## 5 Experiment and Result Analysis of Shaft Frequency Extraction  
### Simulated Experiment  
The performance of the SSDAE-DRN model for shaft frequency extraction is tested in the following four cases: shaft frequency line spectrum missing, second harmonic line spectrum missing, strong leaf frequency line spectrum only, and low signal-to-noise ratio. The results of shaft frequency extraction are shown in Figure 9. The left is the trace map of input data, the right shows the output trace map of the denoising and enhancement part of SSDAE-DRN, and the red dots are the estimated shaft frequency of regression prediction. It is worth noting that the input to the SSDAE-DRN model is the envelope spectrum of the ship radiated noise and that the trace map is made up of the envelope spectra stacked in chronological order. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Training Data** & **Testing Data** & **MAE** \\ \hline Simulated Data & Simulated Data & 0.58 \\ Simulated Data & Real Data & 1.56 \\ Simulated Data + Real Data & Real Data & 1.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance comparison of shaft frequency extraction on fine-tuned model. Figure 8: Processing of real data. (**a**) Input of ship A, shaft frequency: 3.4 Hz; (**b**) output of ship A, shaft frequency:3.4 Hz; (**c**) input of ship B, shaft frequency: 4.0 Hz; (**d**) output of ship B, shaft frequency: 4.0 Hz; (**e**) input of ship C, shaft frequency: 8.6 Hz; (**f**) output of ship C, shaft frequency: 8.6 Hz. It can be seen from Figure 9 that the SSDAE-DRN model can obtain a more accurate shaft frequency estimation in the case of missing shaft frequency line spectra. The estimation error increases in the other three cases but the shaft frequency can still be extracted more accurately. The method proposed in this paper is compared with the method of sequence matching in [13] on simulated data, and the results are shown in Table 3. Compared with sequence matching, the method in this paper has higher accuracy and obvious advantages, especially in the case of missing line spectra and noise interference. The ship shaft frequency extraction algorithm varies in its ability on different data, and the extraction error increases as the true value increases. To obtain the error distribution of this method, the test data are grouped and the shaft frequency is extracted for each group, respectively. Results obtained by two methods are shown in Figure 10 after smoothing. \begin{table}
\begin{tabular}{c c c} \hline
**Type of Ship** & **Method of Sequence Matching** & **Method in This Paper** \\ \hline Normal line spectrum & 2.56 & 0.43 \\ shaft frequency line spectrum missing & 4.15 & 1.28 \\ second harmonic line spectrum missing & 3.17 & 1.54 \\ strong leaf frequency line spectrum only & 5.49 & 1.32 \\ noise interference (0 dB) & 2.82 & 0.48 \\ noise interference (\(-\)10 dB) & 3.73 & 1.25 \\ noise interference (\(-\)20 dB) & 4.85 & 1.79 \\ \hline \end{tabular}
\end{table} 
Table 3: MAE of different methods on simulated data. Figure 10: Frequency distribution of MAE on shaft frequency extraction. Figure 9: Trace map on simulation experiment of shaft frequency extraction. (**a**) Shaft frequency line spectrum missing; (**b**) shaft frequency extraction of (**a**); (**c**) second harmonic line spectrum missing; (**d**) shaft frequency extraction of (**c**); (**e**) strong leaf frequency line spectrum only; (**f**) shaft frequency extraction of (**e**); (**g**) low signal-to-noise ratio; (**h**) shaft frequency extraction of (**g**). Similar to the traditional method, the error of the method in this paper also increases with the increase of the real shaft frequency. Compared with the method of sequence matching, the error growth curve of this method is smoother, and the average absolute error of extraction under the real shaft frequency of 20 Hz is only 2 Hz, which is 3.6 Hz lower than the method of sequence matching. Therefore, the performance of the method in this paper is more stable for ship targets with different speeds.