# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 1 Introduction  
Signal analysis is a broadly explored field of study that has provided fundamental support for significant advancements across various areas since its introduction. This interdisciplinary approach combines principles of mathematics, statistics, and engineering to interpret and manipulate both digital and analog signals for the purpose of estimating relevant information. This process plays a crucial role in a variety of fields, ranging frommedicine to engineering and control systems, facilitating the development and optimization of advanced systems in each area [1; 2]. From the fundamentals established by the Fourier Transform, which enabled the decomposition of signals into frequency components, signal analysis has undergone significant evolution [1; 3]. The Fourier Transform, introduced as a mathematical tool for understanding signal frequency content, has found widespread applications in telecommunications, audio processing, and signal compression [4; 5; 6]. However, it assumes that signals are stationary and linear, limiting its utility for nonlinear and nonstationary signals, which are frequently encountered in real-world scenarios [7; 8]. To address these limitations, new methodologies have emerged, including the Empirical Mode Decomposition (EMD) and the Hilbert-Huang Transform (HHT), which were designed for analyzing nonlinear and nonstationary data [7; 9]. The EMD decomposes a signal into intrinsic mode functions (IMFs) adaptively, which can then be analyzed using the Hilbert Transform to extract instantaneous frequencies and amplitudes [10]. This approach has been applied effectively in power quality assessment, where EMD and the Hilbert Transform are used to detect and classify disturbances such as voltage sags and spikes, demonstrating superior accuracy compared to S-transform methodologies [11]. This methodology has proven effective in applications ranging from geophysics, where it provided new insights into water wave dynamics [10; 12], to engineering, where it has been applied for fault diagnosis and vibration analysis [13; 14; 15; 16]. The robustness of the HHT has been further validated through applications in biomedical engineering, where it has been used to analyze physiological signals such as brain wave activity and heart rate variability [17; 18]. These successes underscore the versatility of EMD and the HHT in handling dynamic and noisy environments, offering a granular view of signal behavior that traditional methods cannot achieve. However, traditional techniques like the Fourier Transform are often limited in their ability to analyze nonlinear and nonstationary signals, which are common in real-world scenarios [7; 8]. Similar to how iterative conformal mapping has been used to simplify complex fluid domains [19], advanced methodologies such as Empirical Mode Decomposition and Variational Mode Decomposition (VMD) have emerged to address these challenges [20]. These approaches are particularly useful in capturing dynamic behaviors and transitions, as seen in the study of instabilities in nonlinear systems like Stokes waves [21]. Further advancements, such as Differential Empirical Mode Decomposition (DEMD), extend the capabilities of EMD by addressing issues like mode mixing and aliasing. DEMD has shown particular promise in applications involving vibration signal analysis, providing superior accuracy in frequency component extraction [22]. Advanced approaches like Empirical Multi-Synchroextracting Decomposition (EMSD) have further enhanced mode decomposition capacity, offering superior precision for dynamic and nonstationary scenarios [23]. Variational Mode Decomposition employs a spectral-domain approach to decompose signals into intrinsic modes. Instead of operating directly in the time domain, VMD optimizes a model based on the spectral distribution of the signals through a cost function [24; 25; 26]. VMD has found success in various fields, including power systems, where it has been employed to detect high-voltage discharges and enhance power quality [27; 28].