# Weak Fault Feature Extraction and Enhancement of Wind Turbine Bearing Based on OCYCBD and SVDD
## 2 Optimized Cyclostationary Blind Deconvolution
### Frequency Weighted Energy Entropy Indicator  
For a given signal \(x(t)=A\cos(\omega t+\phi)\), its analytic signal \(P(t)\) can be expressed as [25]:  
\[P(t)=x(t)+\mathrm{j}H[x(t)] \tag{16}\]  
where \(H[\cdot]\) represents the Hilbert transform. The instantaneous energy of \(x(t)\) is usually represented by the square of the module of the analytic signal \(P(t)\):  
\[E[x(t)]=\left|x(t)+\mathrm{j}H[x(t)]\right|^{2}=A^{2} \tag{17}\]  
Through the above equation, it can be found that only the amplitude information is involved during the calculation process of instantaneous energy of \(x(t)\), but the frequency information is ignored. In order to avoid this drawback, the derivative operation is regarded as the weighting filter, and the frequency weighted energy operator (FWEO) is defined as follows:  
\[\Gamma[x(t)]=\left|\dot{x}(t)+\mathrm{j}H[\dot{x}(t)]\right|^{2}=\dot{x}^{2}( t)+H\big{[}\dot{x}(t)\big{]}^{2}=A^{2}\omega^{2} \tag{18}\]  
where \(\dot{x}(t)\) is the first-order derivative of \(x(t)\). By performing FWEO operation on the given signal, the instantaneous energy signal which contains the amplitude information as well as the frequency information can be gained. For the discrete signal \(x(n)\) with length of \(N\), its FWEO operation is expressed as:  
\[\Gamma[x(n)]=\frac{1}{4}\Big{[}x^{2}(n+1)+x^{2}(n-1)+h^{2}(n+1)+h^{2}(n-1) \Big{]}+\frac{1}{2}\big{[}x(n+1)x(n-1)+h(n+1)h(n-1)\big{]} \tag{19}\]  
where \(h(n)=H[x(n)]\). In the OCYCBD method, an evaluation indicator is needed to fairly assess the quality of deconvolution signal during parameter optimization process. The traditional kurtosis indicator has been frequently used to evaluate the impact feature of time domain waveform. However, this indicator will change sharply if an accidental impact appears in the waveform, which may lead to erroneous assessment due to its sensibility to accidental impact. As a nonlinear instantaneous energy tracking approach, frequency weighted energy operator (FWEO) is in a position to strengthen the impact symptom effectively [19]. And Shannon entropy has a good capacity of reflecting the uniformity of a given time series [26]. Then a new frequency indicator called weighted energy entropy (FWEE), which combining the respective specialties of FWEO and Shannon entropy, is proposed to overcome the drawback of traditional kurtosis indicator. And it is utilized to evaluate the deconvolution signal and guide the parameter optimization process. The expression of FWEE indicator of discrete signal \(x(n)\) is as follows:  
\[\left\{ \begin{array}{l}\mathrm{FWEE}=-\sum\limits_{n=1}^{N}p_{n}lnp_{n}\\ p_{n}=\Gamma[x(n)]/\sum\limits_{n=1}^{N}\Gamma[x(n)]\end{array} \right. \tag{20}\]  
Four simulated signals are constructed to demonstrate the feasibility of this indicator in describing the richness of periodic impact in the time series. The sample frequency and signal length are respectively set to 8192 Hz and 0.1 s. The first simulated signal \(x_{1}(t)\) is constructed as Equation (14), which presents the periodic impact signal of bearing defect without noises. The second simulated signal \(x_{2}(t)\) denotes the Gauss white noises with standard deviation of 0.125. The third simulated signal \(x_{3}(t)\) is mixture of \(x_{1}(t)\) and \(x_{2}(t)\). The fourth simulated signal \(x_{4}(t)\), which presents the superimposed harmonic signal, is expressed as follows:  
\[x_{4}(t)=\cos(60\pi t)+\sin(120\pi t) \tag{21}\]  
Figure 4 illustrates the waveforms of four different simulated signals and their FWEE indicators. It can be seen the FWEE indicator of \(x_{1}(t)\) is the smallest value 5.897, \(x_{2}(t)\), \(x_{3}(t)\) and \(x_{4}(t)\) have larger indicator values, which means FEWW indicator is able to effectively distinguish these four signals due to it can reflect the subtle changes of time series. If the simulated signal contains more prominent periodic impact component, then the regularity of this signal is more distinct and the FWEE indicator is lower. Otherwise, the value of this indicator is higher.