# Impulse Signal Detection for Bearing Fault Diagnosis via Residual-Variational Mode Decomposition
## 2 Proposed Method
### Redefined Kurtosis  
A suitable evaluation criterion of impulsive signals for fault diagnosis is extremely important. One of the most common measurements is kurtosis. Research [27] combined the kurtosis of time-domain data and the kurtosis in the squared envelope spectrum (SES) to evaluate impulse signals. This method is known as ensemble kurtosis (EK). The kurtosis of SES can improve the ability of anti-interference impulses but still inherits the drawbacks of kurtosis. In the current study, kurtosis is redefined and the new kurtosis is applied to EK to build a new index for the evaluation of periodic impulses. Consequently, the new index has stronger power to resist interferences from aperiodic impulses than the traditional EK. Kurtosis can be calculated as:  
\[K=\frac{\frac{\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{4}}{\left(\frac{1}{N}\sum \limits_{i=1}^{N}x_{i}^{2}\right)^{2}}}{\left(\frac{1}{N}\sum\limits_{i=1}^{N}x _{i}^{2}\right)^{2}} \tag{7}\]  
which can be written as:  
\[K=\frac{N\sum\limits_{i=1}^{N}x_{i}^{4}}{\left(\frac{1}{\sum\limits_{i=1}^{N}x _{i}^{2}}\right)^{2}}. \tag{8}\]  
\[\left(\sum\limits_{i=1}^{N}x_{i}^{2}\right)^{2}=\left(x_{1}^{2}+x_{2}^{2}+ \cdots+x_{N}^{2}\right)^{2}\]  
\[=\sum\limits_{i=1}^{N}x_{i}^{4}+\sum\limits_{ \begin{array}{c}i,j=1\\ i\
eq j\end{array} }^{N}x_{i}^{2}x_{j}^{2} \tag{9}\]  
\[K=N\times\frac{1}{1+\frac{\sum\limits_{i,j=1}^{N}x_{i}^{2}x_{j}^{2}/\sum \limits_{i=1}^{N}x_{i}^{4}}} \tag{10}\]  
In this study, \(\sum\limits_{i=1}^{N}x_{i}^{4}\) is assumed constant. If only one element is not zero in \(\{x_{i}\}\), then \(\sum\limits_{i,j=1}^{N}x_{i}^{2}x_{j}^{2}\) will be equal to zero, and the kurtosis of the data will be the largest. The \(i\
eq j\)  
kurtosis of the data will decrease if two elements are not zero in \(\{x_{i}\}\). For large interference impulses, this situation can be set as only one non-zero element in \(\{x_{i}\}\). Thus, the kurtosis will be very large, and our judgment may be distorted by it. To overcome this drawback, we redefine kurtosis as:  
\[K=\frac{N\sum\limits_{i=1}^{N}x_{i}^{4}}{\left(\sum\limits_{i=1}^{N}x_{i}^{2 }\right)^{2}}\times R \tag{11}\]  
where \(R\) is defined as the ratio of the number of elements whose values are above a specific threshold to the data length. Generally, the threshold can be defined as the mean of the data. However, the RMS of \(\left\{x_{i}^{2}\right\}\) is applied in this study because of two reasons. First, each of the data is squared in Equation (10). Thus, the data are reasonably evaluated as \(\left\{x_{i}^{2}\right\}\). Second, the RMS can reflect the large aperiodic impulses more effectively than the mean. The relationship between the RMS and the mean can prove this reason. For a digital signal, their relationship can be written as:  
\[W=\frac{V_{RMS}}{V_{Mean}} \tag{12}\]  
where \(V_{RMS}\) is the RMS, and \(V_{Mean}\) is the mean. Here \(W\) is the waveform factor, which can be defined as:  
\[W=\sqrt{\frac{T}{t}} \tag{13}\]  
where \(T\) is the period of the pulse wave, and \(t\) is the corresponding duration. An example for introducing the two parameters is shown in Figure 1.  
Here \(W\) will increase with the decrease in \(t\), as shown in Figure 1. For some large aperiodic impulses, they can be simulated by pulse waves with a small \(t\) and large \(T\). Thus, \(W\) for the large aperiodic impulses has a big value. On the basis of Equation (12), \(V_{RMS}\) can be calculated as:  
\[V_{RMS}=W\times V_{Mean} \tag{14}\]  
Based on Equation (14), it is easy to understand that RMS can reflect large impulses more efficiently than the mean. For a large RMS, the number of these elements whose values are above it will be small. Thus, the kurtosis defined by Equation (11) can effectively suppress the interferences from aperiodic impulses. In this study, the number of this kind of elements is defined as \(M\). Thus, Equation (11) can be written as:  
\[