# A Deep-Learning-Based Bearing Fault Diagnosis Using Defect Signature Wavelet Image Visualization
## 3 Fault Diagnosis Methodology Using the Defect Signature Wavelet Image
### Deep Convolution Neural Network Structure Specification  
Convolution neural network has several benefits comparable to other feature-learning methodologies. Previously, much as the stacked sparse auto-encoder did, the CNN automatically learns numerous levels of abstract representations from the data via their deep architecture layers. The learning process enables signals of high complexity to be learned to create a high-order representation feature. Secondly, the CNN applies an end-to-end structure for the learning model, hence, a single unique structure has to be optimized and the testing phase only needs a one-pass feed-forward process. Finally, the CNN model is utilized to exploit the spatial characteristic in the constructed DSWI from the sensor data. By using the sparse number of attributes, CNN reduces the number of training parameters when compared to the multi-layer perceptron network (a conventional artificial neural network). In the case of a DSWI from the AE signal, the DCNN defines a spatial architecture as the set of three channels with respect to three channels of the DSWI. A typical case to note is that due to the rolling and sliding mode combination of the roller or ball in bearing, the expected energy which is contained in a fundamental frequency may not totally appear in the frequency range close to the fundamental frequency. Therefore, deploying this information can ameliorate the performance of the fault detection algorithm. Instead of feature extraction stage with the features being designed by experts, it should be considered that the difference of feature-learning method is in the researched work of this paper as here, the feature extraction stage is not employed; thus, the DCNN model is directly implemented on the DSWI of the AE data so that the DCNN has the capacity to learn the features itself. Many optimization constraints, comprising batch normalization, dropout, initialization methods, and leaky rectified linear units, are also used for incorporating into the principal architecture of the DCNN to create better classification performance. A DCNN operates as follows: given an input image consisting of multiple channels, a convolutional layer computes a transformed output as the function of the input, weights and bias parameters, with the difference from the normal artificial neural network being that the adjusted variables of the layer are organized as a sequence of filters and are applied to the convolving operator over the input to produce the output of the convolution layer. Each convolutional layer output is a 3D tensor, which includes a stack of 2-D matrices, the so-called feature maps, which will be utilized as input to feed to the next network layer of the DCNN model. The weight parameters in the filter bank are distributed and shared over the local region of input, which efficaciously exploits the local spatial characteristics, and also diminishes the quantity of optimized parameters. The convolutional operation can be described as:  
\[O_{i}^{(m)}=\varphi\left(\sum_{c=1}^{C}W_{i}^{(c,m)}\circ S_{i-1}^{(c)}+B_{i}^{ (m)}\right) \tag{6}\]  
In this formula, \(i\) stands for the order of the layer as before. The 2D convolution of the input \(S_{i-1}\) and the weight \(W_{i}^{(c,m)}\), which is responsible for yielding \(m^{th}\) output of the map, is presented by the \((\circ)\) operator in the formula. The term \(B_{i}^{(m)}\) represents the bias vector. After that, we apply a nonlinear activation function \(\varphi\) on the sum of convolutions plus a bias vector to obtain the final output. By utilizing a deep architecture, a network with several convolution layers, the model is more robust to complex variations in the data. Thus, if the data naturally describe many variations with high complexity, a deep architecture is necessary. In the case of bearing faults, due to the manifestation of the various faults which are considered here that illustrate a little variation, a reasonably designed deep model suffices.