Instantaneous Frequency & Amplitude Determination

• IFAD is a collection of signal-analysis techniques that assign time-varying amplitude and frequency to oscillatory signals based on specific model assumptions.
• Methods include Hilbert transform formulations, decomposition-driven pipelines, sparse derivative models, and kernel-based derivative estimations tailored for various signal types.
• The effectiveness of these approaches hinges on signal characteristics, with challenges such as non-uniqueness and noise sensitivity prompting careful model selection.

Searching arXiv for recent IFAD-related papers and the cited core references. Instantaneous Frequency and Amplitude Determination (IFAD) denotes the family of signal-analysis procedures that seek to assign, to an oscillatory signal or to its decomposed components, a time-dependent amplitude and a time-dependent frequency. Across the literature represented here, IFAD appears in several mathematically distinct forms: Hilbert/analytic-signal constructions for scalar narrowband signals (Aburakhia et al., 2024); decomposition-based pipelines in which IMFs or narrowband modes are first extracted and then analyzed (Stroeer et al., 2011, Krolak et al., 11 Jun 2026, Cicone et al., 2014); convex sparse coefficient models that recover abrupt amplitude and phase changes around fixed carriers (Ding et al., 2013); local operator-based estimators such as complex Teager–Kaiser formulations (Vaca et al., 21 Jan 2026); geometric formulations in multivariate and three-phase settings (Lilly, 2012, Alshawabkeh et al., 2023, Milano et al., 2022); and kernel, search-based, and high-order time-frequency estimators (Riedel, 2018, See et al., 2015, Li et al., 1 Jun 2026, Shea et al., 2021). A recurring theme is that IFAD is powerful but model-dependent: without structural assumptions on decomposition, analyticity, modulation regularity, or geometry, instantaneous frequency need not be unique (Tavallali et al., 2016).

1. Classical analytic-signal formulation and its scope

The canonical scalar IFAD construction begins from the Hilbert transform and analytic signal. For a real-valued signal $x(t)$, the Hilbert transform is written as

$$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$

and the analytic signal is

$x_a(t) = x(t) + jH\{x(t)\}.$

From this, instantaneous amplitude and instantaneous phase are defined by

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$

$$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$

and instantaneous frequency by

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$

(Aburakhia et al., 2024). The same structure is used in Hilbert–Huang-style pipelines, where each IMF $c(t)$ is converted to an analytic signal

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$

with

$$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$

(Stroeer et al., 2011).

This formulation underlies much of classical IFAD, but the data also make clear that its interpretability is conditional. In the standard AM–FM picture,

$x(t)=A(t)\cos(\phi(t)),$

the analytic signal is meaningful when the signal is effectively mono-component or sufficiently narrowband (Aburakhia et al., 2024). Several papers represented here build on precisely that assumption: kernel estimation for slowly evolving sinusoids (Riedel, 2018), local IMF analysis after EMD/EEMD (Stroeer et al., 2011), and decomposition-based mode analysis after VMD or EMD (Krolak et al., 11 Jun 2026). By contrast, multicomponent or strongly distorted signals produce analytic phases whose derivatives need not correspond to a single physical oscillation (Milano et al., 2022).

The same source that revisits the “five paradoxes” of instantaneous frequency emphasizes that for a scalar signal

$$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$0

one obtains

$$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$1

only under the assumptions enabling

$$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$2

In that mono-component setting, $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$3 and $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$4; outside it, the phase of the analytic signal need not coincide with an intended physical phase (Milano et al., 2022). This suggests that the classical Hilbert definition is best regarded as a model-based IFAD rule rather than a universally intrinsic quantity.

2. Non-uniqueness and identifiability

A central theoretical result in the supplied material is the claim that instantaneous frequency is inherently non-unique. The paper “On the non-uniqueness of the instantaneous frequency” concludes that IF is “non-unique inherently,” that this non-uniqueness persists across adaptive signal-processing methods and even when the physical origin is known, and that “without any a priori assumption about the relationship of the envelope and phase function of an oscillatory signal, there is not any preferred neither best representation of the IF of such oscillatory signal” (Tavallali et al., 2016). Within IFAD, this is the strongest cautionary statement in the corpus.

This non-uniqueness theme reappears in more constructive forms. Sparse Frequency Analysis models a signal as

$$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$5

with time-varying quadrature coefficients $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$6, $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$7, rather than directly as $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$8 (Ding et al., 2013). For each fixed carrier $$H\{x(t)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau,$$9, one may rewrite

$x_a(t) = x(t) + jH\{x(t)\}.$0

where

$x_a(t) = x(t) + jH\{x(t)\}.$1

The method estimates $x_a(t) = x(t) + jH\{x(t)\}.$2, not $x_a(t) = x(t) + jH\{x(t)\}.$3 directly. This means that amplitude and phase are parameterized through quadrature coefficients on a fixed frequency grid, not uniquely recovered as free-form functions (Ding et al., 2013).

