Instantaneous Time-Frequency Atoms

• Instantaneous time–frequency atoms are localized oscillatory components with time-varying amplitude and frequency that enable detailed nonstationary signal analysis.
• They differ from traditional Gabor atoms by allowing adaptive modulation, providing a more flexible framework for representing latent AM–FM structures.
• Extraction methods such as NFMD, dictionary learning, synchrosqueezing, and convex atomic-norm techniques facilitate robust estimation of instantaneous parameters.

Searching arXiv for the cited papers and closely related work on instantaneous time–frequency atoms. Instantaneous time–frequency atoms are elementary oscillatory building blocks used to represent a signal in the coordinates of time and frequency, either as latent AM–FM components or as continuously indexed analysis functions. In Sandoval and De Leon’s formulation, the term refers to components of the form $\psi_k(t)=a_k(t)e^{j\theta_k(t)}$ with time-varying instantaneous amplitude and instantaneous frequency, embedded in a multicomponent latent signal $z(t)$ whose real part is the observed signal $x(t)$ (Sandoval et al., 2015). In adjacent literatures, closely related atom notions appear as sliding-window Fourier modes, phase-adapted dictionary elements, reassigned wavelet or STFT atoms, warped time–frequency atoms, continuous Gabor atoms in atomic-norm formulations, and quadratic chirplets that interpolate among several common analyses (Shea et al., 2021, Hou et al., 2013, Thakur, 2014, Holighaus et al., 2015, Yang et al., 14 Jan 2025, Sandoval et al., 7 Aug 2025). Across these formulations, the common objective is concentration of signal energy along physically or mathematically meaningful instantaneous frequency laws.

1. Hilbert-spectrum definition of instantaneous atoms

In the "Theory of the Hilbert Spectrum," the observed real signal is written as

$x(t)=\Re\{z(t)\},$

with a latent complex representation

$z(t)=\sum_{k=1}^N \psi_k(t).$

Each $\psi_k(t)$ is an instantaneous time–frequency atom, also described as an AM–FM component, of the form

$$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$

where $a_k(t)\ge 0$ is the instantaneous amplitude, $\omega_k(t)=d\theta_k/dt\ge 0$ is the instantaneous frequency, $\phi_k$ is a constant phase reference, $z(t)$0, and $z(t)$1 (Sandoval et al., 2015).

Given $z(t)$2, the instantaneous parameters are defined by

$z(t)$3

This formulation is componentwise: instantaneous frequency is defined on a per-component basis rather than as a property of an arbitrary real signal taken as a whole. The Hilbert spectrum associated with a multicomponent AM–FM model is then

$z(t)$4

Its interpretation is explicit: at each instant $z(t)$5, the magnitude $z(t)$6 sits at frequency $z(t)$7 (Sandoval et al., 2015).

Within this framework, the Hilbert spectrum is presented as the exact time–frequency representation under the chosen decomposition, with no further smoothing or windowing required. The exactness claim is conditional: it holds when the appropriate assumptions on the signal model have been made so that the latent complex components are correctly specified (Sandoval et al., 2015).

2. Quadrature, analyticity, and the departure from Gabor atoms

A central feature of the Hilbert-spectrum theory is the distinction between the observed real part $z(t)$8 and the unobserved quadrature $z(t)$9. In practice one observes only $x(t)$0, so the quadrature must be supplied by a complex-extension rule. Under the classical analytic-signal construction associated with Gabor, Vakman’s three conditions—amplitude continuity, scale-homogeneity, and harmonic correspondence—lead to the unique quadrature $x(t)$1, producing $x(t)$2 and atoms with constant amplitude and constant frequency, i.e. Gabor atoms (Sandoval et al., 2015).

Sandoval and De Leon explicitly relax harmonic correspondence and show that analyticity does not force the Hilbert transform. Their example takes

$x(t)$3

and chooses

$x(t)$4

By the Cauchy–Riemann test, $x(t)$5 remains analytic for any $x(t)$6, although $x(t)$7 violates harmonic correspondence (Sandoval et al., 2015). The paper states the resulting theorem in direct form: if one does not assume harmonic correspondence, there exist analytic extensions $x(t)$8 so that $x(t)$9 remains analytic in the time-complex plane. The Hilbert transform is therefore not unique.

This produces the paper’s main shift in viewpoint. The uncertainty is not treated as a fundamental time–frequency resolution bound; rather, it is shifted into the choice of quadrature and therefore into the choice of signal-model assumptions. The paper states that infinitely many latent signals $x(t)=\Re\{z(t)\},$0 share the same $x(t)=\Re\{z(t)\},$1, and that one resolves this ambiguity by imposing additional assumptions such as constant amplitude, narrowband amplitude, constant instantaneous frequency, bandlimited instantaneous frequency, or the intrinsic-mode-function condition of exactly one extremum pair per half-cycle (Sandoval et al., 2015).

