## 3 Methods
### Overall Framework
#### 3.2.2 Data Generation Module  
The feature generation module is the ACGAN-SN model proposed in this paper, which is an improvement on the ACGAN model in Section 3.1. Considering the feature extraction capability of the CNN model, in the generator and discriminator of ACGAN, convolution and deconvolution are used to form the discriminator and generator. The  
Figure 1: Typical architecture of (**a**) GAN and (**b**) the auxiliary classifier GAN. input of the discriminator is 32 \(\times\) 32 data after the reshape of the 1024-point frequency domain data, and then four convolution layers are used to extract 256-dimensional features, for which each feature size is 2 \(\times\) 2. The output results are reshaped into one-dimensional feature data and then input into two fully connected layers--one fully connected output size is 1, which is used for true and false discrimination. Another fully connected layer with an output size of 10 is used for label prediction. The structure of the generator is symmetrical with the discriminator as a whole. First, the 100-dimensional noise vector is multiplied by the one-dimensional label vector to obtain the 100-dimensional input vector. Then, after full connection and reshape transformation, a 256 \(\times\) 2 \(\times\) 2 three-dimensional matrix is obtained. Four deconvolution operations are performed on the matrix to obtain a three-dimensional matrix of 1 \(\times\) 32 \(\times\) 32. Finally, the reshape operation is used to obtain a one-dimensional vector of 1 \(\times\) 1024, which is the generated data. The specific parameters of the generator and discriminator are shown in Table 1.  
To make the model stable for training, the gradient penalty term [30] is often added to the GAN discriminator. However, the gradient penalty cannot constrain the data in the whole structural space, which leads to the space between the synthetic data and the real data being shifted, the GAN model cannot be trained stably, and the computational speed of the gradient penalty is also a defect. SN is a parameter regularization technique, and the literature [29] has proven that SN can make the discriminator satisfy the Lipschitz constraints, thereby improving the training stability of the model and greatly reducing the computation time compared with the gradient penalty. Assuming that \(\mathbf{\theta}\) is the weight parameter matrix of the model, SN can be expressed by Formula (8):  
\[\mathbf{\theta}^{l}=\frac{\mathbf{\theta}}{\left\lVert\mathbf{\theta}\right\rVert_{2}}= \frac{\mathbf{\theta}}{\max(\mathbf{\theta}^{T}\mathbf{\theta})_{*}} \tag{8}\]  
where \(\left\lVert\mathbf{\theta}\right\rVert_{2}\) is the spectral norm of \(\mathbf{\theta}\). \(\max(\mathbf{\theta}^{T}\mathbf{\theta})_{*}\) is the largest eigenvalue of \(\mathbf{\theta}^{T}\mathbf{\theta}\). \(\mathbf{\theta}^{l}\) is the weight parameter of the \(l\)-th network. Based on SN technology, this paper imposes regularization constraints on each layer of the neural network of the discriminator to realize the stable training process of the model, as shown in Figure 2. Once the model is trained, the generator can be used to synthesize a large amount of fake data. To evaluate the quality of the data, we used three quantitative indicators to analyze the data similarity. They are Pearson Correlation Coefficient (PCC), Cosine Similarity (CS), and Wasserstein distance. Through this similarity measure, the model training process is regulated to obtain the best model parameters.