# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 4 Cases Studies
### Case 4: Estimation of Variable Amplitudes and Frequencies  
Building upon studies that analyze complex nonlinear systems [35; 36; 38; 71], this case study proposes the analysis of a signal composed of four vibratory components with time-varying frequencies and amplitudes, a quadratic trend, and additive Gaussian noise. These components exhibit nonlinear and non-periodic behaviors, which pose a challenge for traditional methods such as the Fast Fourier Transform [7; 37]. The decomposition was performed using Variational Mode Decomposition, Empirical Mode Decomposition, and the hybrid EMD-VMD method. The composite signal \(\mathcal{F}(t)\) is defined as the sum of the following components:  
1. Component \(\mathcal{F}_{1}\): A cosine function with a phase shift applied after \(t=0.5\) s:  
\[\mathcal{F}_{1}(t)= \begin{cases}0&\text{if }t<0.5,\\ 1.5\cos(200\cdot\pi t)&\text{if }t\geq 0.5.\end{cases} \]  
This component is absent during the first half of the time interval. The hybrid EMD-VMD method excels in accurately identifying this absence, thanks to its adaptability to changes in the signal, such as nonstationarity and variations in amplitude and frequency. 2. Component \(\mathcal{F}_{2}\): A cosine function present in the first half of the signal:  
\[\mathcal{F}_{2}(t)= \begin{cases}2\cos(120\cdot\pi t)&\text{if }t<0.5,\\ 0&\text{if }t\geq 0.5.\end{cases} \]  
This component, like \(\mathcal{F}_{1}(t)\), is nonstationary. The hybrid EMD-VMD method allows for the identification and representation of this component, aiding in the analysis of signal behavior at every analyzed time instant. 3. Component \(\mathcal{F}_{3}\): A chirp signal with a smooth frequency variation from 20 Hz to 15 Hz:  
\[\mathcal{F}_{3}(t)=3\cos\left(40\pi t-\frac{\pi}{3}\log(1+e^{30(t-0.45)})\right)\]  
This signal exhibits a continuous transition in its frequency, characteristic of nonstationary systems. While conventional methods struggle to capture these variations, the \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **EMD** & **VMD** & **Hybrid EMD–VMD** \\ \hline \(\Omega_{1}\) & 0.0008 & 0.0391 & 0.0046 \\ \(\Omega_{2}\) & 11.0803 & 8.2764 & 1.4617 \\ \(\Omega_{3}\) & 2921.44 & 5.2769 & 2.24 \\ \(\Omega_{4}\) & 3825.63 & 613.49 & 0.2689 \\ \(\Omega_{5}\) & 1.0404 & 3.8750 & 0.5776 \\ \(\Omega_{6}\) & 552.0700 & 20.4416 & 2.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Mean squared error of frequency estimates. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **EMD Error (\%)** & **VMD Error (\%)** & **Hybrid EMD–VMD Error (\%)** \\ \hline \(\Omega_{1}\) & 0.13\% & 0.94\% & 0.32\% \\ \(\Omega_{2}\) & 8.53\% & 7.39\% & 3.11\% \\ \(\Omega_{3}\) & 90.08\% & 3.81\% & 2.49\% \\ \(\Omega_{4}\) & 74.59\% & 29.85\% & 0.63\% \\ \(\Omega_{5}\) & 1.09\% & 2.12\% & 0.82\% \\ \(\Omega_{6}\) & 20.27\% & 3.92\% & 1.47\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Detailed comparative analysis of percentage errors derived from multiple signal decomposition methods. hybrid EMD-VMD method accurately estimates the instantaneous frequency over time, demonstrating its ability to address the complexity of these systems. 4. Component \(\mathcal{F}_{4}\): A quadratic trend modulated by an exponential decay and a sinusoidal term with a frequency of 6 Hz:  
\[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]  
This component exhibits a significantly varying amplitude over time, characteristic of nonlinear systems. The hybrid method effectively captures this dynamic behavior, demonstrating its suitability for analyzing signals with complex temporal variations. 5. Gaussian Noise \(\mathcal{N}(t)\): White noise with a level equal to 10% of the total amplitude of the signal:  
\[\mathcal{N}(t)=0.1\cdot\mathcal{F}(t)\cdot\mathcal{G}(0,1),\]  
where \(\mathcal{G}(0,1)\) is Gaussian white noise with zero mean and unit variance.