Empirical Fourier Decomposition (EFD)

• Empirical Fourier Decomposition is an adaptive, data-driven technique that segments the Fourier spectrum into narrow-band modes for accurate analysis of nonstationary signals.
• It employs ideal zero-phase bandpass filters to eliminate transition bands and mode mixing, ensuring computational efficiency with an O(R log R) complexity.
• EFD is practically applied in areas like mechanical vibration analysis and biomedical signal processing, outperforming methods such as EWT and EMD in accuracy and speed.

Empirical Fourier Decomposition (EFD) is an adaptive, data-driven signal decomposition technique that leverages Fourier spectrum segmentation combined with ideal zero-phase filtering to yield accurate, consistent modal separation of real-valued, potentially nonlinear and nonstationary signals. EFD is distinct from classical Fourier analysis, Empirical Wavelet Transform (EWT), and Empirical Mode Decomposition (EMD) in its use of spectral-lobe-driven band selection and its avoidance of transition bands and mode mixture. The method has direct applications in fields such as mechanical vibration analysis, biomedical signal processing, and time-frequency analysis, and demonstrates both computational efficiency and robustness, especially with signals containing closely spaced modes (Zhou et al., 2020, Zhou et al., 2019).

1. Mathematical Fundamentals and Formulation

EFD decomposes a real, finite-energy signal $x(t)$ into $K$ narrow-band, physically interpretable “modes”:

$x(t) = \sum_{k=1}^K m_k(t)$

Each $m_k(t)$ is obtained so that its frequency-domain support, $\hat m_k(\omega)$, is restricted to a small interval of $|\omega|$, corresponding to a dominant spectral lobe.

Improved Spectrum Segmentation

• The one-sided spectrum $|\hat x(\omega)|$ on $[0, \pi]$ is constructed.
• The number of desired modes $K$ is set a priori.
• The largest spectral peaks ($\omega=0$, $K$0, and local maxima) are sorted by magnitude, and the top $K$1 are selected: $K$2, with endpoints defined as $K$3 and $K$4.
• For each segment $K$5, a boundary $K$6 is chosen at the local minimum of $K$7 within that interval:

$K$8

[(Zhou et al., 2020), Eq. (26); (Zhou et al., 2019), Algorithm 2.1]

This boundary selection rule assigns one dominant spectral lobe per segment and minimizes the inclusion of noise-related artifacts.

Ideal Zero-Phase Filter Bank

For each segment, a rectangular (ideal) band-pass filter is defined:

$K$9

Each mode $x(t) = \sum_{k=1}^K m_k(t)$0 is reconstructed in the time domain by applying the filter bank in the frequency domain, then inverting via IFFT:

$x(t) = \sum_{k=1}^K m_k(t)$1

This zero-phase filtering completely suppresses transition bands and eliminates adjacent-mode mixing [(Zhou et al., 2020), Sec. 3.2, Fig. 4].

2. Algorithmic Structure and Computational Aspects

The EFD algorithm executes the following core procedure:

1. Compute the FFT of $x(t) = \sum_{k=1}^K m_k(t)$2: $x(t) = \sum_{k=1}^K m_k(t)$3.
2. Perform spectral segmentation based on the sorted magnitudes and local minima to identify $x(t) = \sum_{k=1}^K m_k(t)$4 dominant bands.
3. Construct ideal bandpass filters for each segment.
4. For each $x(t) = \sum_{k=1}^K m_k(t)$5:
   • Mask the spectrum: $x(t) = \sum_{k=1}^K m_k(t)$6.
   • Obtain the $x(t) = \sum_{k=1}^K m_k(t)$7th mode: $x(t) = \sum_{k=1}^K m_k(t)$8.
5. The sum of all $x(t) = \sum_{k=1}^K m_k(t)$9 reconstructs $m_k(t)$0 exactly in the noise-free case.

The pseudocode (from (Zhou et al., 2020, Zhou et al., 2019)) ensures all operations are dominated by $m_k(t)$1 FFT/IFFT calls, leading to a practical complexity of $m_k(t)$2 for input of length $m_k(t)$3 and moderate $m_k(t)$4.

Step	Operation	Complexity
1	FFT	$m_k(t)$5
2	Segmentation (peaks/minima)	$m_k(t)$6
3–5	$m_k(t)$7 masks + IFFTs	$m_k(t)$8

3. Modal Reconstruction, Time-Frequency Representation, and Accuracy

For each mode, analytic representation via the Hilbert transform allows extraction of instantaneous amplitude, phase, and frequency. The time-frequency representation (TFR) is constructed as:

$m_k(t)$9

$\hat m_k(\omega)$0

A TFR plot visualizes $\hat m_k(\omega)$1 per mode, facilitating modal identification even in nonstationary or closely spaced conditions [(Zhou et al., 2020); (Zhou et al., 2019), Sec. 5].

