## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
In VMD-DFA [18] with a fixed threshold of 0.75, there is a risk of including noise-dominant modes in the reconstruction process, particularly when these modes are close to the threshold boundary. On the other hand, VMD-DFA-DWT [19] uses a fixed threshold of 0.25, which might result in discarding signal components that have lower scaling exponents due to mode mixing or interference from noise, leading to information loss. The proposed DFA-based adaptive threshold method (represented by the vertical red dashed line in Figure 9) employs an adaptive approach to overcome the shortcomings of the fixed

Figure 4:0 BLIMFs obtained from VMD of the Bumps signal with 10 dB SNR. threshold methods. In this approach, BLIMFs are selected adaptively based on a threshold derived from the analysis of the scaling exponents, rather than using pre-defined fixed values. For BLIMFs with a scaling exponent greater than the adaptive threshold, SWT is applied to further enhance signal quality by reducing noise. This adaptive threshold mechanism allows the proposed method to selectively apply wavelet-based denoising, making it more robust than the VMD-DFA-DWT [19] approach, particularly in cases where signal and noise characteristics vary significantly. Thus, the proposed method strikes a better balance between retaining useful signal components and minimizing noise. By adopting an adaptive threshold, the proposed method ensures that the mode selection dynamically considers the characteristics of the scaling exponents in each specific case, effectively differentiating between noise and signal components. This approach not only minimizes the inclusion of noise but also helps retain useful signal modes, ensuring a more effective and accurate reconstruction. The DFA-based adaptive threshold method, therefore, offers enhanced robustness in mode selection, leading to improved denoising performance and better conservation of the signal information. Figure 10 shows a comparison between the denoised Bumps signals obtained from different VMD-based denoising methods. The original signal is also plotted in each subfigure. The VMD-DFA and VMD-DFA-DWT methods, as shown in Figure 10a,b, resulted in denoised signals that have more apparent and unwanted spikes. On the other hand, the proposed VMD-based method provides a relatively accurate estimation of the original signals with minimal artifacts (see results presented in Figure 10c). The proposed method effectively recovers all signal information while avoiding artifacts through the selective thresholding of important modes and rejecting coefficients displaying noise-like statistics. Figure 4:1 Plot of scaling exponent \(\alpha\) vs. index of BLIMFs shown in Figure 4:2 (Thresholds obtained via various methods are also shown in the same for clarity of presentation). ### Quantitative Results for Test Signals

This section presents the overall denoising performance for various methods for five test signals (shown in Figure 4) from the viewpoints of following the performance measures. The first performance measure is the normalized mean squared error (NMSE) being defined (in dB) as

Figure 4:3

where \(\mathbf{s}(n)\) is the original signal and \(\mathbf{\hat{s}}(n)\) is the recovered estimated signal. The second performance measure is the MI, which is an information-theoretic measure for quantifying how much information is shared between two or more random variables and is defined as [29]

Figure 4:4

where \(f(z_{1},z_{2})\) denotes the joint probability distribution function of two random variables \(z_{1}\) and \(z_{2}\), and \(f(z_{1})\) and \(f(z_{2})\) denote their respective marginal probability distribution functions. It is important to mention that the NMSE is a dimensionless measure, and when approaching negative infinity, it indicates the reconstruction of the signal with minimal error. On the other hand, MI measures the similarity between the reconstructed and ground truth, and a larger MI value would show better denoising performance.