###### Abstract  
In this work, a new method for denoising signals is developed that is based on variational mode decomposition (VMD) and a novel metric using detrended fluctuation analysis (DFA). The proposed method first decomposes the signal into band-limited intrinsic mode functions (BLIMFs) using VMD. Then, a DFA-based developed metric is employed to identify the 'noisy' BLIMFs (based on their DFA-based scaling exponent and frequency content). The existing DFA-based methods use a single-slope threshold to detect noise, assuming all signals have the same noise pattern and ignoring their unique characteristics. In contrast, the proposed DFA-based metric sets adaptive thresholds for each mode based on their specific frequency and correlation properties, making it more effective for diverse signals and noise types. These predominantly noisy BLIMFs are then denoised using shrinkage techniques in the framework of stationary wavelet transform (SWT). This step allows efficient denoising of components, mainly the noisy BLIMFs identified by the adaptive threshold, without losing important signal details. Extensive computer simulations have been carried out for both synthetic and real electrocardiogram (ECG) signals. It is demonstrated that the proposed method outperforms the state-of-the-art denoising methods and with a comparable computational complexity. empirical mode decomposition; variational mode decomposition; detrended fluctuation analysis; signal denoising; stationary wavelet transform Article  
15 October 202417 November 202417 November 202421 November 202  
applied to isolate the defect echoes. In [10], time-space filtering was applied after EMD to distinguish and remove the dominant noise component by evaluating the correlation and autocorrelation between each signal component and the original signal. In [11], EMD is combined with detrended fluctuation analysis (DFA), and a fixed threshold is used to identify IMFs that contain mostly noise. Expanding on this approach, a modified EMD-DFA method is proposed to improve the extraction of multifractal characteristics in vibration signals. This method introduces a reliable metric based on the scaling exponent to select the relevant IMF components for signal reconstruction, enabling correlation and smoothness measurement without referencing the original signal [12]. The method based on EMD and a three-level stationary wavelet transform (SWT) (EMD-SWT3) processes all IMFs with the assumption that the noise is spread over all of the IMFs [13]. EMD combined with mutual information (MI) (EMD-MI) is developed in [14], which uses the conditional MI criteria to separate IMFs into two groups, noise-dominant and signal-dominant, and only the former group is subjected to thresholding using a shrinkage approach. Existing EMD-based methods suffer from several limitations, including the lack of a mathematical foundation for EMD, the inability to correct errors in the sifting process, difficulties in properly handling noise, and the need to specify hard band limits in the wavelet transform. To address these issues, ensemble EMD (EEMD) has been proposed as an extension of EMD [15]. EEMD introduces a random noise-assisted method, where the original signal is combined with multiple noise realizations to create an ensemble of signals. EMD is then applied to each ensemble member, and the resulting IMFs are averaged to obtain the final decomposition [15]. The introduction of random noise in EEMD helps to overcome the limitations of EMD by reducing end-effect distortions and suppressing mode mixing. However, an alternative approach that offers a fully adaptive method called variational mode decomposition (VMD) has been proposed [16]. While EEMD provides a noise-assisted approach to enhance the performance of EMD, VMD takes a different approach by formulating the decomposition as an optimization problem. VMD is based on the optimization of a specific mathematical problem in order to obtain a set of band-limited IMFs (BLIMFs).