Empirical Mode Decomposition (EMD)

Updated 21 December 2025

Empirical Mode Decomposition (EMD) is an adaptive, data-driven method that decomposes nonlinear, nonstationary signals into intrinsic mode functions (IMFs) and a residual trend.
EMD employs an iterative sifting process using local extrema, envelope interpolation, and stopping criteria to extract meaningful oscillatory components.
EMD is widely applied in biomedical engineering, finance, and geophysics, with variants like EEMD, CEEMDAN, and Forward-PDE addressing challenges such as mode mixing and boundary effects.

Empirical Mode Decomposition (EMD) is an adaptive, data-driven algorithm for analyzing nonstationary and nonlinear time series by decomposing a signal into a finite sum of oscillatory components, termed intrinsic mode functions (IMFs), plus a final monotonic residue. EMD is unique in that it imposes no a priori basis, and its mode extraction is governed by the instantaneous structural properties of the input signal. Originally introduced to facilitate the Hilbert spectrum analysis of complex signals, EMD has evolved into a major tool in signal processing, biomedical engineering, finance, geophysics, and machine learning.

1. Mathematical Foundation: Definition of IMFs and the Sifting Algorithm

An Intrinsic Mode Function (IMF) is defined as any function $c(t)$ that simultaneously satisfies:

Oscillatory symmetry: Across the domain, the number of zero-crossings and extrema differs at most by one.
Local zero-mean envelope: At every $t$ , the mean of the upper envelope (through maxima) and the lower envelope (through minima) is zero:

$m(t) = \frac{e_{\max}(t) + e_{\min}(t)}{2} = 0.$

Given a real-valued input signal $x(t)$ , EMD decomposes it as

$x(t) = \sum_{k=1}^n \mathrm{IMF}_k(t) + r_n(t),$

where $r_n(t)$ is the final residue (trend).

The canonical sifting process is as follows (Nava et al., 2017, Cao et al., 29 Jan 2025, Ram et al., 2015):

Initialization: Set the residual $r_0(t) = x(t)$ .
Extraction of IMFs:
- Identify all extrema of $r_{k-1}(t)$ .
- Interpolate maxima and minima (typically with cubic splines) to obtain the envelopes $e_{\max}(t)$ and $e_{\min}(t)$ .
- Compute the local mean:
$m(t) = \frac{e_{\max}(t) + e_{\min}(t)}{2}.$ - Subtract the mean from the current signal to get the proto-IMF:

$h_1(t) = r_{k-1}(t) - m(t).$ - Iterate this procedure, replacing $r_{k-1}(t)$ with $h_j(t)$ and repeating until the IMF criteria are met within the specified stopping rule. - Once obtained, designate the current $h_j(t)$ as $\mathrm{IMF}_k(t)$ and compute the new residual:

$r_k(t) = r_{k-1}(t) - \mathrm{IMF}_k(t).$ - The extraction proceeds until the residue is monotonic or contains at most one extremum.

Stopping Criteria: Common options include the standard deviation (SD) criterion (Nava et al., 2017, Cao et al., 29 Jan 2025)

$\mathrm{SD} = \frac{\sum_{t}|h_{j-1}(t)-h_j(t)|^2}{\sum_{t}|h_{j-1}(t)|^2} < \varepsilon$

with $\varepsilon$ typically $0.2$–$0.3$, the S-number criterion (IMF conditions hold for S iterations), or Rilling’s two-threshold method (mean envelope excursion thresholding).

2. Interpretations, Theoretical Models, and Extensions

PDE Foundations: Classical EMD is heuristic; however, reformulations such as the forward-PDE model replace envelope averaging with heat-equation-based Gaussian smoothing, thus offering explicit control over frequency selectivity. The forward-PDE EMD employs a heat equation

$\frac{\partial h}{\partial t}(x,t) = a\frac{\partial^2 h}{\partial x^2}(x,t),\ h(x,0) = S(x)$

and uses $h(x,T)$ as the mean envelope, enabling tuning of cutoff frequency and improved separation (Wang et al., 2018).

Operator-Based and Optimization Formulations: EMD has been cast as an optimization problem over constrained function spaces with B-spline bases, leading to null-space pursuit (NSP) and operator-based approaches guaranteeing strong duality and robust mode separation. Hybrid sifting methods combining iterative-slope envelope estimation and operator-based spectral separation further mitigate envelope leakage (Hunhold, 2023).

Energy Preservation and Orthogonality: Classical EMD decompositions can leak energy across modes. Energy-preserving variants (EPEMD) enforce Parseval-type identities by producing linearly independent, nonorthogonal but energy-preserving (LINOEP) IMFs or by Gram–Schmidt orthogonalization in reverse frequency order, yielding truly orthogonal and narrowband components (Singh et al., 2015).

