## 3 Proposed MFFCNN-Based Fault-Diagnosis Method
### Multiscale Transformation  
The so-called multi-scale is actually the sampling of different granularities of the signal. Typically, different features can be observed at different scales to complete different tasks. The multi-scale transformation stage in this paper adopts a parallel multi-branch topology. For a given 1-D signal \(x\) (\(x\in\mathrm{R}^{N\times 1}\)), multiple consecutive signals \(\{y(k)\}\) with different granularities are constructed by a simple process of down-sampling. The representation method of multi-scale down-sampling is shown in Figure 7, and its mathematical description is as follows:  
\[y_{j}^{(k)}=\frac{1}{k}\sum_{i\ =\ (j-1)k+1}^{jk}x_{i},\ 1\ \leq\ j\ \leq\ \frac{N}{k}, \tag{2}\]  
where \(k\) is the length of the non-overlapping window in down-sampling (also called the scale factor), and multiple filtered signals with different scales, respectively (i.e., scale1, scale2, and scale3) can be obtained. Typically, the size of \(k\) is related to the details and trends of feature learning later. Following Figure 7, the original signal was sampled on three different scales to feed into the trunk and branch module (TBM) in the multi-scale feature learning stage. Figure 6: Flow chart of the MFFCNN-based EMA fault-diagnosis system. ### Multiscale Transformation  
After obtaining three granular signals with different scales, each granular signal \(\{y^{(k)}\}\) (\(k\) = 1, 2, 4) will be fed into the TBM separately to learn useful and advanced features. As shown in Figure 8, each TBM consists of three sets of alternately stacked convolution layers and max-pooling layers. The signals pass through three pairs of stacked convolutional layers in parallel (C1\({}^{(k)}\), C2\({}^{(k)}\), and C3\({}^{(k)}\)) and pooling layers (P1\({}^{(k)}\), P2\({}^{(k)}\), and P3\({}^{(k)}\)), learning in an advanced and effective way from multiple granular signals of different time scales' fault characteristics. Specifically, each granular signal uses filters (convolution kernels) of different sizes so that each parallel convolutional layer at the same level can obtain the characteristics of different receptive fields and enhance its capture range of high- and low-frequency features taking into account the overall details of the input signal, thereby improving the diagnostic performance of the model. The first convolutional layers (C1\({}^{(1)}\), C1\({}^{(2)}\), and C1\({}^{(4)}\)) have signal lengths of \(N\), \(N/2\), and \(N/4\), respectively. For each first convolution, the size \(m\) of the corresponding convolution kernel decreases as the value of \(k\) increases, which is beneficial to better extracting useful  
Figure 8: The trunk and branch module. Figure 7: The representation method of multi-scale down-sampling. features. Taking the \(i\)-th element \(a_{i}\) of the \(j\)-th output feature map \(\mathbf{a}_{j}\) of the first convolutional layer ((C1(\({}^{(k)}\)) as an example, the following is obtained:  
\[a_{i}=\sigma\Big{(}\mathbf{w}^{\intercal}y_{i{i}+m-1}+b\Big{)}, \tag{3}\]  
\[\mathbf{a}_{j}=\Big{[}a_{1},a_{2},\cdots,\ a_{i},\cdots,a_{(L-m)/s+1}\Big{]}. \tag{4}\]  
Among them, \(\mathbf{w}\) is the weight vector of the \(j\)-th convolution kernel; \(b\) is the bias term; \(y_{i{i}+m-1}\) is the \(m\)-length sub-signal of the input signal \(y\) starting from the \(i\)-th time step; \(\sigma(\cdot)\) is a nonlinear activation function, namely the rectified linear unit (ReLU); \(s\) is the moving stride of the \(j\)-th convolution kernel on the signal \(y\). In order to increase the sparsity of the model and improve the speed of network training, maximum pooling is used to perform nonlinear down-sampling of the input feature map. Its advantage is that position-independent features can be obtained. Suppose that one input feature map is traversed with the pooling step size \(w\), and one can be calculated for each sliding step \(w\).