The search-based SEIFD method is even more explicit about representation dependence. It models a nonlinear, non-stationary signal as a sum of finite-duration sinusoidal components with parameters $x_a(t) = x(t) + jH\{x(t)\}.$4, yielding a piecewise-constant frequency description over time rather than a continuously varying IF curve (See et al., 2015). This suggests a distinct “segmented IF” interpretation of IFAD, in which frequency is attached to active intervals rather than to a differentiable phase.

A plausible implication of these results is that IFAD is not a single invariant procedure but a collection of admissible constructions whose outputs depend on the signal model, decomposition, and normalization convention. That implication is directly consistent with the non-uniqueness statement in (Tavallali et al., 2016).

3. Decomposition-driven IFAD

A large part of the modern IFAD literature proceeds by first decomposing a signal into narrower modes and then computing amplitude and frequency on each mode.

Hilbert–Huang analysis is the canonical example. In the gravitational-wave comparison framework, the data are whitened, regions of excess power are identified via Bayesian blocks, a low-pass filter is applied, and EEMD is then used “to average over decomposition errors” before Hilbert spectral analysis extracts instantaneous amplitude and frequency (Stroeer et al., 2011). The derived mode-wise quantities are then compared using an uncertainty-weighted statistic

$x_a(t) = x(t) + jH\{x(t)\}.$5

(Stroeer et al., 2011). In this application, IFAD is not only descriptive; it is the basis for timing, coincidence testing, and morphology classification.

Iterative Filtering and its adaptive variant ALIF offer an alternative decomposition stage. IF replaces EMD’s spline-envelope mean by a convolutional moving average

$x_a(t) = x(t) + jH\{x(t)\}.$6

and extracts IMFs by repeated application of $x_a(t) = x(t) + jH\{x(t)\}.$7 (Cicone et al., 2014). Under the stated assumptions on $x_a(t) = x(t) + jH\{x(t)\}.$8, the inner loop converges to

$x_a(t) = x(t) + jH\{x(t)\}.$9

providing an explicit Fourier-domain characterization of the extracted IMF (Cicone et al., 2014). ALIF generalizes this to spatially varying support lengths $A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$0,

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$1

thereby adapting the decomposition scale to the local spacing of extrema (Cicone et al., 2014).

The ringdown analysis paper (Krolak et al., 11 Jun 2026) uses decomposition-first IFAD in a different way. EMD or VMD is first used to isolate approximately monocomponent modes, then amplitude and frequency are extracted either by the Hilbert transform or by a new IFAD method. The mode model is

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$2

with

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$3

The proposed post-decomposition IFAD avoids the Hilbert transform’s finite-record edge effects by interpolating the maxima of $A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$4 to obtain $A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$5, forming

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$6

and then inferring periods from extrema and zero crossings of $A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$7 (Krolak et al., 11 Jun 2026). In this setting, IFAD is explicitly local and geometric rather than integral-transform based.

NFMD furnishes still another decomposition-driven IFAD strategy. It fits, on each local segment, a Fourier model

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$8

minimizing

$A(t) = |x_a(t)| = \sqrt{x(t)^2 + H\{x(t)\}^2},$9

with FFT peak initialization and local optimization of $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$0 (Shea et al., 2021). Mode-wise instantaneous amplitudes are then

$$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$1

while instantaneous frequencies are the local fitted frequencies $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$2 collected across windows (Shea et al., 2021). This suggests a “segmentwise IFAD” interpretation distinct from Hilbert differentiation.

4. Local operators, kernels, and sparse variational formulations

Several papers develop IFAD without relying primarily on decomposition into IMFs.

Sparse Frequency Analysis is a convex variational method tailored to abrupt amplitude or phase changes. The exact problem is

$$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$3

subject to

$$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$4

The noisy version adds quadratic data fidelity (Ding et al., 2013). The method regularizes first differences of the quadrature coefficients, so that “sparse-derivative instantaneous amplitude and phase functions” are represented implicitly through piecewise-constant $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$5, not through smooth AM–FM envelopes (Ding et al., 2013). This is directly relevant to IFAD where phase resets or abrupt envelope changes are of interest.

Kernel estimation treats IF as a nonparametric derivative-estimation problem. For a generic function $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$6, the paper introduces

$$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$7

with bias–variance tradeoff controlled by kernel halfwidth $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$8 (Riedel, 2018). For the analytic phasor $$\theta(t) = \arctan\left(\frac{H\{x(t)\}}{x(t)}\right),$$9, the instantaneous frequency estimate is

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$0

and the optimal bandwidth scaling becomes

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$1

(Riedel, 2018). The paper’s main conceptual contribution is that IF estimation corresponds to derivative estimation of a modulated complex signal rather than mere finite differencing of phase.

The complex Teager–Kaiser approach in power systems develops an exact local identity. With

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$2

the complex TKEO is

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$3

and exactly satisfies

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$4

Since

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$5

the proposed IF estimator is

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$6

(Vaca et al., 21 Jan 2026). Here amplitude is not estimated by the operator alone; rather, envelope derivatives are required to debias IF. This paper therefore contributes to the “AD” part of IFAD indirectly, by making amplitude-curvature contamination of IF explicit.

