# A Deep Learning Method for Bearing Fault Diagnosis through Stacked Residual Dilated Convolutions
## 3 Validation of Proposed SRDCNN Model
### Case Study II: Performance under Different Ambient Noise  
To further investigate the effectiveness of the proposed SRDCNN model under different ambient noise, the original test set adds different signal-to-noise ratios (SNR) noise for testing. The SNR is defined as Equation (5), where \(P_{signal}\) and \(P_{noise}\) represent the power of signal and noise, respectively:  
\[SNR_{dB}=10*\log_{10}(\frac{P_{signal}}{P_{noise}}), \tag{5}\]  
In order to approach the conditions in a practical workshop, additive white Gaussian noise could be mixed with the original vibration signals to form signals with SNR from 2 dB to 10 dB. The time domain waveforms of the ten signal states with different SNR values are shown in Figure 14. Obviously, as the signal-to-noise ratio decreases, the differences between categories become weak but still exist, which is the basis for fault diagnosis. Therefore, it is still feasible to establish a deep learning-based intelligent fault diagnosis model working directly on the time-domain signal.  
Figure 12: Diagram of the LSTM-CNN model (**left**) the training process and (**right**) confusion matrix.  
Figure 13: Diagram the SRDCNN model (**left**) the training process and (**right**) confusion matrix.  
During the experiments, models are trained with dataset C as the training set, which is collected under a load of 2 hp, and then tested with the test set of dataset C adding the noise that makes the SNR 2 dB to 10 dB. The results of these three methods for diagnosing signals with different SNR are shown in Figure 15. It can be seen that after training with 5000 training samples, all the methods achieve over 98% classification accuracy on test samples with over six SNR values, and the classification accuracy of each model exhibits remarkable decrease along with a decreased SNR value.  
From the horizontal point of view, in the test on the same test set with fixed SNR value, the proposed SRDCNN model achieves the highest classification accuracy, the LSTM-CNN model takes the second place, and the 1d-DCNN model shows the worst result. From the vertical perspective, when the SNR value is low enough, that means the noise is very strong. Thus, the accuracy of all the methods decreases as the signal-to-noise ratio decreases. It is obvious that when the SNR value of the test set decreases from 4 dB to 2 dB, the accuracy of the 1d-DCNN model has the sharpest drop among them from 94.6% to 86.8%. The accuracy of the 2d-DCNN model also suffers a 6.4% drop. In contrast, the SRDCNN model and the LSTM-CNN have very small fluctuations in accuracy, down by 4.2% and 4.4%, respectively.  
Figure 14: Diagram of the bearing vibration signal at SNR from 2 dB to 10 dB.  
In summary, the SRDCNN model performs well under noisy environment, especially, it achieves the highest classification accuracy under less strong noise. Additionally, the LSTM-CNN model also has a good performance on denoising effect, and the 1d-DCNN model is over-fitting in the case of high noise. Although the performance of 2d-DCNN is better than 1d-DCNN, there is still a gap compared to SRDCNN and LSTM-CNN. All of these facts indicate that the proposed SRDCNN model is able to extract the essential features and avoid local false features caused by ambient noise.  
In order to visually understand the feature representation of the proposed SRDCNN model, t-SNE [37] is employed to visualize the feature representation of the last fully-connected hidden layer for the test samples. The feature representation distribution of test set C can be visualized in Figure 16. It is clear that the proposed SRDCNN model has learnt the effective features, which can be used to distinguish between the fault categories.  
Figure 16: Feature distribution visualization of noise experiment with SRDCNN model.  
Figure 15: Results of these three methods diagnosing with noisy signal.