[Paper Review] Empirical Fourier Decomposition

Wei Zhou, Zhongren Feng+2 more|arXiv (Cornell University)|Dec 1, 2019

Machine Fault Diagnosis Techniques36 references9 citations

TL;DR

This paper proposes Empirical Fourier Decomposition (EFD), a novel adaptive signal decomposition method for non-stationary and nonlinear signals by combining principles from the Empirical Wavelet Transform (EWT) and Fourier Decomposition Method (FDM). EFD uses enhanced Fourier spectrum segmentation to create adaptive bandpass filters, achieving higher precision, efficiency, and noise robustness than EWT and FDM, especially for closely spaced frequencies and high-frequency noise.

ABSTRACT

In this paper, a novel decomposition method for non-stationary and nonlinear signals is proposed. This method is inspired by the adaptive wavelet filter bank of the empirical wavelet transform (EWT) and Fourier intrinsic band functions (FIBFs) of the Fourier decomposition method (FDM). Therefore, the proposed approach is entitled as empirical Fourier decomposition (EFD). EFD is defined as the adaptive bandpass filter bank, regarded as the adaptive FIBFs based on the segment of the Fourier spectrum. Firstly, an enhanced segmentation technology of the Fourier spectrum based is presented. Secondly, the framework of EFD is established both in a continuous series and a discrete series. Finally, combined with the Hilbert transform, EFD is extended to a time-frequency representation. To verify the effectiveness of EFD, three non-stationary multimode signals, a simulated free vibration, and one real ECG signal are tested. The results manifest that EFD is more effective, compared with EWT and FDM, with higher processing precision, computation efficiency and noise robustness particularly to the closely-spaced frequencies and high-frequency noise.

Motivation & Objective

• Address the limitations of existing decomposition methods like EWT and FDM in handling non-stationary and nonlinear signals with closely spaced frequency components.
• Improve decomposition accuracy and computational efficiency by integrating adaptive filtering with Fourier spectral segmentation.
• Enhance noise robustness, particularly for high-frequency noise, which challenges traditional methods.
• Develop a unified framework for both continuous and discrete signal series to ensure broad applicability.
• Extend the method to time-frequency analysis using the Hilbert transform for improved signal characterization.

Proposed method

• Proposes an enhanced Fourier spectrum segmentation technique to identify intrinsic mode functions (IMFs) adaptively based on spectral peaks and valleys.
• Defines EFD as an adaptive bandpass filter bank derived from segmented Fourier spectra, forming adaptive Fourier Intrinsic Band Functions (FIBFs).
• Establishes a mathematical framework for EFD in both continuous and discrete signal domains to ensure theoretical and practical consistency.
• Applies the Hilbert transform to EFD components to generate a time-frequency representation, enabling detailed signal analysis.
• Uses spectral energy concentration and local maxima detection to guide the segmentation process and improve mode separation.
• Combines the strengths of EWT’s adaptive filtering and FDM’s spectral basis to create a hybrid decomposition approach.

Experimental results

Research questions

• RQ1 Can a hybrid decomposition method combining EWT and FDM principles achieve better performance than either method alone for non-stationary signals?
• RQ2 How does EFD perform in separating closely spaced frequency components compared to EWT and FDM?
• RQ3 To what extent does EFD maintain accuracy and stability under high-frequency noise conditions?
• RQ4 Can EFD be effectively applied to both continuous and discrete signal series with consistent results?
• RQ5 How does the time-frequency representation derived from EFD compare to those from EWT and FDM in terms of resolution and clarity?

Key findings

• EFD demonstrates higher processing precision than EWT and FDM, particularly in resolving signals with closely spaced frequency components.
• The method exhibits superior computation efficiency due to its adaptive spectral segmentation and optimized filtering structure.
• EFD shows enhanced robustness to high-frequency noise, maintaining accurate decomposition where EWT and FDM degrade.
• The time-frequency representation generated via Hilbert transform of EFD components provides clearer and more localized signal features.
• Validation on a simulated free vibration signal and a real ECG signal confirms EFD’s effectiveness in practical, real-world applications.
• The enhanced Fourier spectrum segmentation technique enables more accurate identification of intrinsic modes compared to fixed or heuristic segmentation.