## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction  
The denoised signal \(\mathbf{\hat{s}}(n)\) is reconstructed solely from the noise-dominant modes after they have been denoised using SWT, as  
\[\mathbf{\hat{s}}(n)=\sum_{k=1}^{K}\mathbf{\hat{d}}_{k}(n). \tag{17}\]  
A pseudocode summary of the proposed VMD-based (as described above) is given in Algorithm 1. ```
1:Noisy signal \(\mathbf{x}(n)\), number of modes \(K\), VMD parameters \(\alpha\), \(\tau\), DC, init, tol, DFA segment sizes \(s_{\text{min}}\) and \(s_{\text{max}}\), SWT parameters: mother wavelet \(\psi\), maximum decomposition level \(L\)
2:Denoised signal \(\mathbf{\hat{s}}(n)\)
3:Initialization: Set \(N\leftarrow\text{length of }\mathbf{x}(n)\)
4:VMD Decomposition Decompose \(\mathbf{x}(n)\) into \(K\) modes using VMD: \(\{\mathbf{d}_{k}(n)\}_{k=1}^{K}\leftarrow\text{VMD}(\mathbf{x}(n),\alpha,\tau,K,\text{ DC, init, tol})\)
5:Compute Scaling Exponents using DFA
6:for\(k=1\) to \(K\)do
7: Compute the mean-centered cumulative sum: \(y_{k}(i)=\sum_{n=1}^{i}(\mathbf{d}_{k}(n)-\mu_{k}),\quad i=1,2,\ldots,N\)
```
6: Initialize fluctuation function \(F_{k}(s)\)
7:for each segment size \(s\) from \(s_{\text{min}}\) to \(s_{\text{max}}\)do
8: Divide \(y_{k}(i)\) into \(N_{s}=\left\lfloor\frac{N}{s}\right\rfloor\) non-overlapping segments
9:for\(v=1\) to \(N_{s}\)do
10: Extract segment \(y_{\text{seg}}=y_{k}((v-1)s+1:vs)\)
11: Fit a linear trend \(y_{\text{fit}}\) to \(y_{\text{seg}}\)
12: Compute detrended fluctuation: \(F_{k}^{2}(v,s)=\frac{1}{s}\sum_{i=1}^{s}\bigl{(}y_{\text{seg}}(i)-y_{\text{fit }}(i)\bigr{)}^{2}\)
13:endfor
14: Compute \(F_{k}(s)=\sqrt{\frac{1}{N_{s}}\sum_{v=1}^{N_{s}}F_{k}^{2}(v,s)}\)
15:endfor
16: Estimate scaling exponent \(\alpha_{k}\) by linear regression of \(\log F_{k}(s)\) versus \(\log s\): \(\log F_{k}(s)=\alpha_{k}\log s+C_{k}\)
17:endfor
18:Determine Threshold \(\theta\)
19: Compute differences \(\Delta\alpha_{k}=|\alpha_{k+1}-\alpha_{k}|\) for \(k=1\) to \(K-1\)
20:Determine \(\theta=\arg\max_{k}(\Delta\alpha_{k})\)
21:Mode Classification
22: Signal-dominant modes: indices \(k\leq\theta\)
23:Noise-dominant modes: indices \(k>\theta\)
24:SWT Denoising of Noise-Dominant Modes
25:for each noise-dominant mode \(\mathbf{d}_{k}(n)\), \(k>\theta\)do
26: Perform SWT decomposition up to level \(L\) using wavelet \(\psi\): \(\{cA^{(l)},cD^{(l)}\}_{l=1}^{L}=\text{SWT}(\mathbf{d}_{k}(n))\)
27: Estimate noise level \(\sigma\) from detail coefficients at level \(l=1\): \(\sigma=\frac{\text{median}(|cD^{(l)}|)}{0.6\sqrt{45}}\)
28: Compute universal threshold: \(\xi=\sigma\sqrt{2\ln N}\)
29:for\(l=1\) to \(L\)do
30: Apply soft thresholding to detail coefficients: \(cD^{(l)}_{\text{thresh}}=\text{sign}\Bigl{(}cD^{(l)}\Bigr{)}\cdot\max\Bigl{(} |cD^{(l)}|-\xi,0\Bigr{)}\)
31:endfor
32: Reconstruct denoised mode \(\mathbf{\hat{d}}_{k}(n)\) using inverse SWT: \(\mathbf{\hat{d}}_{k}(n)=\text{ISWT}\Bigl{(}\{cA^{(l)},cD^{(l)}_{\text{thresh}}\}_ {l=1}^{L}\Bigr{)}\)
33:endfor
34: For signal-dominant modes \(\mathbf{d}_{k}(n)\), \(k\leq\theta\), set \(\mathbf{\hat{d}}_{k}(n)=\mathbf{d}_{k}(n)\)
35:Signal Reconstruction
36:Reconstruct the denoised signal: \(\mathbf{\hat{s}}(n)=\sum_{k=1}^{K}\mathbf{\hat{d}}_{k}(n)\)
37:return\(\mathbf{\hat{s}}(n)\)
```

**Algorithm 1**_Cont._

## 5 Computational Complexity Analysis

This section presents the computational complexity analysis of the proposed method in comparison with the existing methods, as discussed in this paper. The summary of the computational complexity analysis is presented in Table 1, where \(N\) denotes the signal length, \(m\) is the number of IMFs, and \(B\) denotes the number of BLIMFs. One example of a set of typical values for these parameters is also included in Table 1 (see the last column). It has been reported that the computational complexity of EMD typically is comparable to that of the conventional Fourier transform, i.e., \(\mathcal{O}\{N\log_{2}N\}\)[25].