The contrast with Gabor atoms is exact and structural. Gabor atoms are

$x(t)=\Re\{z(t)\},$2

so all instantaneous amplitudes and frequencies are constant. Instantaneous atoms in the Hilbert-spectrum sense instead allow both $x(t)=\Re\{z(t)\},$3 and $x(t)=\Re\{z(t)\},$4 to vary in time. The triangle-wave example in the paper makes the model dependence concrete: the same signal can be represented as a harmonic sum under harmonic correspondence, as a single pure AM component with constant $x(t)=\Re\{z(t)\},$5, as a single pure FM component with $x(t)=\Re\{z(t)\},$6, or as one wideband AM–FM atom $x(t)=\Re\{z(t)\},$7; each choice corresponds to a different quadrature, and only the harmonic-sum case coincides with $x(t)=\Re\{z(t)\},$8 (Sandoval et al., 2015).

3. Major atom constructions in related time–frequency frameworks

The term “instantaneous time–frequency atom” is used most explicitly in the Hilbert-spectrum setting, but closely allied constructions recur in several neighboring frameworks. These constructions differ in whether the atom is treated as a latent component, a dictionary element, or an analysis template.

Framework	Atom form	Role
Hilbert spectrum	$x(t)=\Re\{z(t)\},$9	Latent AM–FM component
NFMD	$z(t)=\sum_{k=1}^N \psi_k(t).$0	Recovered mode in sliding windows
Dictionary learning	$z(t)=\sum_{k=1}^N \psi_k(t).$1	Phase-adapted sparse atom
Synchrosqueezing	$z(t)=\sum_{k=1}^N \psi_k(t).$2 and $z(t)=\sum_{k=1}^N \psi_k(t).$3	Reassigned time–frequency atom
Warped representation	$z(t)=\sum_{k=1}^N \psi_k(t).$4	Continuously indexed warped atom
AST-STF	$z(t)=\sum_{k=1}^N \psi_k(t).$5	Continuous-parameter Gabor atom
Quadratic chirplet unification	$z(t)=\sum_{k=1}^N \psi_k(t).$6	Two-parameter chirplet template

In nonstationary Fourier mode decomposition, the assumed signal model is

$z(t)=\sum_{k=1}^N \psi_k(t).$7

with instantaneous frequency $z(t)=\sum_{k=1}^N \psi_k(t).$8. The explicit target is recovery of the atom set $z(t)=\sum_{k=1}^N \psi_k(t).$9 together with the instantaneous mean $\psi_k(t)$0 (Shea et al., 2021).

In the dictionary-learning approach to sparse time–frequency decomposition, the real signal is modeled as

$\psi_k(t)$1

and the atom family is built in the $\psi_k(t)$2-coordinate. The paper defines instantaneous-frequency atoms

$\psi_k(t)$3

so that each IMF $\psi_k(t)$4 is sparse on $\psi_k(t)$5 (Hou et al., 2013).

In Synchrosqueezing, the underlying analysis functions are classical wavelet or STFT atoms, but the reassignment step concentrates them near local instantaneous frequencies. The continuous-wavelet atom is

$\psi_k(t)$6

and after squeezing the construction yields an approximate instantaneous atom $\psi_k(t)$7 concentrated near time $\psi_k(t)$8 and frequency $\psi_k(t)$9 (Thakur, 2014).

In warped time–frequency analysis, Holighaus, Wiesmeyr, and Balazs define

$$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$0

with a warping function $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$1. The same framework recovers classical short-time Fourier atoms when $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$2 and wavelet-type systems when $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$3 (Holighaus et al., 2015).

In atomic-norm soft thresholding for sparse TF representation, the atomic set is a continuous dictionary of Gabor atoms

$$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$4

with $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$5, $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$6, and $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$7 (Yang et al., 14 Jan 2025).

Finally, the quadratic-chirplet unification defines

$$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$8

whose instantaneous frequency is $$\psi_k(t)=a_k(t)\exp\!\Bigl[j\bigl(\int^t \omega_k(\tau)\,d\tau+\phi_k\bigr)\Bigr] =a_k(t)e^{j\theta_k(t)} =s_k(t)+j\sigma_k(t),$$9, and whose Wigner–Ville distribution is a tilted Gaussian ridge along $a_k(t)\ge 0$0 (Sandoval et al., 7 Aug 2025).

