# Noise Reduction Based on a CEEMD-WPT Crack Acoustic Emission Dataset
## 3 Improved CEEMD-WPT Noise Reduction Algorithm
### Principal Interval Coefficient  
At present, the main quantitative indicators of the noise reduction effect include the signal-to-noise ratio (SNR), root mean square error (RMSE), and noise reduction error ratio (dnSNR), etc. Their calculation methods are shown in Equations (12)-(14). Because pure signal or noisy signal is unknown in the real signal, the SNR and RMS error cannot measure the noise reduction effect of the real signal. The noise reduction error ratio only focuses on the ratio of signal power before and after noise reduction, so it can be used to quantify the noise reduction effect of the real signal. However, it does not pay attention to the specific components of the eliminated signal, even if the eliminated signal is effective;  
Figure 1: Flow chart of improved CEEMD-WPT noise reduction method.  
it can produce a better value, so the quantization reliability of the noise reduction effect is poor.  
\[SNR=10\lg\left[\begin{array}{c}\frac{\sum\limits_{t=1}^{n}{Y(t)}^{2}}{\sum \limits_{t=1}^{n}{\left({Y(t)-\mathrm{y}(t)}\right)}^{2}}\\ \\ \end{array}\right] \tag{12}\]  
\[RMSE=\sqrt{\frac{1}{n}{\sum\limits_{t=1}^{n}{\left({Y(t)-\mathrm{y}(t)} \right)}^{2}}} \tag{13}\]  
\[\mathrm{dn}SNR=10\lg\left[\begin{array}{c}\frac{\sum\limits_{t=1}^{n}{x(t) }^{2}}{\sum\limits_{t=1}^{n}{y(t)}^{2}}\\ \\ \end{array}\right] \tag{14}\]  
In the above equation, \(n\) is the signal length, \(Y(t)\) is the pure signal, \(x(t)\) is the signal before noise reduction, and \(y(t)\) is the signal after noise reduction.  
In order to solve the problem of the noise reduction effect of the real signal not being quantified reliably, this paper designs a new quantization parameter \"principal interval coefficient (PIC)\" according to the fact that the noise is mostly concentrated in the high-frequency band. The practical significance of the principal interval coefficient is the ratio of the frequency interval occupied by the top 90% amplitude to the total frequency interval on the spectrum diagram. The specific calculation method is shown in Equation (15).  
\[PIC=\frac{\left({0.9\sum\limits_{i=1}^{n}{A_{i}}}\right)^{*}}{H} \tag{15}\]  
In the above equation, \(PIC\) is the main interval coefficient, \(A_{i}\) is the amplitude corresponding to frequency \(i\) in the spectrum diagram, \(n\) is the maximum frequency, \(H\) is the total frequency interval, and the (\(x\))* function is the frequency interval corresponding to \(x\).  
The quantization principle of the PIC on the signal noise reduction effect is that the effective signal is mainly concentrated in the low-frequency region. When the effective signal content in the signal is higher, the frequency is more concentrated in the low frequency, and the frequency interval occupied by the top 90% amplitude will also decrease, ultimately reducing the PIC index. On the contrary, when there is a lot of noise in the signal, the PICindex will rise. Therefore, the noise reduction effect of the algorithm and the amount of noise contained in the signal can be judged according to the PIC index of the signal before and after noise reduction.  
Since the parameters needed to calculate the PIC can be obtained directly from the signal, it can be used to quantify the noise reduction effect of the real signal, which is more practical than the SNR and RMSE. Compared with the dnSNR, the PIC is calculated based on the distribution characteristics of the effective signal and noise, so the quantization results are more reliable. The reliability of the quantization effects of the dnSNR and the PIC is verified in Section 4.