Intrinsic Mode Function (IMF) Overview

• Intrinsic Mode Function (IMF) is an oscillatory component defined by envelope conditions where the differences between zero-crossings and extrema are at most one, enabling an amplitude–frequency representation.
• Various extraction techniques—including EMD, iterative filtering, and masking—address challenges like mode mixing and energy leakage while ensuring adaptive decomposition.
• IMFs support Hilbert transform analysis, providing meaningful instantaneous amplitude and frequency for adaptive time–frequency representation in complex signal processing.

An intrinsic mode function (IMF) is an oscillatory component used as an adaptive basis for nonlinear and non-stationary data analysis. In the original Empirical Mode Decomposition (EMD) of Huang et al. (1998), an IMF is a real-valued function whose number of zero-crossings and local extrema differ at most by one, and whose upper and lower envelope mean is zero; under these conditions, every IMF admits an amplitude–frequency form $s(t)=a(t)\cos(\theta(t))$ with $a(t)>0$ and $\theta'(t)>0$ (Hirsh et al., 2018, Cicone et al., 2022). Subsequent work treats IMFs as narrow-band AM–FM atoms, as outputs of iterative filtering operators, as modes with separated spectrum in neural or variational decompositions, and as components for which Hilbert-based instantaneous frequency is meaningful (Hou et al., 2013, Yiting et al., 2021, Yuhang et al., 2024).

1. Classical definition and later formalizations

In the classical EMD framework, a function $c(t)$ is an IMF if two conditions hold over the full data span: the number of zero-crossings and the number of extrema differ at most by one, and at every time $t$ the mean of the upper envelope through local maxima and the lower envelope through local minima is zero. With cubic-spline envelopes $u(t)$ and $\ell(t)$, this is written as

$m(t)=\frac{u(t)+\ell(t)}{2}=0.$

The same pair of conditions appears across EMD-based expositions, including speech denoising, entropic EMD, multivariate extensions, and power-signal analysis (Kemiha, 2014, Ram et al., 2015, Yuhang et al., 2024).

A second, more explicit AM–FM formulation writes an IMF as

$a(t)\cos(\theta(t)),$

where $a(t)>0$, $a(t)>0$0 is a $a(t)>0$1 phase, and $a(t)>0$2. In the data-driven sparse time-frequency formulation, both $a(t)>0$3 and $a(t)>0$4 are required to lie in a low-frequency subspace

$a(t)>0$5

with $a(t)>0$6 and $a(t)>0$7; elements of this space are guaranteed to be smoother than $a(t)>0$8 (Hou et al., 2013).

Other frameworks sharpen the definition further. In Neural Mode Decomposition, an IMF $a(t)>0$9 is an AM–FM component

$\theta'(t)>0$0

with $\theta'(t)>0$1, $\theta'(t)>0$2, and slowly varying amplitude and instantaneous frequency relative to the carrier oscillation. In the orthogonal decomposition framework for finite discrete signals, an intrinsic mode is defined through the sign of the instantaneous frequency: a function $\theta'(t)>0$3 in the interpolation function space is an intrinsic positive-frequency mode if $\theta'(t)>0$4 for all $\theta'(t)>0$5, and an intrinsic negative-frequency mode if $\theta'(t)>0$6 for all $\theta'(t)>0$7 (Yiting et al., 2021, Li et al., 2024).

Within Iterative Filtering, the notion of IMF is operationalized differently: an IMF is the output of a linear “variation-of-moving-average” operator applied until convergence, and numerically each extracted component satisfies the two EMD conditions in a discrete sense and contains one “intrinsic” oscillatory scale (Cicone et al., 2022). This suggests that the term “IMF” is used in both a classical envelope-based sense and in stricter narrow-band or instantaneous-frequency-based senses.

2. Extraction procedures and decomposition algorithms

The classical extraction mechanism is the EMD sifting process. Starting from a signal $\theta'(t)>0$8, one initializes the residual $\theta'(t)>0$9 and, for each mode, repeatedly identifies local maxima and minima of the current iterate, interpolates them to form upper and lower envelopes, computes the local mean, subtracts that mean, and stops when the candidate satisfies the IMF conditions or a stopping criterion such as the standard-deviation criterion

$c(t)$0

After $c(t)$1 IMFs have been extracted, the decomposition has the form

$c(t)$2

where $c(t)$3 is a final residual or trend (Kemiha, 2014, Ram et al., 2015).