Quantitative accuracy metrics include mode-wise RMSE, TFR RMSE, and a consistency measure $\hat m_k(\omega)$2. Across a suite of synthetic and real signals, EFD achieves either the lowest or second-lowest RMSE relative to EMD, EWT, and VMD, with particular advantages in situations involving closely spaced or nonstationary modes (see (Zhou et al., 2020), Tables 2–6 and Figs. 14–15). For instance, in decomposing Sig3 and Sig4, EFD's RMSE is $\hat m_k(\omega)$3 and $\hat m_k(\omega)$4, outperforming EWT and FDM variants.

4. Comparative Advantages and Limitations

Advantages:

• Accurate mode separation: EFD consistently yields high-accuracy decompositions even for nonstationary signals and signals with closely spaced spectral components (Zhou et al., 2020, Zhou et al., 2019).
• No mode mixing: Zero-phase, non-overlapping filters eliminate the interband leakage present in EWT and EMD.
• Deterministic and consistent: Outcomes are free from the low-to-high/high-to-low ambiguity observed in the original FDM/LTH and FDM/HTL scans.
• Computational efficiency: EFD dominates EWT, FDM, VMD, and EMD in runtime, achieving $\hat m_k(\omega)$50.016 s (versus VMD/EMD $\hat m_k(\omega)$60.1–1.4 s) for benchmark signals [(Zhou et al., 2020), Table 6; (Zhou et al., 2019)].

Limitations:

• Prescribed mode count: The number of modes $\hat m_k(\omega)$7 must be defined or robustly estimated; overestimation may introduce near-zero energy modes, and underestimation causes missed modal content.
• Gibbs artifacts: Discrete ideal filtering introduces Gibbs oscillations at segment boundaries; mild windowing or tapering can reduce this, at the cost of slightly fuzzier separation.
• Low SNR: Peak and minimum detection in segmentation may become unreliable at very low SNR, thereby necessitating noise-robust variants or threshold optimization.

5. Experimental Evaluation and Applications

Numerical experiments using synthetic nonstationary, multimode, and closely spaced signals, as well as real biosignals (e.g., ECG), confirm EFD's consistent separation accuracy and its superior TFR fidelity (Zhou et al., 2020, Zhou et al., 2019). Notable application domains include:

• Mechanical systems: Modal analysis, vibration diagnostics (e.g., gear or shaft monitoring).
• Biomedical engineering: Component separation in ECG and EEG recordings, extraction of physiological oscillations.
• Acoustics and seismology: Signal discrimination in noisy, multimodal waveforms.
• Speech and music: Harmonic/pitch detection, time-varying formant analysis.

In a direct benchmark, EFD outperforms FDM and EWT in extracting underlying heartbeat features in real ECG, providing more compact and physiologically meaningful TFR visualization (Zhou et al., 2019).

6. Relationships to FDM, EWT, and Future Directions

EFD refines the Fourier Decomposition Method (FDM) (Singh et al., 2015) by resolving its ambiguity splitting (between low-to-high and high-to-low scans) and tempering FDM's oversensitivity to noise with a more robust segmentation and mode selection. Compared to EWT, EFD’s use of hard-edged, non-overlapping filters eliminates transition band-induced interference.

Areas for further research include:

• Adaptive mode number detection: Algorithmic estimation of $\hat m_k(\omega)$8 via spectral heuristics or machine learning.
• Noise-robust peak detection: Enhanced minima/maxima-finding for segmentation in low-SNR regimes.
• Extension to multivariate and 2D signals: The development of multichannel or image-domain EFD is a current research frontier (Zhou et al., 2019).
• Windowed and smooth-filter variants: Investigation of windowing strategies to control Gibbs effects without reintroducing mode mixing.
• Hybridization with other methods: Integration with Hilbert or wavelet-based approaches for signals with heavy cross-component overlap.

7. Summary Table: EFD vs. Related Methods

Method	Mode Mixing	Consistency	Runtime	End Effects	Key Limitation
EFD	None	Consistent	Fast	Minor	Preset $\hat m_k(\omega)$9, Gibbs
EWT	Present	Consistent	Moderate	Mild	Transition leakage
FDM	None	Inconsistent	Slower	None	LTH/HTL split, noise
EMD/VMD	Yes	Moderate	Slowest	Strong	Mode-mixing, cost

EFD thus offers a mathematically precise, adaptive, and computationally efficient solution for modal decomposition of complex signals, with mathematical and empirical evidence validating its performance across domains (Zhou et al., 2020, Zhou et al., 2019, Singh et al., 2015).