Taken together, these formulations suggest that the word “atom” has at least three stable meanings in the literature: a latent oscillatory component, a sparse dictionary element, or a continuously indexed analysis kernel. The overlap lies in the use of localized AM–FM structure and explicit parameterization of amplitude, phase, or frequency.

4. Extraction and estimation procedures

The computational problem is to determine atom parameters from data. The methods represented here differ sharply in whether they solve local nonlinear least-squares problems, phase-adaptive sparse coding problems, convex atomic-norm programs, or reassignment-based reconstructions.

In NFMD, the signal is tiled into overlapping windows $a_k(t)\ge 0$1, and on each segment a locally linear-phase model is imposed:

$a_k(t)\ge 0$2

The coefficients and local frequencies are estimated by minimizing squared error on each segment. The paper’s Algorithm FMD alternates least-squares updates of amplitudes with gradient descent on frequencies, while Algorithm NFMD propagates the previous window’s frequency estimate as a warm start. The assembled outputs are the instantaneous frequency vectors $a_k(t)\ge 0$3 and amplitudes $a_k(t)\ge 0$4 (Shea et al., 2021).

The dictionary-learning method instead solves a non-convex sparse-coding problem in which both coefficients and phases are unknown:

$a_k(t)\ge 0$5

The proposed solver is a Gauss–Newton–type alternating iteration. At fixed phases, it solves a linearized $a_k(t)\ge 0$6 problem involving both cosine and sine dictionaries; then it reconstructs envelopes $a_k(t)\ge 0$7 and $a_k(t)\ge 0$8 and updates phase derivatives through a projected Gauss–Newton step. The linearized sparse-coding subproblem is solved by an Augmented Lagrangian Multiplier method, and the paper emphasizes a “Sweeping ALM” whose subproblems admit shrinkage-form updates and whose cost is accelerated by the fast wavelet transform (Hou et al., 2013).

AST-STF models the signal as a sum of continuous-parameter Gabor atoms and promotes sparsity via the atomic norm

$a_k(t)\ge 0$9

Given noisy measurements $\omega_k(t)=d\theta_k/dt\ge 0$0 and a de-window operator $\omega_k(t)=d\theta_k/dt\ge 0$1, the primal problem is

$\omega_k(t)=d\theta_k/dt\ge 0$2

The dual problem yields a dual certificate $\omega_k(t)=d\theta_k/dt\ge 0$3, and the dual polynomial

$\omega_k(t)=d\theta_k/dt\ge 0$4

achieves its maximum exactly at the true support points $\omega_k(t)=d\theta_k/dt\ge 0$5. The semidefinite-lifted formulation is solved by ADMM, with updates for $\omega_k(t)=d\theta_k/dt\ge 0$6, PSD-cone projection by eigen-decomposition, and dual ascent for the Lagrange multiplier (Yang et al., 14 Jan 2025).

Synchrosqueezing begins from continuous wavelet or STFT coefficients, computes a phase-transform frequency estimator,

$\omega_k(t)=d\theta_k/dt\ge 0$7

and reassigns coefficients from scale to frequency through the squeezing operator

$\omega_k(t)=d\theta_k/dt\ge 0$8

Component reconstruction is then performed by integrating $\omega_k(t)=d\theta_k/dt\ge 0$9 over a narrow band around each instantaneous-frequency curve (Thakur, 2014).

These procedures represent different operational definitions of atom recovery. NFMD estimates piecewise-constant local amplitudes and frequencies; dictionary learning updates the phase coordinate itself; AST-STF solves an off-the-grid convex sparse inverse problem; Synchrosqueezing sharpens an existing linear representation by reassignment.

5. Theoretical properties: exactness, sparsity, invertibility, and performance

The Hilbert-spectrum theory states that with appropriate assumptions on the signal model, the instantaneous amplitude and instantaneous frequency can be obtained exactly, so that exact representation in time–frequency coordinates can be achieved. The limitation is transferred to selecting the correct quadrature rather than to a fixed time–frequency bound (Sandoval et al., 2015).

Synchrosqueezing provides a different theory of exactness. For signals in the class $\phi_k$0, with $\phi_k$1, $\phi_k$2, slow variation conditions

$\phi_k$3

and well-separated instantaneous frequencies, the theory states that the estimated frequency $\phi_k$4 lies within $\phi_k$5 of the true $\phi_k$6 in the relevant wavelet bands, while reconstructed modes differ from $\phi_k$7 by at most $\phi_k$8 in sup-norm. The same article states deterministic perturbation and stochastic-noise stability results, and describes Synchrosqueezing as sparse and invertible (Thakur, 2014).

