Intrinsic Mode Functions (IMFs)

• Intrinsic Mode Functions are data-adaptive, zero-mean oscillatory components that satisfy specific extrema and envelope symmetry criteria, enabling meaningful instantaneous frequency analysis.
• They are extracted using Empirical Mode Decomposition (EMD) and its multivariate extensions, which iteratively sift signals to isolate intrinsic oscillatory modes.
• IMFs play a crucial role in adaptive signal analysis across fields such as biomedical, geophysical, and financial time series by enhancing noise reduction and trend extraction.

An Intrinsic Mode Function (IMF) is a data-adaptive, zero-mean, narrow-band oscillatory mode extracted from a signal via Empirical Mode Decomposition (EMD) or related algorithms. An IMF satisfies two essential criteria: over its domain, the number of extrema and the number of zero crossings differs by at most one, and at every point, the mean of its upper (maxima) and lower (minima) envelopes is zero. These properties ensure that each IMF is amenable to Hilbert analysis, supports meaningful instantaneous frequency assignments, and represents a physically interpretable oscillatory component even in nonlinear and nonstationary data (Ram et al., 2015, Riscos et al., 14 Dec 2025, Hirsh et al., 2018).

1. Mathematical Definition and Properties

Formally, a real-valued function $c(t)$ is an IMF if:

1. $|N_{\mathrm{ext}} - N_{\mathrm{zc}}| \leq 1$, where $N_{\mathrm{ext}}$ is the number of extrema and $N_{\mathrm{zc}}$ is the number of zero crossings.
2. At every $t$, $m(t) = \frac{1}{2}[e_{\max}(t) + e_{\min}(t)] = 0$, where $e_{\max}(t)$ and $e_{\min}(t)$ are the upper and lower envelopes interpolated through local maxima and minima, respectively.

These conditions enforce local symmetry and quasi-monocomponent oscillation. For multidimensional or image data, the envelope symmetry extends along relevant axes, and more complex geometric constraints may define local zero-mean oscillations (Schmitt et al., 2014, Islam et al., 2022, Islam et al., 2022).

IMFs admit an analytic representation via the Hilbert transform: $$H[c](t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^{\infty} \frac{c(\tau)}{t-\tau}\, d\tau,$$ enabling the definition of instantaneous amplitude $A(t)$ and phase $\phi(t)$ via $c(t) + j H[c](t) = A(t)\, e^{j\phi(t)}$, and hence the instantaneous frequency $\omega(t)=d\phi/dt$.

2. Extraction Methodologies

A. Empirical Mode Decomposition (EMD)

EMD is a fully data-driven algorithm that decomposes a signal $x(t)$ into a sum of IMFs $\{c_i(t)\}$ plus a monotonic residual $r(t)$: $x(t) = \sum_{i=1}^{n} c_i(t) + r(t).$ The extraction (sifting) process for each IMF involves:

• Detection of local maxima and minima of the current residual.
• Construction of upper and lower envelopes via spline interpolation.
• Computation of the local mean $m(t)$ and subtraction from the signal.
• Iterative repetition until IMF criteria are satisfied.

Sifting is typically terminated when $|N_{\mathrm{ext}} - N_{\mathrm{zc}}| \leq 1$ and the mean envelope is sufficiently small, or a standardized deviation threshold is met (Ram et al., 2015, Riscos et al., 14 Dec 2025, Kemiha, 2014, Fosso et al., 2017).

B. Multivariate and Multidimensional Extensions

Multivariate EMD (MEMD) and genuine/pseudo 2-D EMD generalize the envelope symmetry and extrema criteria to multichannel and image domains, extracting scale-aligned IMFs that respect joint-channel dynamics or 2D oscillatory geometry (Islam et al., 2022, Islam et al., 2022, Schmitt et al., 2014).

C. Alternative Decomposition Approaches

Algorithms such as Variational Mode Decomposition (VMD), Fast Intrinsic Mode Decomposition (FIMD), and sawtooth transform-based methods accelerate or regularize IMF extraction using optimization, median-based spline fitting, or monotonic time transformations. VMD produces IMFs by minimizing bandwidth in the analytic signal domain, with explicit frequency localization (Yuhang et al., 2024, 0808.2827, 0710.3170).

