## 4 Results
### Analysis  
The experimental content is divided into two parts: (1) comparing the performance of the DFRT model with other models; (2) verifying whether HOS can improve the recognition accuracy. Both parts use classification accuracy as the evaluation metric, and parameters such as the optimizer and the number of training epochs are the same, except for the different contents being compared. To validate the effectiveness of the DFRT model for underwater acoustic signal modulation classification, this paper conducts comparative experiments using three models: traditional CNN, CNN-Transformer, and DFRT. The advantages and disadvantages of each model are summarized in Table 3, and the accuracy results are presented in Figure 6. As the CNN relies only on simple convolutional and fully connected layers, it struggles to effectively capture the complex features of underwater acoustic signals. In contrast, the DFRT optimizes gradient propagation through a residual structure and combines the long-range dependency modeling capability of Transformer, significantly enhancing multi-scale feature extraction and cross-domain correlation analysis abilities. This allows it to exhibit superior accuracy across all distance ranges. In short-distance ranges (1, 5, 10 km), the DFRT model achieves an average recognition accuracy of 98.89%, representing a 2.68% improvement over the other two models. In long-distance ranges (15, 18 km), the DFRT model still maintains an accuracy of 95.84%, which is 6.67% higher than the suboptimal CNN-Transformer model (89.91%). \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Advantages** & **Disadvantages** \\ \hline \multirow{8}{*}{**CNN**} & **(1) Inadequate long-range dependency** \\  & **modeling**: Fails to effectively capture global \\  & **(1) Simple architecture:** Characterized by a small parameter size and high computational efficiency, making it suitable for lightweight scenarios. \\ \cline{1-1}  & **(2) Local feature sensitivity:** Capable of capturing local structures in TFR (e.g., frequency hopping patterns) with reasonable accuracy. \\ \cline{1-1}  & **(1) Simple architecture:** Characterized by a small parameter size and high computational efficiency, making it suitable for lightweight scenarios. \\ \cline{1-1}  & **(1) Simple architecture:** Characterized by a small parameter size and high computational efficiency, making it suitable for lightweight scenarios. \\ \cline{1-1}  & **(2) Local feature sensitivity:** Capable of capturing local structures in TFR (e.g., frequency hopping patterns) with reasonable accuracy. \\ \cline{1-1}  & **(3) Gradient degradation issue:** Prone to gradient vanishing in deep networks, limiting the ability to learn complex features. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The advantages and disadvantages of each model. Secondly, to verify the effectiveness of HOS in enhancing signal features, this paper conducts comparative experiments based on the DFRT model using TFR and TFR-HOS features. The experimental results are shown in Figure 7. In short-distance ranges, the accuracy of the two approaches is similar, but in long-distance ranges, the input of mixed