A distinct line of work formulates IMF extraction as a sparse nonlinear optimization problem. Given a signal $c(t)$4, the goal is to represent it as

$c(t)$5

with the smallest possible $c(t)$6, over the dictionary

$c(t)$7

This leads to the sparsest-decomposition problem $c(t)$8, an $c(t)$9-type problem over an uncountable dictionary, which is solved approximately by a matching-pursuit-style, $t$0-regularized nonlinear least-squares iteration. At each step one solves

$t$1

subject to $t$2 and $t$3 (Hou et al., 2013).

Several fast alternatives replace repeated envelope sifting by more direct constructions. The sawtooth-transform method maps the original data to a piecewise-linear sawtooth function, constructs upper and lower envelopes by straight-line connection of extrema in sawtooth space, defines the IMF candidate as the difference between the sawtooth and the envelope mean, and maps the result back to the original time axis. It finds each IMF in a single pass over the data, with no inner sifting loop (0710.3170). Fast IMD based on median-point refinement initializes the residue by straight-line interpolation through extrema of the first derivative, then refines the residue by cubic splines through median control points derived from extrema of the current IMF (0808.2827). The Equivalent Effect Function framework instead interpolates cumulative integral values at selected control points, differentiates the spline, and uses the derivative as a smooth trend, with the residual acting as an IMF-like fluctuation (Lu, 2011).

These algorithms share the same reconstructive structure—signal equals sum of extracted modes plus a residual—but they differ sharply in the mechanism by which the “intrinsic” oscillatory component is isolated.

3. Hilbert analysis and time–frequency representation

A principal reason for imposing IMF structure is that it supports meaningful Hilbert analysis. Given an IMF $t$4, its Hilbert transform is

$t$5

and the corresponding analytic signal is

$t$6

The instantaneous amplitude and phase are

$t$7

and the instantaneous frequency is $t$8 (Hirsh et al., 2018).

In the Hilbert-spectrum formulation, if IMFs $t$9 are transformed to analytic signals

$u(t)$0

then each mode yields instantaneous amplitude $u(t)$1 and instantaneous frequency $u(t)$2. The Hilbert spectrum is the time–frequency–energy distribution

$u(t)$3

and the marginal spectrum is

$u(t)$4

These quantities are used for adaptive time–frequency–energy analysis of nonlinear and nonstationary signals (Singh et al., 2015).

The IMFogram provides a related time–frequency representation built from IMF decompositions obtained by Iterative Filtering. For each IMF, local amplitude and local frequency are computed over time windows, then assembled into a matrix by adding the local amplitude of IMF $u(t)$5 in window $u(t)$6 into the row corresponding to its local frequency. For a class of piecewise stationary signals, and as the Fast IF stopping tolerance $u(t)$7, the entry-wise square of the IMFogram converges to the usual STFT spectrogram of the windowed data matrix: $u(t)$8 This establishes a direct link between adaptive IMF-based analysis and a classical time–frequency representation (Cicone et al., 2022).

The common theme is that the IMF is not merely a decomposition artifact; it is the unit from which instantaneous amplitude, instantaneous frequency, Hilbert spectra, and related adaptive time–frequency objects are constructed.

4. Dynamical-systems interpretation and nonlinearity indices

An IMF can also be interpreted through a second-order ordinary differential equation. If

$u(t)$9

then a direct calculation shows that $\ell(t)$0 satisfies

$\ell(t)$1

with

$\ell(t)$2

and

$\ell(t)$3

In general, $\ell(t)$4 and $\ell(t)$5 depend nonlinearly on $\ell(t)$6 because $\ell(t)$7 and $\ell(t)$8 are linked to $\ell(t)$9 (Hou et al., 2013).

To expose structure, one may freeze $m(t)=\frac{u(t)+\ell(t)}{2}=0.$0 and $m(t)=\frac{u(t)+\ell(t)}{2}=0.$1 locally in time and absorb external forcing into $m(t)=\frac{u(t)+\ell(t)}{2}=0.$2, obtaining an autonomous equation

$m(t)=\frac{u(t)+\ell(t)}{2}=0.$3

Writing $m(t)=\frac{u(t)+\ell(t)}{2}=0.$4 gives the conservative form

$m(t)=\frac{u(t)+\ell(t)}{2}=0.$5

Multiplying by a smooth compact-support test function $m(t)=\frac{u(t)+\ell(t)}{2}=0.$6 and integrating by parts yields the weak form

$m(t)=\frac{u(t)+\ell(t)}{2}=0.$7

The functions $m(t)=\frac{u(t)+\ell(t)}{2}=0.$8 and $m(t)=\frac{u(t)+\ell(t)}{2}=0.$9 are then expanded in a polynomial basis up to order $a(t)\cos(\theta(t)),$0: $a(t)\cos(\theta(t)),$1 Substitution produces a linear system in the coefficients $a(t)\cos(\theta(t)),$2, and sparsity is promoted by solving an $a(t)\cos(\theta(t)),$3-regularized normal-equations problem with compactly supported bump functions $a(t)\cos(\theta(t)),$4 (Hou et al., 2013).