Warped time–frequency representations establish a frame-theoretic version of atomic decomposability. Under the conditions stated on the prototype $\phi_k$9 and the derivative $z(t)$00, the family $z(t)$01 is a tight continuous frame in $z(t)$02 and satisfies a Moyal identity. The canonical synthesis formula is

$z(t)$03

for $z(t)$04. The same work further states sufficient conditions for subsampled warped systems to form atomic decompositions and Banach frames for generalized coorbit spaces (Holighaus et al., 2015).

AST-STF gives explicit numerical benchmarks for localization. For a multi-component test signal sampled at $z(t)$05 with $z(t)$06, time–frequency concentration was measured by the third-order Rényi entropy. In the noise-free case the reported entropies were $z(t)$07 for STFT, $z(t)$08 for reassignment, $z(t)$09 for STFT+$z(t)$10 sparse reconstruction, and $z(t)$11 for AST-STF. With added white noise at $z(t)$12, the reported Rényi entropies were $z(t)$13, $z(t)$14, $z(t)$15, and $z(t)$16, respectively, and the reported RMSE values were $z(t)$17 for STFT+$z(t)$18 sparse reconstruction and $z(t)$19 for AST-STF. In a real bat-echolocation example with $z(t)$20 samples, AST-STF achieved a Rényi entropy of $z(t)$21 versus $z(t)$22–$z(t)$23 for the other methods (Yang et al., 14 Jan 2025).

Related extraction methods also report signal-specific performance. NFMD is described as more robust than HHT under added noise at $z(t)$24, as locking onto an abrupt frequency jump at $z(t)$25, and as isolating mean shifts without cross-mixing. Its experimental example on cantilever-based electrostatic force microscopy uses the extracted instantaneous mean $z(t)$26 to fit a model and recover $z(t)$27 across experiments (Shea et al., 2021). The dictionary-learning method reports accurate recovery of two intersecting ridges with frequency error $z(t)$28, perfect separation of outliers with reconstruction error $z(t)$29, and theoretical mode frequencies recovered to within $z(t)$30 relative error in a two-degree-of-freedom ODE example (Hou et al., 2013).

6. Unification and recurring conceptual distinctions

The most explicit unifying construction is the two-parameter quadratic chirplet family $z(t)$31. The paper "Unifying Common Signal Analyses with Instantaneous Time-Frequency Atoms" states that time-domain analysis, frequency-domain analysis, the fractional Fourier transform, the synchrosqueezed short-time Fourier transform, and the synchrosqueezed short-time fractional Fourier transform can all be viewed as decompositions into AM–FM components using specialized or limiting forms of a quadratic chirplet as a template (Sandoval et al., 7 Aug 2025).

The specializations are given concretely. Taking $z(t)$32 and $z(t)$33 yields time-domain evaluation through $z(t)$34 and $z(t)$35. Taking $z(t)$36 and $z(t)$37 yields frequency-domain analysis. Setting

$z(t)$38

recovers the FRFT kernel in the $z(t)$39-variable. Fixing $z(t)$40 and finite $z(t)$41 yields the Gaussian-window STFT, while keeping general $z(t)$42 gives a short-time FRFT. The same paper organizes these analyses as points in a two-dimensional $z(t)$43 continuum, with the line $z(t)$44 ranging from pure time to pure frequency and the curve $z(t)$45 tracing the FRFT family (Sandoval et al., 7 Aug 2025).

This unification does not erase substantive differences among atom concepts. One recurrent misconception is that instantaneous time–frequency atoms must be identical to Hilbert-transform analytic components; the quadrature non-uniqueness theorem in the Hilbert-spectrum theory directly rejects that conclusion (Sandoval et al., 2015). Another misconception is that the atom family must be fixed in advance: the Hilbert-spectrum approach prescribes a decomposition model of adaptive AM–FM atoms, the dictionary-learning method adapts the basis as part of the optimization, and AST-STF uses an off-the-grid continuous dictionary rather than a fixed frequency grid (Hou et al., 2013, Yang et al., 14 Jan 2025). A further misconception is that nonstationary extraction necessarily imposes smoothing in time; NFMD explicitly states that it uses no smoothness penalty on $z(t)$46 or $z(t)$47, so abrupt changes are not smoothed over (Shea et al., 2021).

A plausible implication is that “instantaneous time–frequency atom” functions less as the name of one canonical object than as a family resemblance term for representations that localize amplitude and phase information along instantaneous spectral structures. In one branch, the atom is a latent AM–FM constituent whose exactness depends on model assumptions; in another, it is an analysis atom embedded in a frame, dictionary, or convex atomic set; and in a third, it is a chirplet or reassigned wavelet/STFT kernel that unifies or sharpens classical transforms. What remains invariant across these branches is the aim of expressing nonstationary behavior through localized oscillatory elements whose time and frequency coordinates are themselves part of the representation.