3. Functional Role of IMFs in Signal Representations

Decomposition into IMFs facilitates adaptive, physically-meaningful signal analysis in several domains:

• Each IMF isolates a single timescale, supporting instantaneous time-frequency analysis via the Hilbert transform.
• The set $\{c_i\}$ is nearly orthogonal and complete, allowing perfect or near-perfect reconstruction of the original signal when combined with the residue (Riscos et al., 14 Dec 2025, Hirsh et al., 2018).
• The hierarchy of IMFs indexes intrinsic scale: lower-order IMFs capture high-frequency detail or noise, higher-order IMFs encapsulate trends or slow variations.

Schematic representation relating IMFs to signal decomposition:

Feature	IMF property	Extraction context
Envelope symmetry	Mean of upper/lower envelopes = 0	All time, all (multi)dimensions
Zero crossing/extrema	Counts differ by $\leq 1$	All time, all (multi)dimensions
Analytic representation	Hilbert transform well-posed	Adaptivity for instantaneous freq
Residual	Monotonic or trend-like	Decomposition completion

4. Mode Mixing and Its Mitigation

Mode mixing is a key challenge: a single IMF may blend oscillations from disparate frequencies, or a single scale may be split across multiple IMFs, impairing the physical interpretability of the decomposition and the clarity of the Hilbert spectrum (Ram et al., 2015, Fosso et al., 2017). This typically arises from intermittent events or noise bursts causing complex envelope behavior.

Mitigation strategies include:

• Entropic-EMD: Incorporation of Permutation Entropy (PE) to quantify local complexity, dynamically partitioning the signal and applying sifting only within high-entropy segments. This reduces mode mixing by adaptively isolating intermittent high-frequency components (Ram et al., 2015).
• Masking techniques: Use of engineered masking signals to move nearby frequencies out of the mixing regime, as prescribed by theoretical mode-mixing boundaries (Fosso et al., 2017).
• Noise-assisted approaches (e.g., CEEMDAN): Injection of white noise at each sifting stage with ensemble averaging, which reduces the bias introduced by residual noise during IMF extraction (Riscos et al., 14 Dec 2025).

5. Signal Reconstruction, Orthogonality, and Stopping Criteria

The sum of all extracted IMFs and the final residual reconstructs the original signal: $x(t) = \sum_{i=1}^{n} c_i(t) + r(t).$ The extraction continues until the residue $r(t)$ is monotonic or contains at most one extremum. In practice, thresholds on the minimum number of extrema, minimum energy, or maximum mode count are also used (Riscos et al., 14 Dec 2025, Fosso et al., 2017, Kemiha, 2014).

Orthogonality of IMFs is empirical but approximate: extracted modes are nearly orthogonal under the inner product, promoting interpretability and minimizing redundant representation (Riscos et al., 14 Dec 2025). However, perfect orthogonality is not guaranteed due to the adaptive, nonlinear nature of the decomposition.

6. Applications and Illustrative Examples

IMFs are widely used in:

• Nonstationary time-frequency analysis of signals in physics, engineering, biomedicine (EEG, ECG), earth sciences, and finance (Riscos et al., 14 Dec 2025, Islam et al., 2022, Islam et al., 2022).
• Adaptive feature extraction for classification and prediction tasks (e.g., emotion detection from EEG using high-oscillation IMFs) (Islam et al., 2022).
• Local topological charge analysis in electromagnetic fields—IMFs enable high-resolution spectral and spatial discrimination unattainable by Fourier analysis (Hui et al., 2015).
• Real-time denoising and trend extraction, leveraging noise-robust IMFs to filter or reconstruct clean signal components (Kemiha, 2014, 0808.2827).
• Image decomposition and local spectral analysis (2-D Prony–Huang transform), with IMFs enabling localized orientation and frequency estimation (Schmitt et al., 2014).

IMFs have been instrumental in revealing nonstationary oscillatory patterns, enabling robust adaptive filtering, and supporting interpretable decomposition for both single-channel and multivariate signals. Advanced algorithms mitigate traditional limitations such as mode mixing and envelope distortion, cementing IMFs as central objects in nonlinear and nonstationary signal analysis (Ram et al., 2015, Riscos et al., 14 Dec 2025, Fosso et al., 2017, Yuhang et al., 2024, Schmitt et al., 2014).