This identification step leads to explicit nonlinearity indices,

$a(t)\cos(\theta(t)),$5

where $a(t)\cos(\theta(t)),$6 is a small pruning threshold. The pair $a(t)\cos(\theta(t)),$7 measures the highest polynomial degree appearing in the damping and restoring-force terms. If $a(t)\cos(\theta(t)),$8 and $a(t)\cos(\theta(t)),$9, the ODE is linear; larger values indicate stronger nonlinearity.

The numerical examples are explicit. For the Van der Pol oscillator

$a(t)>0$0

the algorithm recovers nonzero $a(t)>0$1, $a(t)>0$2, and $a(t)>0$3 exactly, both in the noise-free and in the 10% noise cases. For the Duffing oscillator

$a(t)>0$4

it finds $a(t)>0$5 and $a(t)>0$6 reliably, and for a time-varying transition from Van der Pol to Duffing type it tracks the changing polynomial form of $a(t)>0$7 and $a(t)>0$8 (Hou et al., 2013).

5. Mode mixing, energy leakage, and orthogonality

A central limitation of classical EMD is mode mixing: an IMF may contain oscillations of widely disparate scales, which can hinder physical interpretation. Entropic EMD addresses this by computing Permutation Entropy on the signal or on its sign-gradient, identifying high-entropy intervals where $a(t)>0$9, sifting those intervals separately, reinserting the pertinent IMFs, and finally performing EMD on the repaired series. In the synthetic example described in that work, the procedure yields two final IMFs—a clean high-frequency burst and a background low-frequency oscillation—and a near-zero residue. The reported quantitative improvement is that the mode-mixing index falls by over 80 % compared to classical EMD, and the instantaneous frequency estimate of each IMF matches the true components within 2 % error (Ram et al., 2015).

A second remedy is the masking-signal method based on the Rilling–Flandrin two-tone separation map. For

$a(t)>0$00

the map organizes behavior in the plane of frequency ratio $a(t)>0$01 and amplitude ratio $a(t)>0$02. Region I, $a(t)>0$03, yields automatic separation; Region II, $a(t)>0$04, is a gray zone; Region III, $a(t)>0$05, yields unavoidable mode mixing. The masking strategy chooses a tone $a(t)>0$06 so that $a(t)>0$07 and $a(t)>0$08, forms $a(t)>0$09 and $a(t)>0$10, performs EMD on both, and averages the first IMFs: $a(t)>0$11 In the three-tone example

$a(t)>0$12

a masking tone with $a(t)>0$13 and $a(t)>0$14 Hz produces clean 24 Hz and 30 Hz IMFs with near-perfect agreement, $a(t)>0$15, with pure sinusoids (Fosso et al., 2017).

A separate issue is that classical IMFs are only approximately orthogonal, so energy leaks among modes. Energy Preserving EMD addresses this in two ways. The first produces linearly independent, non-orthogonal yet energy-preserving IMFs satisfying

$a(t)>0$16

The second applies Gram–Schmidt from lowest frequency to highest frequency, producing reverse-order orthogonal IMFs that preserve IMF properties and satisfy

$a(t)>0$17

Reported simulations show that classical EMD yields small but nonzero energy leakage indexed by

$a(t)>0$18

whereas EPEMD drives $a(t)>0$19 to machine zero $a(t)>0$20 and $a(t)>0$21; reverse-order GSOM yields zero $a(t)>0$22 and no negative instantaneous frequency (Singh et al., 2015).

Iterative Filtering provides a different energy result. With double-convolution filters, the decomposition

$a(t)>0$23

is energy-conserving in Fourier–$a(t)>0$24 energy: $a(t)>0$25 and produces no unwanted oscillations because for every frequency $a(t)>0$26,

$a(t)>0$27

Orthogonal mode decomposition for finite discrete signals goes further by extracting each mode as an orthogonal projection onto a narrow-band subspace, yielding exact orthogonality by construction and uniqueness