# Wind Turbine Gearbox Fault Diagnosis Based on Improved EEMD and Hilbert Square Demodulation
## 3 The Proposed Method  
### Criterion of Resampling Frequency Selection  
To examine the effect of sampling frequency, a simulated signal with transient impulse was constructed as shown in Equation (12). The reason for using this kind of simulated signal was that impulses would appear in the acceleration signals when gear tooth damage or a bearing fault occurred in the gearbox of a wind turbine. The simulated signal, its spectrum and components, are shown in Figure 1.  
\[ \begin{array}{l}x_{1}(t)=e^{-400t_{1}}\sin(2\pi 800t),\ t_{1}=\text{mod}(t,1/33) \\ x_{2}(t)=\sin(2\pi 180t)\\ x_{3}(t)=\sin(2\pi 50t)\\ x_{4}(t)=0.16 random(n,1),n=length(t)\\ x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)+x_{4}(t)\end{array}  \tag{12}\]  
Figure 1: _Cont._  
To facilitate quantitative evaluation, three quantities were utilized to analyze the performance of the EEMD. These were the successive IMF orthogonality (SIO), the IMF coherence (IC) and the residual energy (RE). The last two measures were only utilized when the number of components in the decomposition was known. The SIO was a measure equivalent to that stated in [17], but was only implemented on consecutive IMFs. This measure was defined as:  
\[\text{SIO}=\sum_{i=1}^{I-1}\frac{1}{N}\sum\text{IMF}_{i}(n)\text{IMF}_{i+1}(n) \tag{13}\]  
where \(I\) was the number of IMFs, \(N\) the length of signal and \(n\) the discrete time. The IC was the measure of correlation between the valid or expected number of components (ENC) of the decomposition. This measured the ability of the EMD to decompose the signal into physically meaningful components and was defined as:  
\[\text{IC}=\frac{1}{\text{ENC}}\sum_{i}^{\text{ENC}}cor(\text{IMF}_{i}(n),\text {s}_{i}(n)) \tag{14}\]  
where \(s_{i}(n)\) was the source component of the raw signal corresponding to \(\text{IMF}_{i}(n)\). The RE was the energy in the IMFs that were outside the expected number of components. This was another measure of the interpretability of EEMD and was defined as:  
\[\text{RE}=\frac{1}{N}\left(\text{s}(n)^{2}-\sum_{i}^{\text{ENC}}\sum_{n=1}^{N }\text{IMF}_{i}(n)^{2}\right) \tag{15}\]  
where \(\text{s}(n)\) was the signal under analysis. According to the [PERSON] criteria, the sampling frequency must be more than twice the maximum frequency component of the signal; therefore, the sampling frequency of the simulated signal varied from 1650 Hz to 14,100 Hz by steps of 50 Hz. As shown in Figure 2, these performance measure results present a periodic variation trend with the increase of sampling frequency. As the value of sampling frequency was about 2400 Hz, 4800 Hz and 9200 Hz, which corresponded to 3-times, 6-times and 12-times the maximum frequency of the simulated signal respectively, SIO and RE became the local minimum value, while IC became the local maximum value. When SIO and RE take the minimum value, decomposition results had a good orthogonality and maintained the integrity of the information of the raw signal. When IC take the maximum value, the decomposition result was physically meaningful, which meant the EEMD decomposition results were relatively satisfactory. From the  
Figure 1: Simulated signal and its components. (**a**) Waveform of the simulated signal; (**b**) its spectrum; (**c**) source components of the simulated signal. view of the information integrity, accuracy and efficiency of the decomposition results, the resampling frequency should be selected as about 12-times in the maximum frequency of a vibration signal. The decomposition results of the simulated signal with different sampling frequencies of 9450 Hz and 12,100 Hz are illustrated in Figure 3.  
Figure 3: EEMD decomposition result with different sampling frequencies. (**a**) Decomposition result of EEMD with a sampling frequency of 9450 Hz; (**b**) decomposition result of EEMD with a sampling frequency of 14,100 Hz. Figure 2: The effect of sampling on ensemble empirical mode decomposition (EEMD) performance. ISO, successive IMF orthogonality; RE, residual energy; IC, IMF coherence.