# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 3 System Model
### Sparse Jamming Model  
In our scenario, the jamming vector \(\mathbf{e}\) is assumed as a spare vector because practical jammers are modeled as band-limited or time-limited jammers. In fact, a jamming method that interferes with every time-frequency symbol simultaneously would be a powerful jamming method. However, in the proposed covert communication system, the jammer cannot be sure of the exact time-frequency band of the target signal because the proposed system is intended to provide undetectable characteristics of the transmitted waveform, as shown in Section 4. Several studies on the energy optimization of jammers, including [22; 23], have indicated that a jamming policy without a precise signal power estimation is not optimal. To achieve energy efficiency, the jammer must acquire the time-frequency band of the target communication. Without knowing the codebook and exact time-frequency band of target signals, the jammer can have two options--blind barrage jamming and sparse jamming. On the one hand, we consider a barrage jammer that aims to attack a target signal with its signal bandwidth when the time duration is unknown. The jammer cannot launch an attack with a wider bandwidth and longer time duration than those of the target communication signal. Consider \(B\) as the bandwidth of the target signal and \(D\) as the time duration of the target signal. Obviously, the bandwidth and time duration of the attack signal become \(B+\Delta B\) and \(D+\Delta D\) for \(\Delta B\geq 0\) and \(\Delta D\geq 0\), respectively. Here, it can be observed that the jammer would expend an excessive amount of energy due to the amount of excessive time and bandwidth \(O(\Delta B\Delta D)\). Nevertheless, the attack by the jammer may still remain ineffective, provided that the time and frequency band of the target signal is unknown. The only option that the barrage jammer has is to spread jamming power over a wide range of time-frequency regions with large \(\Delta B\) and \(\Delta D\). However, the power budget of a practical jammer is not infinite. Thus, to launch a barrage jamming attack, the energy density per unit time-frequency region shall remain low. In such a case, the effective jamming energy, i.e., the actual jamming energy spent on the target time-frequency region, shall remain small, and the jamming becomes unsuccessful. On the other hand, the jammer can choose a sparse jamming approach, aiming to reduce the inefficiency of barrage jamming and concentrate the jamming energy onto a specific time-frequency band. Many jammers therefore aim to identify the time-frequency band of the ongoing communications. Once this is identified, an attack can be launched in the form of partial-band noise jamming, such as pulse jamming, as discussed in [9], and reactive jamming, which is discussed in [10]. Thus, the jammers can concentrate their jamming power on a small, selected part of the entire time-frequency band. What does this mean to the proposed covert anti-jamming communication system? The proposedsystem is supposed to provide undetectable characteristics of the transmitted waveform, which results in two scenarios. On the one hand, jammers cannot detect the presence of the signal or its time-frequency band at all. This is the case when [PERSON] is physically located far from the transmitter. In this case, the jammers do not launch any jamming attacks. On the other hand, in the case in which [PERSON] is close to the transmitter unit, the signal is partially detectable. It can be recalled that our signals are wideband Gaussian signals. Thus, there are a few time-frequency bands of which the energy levels pierce through the noise floor (refer to the illustration in Figure 5). [PERSON] can be deceived that the ongoing communication exists only on those detected time-frequency bands; consequently, [PERSON] will aim to launch precise jamming attacks at the detected time-frequency bands. The previous discussion explains why we have modeled the jamming attacks to be sparse in the time-frequency domain.