# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 4 Cases Studies
### Case 4: Estimation of Variable Amplitudes and Frequencies
The composite signal is then given by  
\[\mathcal{F}(t)=\mathcal{F}_{1}(t)+\mathcal{F}_{2}(t)+\mathcal{F}_{3}(t)+ \mathcal{F}_{4}(t)+\mathcal{N}(t).\]  
The composite signal \(\mathcal{F}(t)\) is shown in Figure 19, highlighting its complexity with time-varying frequencies and amplitudes, as well as additive noise. The FFT spectrum of the composite signal is shown in Figure 20. While FFT captures the dominant frequencies, it fails to reflect the time-varying characteristics of the signal components. Figure 19: Composite signal \(\mathcal{F}(t)\) consisting of vibratory components, a quadratic trend, and additive Gaussian noise. Figure 20: Frequency spectrum of the composite signal obtained using FFT. The comparison between the original components and the reconstructed modes for EMD, VMD, and the hybrid EMD-VMD method is presented in Figure 21. Each subfigure corresponds to a component: \(\mathcal{F}_{1}\), \(\mathcal{F}_{2}\), \(\mathcal{F}_{3}\), and \(\mathcal{F}_{4}\). The results depicted the ability of the hybrid method to accurately reconstruct the original signal components. The reconstruction errors for each IMF, shown in Figure 22, are calculated as:  
\[e_{\mathcal{F}_{i}}=\mathcal{F}_{i}-\widehat{\mathcal{F}_{i}},\]  
where \(\mathcal{F}_{i}\) and \(\widehat{\mathcal{F}_{i}}\) denote the original and estimated components, respectively. Figure 21: Comparison of original components \(\mathcal{F}_{i}(t)\) with the corresponding modes \(\widehat{\mathcal{F}}_{i}(t)\) reconstructed using EMD, VMD, and the hybrid EMD-VMD method: (**a**) \(\mathcal{F}_{1}\), (**b**) \(\mathcal{F}_{2}\), (**c**) \(\mathcal{F}_{3}\), (**d**) \(\mathcal{F}_{4}\). \[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]0 _Cont._  
The instantaneous frequencies estimated by each method are shown in \[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]1 The hybrid method accurately tracks the time-varying nature of the frequencies, highlighting its robustness in handling nonstationary components. Unlike EMD, which struggles to resolve overlapping frequencies, and VMD, which requires predefined parameters that limit its adaptability, the hybrid method dynamically adapts to the frequency variations present in the signal. This capability is particularly evident in components with smooth transitions, such as \(\widehat{\mathcal{F}}_{3}(t)\), where the hybrid approach demonstrates superior accuracy in reconstructing the instantaneous frequency trajectory. \[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]2 Instantaneous frequencies \(\widehat{\Omega}_{i}\) estimated for each IMF using EMD, VMD, and the hybrid EMD–VMD method: (a) \(\widehat{\mathcal{F}}_{1}\), (b) \(\widehat{\mathcal{F}}_{2}\), (c) \(\widehat{\mathcal{F}}_{3}\), (d) \(\widehat{\mathcal{F}}_{4}\). \[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]3 Error analysis for each IMF reconstructed using EMD, VMD, and the hybrid EMD–VMD method. The instantaneous amplitudes for each component are presented in \[\mathcal{F}_{4}(t)=\left(\gamma t^{2}e^{-t/5}+1\right)\sin(2\pi\cdot 6t).\]4 The hybrid method provides accurate amplitude tracking, further validating its effectiveness. Notably, for components like \(\bar{\mathcal{F}}_{4}(t)\), where the amplitude varies significantly due to modulation, the hybrid method captures these variations with precision, surpassing the limitations of both EMD and VMD in scenarios involving rapid amplitude fluctuations. This accuracy underscores the method's ability to reconstruct the original signal components faithfully, even under challenging conditions such as additive Gaussian noise and nonstationarity. Overall, the hybrid EMD-VMD method proves to be a robust and versatile tool for analyzing complex signals with time-varying frequencies and amplitudes. It not only excels in tracking the reference trajectories of instantaneous frequencies and amplitudes but also minimizes reconstruction errors.