# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 3 Proposed Method of Fast Empirical Wavelet Transform
### Trend Estimation Method Based on the Key Function  
In the proposed method, the step of trend estimation based on the key function can be used to reconstruct the trend of the spectrum. This method inherits the theoretical basis of the Fourier transform. It is better than an interpolation algorithm in terms of computation time; the algorithm is more effective than the traditional empirical wavelet transform. Figure 3 shows the process of obtaining trend components. Step 1: Obtain the spectrum of the measured signal using the FFT algorithm, as shown in Figure 3a. \[Y(f)=FFT(y(t)) \tag{10}\]  
Step 2: Calculate the Fourier transform function of \(Y(f)\). Figure 3b,d show the key function. \[K(f)=FFT(Y(f)) \tag{11}\]  
Step 3: The components of key functions are closely related to the spectrum. Different functions can be obtained through low-pass filtering of key functions. The blue line in Figure 3c can be obtained by inverse FFT (iFFT) low-pass filtering with B\(1=30\) as the boundary. This blue line approximates the trend of the spectrum, called the trend component:  
\[T_{c}(f)=iFFT(K_{B}(f)) \tag{12}\]  
where \(K_{B}(f)\) is the left part of boundary B in \(K(f)\) and \(T_{c}(f)\) is a trend component. The purple line in Figure 3e can be obtained by inverse FFT low-pass filtering with \(\mathrm{B1}=300\) as the boundary. Compared with the above results, it is obvious that different trend components can be reconstructed by inverse FFT with different frequency components. Therefore, it is necessary to study how the number of reconstructed points affects trend components. It can be seen that as the number of reconstructed points increases, the trend components become more and more complex; different trend components will have different division results. When the number of reconstructed points is smaller than the first peak value of the key function, such as \(\mathrm{B1}\), the trend component is gentle and fluctuates very little, approximates a straight line, and cannot clearly be seen from the spectrum fluctuation trend. One minimum divides the spectrum into two areas. The first area contains a large amount of information and does not separate the corresponding frequency components in the spectrum. Thirty points are used to reconstruct the trend component (blue line), where the minimum values are regarded as the boundaries (black line). The key clusters in the spectrum are successfully extracted. Compared with the red line, the trend component corresponding to the blue line is divided into more bands, and the spectrum information contained in each band is more representative and readable. If the number of reconstructed points is increased to \(\mathrm{B5}\), \(\mathrm{B6}\), \(\mathrm{B7}\), the trend component will become more complex and the number of minimum points will increase rapidly. The boundary divides the spectrum into more ineffective components. Comparing the seven trend components, the trend component can respond to the trend of spectral fluctuation when the number of reconstructed points is \(\mathrm{B3}=30\). In order to accurately determine the optimal frequency component in the key function, the intervals of five frequency components  
Figure 3: A flowchart for calculating trend components: (**a**) spectrum of the signal; (**b**) key function of the spectrum with \(\mathrm{B1}=30\); (**c**) trend component of the key function with \(\mathrm{B1}=30\); (**d**) key function of the spectrum with \(\mathrm{B2}=300\); (**e**) trend component of the key function with \(\mathrm{B2}=300\). are \(\text{B1}=20,\text{B2}=25,\text{B3}=30,\text{B4}=35,\text{B5}=40\). To calculate trend components, as in Figure 4b, on the premise of 20 points spanning the front and back, the trend components are almost the same; the boundary sequence of trend minima is also similar. Therefore, it is meaningless to find a more accurate frequency component; it allows the number of reconstructed points to fluctuate in a small range.