\beta_{k}\) is the modulation index of the \(k\)-th carrier. 3. Chirp jamming: Its instantaneous frequency periodically sweeps over a predefined frequency band within a short time interval and resets at the end of each cycle. Common forms include linear chirp jamming and sinusoidally modulated chirp jamming, defined as follows [20]: \[z_{cp}[n]=Pp[n]\otimes\exp\left(j\biggl{(}2\pi f_{ch}nT+\pi a\frac{f_{max}-f_{ min}}{T_{supp}}\theta_{c}\biggr{)}\right)\] (3) \(f_{ch}\) is the initial sweep frequency, \(f_{min}\) and \(f_{max}\) define the lower and upper bounds of the sweep bandwidth, respectively, \(T_{supp}\) is the time taken to sweep from \(f_{min}\) to \(f_{max}\), \(a\in\{+1,-1\}\) is a random variable used to determine the sweep direction, and \(\theta_{c}\) is the initial phase of the chirp jamming signal. 4. Narrowband jamming: This refers to the injection of narrowband interference components with power significantly higher than the useful GNSS signal within the main frequency band. Its bandwidth is typically narrower than the main lobe bandwidth of the GNSS signal. The spectral expression is given by [20] \[\Omega(f)=\left\{ \begin{aligned} \mu,|f-f_{0}|<B/2\\ 0,\text{otherwise}\end{aligned} \right.\] (4) Here, \(\mu\), \(f_{0}\), and \(B\) represent the amplitude, center frequency, and bandwidth of the narrowband jamming signal, respectively. By passing a stationary random process through a band-pass filter, the time-domain expression of the interference, \(\omega[n]\), is obtained. Its convolution with the pseudocode sequence of the navigation signal, \(p[n]\), is given by \(z_{nb}[n]=Pp[n]\otimes\omega[n]\). 5. Pulse jamming: This type of interference periodically generates multiple pulses within a fixed repetition interval, each with a certain duty cycle. It is characterized by short duration, significant spectral spreading, and high peak power.