# BearingCog: A Bearing Fault Diagnosis Method under Variable Operational Conditions
## 3 Methodology
### Signal Preprocessing  
VMD is a popular technique for signal decomposition. Different from EMD, VMD is skilled at decomposing an objective signal into several certain modes, which are annotated as \(u_{k}\). Each mode has specific sparsity properties in the frequency domains. To be specific, the obtained mode, \(u_{k}\), is concentrated around a central frequency while the corresponding bandwidth is chosen by utilizing the H1 Gaussian smoothness. The detailed steps are listed as follows: Firstly, the Hilbert transform is applied to each mode \(u_{k}\), obtaining the corresponding analytic expression named \(u_{k}^{+}\). The calculation is expressed in Equation (1), where \(\delta(t)\) represents the Dirac function,  
\[u_{k}^{+}(t)=\left(\delta(t)+\frac{j}{\pi t}\right)\times u_{k}(t). \tag{1}\]  
Then, an exponential tuned term is utilized to shift each mode's spectrum to the respective baseband. After, the bandwidth is estimated based on the H1 Gaussian smoothness of the demodulated signal by the squared L2-norm of the gradient [24]. The decomposition process is summarized as solving an optimization problem in Equation (2). \[\min\left\{\sum_{k}\left\|\hat{u}_{t}[\left(\delta(t)+\frac{j}{\pi t}\right) \otimes\mu_{k}(t)]e^{-i\omega_{k}t}\right\|_{2}^{2}\right\},\ \text{s.t.}\sum_{k}\mu_{k}=f(t). \tag{2}\]  
where \(f(t)\) represents the original signal, \(\mu_{k}\) represents the \(k\)th intrinsic mode function (IMF) of the decomposed signal, \(k\) represents the number of the decomposed number, \(\omega_{k}\) represents the center frequency of the decomposed mode, respectively. The symbol \(\otimes\) represents the convolution operation. For the purpose of solving the minimization problem, a penalty term and a Lagrange multiplier are imported. Then the optimization problem can be converted into a variational problem, which is shown in Equation (3). Figure 1: Main framework of the proposed BearingCog. \[\left\{ \begin{array}{l}\mathcal{L}(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha \sum\limits_{k=1}^{K}\left\|\hat{\vartheta}\Big{[}\Big{(}\delta(t)+\frac{j}{nt} \Big{)}*u_{k}(t)\Big{]}e^{-j\omega_{k}t}\right\|_{2}^{2}\\ +\Big{\|}f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\Big{\|}_{2}^{2}+\left\langle\lambda (t),f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\right\rangle\end{array} \right\}. \tag{3}\]  
Based on the study in [25], the alternate direction method of multipliers is adopted to determine the saddle point of the augmented Lagrange in a sequence of iterative suboptimizations. The complete VMD algorithm is concluded in Algorithm 1. ```
Initialize \(\{\theta_{k}^{1}\},\{\omega_{k}^{1}\},0\gets n\) repeat \(n\gets n+1\) for\(k=1\): \(K\)do  Update \(\{\mu_{k}^{1}\}\) for all \(\omega\geq 0\) \(\left\{\hat{\mu}_{k}^{n+1}\right\}\leftarrow\frac{f(\omega)-\sum_{i\in k} \theta_{k}^{n+1}(\omega)-\sum_{i\in k}\theta_{i}^{n}(\omega)+(\lambda^{n}( \omega)/2)}{1+2\alpha(\omega-\omega_{k}^{n})^{2}}\) Update \(\omega:\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega\left\|\hat{\mu}_{k}^{n+1 }(\omega)\right\|^{2}d\omega}{\int_{0}^{\infty}\left\|\hat{\mu}_{k}^{n+1}( \omega)\right\|^{2}d\omega}\) endfor  Dual ascent for all \(\omega\geq 0\) \(\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda}^{n}(\omega)+\tau\Big{(} \hat{f}(\omega)-\sum\limits_{k}\theta_{k}^{n+1}(\omega)\Big{)}\) Until convergence \(:\sum\limits_{k}\left\|\hat{\mu}_{k}^{n+1}-\theta_{k}^{n}\right\|_{2}^{2}/ \left\|\hat{\mu}_{k}^{n}\right\|_{2}^{2}<\epsilon\)
```  
**Algorithm 1**Complete optimization of VMD  
By performing the algorithm, the decomposed mode is estimated and updated by Wiener filtering.