# A Two-Stage Framework for Time-Frequency Analysis and Fault Diagnosis of Planetary Gearboxes
## 2 Preliminaries  
### U-Net  
In 2015, [PERSON] et al. [26] first proposed the U-net model for the problem of cell image segmentation. As an important variant of fully convolutional neural network, U-net does not contain fully connected layers but uses symmetrically arranged convolutional layers to compress and reconstruct feature information, thereby exploiting context information at multiple scales and generating pixel-level image segmentation results. So far, U-net and its variants have seen the most applications in the field of medical image processing [27] and are expanding into machine vision [28], image-based fault detection [29], etc. Figure 1 illustrates the modified U-net architecture used in this paper. Figure 1: U-net structure. The left half of the U-net is the contracting path, also known as the encoding path, which consists of four repeated contracting operators. The contracting operator has the typical structure of a convolutional neural network, including a 3 \(\times\) 3 convolutional layer, a batch normalization layer, followed by a rectified linear unit (ReLU), and a 2 \(\times\) 2 pooling layer. The right half of the U-net is the expanding path, also known as the decoding path. Its overall structure is very similar to the contracting path, except with its more complex expanding operators. Some of the most crucial changes include deconvolution instead of convolution and the addition of concatenation from the contracting path. Based on the above design, U-net presents a simple and elegant U-shaped structure, which is also the origin of its name. In order to explain the U-net structure in more detail, several critical components in the network are described next:  
1. The convolutional layer is composed of a set of convolutional kernels that can be set high and wide. The learnable convolutional kernel enables the convolutional layer to generalize the feature information in the input and map it into a new feature space. In addition, the setting of shared weights enables the convolutional layer to have lower computational complexity than the fully connected layer. For each convolutional kernel, its output can be expressed as: \[O_{i}=f\left(\sum_{j}\omega_{ij}\times X_{j}+b_{i}\right)\] (1) where \(X_{j}\) is the \(j\)th input channel, \(O_{i}\) is the \(i\)th channel of the feature map, and \(f(\bullet)\) is the activation function. \(\omega_{ij}\) and \(b_{i}\) denote convolutional weight and bias term respectively, both of which are trainable parameters. 2. Deconvolution is also known as up-convolution or transposed convolution. Its function in the U-net model is to increase the resolution of feature maps rather than computing the true inverse of convolution. In order to obtain an appropriate expansive capability matching the contracting path, we use a 2 \(\times\) 2 deconvolutional kernel and set the deconvolutional stride to 2. 3. Activation functions enable convolutional neural networks to model nonlinear mapping hidden in data. Common activation functions include Sigmoid, Tanh, ELU, ReLU [30], etc. As a non-saturated activation function, ReLU can alleviate the problems of gradient vanishing and exploding while accelerating model learning. Therefore, ReLU is selected as the activation function in the contracting operator and expansive operator. 4. Batch normalization is an effective tool to deal with the problem of feature distribution drift during batch training. Severe feature distribution drift will reduce the stability of neural network training and aggravate the over-fitting issue, which is particularly obvious in deep neural networks. Batch normalization transforms the distribution of neuron activations into the standard normal distribution.