# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 2 Evolution of the Hilbert-Huang Transform  
Signal analysis has seen significant advancements since the introduction of the Fourier Transform, which enabled the decomposition of functions into sums of sines and cosines, facilitating signal analysis in terms of frequency components. This method laid the groundwork for subsequent developments, such as the Fast Fourier Transform, which revolutionized signal processing and found applications in a wide range of disciplines, including electrical engineering, economics, and medicine [1, 3, 6, 8]. However, the Fourier Transform has limitations when applied to nonlinear and non-stationary signals due to its assumption of stationarity [6]. These limitations led to the development of adaptive methods to address the complexity of nonlinear systems. One such advancement is Empirical Mode Decomposition, which decomposes signals into intrinsic mode functions that adapt to the local characteristics of the signal. The IMF is defined as a function that meets two conditions, where the number of extrema and the number of zero-crossings differ by no more than one, and the mean of the upper and lower envelopes is zero at any point. The IMF is essential in signal decomposition, enabling adaptive analysis of nonstationary and nonlinear signals [7]. This method offers a comprehensive representation of signal components across time and frequency domains. The EMD process includes identifying local extrema, constructing upper and lower envelopes, and iteratively subtracting their mean to yield the IMF. This ensures an accurate decomposition of harmonic vibratory components while maintaining the fundamental characteristics of the original signal. Further refinements to EMD were made to improve its mathematical framework while maintaining its adaptability [9]. Additionally, EMD has proven effective in separating oscillation modes into stationary sub-signals, making it valuable for analyzing complex, nonlinear, and nonstationary data [12]. EMD has also been applied in various fields, including image compression, where it has been compared to wavelet packet decompositions [24]. Despite its strengths, EMD faces challenges, particularly the mode mixing problem, which can hinder the clear separation of signal components. To address this, an enhanced methodology called Ensemble Empirical Mode Decomposition (EEMD) was developed, specifically designed to overcome mode mixing [25]. EEMD enhances signal feature extraction and has proven effective in decomposing signals into local periodic oscillations [13]. The Hilbert Transform also plays a crucial role in signal analysis by converting real signals into their analytic form, allowing for the better estimation of instantaneous envelope and phase [25]. Its applications span various fields, including theoretical physics, engineering, and modulated signal analysis [28, 29]. The Hilbert-Huang Transform integrates EMD with the Hilbert Transform to provide a unified approach to signal analysis [7, 10]. This method enables the adaptive decomposition of signals into intrinsic components, which can then be analyzed in detail using the Hilbert Transform. The HHT has proven to be particularly effective in analyzing nonlinear and nonstationary signals, surpassing traditional methods such as the Fourier Transform in these contexts [39, 40]. The Hilbert-Huang Transform has been widely used in various applications, such as fault diagnosis, vibration analysis, and structural health monitoring, where it enables early fault detection and improves machinery performance [30; 41; 42]. Additionally, it has been applied in diverse areas, including the analysis of animal vocalizations and the study of macroeconomic indicators [43; 44]. Further advancements in signal decomposition have been achieved with the introduction of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), which improves the separation of signal components by addressing mode mixing through the use of adaptive noise [45].