# A New Improved Multi-Sequence Frequency-Hopping Communication Anti-Jamming System
## 3 System Model and Problem Formulation
### Problem Statement  
According to the system model description, the receiver must detect the communication frequency of the user signal at each time slot, which is actually to identify whether there is a user signal on each channel, where \(f\in F\). Assuming that in time slot k, the user selects frequency \(f_{k}^{ll}\) for communication, the power of the user's communication signal can be expressed as \(P_{U}=\int_{-b_{U}/2}^{b_{U}/2}S(f)df\), where \(b_{U}\) is the bandwidth of the user's baseband signal, and \(S(f)\) is the low-pass equivalent power spectral density (PSD) of the user's bandpass signal. After the intelligent jammer perceives the user's communication decision in time slot k, it selects the jamming power \(P_{J}\) and jamming frequency \(f_{k}^{l}\) to interfere with the user's reactive jamming signal. In order to ensure the effectiveness of jamming performance, the intelligent jammer will inevitably emit reactive jamming signals with a higher power than the user signal to the target frequency. Of course, a non-intelligent jammer will also emit non-intelligent jamming signals with a power \(P_{j}\) and a jamming frequency \(f_{k}^{j}\). Through the above settings, the instantaneous environmental state of time slot k can be represented as \(s_{k}=(s_{k}^{1},s_{k}^{2},s_{k}^{3},\ldots,s_{k}^{m})\), where \(s_{k}^{m}\) denotes the received power of a signal with a frequency of \(m\) in the time slot k; (1) refers to its specific form. In Equation (1), \(g_{I}\) and \(g_{j}\) represent the transfer function of intelligent reactive jamming signals and non-intelligent jamming signals from the jammer to the receiver, respectively, while \(g_{U}\) represents the transfer function of the communication signal from the transmitter to the receiver. In addition, \(n(f)\) is the PSD function of noise; \(\delta(x)\) is the indicator function. When \(x\) is a true value, \(\delta(x)=1\), and vice versa; then, \(\delta(x)=0\). Given the dynamic and uncertain nature of the environment, numerous unknown parameters, such as \(g_{I}\) and \(g_{j}\), make it impractical to directly determine the communication frequency based on \(S_{k}\). In order to utilize reactive jamming in complex jamming environments, although it is difficult to obtain intelligent jamming signal decisions, the decision of intelligent reactive jamming may be correlated with the frequency of the user's signal [6]. So, we define the environmental state as \(S_{k}=\{s_{k},s_{k-1},\ldots,s_{k-l+1}\}\), where the backtracking time length is \(l\). \(S_{k}\) is a two-dimensional matrix of size \(l\times M\). Its thermodynamic image is called a spectral waterfall plot, which includes both time-domain and frequency-domain information. As shown in Figure 2, we can see from the spectrum waterfall that there is a certain regularity between the frequency decision of reactive jamming and the user's communication frequency. Figure 2 illustrates the spectrum waterfall of a complete communication stream in a complex jamming environment. This communication stream includes intelligent reactive jamming, as well as non-intelligent frequency sweeping jamming and multi-tone jamming. The interval between each white line represents the communication time required for one hop. From Figure 2, we can observe that the intelligent jammer is capable of sensing the user's communication frequency in each time slot and transmitting intelligent reactive jamming signals. When the reactive jamming signal synchronizes with the user signal, it can be assumed that each frequency hop encounters the same-frequency reactive jamming. In this scenario, the presence of reactive jamming increases the power of each hop signal, thereby enhancing frequency-detection performance based on energy detection. However, the channel state in real-world environments is constantly changing dynamically, and not all jamming signals overlap with the user signals at all times.