## 4 Our Proposed Method
### Proposed Semi-Supervised AMC Method Based on Consistency Regularization and Pseudo Labeling
The unsupervised loss \(L_{ul}\) can be expressed as:  
\[L_{ul}=\frac{1}{N-L}\sum_{j=1}^{N-L}\mathbb{A}(\max(p(y|\alpha(x_{ul}^{j})))\geq \tau)L_{ce}(\hat{q}_{j},q_{j}), \tag{10}\]  
where \(N-L\) represents the number of unlabeled modulation signal samples. \(\tau\) denotes the confidence threshold. \(\mathbb{A}(\cdot)\) is a function. It takes the value of 1 if the maximum class probability is greater than the threshold, otherwise, it takes the value of 0. The value of the confidence threshold directly affects the quality of pseudo-labels. Setting the threshold too high results in a small number of samples used for training, and setting the threshold too low results in the introduction of a lot of erroneous information. In this paper, weak data augmentation consists of a standard flip-and-shift operation, with a 50% probability of flipping and a 12.5% probability of shifting. Employing only weak data augmentation could lead to overfitting of the training process and failure to extract crucial features. Strong data augmentation can cause severe distortion of signal images but still retains sufficient features to recognize the modulation schemes. Strong data augmentation applies the Randaugment [45] augmentation strategy and cutout [46] augmentation. The Randaugment methodology represents a variant of the Autoaugment [47] strategy, which employs a random sampling strategy to reduce the network's dependence on the degree of coupling between augmentations. Specifically, there is a list of 14 augmented techniques in Randaugment, along with a range of augmented magnitudes. N augmented methods are randomly selected from this list. A random magnitude M is chosen. Subsequently, the chosen augmented techniques and magnitude are implemented on the training signal image, where each augmented technique has a 50% probability of being utilized. The cutout strategy encourages the model to learn robust features by randomly masking a portion of the training image. It can successfully avert interference from noise or unexpected features. The combination of Randaugment and cutout effectively suppresses the noise introduced by Randaugment, thereby further enhancing the model's understanding of key features [48]. The overall loss of model training is the sum of supervised and unsupervised losses. The overall loss can be expressed as:  
\[L_{total}=L_{s}+\lambda_{u}L_{ul}, \tag{11}\]  
where \(\lambda_{u}\) denotes the weight value of the unsupervised loss. In experiments, the value of this parameter is usually set to 1.0. After calculating the total training loss through forward propagation, back-propagation (BP) is performed. The goal of back-propagation is to compute the gradient of the model's loss function with respect to model parameters. The gradient represents the rate of change in the loss function for each model parameter. It specifies the direction that causes the loss function to fall the fastest in the model parameter space. By continuously updating the model parameters, the model's predictions gradually approach the ground truth. In this paper, we select the wide residual network (WRN) [49] as the backbone network for extracting features from time-frequency images. The WRN has residual blocks and jump connections. Each residual block is composed of multiple convolution layers and batch normalization layers. The WRN retains the advantages of residual networks in preventing gradient vanishing.