5. Geometric and multivariate generalizations

A major strand of the literature generalizes IFAD from scalar analytic signals to multivariate trajectories and geometric invariants.

For trivariate oscillations, the analytic vector representation is

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$7

which defines a canonical time-varying ellipse (Lilly, 2012). The joint instantaneous amplitude, frequency, and bandwidth are

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$8

$F(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}$9

$c(t)$0

(Lilly, 2012). The central identity is

$c(t)$1

so circulation equals instantaneous frequency times squared instantaneous amplitude (Lilly, 2012). This is a geometric reinterpretation of IFAD rather than a new estimator.

In three-phase systems, the Clarke transform plays the role that the analytic signal plays in the scalar case. For balanced voltages,

$c(t)$2

and

$c(t)$3

Thus

$c(t)$4

(Milano et al., 2022). This directly supplies joint amplitude and IF in balanced three-phase settings without invoking a Hilbert transform.

Affine differential geometry extends this idea to unbalanced systems. For a planar voltage trajectory $c(t)$5, the estimator proposed is

$c(t)$6

It is exact for stationary unbalanced three-phase voltages whose Clarke-plane trajectory is an ellipse, and it extends to single-phase systems via the embedding

$c(t)$7

(Alshawabkeh et al., 2023). This is an IF-focused contribution; amplitude is only implicit in the ellipse geometry.

Quaternion Fourier analysis generalizes analytic-signal IFAD to complex-valued signals. The hypercomplex representation is

$c(t)$8

with one-sided quaternion Fourier spectrum

$c(t)$9

Through a quaternion polar form

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$0

one obtains a complex envelope $z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$1, a phase $z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$2, and instantaneous frequency

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$3

(Bihan et al., 2012). This suggests that IFAD can be extended beyond real-valued signals provided the notion of one-sidedness is reformulated in quaternion terms.

6. Failure modes, stochastic pathologies, and current directions

The corpus is unusually explicit about the limitations of IFAD.

Noise corruption of EMD is a major failure mechanism. In noisy data, EMD produces “transition IMFs” containing both signal and noise, and these contaminated modes invalidate Hilbert-based IF because the modes are no longer nearly monochromatic (Kaslovsky et al., 2010). Since the overall IF estimate is often an amplitude-weighted combination of IMF IFs, transition IMFs can dominate and corrupt the result (Kaslovsky et al., 2010). A plausible implication is that amplitude estimates derived from the same analytic modes are likewise compromised, although that point is more strongly suggested than explicitly theoremized.

The stochastic theory of Gaussian processes adds another caution. For a proper mean-square differentiable complex Gaussian process $z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$4, the fixed-time IF is

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$5

when $z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$6, and $z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$7 at zeros (Wahlberg et al., 2010). The paper shows that at a fixed time the IF has either zero or infinite variance, and that for harmonizable processes

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$8

(Wahlberg et al., 2010). Thus the mean IF is a normalized first-order frequency moment of the Wigner spectrum, but the pointwise phase derivative can be intrinsically heavy-tailed.

Recent work also points toward higher-order time-frequency methods. HSWCT models components as

$z(t)=c(t)+i\tilde c(t)=IA(t)e^{i\phi(t)},$9

with polynomial phase behavior over short intervals, and produces arbitrary-order IF and chirprate reassignment operators $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$0 and $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$1 (Li et al., 1 Jun 2026). The main theorem provides explicit approximation bounds for IF and chirprate estimation under the class $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$2, and experiments show improved performance for strongly modulated signals and crossing IF curves (Li et al., 1 Jun 2026). Amplitude recovery enters via mode reconstruction rather than a dedicated direct estimator.

Application-specific IFAD systems continue to diversify. In rolling-bearing monitoring, Hilbert-derived $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$3 and $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$4 are used to form IAFM, IAFC, and IEFD representations, from which six discriminative features $$IA(t)=|z(t)|,\qquad IF(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}$$5 are extracted (Aburakhia et al., 2024). In multipoint fiber sensing, wavelet-based decomposition of superposed wavelength-shift traces yields per-sensor vibration amplitude, frequency, and phase, though the paper is more an application architecture than a general IFAD theory (Chatterjee et al., 2021).

Taken together, these works show that IFAD is not a monolithic doctrine but a technically heterogeneous field. Its common core is the attempt to assign local oscillatory descriptors—amplitude, phase, frequency, sometimes chirprate—to signals or components. Its main fault lines are equally clear: decomposition quality, local versus global construction, smooth versus sparse modulation priors, scalar versus multivariate geometry, and uniqueness versus representation dependence. The strongest general conclusion supported by the present record is therefore not that IFAD has a single definitive formulation, but that meaningful instantaneous amplitude and frequency require an explicit signal model, and that different IFAD frameworks recover different local invariants under different assumptions (Tavallali et al., 2016).