## 4 Vibration Signal Denoising  
### Cycbd  
The objective of CYCBD is to extract fault excitations from a noise-contaminated raw signal while filtering out noise components. The specific steps of CYCBD are as follows.  
The rolling bearing signal can be represented \(x\), and its transmission and deconvolution process can be expressed using the convolution of a Linear Time-Invariant (LTI) system as follows:  
\[s=x\otimes h\approx s_{0} \tag{1}\]  
where \(s\) represents the result after deconvolution, \(\otimes\) denotes the convolution operation, and \(h\) is the Finite Impulse Response (FIR) inverse filter. The above equation can be expressed in matrix form as \(s=Xh\), as shown below.  
\[\begin{bmatrix}s_{1}\\ s_{2}\\ s_{3}\\ \vdots\\ s_{n}\end{bmatrix}=\begin{bmatrix}x_{1}&0&0&\cdots&0\\ x_{2}&x_{1}&0&\cdots&0\\ x_{3}&x_{2}&x_{1}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{n}&x_{n-1}&x_{n-2}&\cdots&x_{1}\end{bmatrix}\begin{bmatrix}h_{1}\\ h_{2}\\ h_{3}\\ \vdots\\ h_{n}\end{bmatrix} \tag{2}\]  
The periodicity of the bearing vibration signal can be described as a cyclostationary process [28]. The cyclic frequency, which reflects the periodic fluctuations in signal energy, is defined as follows:  
\[\alpha=\frac{k}{T_{s}} \tag{3}\]  
where \(k=1,2,3,\ldots,N\), \(N\) denotes the length of the inverse filter, and \(T\) is the rotation period of the shaft.  
CYCBD aims to maximize the second-order cyclostationarity indicator (\(ICS_{2}\)) and is solved using an iterative eigenvalue decomposition algorithm. The general expression for second-order cyclostationarity is shown in the equation below:  
\[ICS_{2}=\frac{\sum_{k>0}\left\lvert\frac{k_{s}^{2}}{k}\right\rvert^{2}}{\left \lvert\frac{k_{s}^{2}}{k}\right\rvert^{2}} \tag{4}\]  
\[c_{s}^{k}=\frac{1}{L-N+1}\sum_{l=N-1}^{L-1}\lVert s_{l}\rVert^{2}e^{-j2\pi al} =\frac{E^{H}\lVert s\rVert^{2}}{L-N+1} \tag{5}\]  
\[c_{s}^{0}=\frac{\lVert s\rVert^{2}}{L-N+1} \tag{6}\]  
in the formula, \(j\) represents the imaginary unit, the superscript \(H\) denotes the conjugate transpose of a matrix, and \(s_{l}\) is the value of the signal \(s\) at time \(l\). The matrix \(E\) is defined as \(E=[e_{1},\ldots,e_{k},\ldots,e_{K}]\), and \(L\) represents the length of the signal \(s\).  
\[e_{k}=\begin{bmatrix}e^{-j2\pi\frac{k}{L_{s}}(N-1)}\\ \vdots\\ e^{-j2\pi\frac{k}{L_{s}}(L-1)}\end{bmatrix} \tag{7}\]According to the above types, \(ICS_{2}\) can be re-expressed as:  
\[ICS_{2}=\frac{|s|^{2}\text{H}EE^{\text{H}}|s|^{2}}{|s|^{2}\text{H}s|^{2}} \tag{8}\]  
Let \(P|s|^{2}\) denote the signal containing the periodic components of \(|s|^{2}\); the periodic components of the signal can be expressed as follows:  
\[P[|~{}~{}s|^{2}~{}]=\frac{1}{L-N+1}\sum_{k}e_{k}\Big{(}e_{k}^{\text{H}}|~{}~{} s|^{2}~{}\Big{)}=\frac{EE^{\text{H}}|~{}s|^{2}}{L-N+1} \tag{9}\]  
By substituting the above formula into \(ICS_{2}\) solution process, we can obtain:  
\[ICS_{2}=\frac{\ell^{4}\chi^{\text{H}}\text{W}Xh}{\ell^{4}Xh}=\frac{\ell^{4} \text{H}E_{\text{W}Xh}}{\ell^{4}R_{XX}h} \tag{10}\]  
and the weighting matrix W can be expressed as:  
\[W=diag\Bigg{(}\frac{P(|s|^{2})}{s^{H}s}\Bigg{)}(L-N+1)=\Bigg{[}\begin{array}[ ]{cc}\ddots&0\\ &P(|~{}s|^{2})\\ 0&\ddots\end{array}\Bigg{]}\frac{L-N+1}{\sum_{l=-1}^{L-1}|s|^{2}} \tag{11}\]  
In the formula, diag is an operator used to generate diagonal matrices, and \(R_{X\text{W}X}\) and \(R_{XX}\) represent the weighted correlation matrix and the correlation matrix, respectively. Solving for the optimal filter is transformed into solving for the maximum eigenvalue \(\lambda\) and its corresponding eigenvector of \(R_{X\text{W}X}h=R_{XX}h\lambda\).  
CYCBD will continue to iterate until the convergence condition is met. When \(ICS_{2}\) is maximized, the corresponding \(h\) is the ideal inverse filter, and the signal obtained after filtering is the noise-reduced signal.