## 2 Vibration Signal Data Processing and Feature Enhancement
### Data Expansion  
As can be seen from Table 1, the samples of each type of bearing faults are too small, so it is necessary to introduce data expansion processing techniques to split the data. Since the dataset is one-dimensional, this paper utilizes the Fast Fourier Transform (FFT) to convert the data into two-dimensional frequency domain signals and then uses sliding window panning to divide the signals. The total number of samples is maximized and the computational effect of the neural network is enhanced while preserving the temporal vibration signal coherence as well as the signal characteristics. The Fourier Transform transforms a function from the time domain to the frequency domain. The Discrete Fourier Transform (DFT) is an application of the Fourier Transform to discretize the Fourier Transform. It can decompose a signal into a series of sine and cosine sums for spectrum analysis, filtering, signal processing, etc., while the Fast Fourier Transform (FFT) is an efficient way of solving the Discrete Fourier Transform. The DFT formula is as follows:  
\[X(k)=\sum\left[x(n)\times e^{\frac{(-j\sim 2\pi kn)}{N}}\right] \tag{1}\]  
where \(N\) denotes the number of sampling points, \(x(n)\) denotes the input signal, and \(k\) represents the frequency domain index. The FFT splits the DFT computation into several small-scale DFT computations and then recombines them and obtains the overall result by butterfly operation. The formula is as follows:  
\[X(k)=E(k)+W_{k}^{N}\times 0[k] \tag{2}\]  
\[X\bigg{(}k+\frac{N}{2}\bigg{)}=E(k)-W_{k}^{N}\times 0[k] \tag{3}\]  
\(E[k]\) is the input elements at even index positions, \(0[k]\) is the input elements at odd index positions. \(W_{k}^{N}\) is the rotation factor, and \(N\) is the total number of input elements. The rotation factor \(W_{k}^{N}\) is  
\[W_{k}^{N}=e^{\frac{2\pi i\hbar n}{N}} \tag{4}\]  
where \(n\) is the subscript of the sequence, k is the subscript of the frequency, and \(N\) is the sequence size. In arithmetic code, the rotation factors are usually stored in complex form, and the size of the rotation factors is automatically adjusted by calling the data. The FFT algorithm computes the DFT of these components recursively by decomposing the signal into parity components and then combining the results to obtain the final spectrum. Using the FFT to convert one-dimensional values into Fourier-transformed frequency domain values can provide higher accuracy for periodic signals or signals that contain multiple frequency components. It can provide more accurate frequency \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Bearing Type**} & **Motor Load** & **Motor SPEED** & **Fault Diameter** & \multirow{2}{*}{**Fault Location**} & \multicolumn{2}{c}{**Sample Set**} \\  & **(Horsepower)** & **(r/min)** & & **(mm)** & & **(Number of Data Points Per** \\  & & & & & & **Sample/Number of Samples)** \\ \hline  & & & & Inner ring & 122,100/1 \\  & & & 0.1778 & Rolling element & 122,100/1 \\  & & & & Outer ring & 122,100/3 \\  & & & & Inner ring & 122,100/1 \\ Drive end bearing & & & 0.3556 & Rolling element & 122,100/1 \\ (6205-2 RS JEM & 1 & 1772 & & Outer ring & 122,100/1 \\ SKF) & & & 0.5334 & Rolling element & 122,100/1 \\  & & & & Outer ring & 122,100/3 \\  & & & & Inner ring & 122,100/1 \\  & & & 0.7112 & Rolling element & 122,100/1 \\  & & & & Outer ring & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sample set of CWRU bearing failure data. components and amplitude values, since noise signals have a certain anti-interference ability. Taking the 0.1778 mm fault diameter inner ring fault point as an example, the time-frequency domain graph obtained after the transformation is as Figure 1.  
Since the signal is a long sequence signal and the data samples are too small, the signal must be processed by window function segmentation after the FFT transform. These windows usually have overlapping parts.