## 3 The Preliminaries
### The Deep Convolutional Neural Network Architecture  
DCNA comprises hidden layers (called convolutional layers), pooling layers, and fully connected layers [40; 41]. The convolutional layer performs feature extraction from the input image data through a kernel-filter-based convolutional process; then, the pooling layer implements the down-sampling process. The pooling layer helps to reduce computational complexity and to recognize the learned extracted features. In addition, a variety of constraint-optimizing layers, such as rectified linear units (ReLU), dropout, and normalization, are integrated into the DCNA for classification improvement [50]. Afterward, the fully connected layer uses weighted-base wiring to connect the output of the final convolutional or pooling layer for transferring information to the classification layer, which outputs the likelihood decision for classifying the fault types, normally using a SoftMax function [51]. Figure 5 demonstrates the general structure of the DCNA. The convolutional layer (Cv) is responsible for the latent feature engineering processing. The Cv performs feature mapping through its layers for the extraction of representative attributes from input images that contain key information about gear states. To demonstrate the feature mapping process, we consider two consecutive layers: \(j^{\text{th}}\) and \((j+1)^{\text{th}}\) convolutional layers. There are \(k\) filters (or kernels), with the size of \(m\times n\), which are utilized for extracting features from the output of the \(j^{\text{th}}\) layer. The output space of the \(j^{\text{th}}\) layer, with dimensions of \(m\times n\), is locally swept to convolve with each filter of \(D\times R\) size, using w training weights for adjustment. Then, each result, which corresponds to a single kernel, is added in scale computation with bias b, and functionalized by activation functions of nodes in the \((j+1)^{\text{th}}\) layer, these are normally non-linear functions, such as the rectified linear unit function (ReLU), used to perform non-linear feature mapping through layers. Assuming that the parameter used in the convolutional calculation is a unity, then feature space with a dimension of \((m-D+1)\times(n-R+1)\) is formed corresponding to each filter. In general, the \(i\)-th feature mapping space \((fms)\) of the convolutional layer \(k\) can be formulated as follows:  
\[fms_{i}^{k}=RL\left(\sum_{r\in A^{k-1}}fms_{r}^{k-1}\varovec{w}_{i}^{k}+b_{i}^{k }\right), \tag{8}\]  
with \(RL\) as the ReLU function: \(RL(x)\ =\max(0,x)\). Where, \(w_{i}^{k}\) and \(b_{i}^{k}\) are the sets of weights and bias for the \(i^{\text{th}}\) filter in layer k, \(\varovec{\odot}\) indicates the convolution operator, \(A^{k-1}\) denotes all feature mapping spaces in the \((k-1)^{\text{th}}\) layer. The feature spaces become more separable as it goes from lower convolutional layer to bottleneck layer network. Typically, the pooling layer (Pm) is used next to each convolutional layer for the down-sampling process. It scans the whole range of a feature mapping space sequentially, and then applies the pooling operation on a defined pooling region by a non-overlapping searching method. The pooling operation that is most commonly used is the mean average, or maximum value in the defined pooling area [41]. Figure 5: Description of typical DCNA. Usually, many incorporated pairs of convolutional and pooling layers are employed in DCNA. After the final convolutional layer or pooling layer, several fully connected layers (Fc) are used to expand deep representation feature mapping spaces, as well as the concatenation of feature mapping spaces into a feature vector. Finally, the represented feature vectors are provided as an input to non-linear nodes for classifying the features into their corresponding categories (the fault states of a gearbox). The SoftMax function is typically used as the final activation function in the classification layer for classifying the input data into their corresponding categories.