## 2 Feature Extraction Method--Dual Time-Frequency Images Fusion
### Establishment of Dual Time-Frequency Images Fusion  
According to the above part, it can be seen that the principles of the two time-frequency transformation methods are different, so the extracted features are complementary in the dual time-frequency images. To extract more features and improve the diagnosis accuracy of compressor manufacturing defects, this paper proposes a dual time-frequency image fusion method, and its structure is shown in Figure 1. The inputs of two parallel convolution and pooling layers are different time-frequency images, and the output is the one-hot label of the input image. The CNN has been successfully applied in various aspects such as image processing [25] and speech recognition [26]. Figure 1 shows that the proposed method structure consists of signal preprocessing and CNN. In the signal preprocessing part, CWT and HHT are performed on the same raw vibration signal to obtain two time-frequency images, which reshape to be 128 \(\times\) 128 pixels. The CNN comprises two parallel convolution and pooling modules, two fully connected layers and an output layer. First, two parallel convolution and pooling modules extract features from dual time-frequency images, respectively. Feature maps can be obtained in the convolutional layer by convolving the input images with multiple convolutional kernels. The results are input into the activation function. The formula is as follows:  
\[a_{c,k}=f(X*W_{c,k}+b_{c,k}) \tag{11}\]  
Here, \(a_{c,k}\) represents the output of the \(k\)-th convolutional surface of the convolutional layer; \(W_{c,k}\) represents the weight matrix corresponding to the \(k\)-th convolution kernel; \(X\) represents the input matrix; \(b_{c,k}\) represents the bias term; and \(f\) represents the activation function. We chose the most widely used activation function ReLU. For the pooling layer, this paper chooses the maximum pooling. The formula is as follows:  
\[a_{n,k}=\max(a_{ij})i,j=1,2,\ldots,n \tag{12}\]  
Here, \(a_{n,k}\) represents the \(k\)-th pooling layer, \(a_{ij}\) represents the block of the output matrix of the previous convolutional layer and \(n\) represents the dimension of sub\(-\)block of the output matrix of the previous convolutional layer. The pooling layer often follows the convolution layers. The pooling layers play the role of quadratic feature extraction, feature dimensionality reduction and limited feature selection. Secondly, two parallel convolution and pooling modules extract CWT time-frequency image and HHT time-frequency image features, which spliced in the first fully connected layer. Therefore, this process shows realistic feature fusion. The calculation formula of the fully connected layer is as follows:  
\[a_{fc}=f(a_{n}W_{fc}+b_{fc}) \tag{13}\]  
Here, \(a_{fc}\) represents the output; \(a_{n}\) represents input; \(W_{fc}\) represents the weight matrix; \(b_{fc}\) represents the bias term; and \(f\) represents the ReLU activation function. The fusion stage is implemented by feeding all the extracted dual time-frequency images features into a full-connected layer where the process of feature fusion is realized. Figure 1: Structure of the proposed method. The fully connected layer passes the output value to the output layer for classification. The expression of the softmax function of the output layer is as follows:  
\[\text{softmax}(y)_{i}=\left.e^{y_{i}}\middle/\sum_{j=1}^{n}e^{y_{i}}\right. \tag{14}\]  
Here, \(y_{i}\) represents the \(i\)-th output in the fully connected layer; \(\text{softmax}(y)_{i}\) represents the probability of the corresponding manufacturing defect types corresponding to the \(i\)-th outputs. The method of \"Dropout\" [27] is introduced within the fully connected layer to prevent over-fitting during classification. During training, some neuron in the hidden layer is stopped. This improves the network's capability to generalize and prevents overfitting. According to the above section, the proposed method consists of four convolutional layers and pooling layers, two connected layers and a Softmax layer.