Empirical Mode Decomposition and Hilbert Huang Transform for Impedance Calculations

I am looking to use the emd/vmd and hht functions to compute impedance of a nonlinear time varying system. The approach is to divide the voltage and current time series that have been transformed to the frequency domain through Ohm’s law. Unfortunately, the emd/hht are tied to instantaneous frequencies rather than spectral frequencies (they can be different), and furthermore the intrinsic mode functions are not sinusoidal. The hht does appear to output a frequency vector f that seems to be different than the instantaneous frequencies, but don’t think this is useable. Is there still a way to compute impedance? There are also transforms like the empirical Fourier decomposition, the nonlinear STFT, and the nonlinear FFT, but not sure if they help since they are tied to specific PDEs/still use instantaneous frequencies. Please note I’ve already implemented an FFT approach and sliding FFT as well for time varying nature, but I want to account for nonlinear behavior.

Answer 1

[Namnendra]

Hi Centauri,

Computing the impedance of a nonlinear and time-varying system poses unique challenges, especially when using methods like Empirical Mode Decomposition (EMD), Variational Mode Decomposition (VMD), or Hilbert-Huang Transform (HHT). The crux of the problem lies in the nature of the signals and the analysis methods: traditional impedance calculation relies on steady-state, sinusoidal signals, whereas EMD/HHT focus on instantaneous frequencies and can deal with non-sinusoidal, non-stationary signals.

Understanding the Challenge

1. Instantaneous vs. Spectral Frequencies: Instantaneous frequency, as derived from HHT or EMD, represents the frequency of a signal at any given instant, which is ideal for analyzing non-stationary signals. Spectral frequency, on the other hand, refers to the frequency components obtained through Fourier-based transforms, assuming stationarity. This fundamental difference complicates the direct application of Ohm's law (V/I) in the frequency domain for impedance calculation in nonlinear, time-varying systems.

2. Non-Sinusoidal IMFs: The Intrinsic Mode Functions (IMFs) obtained from EMD or VMD are not necessarily sinusoidal, which means traditional methods of impedance calculation that assume sinusoidal waveforms may not apply directly.

Proposed Solutions

Despite these challenges, there are a few approaches you could consider:

1. Time-Domain Impedance Calculation: One straightforward method is to calculate impedance in the time domain rather than transforming to the frequency domain. This involves using the instantaneous voltage and current waveforms directly:

 Z_instantaneous = V_instantaneous ./ I_instantaneous;

This gives you an instantaneous impedance waveform, which can then be analyzed further for time-varying characteristics.

2. Hilbert Spectral Analysis: Use HHT to obtain the instantaneous frequency and amplitude of both voltage and current signals. Then, compute the impedance for each IMF component separately. This approach acknowledges the time-varying nature of both amplitude and frequency, which could be more appropriate for nonlinear systems.

3. Empirical Fourier Decomposition (EFD): If available, EFD can be used to decompose the signal into its Fourier components, which might provide a clearer separation of frequency components compared to EMD, especially for analyzing signals with harmonic content. The challenge remains in how to effectively use these components for impedance calculation due to the non-stationary nature of the signals.

4. Adaptive Filtering or Machine Learning Models: Given the complexity and nonlinearity of the system, consider using adaptive filtering techniques or machine learning models to estimate the impedance based on the nonlinear characteristics of the voltage and current signals. This could involve training a model on a dataset of voltage, current, and corresponding impedance measurements to predict impedance under varying conditions.

Implementation Tips

- Signal Preprocessing: Preprocess signals (e.g., detrending, denoising) before applying EMD/VMD/HHT to improve the quality of the IMFs and the accuracy of the subsequent analysis.

- Combining Methods: Consider combining EMD/VMD with other transforms (e.g., STFT, wavelet) to capture both the non-stationary and nonlinear characteristics of the signals. This hybrid approach might offer a more comprehensive analysis framework.

- Validation: Validate the impedance calculations using known benchmarks or simulations. This could involve comparing the time-varying impedance results with those obtained from more traditional methods under conditions where both should yield similar outcomes.

Conclusion

While traditional impedance calculation methods may not directly apply to nonlinear, time-varying systems analyzed with EMD/HHT, alternative approaches focusing on time-domain calculations, component-wise analysis, or advanced signal processing techniques offer potential pathways. The key is to carefully consider the nature of the signals and the information provided by methods like EMD and HHT to adapt the impedance calculation process accordingly.

I hope the above information helps you in approaching the task.

Thank you.