# Weak Fault Feature Extraction and Enhancement of Wind Turbine Bearing Based on OCYCBD and SVDD
## 2 Optimized Cyclostationary Blind Deconvolution  
### Theoretical Background of Cyclostationary Blind Deconvolution  
The purpose of blind deconvolution is to separate the input signal source \(s_{0}\) from the collected signal \(x\) by constructing an inverse filter:  
\[s=x*h=(s_{0}*g)*h\approx s_{0} \tag{1}\]  
where \(g\) represents the frequency response function of an unknown system, \(h\) denotes the inverse filter, \(s\) refers to the estimated signal source and \(*\) indicates the convolution operation. The convolution operation for discrete signal can be described as the following matrix form:  
\[s=Xh \tag{2}\]  
\[\left[\begin{array}{c}s[L-1]\\ \vdots\\ s[N-1]\end{array}\right]=\left[\begin{array}{ccc}x[L-1]&\cdots&x[0]\\ \vdots&\ddots&\vdots\\ x[N-1]&\cdots&x[N-L-2]\end{array}\right]\left[\begin{array}{c}h[0]]\\ \vdots\\ h[L-1]\end{array}\right] \tag{3}\]  
where N and L are the length of \(s\) and \(h\), respectively.  
A new deconvolution algorithm based on cyclostationarity maximization is proposed in reference [15]. In this algorithm, the cyclic frequency is defined as:  
\[a=1/T_{s} \tag{4}\]  
where \(T_{s}\) is the cycle related to fault occurrence rate.  
The novel indicator called second-order cyclostationarity to drive the deconvolution process is described as follows:  
\[ICS_{2}=\frac{\sum_{k>0}|c_{s}^{k}|^{2}}{|c_{s}^{0}|^{2}} \tag{5}\]  
with  
\[c_{s}^{k}=\left\langle|s|^{2},\mathrm{e}^{\mathrm{i}2\pi kam}\right\rangle= \frac{1}{N-L+1}\sum_{n=L-1}^{N-1}|s[n]|^{2}\mathrm{e}^{-\mathrm{i}2\pi kam} \tag{6}\]  
\[c_{s}^{0}=\frac{\left\|s\right\|^{2}}{N-L+1} \tag{7}\]Equations (6) and (7) can be further expressed as following matrix form:  
\[c_{s}^{k}=\frac{\mathrm{E}^{\mathrm{H}}|s|^{2}}{N-L+1} \tag{8}\]  
\[c_{s}^{0}=\frac{s^{\mathrm{H}}s}{N-L+1} \tag{9}\]  
where \(|s|^{2}=\left[\mathrm{[s}[L-1]]^{2},\cdots,\mathrm{[s}[N-1]]^{2}\right]^{ \mathrm{T}}\), \(\mathrm{E=}[\mathrm{e}_{1}\cdots\ \mathrm{e}_{k}\cdots\ \mathrm{e}_{K}]\), \(\mathrm{e}_{k}=\left[\mathrm{e}^{-\mathrm{j}2\pi\mathrm{k}a(L-1)}\cdots \mathrm{e}^{-\mathrm{j}2\pi\mathrm{k}a(N-1)}\right]^{\mathrm{T}}\), and \(k\) denotes the sample index.  
Based on above Equations, Equation (5) can be expressed as:  
\[ICS_{2}=\frac{|s|^{2\mathrm{H}}\mathrm{EE}^{\mathrm{H}}|s|^{2}}{|s^{\mathrm{H}} s|^{2}} \tag{10}\]  
The signal which contains the periodic component \(|s|^{2}\) corresponding to all the cyclic frequencies of interest \(ka\), can be described as:  
\[\Re[s]=\frac{1}{N-L+1}\sum_{k}\mathrm{e}_{k}(\mathrm{e}_{k}^{\mathrm{H}}|s|^{ 2})=\frac{\mathrm{EE}^{\mathrm{H}}|s|^{2}}{N-L+1} \tag{11}\]  
Substituting Equations (2) and (11) into Equation (10), and the obtained final outcome is:  
\[ICS_{2}=\frac{h^{\mathrm{H}}\mathrm{X}^{\mathrm{H}}W\mathrm{X}h}{h^{\mathrm{H} }\mathrm{X}^{\mathrm{H}}\mathrm{X}h}=\frac{h^{\mathrm{H}}R_{X\mathrm{W} \mathrm{X}}h}{h^{\mathrm{H}}R_{X\mathrm{X}}h} \tag{12}\]  
where the expression of weighting matrix \(W\) is:  
\[W=\mathrm{diag}\bigg{(}\frac{\Re[|s|^{2}]}{s^{\mathrm{H}}s}\bigg{)}(N-L+1)= \left[\begin{array}{cc}\ddots&&0\\ &\Re[|s|^{2}]\\ 0&&\ddots\end{array}\right]\frac{(N-L+1)}{\sum_{n=L-1}^{N-1}|s|^{2}} \tag{13}\]  
Equation (12) is the core of CYCBD. By solving Equation (12) through Equation (13), the deconvolution signal exhibiting the maximum second-order cyclostationarity can be extracted according to the cyclic frequency.