# Fault Diagnosis for a Bearing Rolling Element Using Improved VMD and HT
## 2 Variational Mode Decomposition  
The VMD method is a completely non-recursive and adaptive signal processing method [81]. In the VMD method, the intrinsic mode function (IMF) is defined as an amplitude modulation-amplitude frequency signal, and its expression is described as follows:  
\[u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)) \tag{1}\]  
where the phase, \(\phi_{k}(t)\), is a non-decreasing function, \(\phi_{k}^{\prime}(t)\geq 0\); the envelope, \(A_{k}(t)\geq 0\), and the change of \(A_{k}(t)\) and the instantaneous frequency, \(\omega_{k}(t)=\phi_{k}^{\prime}(t)\), are much slower than the phase, \(\phi_{k}(t)\). In order to evaluate the band width of the modal, the following scheme is described as follows:  
(1) For each modal function, \(u_{k}(t)\), the Hilbert transform is used to obtain the unilateral spectrum of the analytic signal:  
\[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t) \tag{2}\]where \(\delta(t)\) is the Dirichlet distribution function; * represents the convolution. (2) The exponential term, \(e^{i\omega_{k}t}\), is added, the estimated center frequency adjusted, and each modal spectrum is transferred to the base band:  
\[\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j\omega_{k}t} \tag{3}\]  
where \(\omega_{k}\) represents the center frequency. (3) The Gauss smoothing of the signal is demodulated. That is to say, the norm of the gradient, \(L^{2}\), estimates the band width:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum_{k}\|\partial_{t}\bigg{[}\left( \delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\bigg{]}e^{-j\omega_{k}t}\bigg{\|} \begin{array}{c}2\\ 2\end{array} \right\} \tag{4}\]  
where \(\{u_{k}\}=\{u_{1},u_{2},\ldots,u_{K}\}\) and \(\{\omega_{k}\}=\{\omega_{1},\omega_{2},\ldots,\omega_{K}\}\) represent the \(K\)-modal function and the center frequency of each modal function. To transform the constraint problem into an unconstrained problem, a quadratic penalty term and Lagrangian multiplier are introduced. The quadratic penalty term is used to guarantee the reconstruction accuracy of the signal. The Lagrangian multiplier is used to guarantee the strictness of the constraint. The augmented Lagrangian is described as follows:  
\[L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum_{k}\|\partial_{t}\bigg{[}\left( \delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\bigg{]}e^{-j\omega_{k}t}\bigg{\|} \begin{array}{c}2\\ 2\end{array} +\left\|f(t)-\sum_{k}u_{k}(t)\right\| \begin{array}{c}2\\ 2\end{array} +\left\langle\lambda(t),f(t)-\sum_{k}u_{k}(t)\right\rangle \tag{5}\]  
where \(\alpha\) represents a penalty term; \(\lambda\) represents a Lagrangian multiplier. The value of the modal function, \(u_{k}^{n+1}\), is described as follows:  
\[u_{k}^{n+1}=\underset{u_{k}\in X}{\operatorname{argmin}}\left\{\alpha\| \partial_{t}\bigg{[}\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\bigg{]}e^{ -j\omega_{k}t}\bigg{\|} \begin{array}{c}2\\ 2\end{array} +\left\|f(t)-\sum_{i}u_{i}(t)+\frac{\lambda(t)}{2}\right\| \begin{array}{c}2\\ 2\end{array} \right\} \tag{6}\]  
The equidistance transformation of Parseval/Plancherel Fourier is used to transform Equation (6) into the frequency domain:  
\[\hat{u}_{k}^{n+1}=\underset{\hat{u}_{k},u_{k}\in X}{\operatorname{argmin}} \left\{\alpha\|j(\omega)[(1+\operatorname{sng}(\omega+\omega_{k}))\hat{u}_{k}( \omega+\omega_{k})]\| \begin{array}{c}2\\ 2\end{array} +\left\|\hat{f}(\omega)-\sum_{i}\hat{u}_{i}(\omega)+\frac{\hat{ \lambda}(t)}{2}\right\| \begin{array}{c}2\\ 2\end{array} \right\} \tag{7}\]  
The \(\omega\) in the first item is replaced by \(\omega-\omega_{k}\) to obtain the following expression:  
\[\hat{u}_{k}^{n+1}=\underset{\hat{u}_{k},u_{k}\in X}{\operatorname{argmin}} \left\{\alpha\|j(\omega-\omega_{k})[(1+\operatorname{sng}(\omega))\hat{u}_{k} (\omega)]\| \begin{array}{c}2\\ 2\end{array} +\left\|\hat{f}(\omega)-\sum_{i}\hat{u}_{i}(\omega)+\frac{\hat{ \lambda}(t)}{2}\right\| \begin{array}{c}2\\ 2\end{array}