# Method for Recognition of Communication Interference Signals under Small-Sample Conditions
## 3 Data Enhancement Network
### Adversarial Loss Function Design  
The loss functions for the original GAN network's generator and discriminator are as follows:  
\[\min_{\theta G}\max_{\theta D}V(G,D)=E_{\mathbf{x}\sim P_{\mathbf{z}}}[\log D( \mathbf{x})]+E_{\mathbf{z}\sim P_{\mathbf{z}}}[\log(1-D(G(\mathbf{z})))] \tag{4}\]  
where \(P_{\mathbf{x}}\) and \(P_{\mathbf{z}}\) represent the sample distributions of real data \(\mathbf{x}\) and random noise \(\mathbf{z}\) respectively. \(D\) is the discriminator, and \(G\) is the generator. During the optimization process, the optimization goal of the discriminator \(D\) is for the probability value of \(D(\mathbf{x})\) to approach 1 and for the probability value of \(D(G(\mathbf{z}))\) to approach 0. Conversely, the optimization goal of the generator \(G\) is for the probability value of \(D(G(\mathbf{z}))\) to approach 0. Therefore, the training process of GAN is transformed into a Minimax problem, as shown in Equation (4). Additionally, Equation (4) can be derived to transform into the Jensen-Shannon Divergence, so the actual computed loss function is:  
\[\min_{\theta G}\max_{\theta D}V(G,D)=2 JS(P_{\mathbf{x}}\|P_{\mathbf{z}})-2\log 2 \tag{5}\]  
where \(JS(P_{\mathbf{x}}\|P_{\mathbf{z}})\) denotes the computation of the Jensen-Shannon Divergence between the distributions of real data \(\mathbf{x}\) and random noise \(\mathbf{z}\). Due to the discrete nature of the Jensen-Shannon divergence, the phenomenon of gradient vanishing can occur during the optimization process [16]. However, Wasserstein distance can provide a smoother reflection of the differences between two distributions. Therefore, using a Wasserstein distance metric instead of Jensen-Shannon divergence  
Figure 2: Time–frequency plots of six types of jamming signals. to design a loss function can effectively alleviate the issue of gradient vanishing. The Wasserstein distance is defined as shown below:  
\[W\big{(}P_{\mathbf{x}},P_{\mathbf{y}}\big{)}=\underset{\gamma\sim\Pi(P_{\mathbf{x}},P_{\mathbf{y} })}{\inf}E_{(\mathbf{x},\mathbf{y})\sim\gamma}\big{[}\|\mathbf{x}-\mathbf{y}\|\big{]} \tag{6}\]  
where \(\gamma\sim\prod\big{(}P_{\mathbf{x}},P_{\mathbf{y}}\big{)}\) represents the joint distribution set of real samples \(\mathbf{x}\) and generated samples \(\mathbf{y}\), while inf represents the infimum, which is the minimum value among all possible joint distribution sets. Therefore, the Wasserstein distance focuses on finding the expected value within the joint distribution that minimizes \(E_{(\mathbf{x},\mathbf{y})\sim\gamma}\big{[}\|\mathbf{x}-\mathbf{y}\|\big{]}\). However, during the design process of GANs based on the Wasserstein distance, using weight clipping to enforce the discriminator to be in the 1-Lipschitz function space can lead to the phenomenon of gradient explosion [17]. Instead of weight clipping, gradient penalty, which satisfies the 1-Lipschitz function condition, can effectively alleviate gradient explosion and stabilize training. The gradient penalty can be shown as:  
\[gp=\lambda\times\underset{\widetilde{\mathbf{x}}\sim P_{\widetilde{\mathbf{x}}}}{\inf} \Big{[}\big{(}\|\
abla_{\widetilde{\mathbf{x}}}D(\widetilde{\mathbf{x}})\|_{2}-1\big{)} ^{2}\Big{]} \tag{7}\]  
where \(\widetilde{\mathbf{x}}=\epsilon\mathbf{x}+(1-\epsilon)\mathbf{y},~{}\epsilon~{}\sim\text{ Uniform}[0,1],\widetilde{\mathbf{x}}\) represents uniform sampling between the distributions of real sample \(\mathbf{x}\) and generated sample \(\mathbf{y}\); \(\
abla_{\widetilde{\mathbf{x}}}D(\widetilde{\mathbf{x}})\) denotes the gradient computation of the discriminator's output with respect to \(\widetilde{\mathbf{x}}\); \(\|\
abla_{\widetilde{\mathbf{x}}}D(\widetilde{\mathbf{x}})\|_{2}\) represents the L2 norm of \(\|\cdot\|\).