## 4 Training and Fine-Tuning of SSDAE-DRN
Experiments are carried out on networks with different activation functions and batch normalization, and the cost function curves are shown in Figure 6.  
When using ELU function activation and BN layer for batch standardization, the network cost function curve is smoother and the convergence value is smaller, which has better reconstruction performance of the original data. After the model training is completed, the simulated modulation spectrum is input into the pre-trained SSDAE-DRN model, and the reconstructed results are shown in Figure 7.  
Figure 5: Unsupervised pre-training diagram. Figure 6: Cost function curves. The shaft frequencies corresponding to the input data shown in Figure 7 are 4.5 Hz, 6.5 Hz, and 8.5 Hz, respectively, and these non-integer shaft frequency samples are not included in the training data. The comparison between the output data and the input data shows that the harmonic line spectrum is significantly enhanced, the continuous spectrum envelope is effectively suppressed, the submerged or missing shaft frequency line spectrum is recovered, and the non-harmonic line spectrum is effectively eliminated. Since the number of effective line spectra of the data used in the training model is 7, the higher harmonic line spectra of the output are inevitably enhanced, but this does not hinder the subsequent task of shaft frequency extraction. Input the modulation spectrum data of the radiated noise of three ships into the model, and the reconstruction results are shown in Figure 8. It can be seen that the SSDAE-DRN model has a good generalization ability and denoising effect on the real data. In particular, Figure 8d,f reflect the ability of the model to accurately restore the axial shaft frequency line spectrum and the harmonic line spectrum between the shaft frequency and the leaf frequency line spectrum, which is based on the leaf frequency line spectrum and the higher harmonics information. This shows that the model possesses the sensitivity of line spectrum components and harmonic relationships and is not affected by the change of position. Then, the simulation data and some real historical data are mixed to form a new dataset, which is used to fine-tune the regression part of SSDAE-DRN so that the model has the capability of shaft frequency extraction. The real data comes from 2000 samples of two cooperative ships in the Qiandao Lake at different speeds, which enables the model to effectively extract line spectrum features, better learn continuous spectrum envelope features and noise distribution rules, and improve the shaft frequency extraction performance of the network. A total of 1500 real samples are randomly selected to fine-tune the SSDAE-DRN model, and 500 samples are used to test the performance of the model. During fine-tuning, the pre-trained SSDAE model parameters are frozen and will not participate in the adjustment of the regression. The experimental results of the model with different combinations of the dataset are shown in Table 2.  
Figure 7: Processing of simulated data. (**a**) Input shaft frequency: 4.5 Hz; (**b**) output shaft frequency: 4.5 Hz; (**c**) input shaft frequency: 6.5 Hz; (**d**) output shaft frequency: 6.5 Hz; (**e**) input shaft frequency: 8.5 Hz; (**f**) output shaft frequency: 8.5 Hz. Using simulated data to fine-tune the model, the MAE is validated on the simulated data at 0.58 Hz while it is increased by nearly 1 Hz when validated on the measured data. This is mainly due to the fact that the real data are not used in the training of the SSDAE-DRN model, so the model has limited denoising and enhancement effect on real data. However, the regression part of the SSDAE-DRN model has feature extraction capability. Real data are added to fine-tune the model so that it can effectively learn the reconstruction features and improve the accuracy of shaft frequency extraction. Then the model is fine-tuned with both simulated and real data and validated with real data. The MAE is reduced by 0.33 Hz, and the model's performance is improved.