3. Algorithmic Variants and Pitfalls

Mode Mixing: EMD inherently suffers from mode mixing—conflation of disparate scales into a single IMF or splitting of a physical mode across IMFs. Primary causes include intermittent mode occurrence and proximity of signal frequencies. The "mode-mixing map" formalizes boundaries under which EMD can (or cannot) separate tones, depending on amplitude and frequency ratios (Fosso et al., 2017). Remedies include

Noise-assisted methods: Ensemble EMD (EEMD) adds white noise and averages over realizations to disambiguate modes. CEEMDAN (complete EEMD with adaptive noise) further improves mode separation and exact signal reconstruction at higher computational cost (Luukko et al., 2017, Spinosa et al., 2021).
Permutation Entropy–guided segmentation: Entropic EMD uses local complexity measures to target intermittent, mixed-complexity segments for separate decomposition, preventing low-frequency intrusion into high-frequency IMFs and increasing modal orthogonality (Ram et al., 2015).
Masking techniques: Addition of specific masking sinusoids can force the separation of closely spaced frequencies by intentionally reversing mode-mixing conditions (Fosso et al., 2017).

Loss of Orthogonality: IMFs are not generally orthogonal. The orthogonality index quantifies cross-mode energy: $\mathrm{OI} = \sum_{j \ne k} \frac{\langle c_j, c_k\rangle}{\|c_j\|\|c_k\|},$ and significant nonzero values indicate interference (Santander et al., 2019).

End Effects and Modal Aliasing: Near signal endpoints, boundary effects corrupt envelope fitting, leading to spurious features that propagate inward and degrade forecasting or reconstruction (Safari et al., 2020).

4. Algorithmic Variants: Noise-Assisted, Multidimensional, and Application-Specific EMD

Variant	Key Features	Application Strength
Standard EMD	Data-driven, adaptive, single-channel	Generic nongaussian, nonstationary data
EEMD	Ensemble average, mitigates mode mixing	Robustness to noise, intermittent artifacts
CEEMDAN	Complete, adaptive noise, exactness	High accuracy, enhanced mode separation
Serial-EMD	1D serialization of multidim. signals	Real-time multidimensional decomposition
Entropic EMD	Entropy-based segmentation	Intermittent nonstationarity, biomedical
Forward-PDE	Heat-equation mean, explicit cutoff	Theoretical analysis, parametric tuning
EPEMD	Parseval energy-preserving, orthogonal	Instantaneous energy/frequency accuracy

EEMD and CEEMDAN often require parameter tuning (noise std, ensemble size). Serial-EMD leverages blockwise serialization for orders-of-magnitude speedups in 2D/3D settings, outperforming traditional bidimensional methods for both time and classification accuracy (Zhang et al., 2021).

5. Practical Applications and Performance

Biomedical Signal Processing: Applications span artifact removal in EEG via machine-learned envelope interpolation—yielding improved SNR and fidelity in reconstructed neural oscillations (Rakhmatulin, 22 Sep 2024), SSVEP-based cross-stimulus transfer learning using IMF frequency remapping (Cao et al., 29 Jan 2025), denoising of EMG and speech via thresholded IMF reconstruction (Kemiha, 2014), and water-entry force denoising using EEMD with interval thresholding (Spinosa et al., 2021).

Financial and Geophysical Time Series: Adaptive scale-dependent cross-correlation enables the detection of rich, heterogeneous dependencies and lead-lag effects across asset returns (Nava et al., 2017). EMD enables decoupling and recovery of underlying components in turbulent flows (Mazellier et al., 2010).

Forecasting and Resource Allocation: EMD-based decomposition of load and renewable time series components yields improved modeling but is sensitive to end effects in real time. Decomposition-based predictive resource allocation in wireless communications reduces interference prediction RMSE by 20–25%, delivering two to three orders of magnitude lower outage compared to state-of-the-art resource allocation algorithms (Jayawardhana et al., 2023).

Multiscale Analysis and Time-Frequency Representation: IMFs extracted via EMD or EPEMD support robust Hilbert spectrum estimation for visualizing and analyzing instantaneous frequency and energy trajectories.

6. Recommendations, Limitations, and Best Practices

Noise-assisted variants (CEEMDAN) are recommended for minimal mode mixing, provided moderate noise (e.g., 0.2× signal std) and sufficiently large ensemble size (e.g., 500) are used, with careful stopping rules (e.g., Rilling's two-threshold criterion).
Modal content inspection: Always inspect IMF orthogonality or structure in the time–frequency domain; low reconstruction error alone does not guarantee physical interpretability (Santander et al., 2019).
Parameter tuning: For application-specific scenarios (biomedical, financial, mechanical), tuning of window/segment size, IMF selection, noise injection, and sifting thresholds is critical.
Limitations: Mode mixing (especially with closely spaced frequencies), lack of uniqueness, sensitivity to spline interpolation and endpoints, and increased computational demand in ensemble approaches are unresolved challenges. PDE-based and operator-based variants offer theoretical improvements but can require more domain-specific tuning (Wang et al., 2018, Hunhold, 2023).
Boundary-handling and windowing: In online or real-time settings, windowing strategies, endpoint extension (mirror, polynomial), and noise-assisted or ensemble methods help mitigate boundary artifacts (Safari et al., 2020).
Orthogonality and energy preservation: For applications requiring high energy fidelity and true time-frequency localization, EPEMD or related orthogonalization approaches are preferred (Singh et al., 2015).