# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 3 Hybrid Methodology EMD-VMD
### Step 1: Estimation of IMFs Using EMD  
Empirical Mode Decomposition decomposes the signal into a set of intrinsic mode functions by identifying the inherent oscillatory modes present in the signal with multiple harmonic vibratory components. Each IMF is a fundamental oscillatory component within the signal, allowing it to be broken down into functions that represent oscillatory characteristics at different scales or frequencies. As shown in Figure 2, the signal exhibits multiple harmonic vibratory components, each representing distinct oscillatory characteristics that can be analyzed through the Empirical Mode Decomposition process. This is essential for analyzing nonstationary and nonlinear signals, in which frequencies and amplitudes vary over time. The process of obtaining the IMFs involves several key steps including the identification of local extrema, the construction of upper and lower envelopes, the mean envelope calculation, the IMF estimation, and the iteration with residue calculation. Identification of Local Extrema: The first step in EMD is to identify all local maxima and minima of the signal \(x(t)\), as depicted in Figure 3. These extrema are mathematically defined by the following sets:  
\[T_{\text{max}}=\{t\in\mathbb{R}\mid x(t)>x(t-\Delta t)\text{ and }x(t)>x(t+\Delta t)\},\]  
\[T_{\text{min}}=\{t\in\mathbb{R}\mid x(t)<x(t-\Delta t)\text{ and }x(t)<x(t+\Delta t)\}.\]  
Here, \(T_{\text{max}}\) represents the times at which \(x(t)\) reaches a local maximum, and \(T_{\text{min}}\) represents the times at which \(x(t)\) reaches a local minimum. The Figure 3 corresponding values of the signal \(x(t)\) at these times are used to form the sets of ordered pairs:  
\[X_{\text{max}}=\{(t,x(t))\mid t\in X_{\text{max}}\},\quad X_{\text{min}}=\{(t, x(t))\mid t\in X_{\text{min}}\}.\]  
Figure 3: Detected maxima \(x_{\text{max}}\) and minima \(x_{\text{min}}\) in the signal \(x(t)\) with multiple harmonic vibratory components. Figure 2: Signal with multiple harmonic vibratory components. These pairs, \(X_{\max}\) and \(X_{\min}\), serve as the nodes for constructing the upper and lower envelopes of the signal. Construction of Upper and Lower Envelopes: Once the local extrema are identified, the next step is to construct the upper \(e_{\max}(t)\) and lower \(e_{\min}(t)\) envelopes of the signal using cubic spline interpolation. For the sets of maxima \(X_{\max}=\{(t_{i},x_{\max}(t_{i}))\}\) and minima \(X_{\min}=\{(t_{i},x_{\min}(t_{i}))\}\), the cubic spline interpolation ensures that each segment between successive extrema is smoothly connected:  
\[S_{i}(t)=a_{i}+b_{i}(t-t_{i})+c_{i}(t-t_{i})^{2}+d_{i}(t-t_{i})^{3}\]  
The coefficients \(a_{i},b_{i},c_{i}\), and \(d_{i}\) are determined by the following conditions:  
* The spline passes through all data points: \[S_{i}(t_{i})=x_{\max}(t_{i}),\quad S_{i}(t_{i+1})=x_{\max}(t_{i+1})\]
* Continuity of the first and second derivatives at each interior point: \[\dot{S}_{i}(t_{i+1})=\dot{S}_{i+1}(t_{i+1}),\quad\ddot{S}_{i}(t_{i+1})=\ddot{ S}_{i+1}(t_{i+1})\]
* Natural spline boundary conditions (second derivative is zero at endpoints): \[\ddot{S}(t_{1})=0,\quad\ddot{S}(t_{n})=0\]  
Solving this system yields the spline functions \(S_{i}(t)\) for each interval, which together form the upper envelope \(e_{\max}(t)\). The lower envelope \(e_{\min}(t)\) is constructed similarly using the minima. Figure 4 illustrates the identified upper \(e_{\max}(t)\) and lower \(e_{\min}(t)\) envelopes of the signal, constructed from local maxima and minima. Mean Envelope Calculation: The mean of the upper and lower envelopes is computed to estimate the local mean of the signal:  
\[T_{\text{max}}=\{t\in\mathbb{R}\mid x(t)>x(t-\Delta t)\text{ and }x(t)>x(t+\Delta t)\},\]0  
Figure 5 presents the use of envelopes to capture the oscillation range in the signal, and these are used to calculate the mean envelope \(m(t)\).