## 2 Parameter Optimization VMD-NLMS Principle  
### Principle of the VMD Algorithm  
Variational mode decomposition continues the concept of empirical mode decomposition [28]. The algorithm employs a fully non-recursive modal decomposition method to avoid modal aliasing in the signal decomposition process. This method, known as variational mode decomposition (VMD), calculates the intrinsic mode function (IMF) component, where the IMF is defined as the amplitude modulation function and can be expressed as follows:  
\[u_{k}(t)=A_{k}(t)\cos\varphi_{k}(t) \tag{1}\]  
where \(u_{k}(t)\) is the k-th IMF component, which is a narrowband signal with time-varying amplitude and frequency, \(A_{k}(t)\) is the instantaneous amplitude of the k-th IMF, \(\varphi_{k}(t)\) is the instantaneous phase, and the instantaneous frequency is \(\omega_{k}(t)=\varphi_{k}{{}^{\prime}}(t)\). The VMD algorithm constructs a constrained variational problem to decompose the signal into IMFs. The goal is to minimize the sum of the bandwidths of all IMFs, while ensuring that the sum of the IMFs equals the original signal. The variational problem is formulated as:  
\[\left\{ \begin{array}{l}\min\limits_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_ {k=1}^{K}\left\|\partial_{t}\bigg{\{}\bigg{[}\delta(t)+\frac{j}{\pi t}\bigg{]}*u_ {k}(t)\bigg{\}}e^{-j\omega_{k}t}\right\|_{2}^{2}\right\}\\ \sum\limits_{k=1}^{K}u_{k}=f\end{array} \right. \tag{2}\]  
where \(\delta(t)\) is the Dirac delta function, \({}^{*}\) is a convolution symbol, \(f\) is the original signal to be decomposed, \(u_{k}(t)\) is the k-th IMF component, \(\omega_{k}\) is the center frequency of the k-th IMF, \(\partial_{t}\) is the partial derivative with respect to time, and \(j\) is the imaginary unit. The above constraint is transformed into an unconstrained variational problem. By introducing a quadratic penalty factor \(\alpha\) and a Lagrange multiplier operator \(\lambda\), an extended Lagrange function is constructed, which is expressed as:  
\[ \begin{array}{l}L(\{u_{k}\},\{\omega_{k}\},\lambda)=\sum\limits_{k=1}^{K} \left\|\partial_{t}\bigg{\{}\bigg{[}\delta(t)+\frac{j}{\pi t}\bigg{]}*u_{k}(t )\bigg{\}}e^{-j\omega_{k}t}\right\|_{2}^{2}\\ +\bigg{\langle}\lambda(t),x(t)-\sum\limits_{k=1}^{K}u_{k}(t)\bigg{\rangle} \end{array}  \tag{3}\]  
where \(\alpha\) is the quadratic penalty factor, \(\lambda(t)\) is the Lagrange multiplier, \(\lambda\) promises rigor \(\alpha\) guarantees the reconstruction accuracy, \(\{u_{k}\}=\{u_{1},u_{2},\ldots,u_{k}\}\) obtains the \(K\) modal components for the decomposition, and \(\{\omega_{k}\}=\{\omega_{1},\omega_{2}\ldots,\omega_{k}\}\) is the frequency center of each component. The variational problem is solved using the alternate direction method of multipliers (ADMM). The updates for \(u_{k}\), \(\omega_{k}\), and \(\lambda\) are performed iteratively in the frequency domain. The update of \(u_{k}\), \(\omega_{k}\), and \(\lambda\) is as follows:  
\[\left\{ \begin{array}{l} u_{k}^{n+1}(\omega)=\frac{\hat{f}( \omega)-\sum\
olimits_{i\
eq k}\hat{u}_{i}(\omega)+\frac{\hat{\lambda}(\omega )}{2}}{1+2\alpha(\omega-\omega_{k})^{2}}\\ \omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega|u_{k}(\omega)|^{2}d \omega}{\int_{0}^{\infty}|u_{k}(\omega)|^{2}d\omega}\\ \lambda^{\hat{n}+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\bigg{(}\hat{f}( \omega)-\sum\limits_{k=1}^{K}\hat{\underset{k}{\hat{n}}}^{n+1}(\omega)\bigg{)} \end{array} \right. \tag{4}\]  
where \(u^{n+1}\), \(\hat{f}(\omega)\), and \(\hat{\lambda}(\omega)\) are, respectively, \(u_{k}^{n+1}(t)\), \(f(t)\), and \(\lambda(t)\) Fourier transform, \(\omega_{k}^{n+1}\) is the updated center frequency of the k-th IMF at iteration n + 1, \(\left|u_{k}(\omega)\right|^{2}\) is the power spectral density of the k-th IMF, and \(\tau\) is the noise tolerance parameter, which controls the update rate of the Lagrange multiplier.