## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
For the previous method [21], DFA is used to calculate the slope of the scaling exponents of each IMF. The threshold is calculated using (9), which computes the amount of frequency or fluctuation change that causes the biggest loss in long-range correlations or maximum rise in noise content. In the case of a Bumps signal with an SNR of 10 dB, the steepest slope is achieved between IMF2 and IMF3 (\(\Delta\alpha_{2}=0.7766\)), indicating the greatest frequency rise. Therefore, it is concluded that the first three modes belong to the noisy group, and later modes (IMF4 till the residual) constitute the signal-dominant group. This clear segregation between the two groups is depicted by a vertical dash-dotted line in Figure 6. Notably, IMF3 is included in the signal reconstruction, which was discarded by the fixed threshold-based method. Upon examining Figure 5, one can visually observe that IMF3 contains a combination of features that suggest it is not purely noise. By including IMF3, the adaptive approach makes a more informed decision to preserve potentially im

Figure 6: Plot of scaling exponents \(\alpha\) vs. index of IMFs shown in Figure 5 (threshold obtained via various methods are also shown in the same for clarity of presentation). portant signal components, thus improving the overall denoising process. This adaptability ensures that the reconstructed signal retains essential information while minimizing the noise. Specifically, by calculating the largest slope change, the adaptive method is able to better pinpoint the transition from noise-dominant to signal-dominant IMFs. This results in more reliable denoising and improved SNR for the reconstructed signal, especially when the characteristics of the noise or signal are unknown or vary significantly. While the adaptive thresholding method is more effective in preserving potentially important signal components, the selected modes may have some noise present therein. To address this, wavelet denoising is subsequently performed via SWT to further eliminate these noise traces and enhance the overall quality of the reconstructed signal. After thorough experimentations, _sym8_ is selected as a mother wavelet for SWT, and EMD is performed for 10 levels of decomposition. The mother wavelet has been chosen for its suitability in handling signals with diverse characteristics, while the choice of 10 decomposition levels struck a balance between capturing fine-scale details and avoiding the potential drawbacks associated with excessive decomposition. The visual comparison of denoised Bumps signals from various EMD-based denoising methods is presented in Figure 7. As observed in Figure 7a,b, EMD-SC and EMD-MI methods, respectively, provide estimates that are quite near to the actual signals, but they both have artifacts. Thus, their ability to extract signal details is overshadowed. Figure 7c exhibits how EMD-DFA essentially recovers the variances while eliminating noise. Nevertheless, it is unable to detect subtle fluctuations, particularly in the second half of the denoised signal. Figure 7d illustrates that EMD-SWT3 discovered more obvious (unwanted) spikes in the denoised signal. In addition, Figure 7a,b,d depict amplitude-wise differences in the reconstructed signal against the original noiseless signal. The EMD-SC method employs strict thresholding to remove the small amplitude components or noise from the signal. As a result, the low-amplitude details present in the original signal can be lost during the reconstruction process. This loss of information contributes to the observed variations in amplitudes between the reconstructed and original signals. The strict thresholding selectively removes certain components based on their amplitudes, potentially causing a deviation in the amplitudes of the reconstructed signal from the noiseless original signal. Similarly, in the EMD-MI method, the utilization of MI for amplitude estimation introduces its own set of challenges. MI estimates the relationship between variables based on their statistical dependence.