# Deep-Learning-Based Recovery of Frequency-Hopping Sequences for Anti-Jamming Applications
## 3 Proposed FH Sequence Recovery Method
### Designed Network Structure
\tag{3}\]  
where \(\mathbf{W}_{iz}\) and \(\mathbf{W}_{hz}\) are the updated gate's weight coefficients, and \(b_{iz}\) and \(b_{hz}\) are the bias coefficients. \(\mathbf{W}_{ir}\) and \(\mathbf{W}_{hr}\) are the reset gate's weight coefficients, and \(b_{ir}\) and \(b_{hr}\) are the bias coefficients. \(\mathbf{W}_{in}\) and \(\mathbf{W}_{hn}\) are the weight coefficients of the candidate hidden layers, \(b_{in}\) and \(b_{hn}\) are the bias coefficients, \(\sigma\) is the Sigmoid function, \(x_{t}\) is the input at time \(t\), and \(h_{t}\) is the output at time \(t\). Classifier design  
In this paper, the problem of frequency-hopping sequence estimation is regarded as a sequence classification problem. Assuming that the length of the frequency-hopping sequence is \(N\), and the number of frequency bins is \(L\), then the number of all possible classes is \(L^{N}\), so it can be solved by using a single classifier with \(L^{N}\) classes. However, as the sequence length and the number of frequency bins increase, the number of categories increases exponentially, which increases the time and space complexity of the network. Secondly, the number of training samples required to train this classifier will be much larger than \(L^{N}\). It is uneconomical to generate such a large number of training samples because it is difficult to converge in a limited time. Finally, when the length of the frequency-hopping sequence changes, the number of categories will change, at which time it is necessary to change the output size of the classifier and retrain the network. In order to solve the above problems, we input the time-frequency diagram of the received signal into the network according to the frequency-hopping period and use a classifier with \(L\) categories to process the output of the GRU. The output at each moment is used as the frequency-hopping sequence value of each hop. This not only can ensure that the network converges in a limited time but also adapt to the change in the length of the input signal. When signals of different lengths are input to the network, the network can obtain the frequency-hopping sequence of the corresponding length through the GRU. Let the signal received by the receiver be \(y(t)=[y_{1},y_{2},y_{3},\cdots y_{N}]\), where \(y_{i}\) represents the signal sequence of the \(i\)-th hop. After STFT processing, a time-frequency matrix  
Figure 5: Schematic diagram of GRU. \(M=[m_{1},m_{2},m_{3},\cdots,m_{N}]\) can be obtained, where \(m_{i}\) is the time-frequency matrix of the \(i\)-th hop signal. The output of the network is the estimated frequency-hopping sequence \(\hat{S}=[\hat{s}_{1},\hat{s}_{2},\hat{s}_{3},\cdots,\hat{s}_{N}]\). The optimization of the network can be expressed as  
\[\underset{Q}{\text{min}}||\hat{S}-S||_{1},\hat{S}=F(M:Q) \tag{4}\]  
where \(Q\) represents the model parameters of the network, and \(F(\cdot:Q)\) represents the functional mapping of the network from input to output. As we treat the problem as a classification problem, we adopt the cross entropy as the loss function in training the network  
\[\xi=-\frac{1}{N_{B}}\sum_{i=1}^{N_{B}}\sum_{t=1}^{I}\sum_{k=1}^{q}\mathit{ label}_{itk}\log(p_{itk}) \tag{5}\]  
where \(N_{B}\) represents the number of samples in the batch, \(p_{itk}\) represents the output probability of the \(k\)-th class of the \(i\)-th sample at the \(t\)-th time of the classifier, and \(\mathit{label}_{itk}\) represents the \(k\)-th label of the frequency-hopping sequence value corresponding to the \(t\)-th hop of the \(i\)-th sample.