Instantaneous Spectrum Analysis

Updated 8 July 2026

Instantaneous Spectrum (IS) is a collection of time-local frequency representations that assign energy to instantaneous frequency trajectories, fundamental in AM–FM analysis.
It includes formulations like the Hilbert-spectrum, synchrosqueezing, and quadratic-chirplet approaches, each offering unique computational benefits.
These techniques enable robust signal decomposition in both scalar and multivariate settings, linking spectral moments directly to physical dynamics.

Instantaneous Spectrum (IS) denotes a class of time-local frequency representations for nonstationary signals, and, in some measurement systems, a single-shot physical mapping from frequency to space. In AM–FM signal models, IS assigns amplitude or energy to instantaneous-frequency trajectories; in synchrosqueezing, it is a concentrated and invertible reassignment of a time–frequency transform; in sparse and nonperiodic formulations, it is a sharp spectrum assembled from locally estimated modes; and in analog microwave analyzers, it is realized as an optical image of resonant response in a magnetic-field gradient (Sandoval et al., 2015, Thakur, 2014, Sandoval et al., 7 Aug 2025, Chipaux et al., 2015).

1. Core definitions and formal variants

For a nonstationary signal written as

$f(t)=\sum_{k=1}^K A_k(t)e^{2\pi i\phi_k(t)},$

with slowly varying amplitudes $A_k(t)>0$ and strictly increasing phases $\phi_k(t)$ , the instantaneous frequency (IF) of the $k$ th mode is $\phi_k'(t)$ . In this setting, the instantaneous spectrum at time $t$ is a distribution $S_f(t,\omega)$ that concentrates the energy of $f$ near the curves $\omega=\phi_k'(t)$ ; operationally, it answers the question “at time $t$ , which frequencies $A_k(t)>0$ 0 carry energy in $A_k(t)>0$ 1?” (Thakur, 2014).

A different but closely related formulation arises in the Hilbert-spectrum framework. Starting from a multicomponent latent signal

$A_k(t)>0$ 2

the instantaneous spectrum is defined by depositing the amplitude $A_k(t)>0$ 3 at the instantaneous frequency $A_k(t)>0$ 4,

$A_k(t)>0$ 5

This representation is described as exact and free of time–frequency smoothing once a particular AM–FM decomposition has been fixed (Sandoval et al., 2015).

A third formulation is atom-based. In the quadratic-chirplet framework, one defines a local transform

$A_k(t)>0$ 6

and then sets

$A_k(t)>0$ 7

Here the IS is the squared magnitude of a local analysis coefficient indexed by time $A_k(t)>0$ 8, frequency $A_k(t)>0$ 9, chirp-rate parameter $\phi_k(t)$ 0, and window-spread parameter $\phi_k(t)$ 1 (Sandoval et al., 7 Aug 2025).

Framework	IS expression	Distinguishing feature
Hilbert spectrum	$\phi_k(t)$ 2	Exact AM–FM deposition
Synchrosqueezing	$\phi_k(t)$ 3	Reassigned, sparse, invertible
Sparse Frequency Analysis	$\phi_k(t)$ 4	TV-sparse amplitudes on a discrete grid
Quadratic-chirplet framework	$\phi_k(t)$ 5	Unifies common analyses

Taken together, these definitions suggest that IS is not a single universal object but a family of representations whose exact form depends on the signal model, transform, or measurement principle.

2. Exact AM–FM and Hilbert-spectrum formulation

In the Hilbert-spectrum theory, one begins with a real signal $\phi_k(t)$ 6 and seeks a complex extension

$\phi_k(t)$ 7

Each component is written as

$\phi_k(t)$ 8

where $\phi_k(t)$ 9 is the instantaneous amplitude, $k$ 0 is the instantaneous phase, the quadrature $k$ 1 is not fixed a priori, and the instantaneous frequency is assumed nonnegative:

$k$ 2

The instantaneous parameters are then defined exactly by

$k$ 3

Under this construction, the Hilbert or instantaneous spectrum is

$k$ 4

The central claim of this framework is that, with appropriate assumptions on the multicomponent AM–FM model, one obtains exact instantaneous amplitudes and instantaneous frequencies rather than a smeared time–frequency display (Sandoval et al., 2015).

A major conceptual feature of this theory is its treatment of uncertainty. Classical time–frequency methods such as STFT and wavelets are described as obeying a Heisenberg trade-off, whereas the AM–FM model shifts uncertainty into the model assumptions and the quadrature choice. In particular, the paper states that if one does not assume harmonic correspondence, there exist infinitely many choices of quadrature $k$ 5 for which the complex extension remains analytic in the complex-time sense; correspondingly, no universal linear operator can recover the “true” quadrature for all possible latent signals. Exactness is therefore conditional on adopting an appropriate component model rather than on applying a universal transform (Sandoval et al., 2015).

The practical computation described in this literature is the HSA–IMF algorithm. It proceeds by decomposing $k$ 6 into Intrinsic Mode Functions via Empirical Mode Decomposition, enforcing carried assumptions to obtain a unique amplitude estimate for each IMF, removing the amplitude to obtain a real frequency-modulated signal, normalizing to unit amplitude and recovering a quadrature, differentiating the recovered phase to obtain $k$ 7, and finally assembling

$k$ 8

The paper characterizes this as returning the exact instantaneous spectrum under the AM–FM component assumptions (Sandoval et al., 2015).

3. Transform-based, sparse, and nonperiodic computational constructions

Synchrosqueezing provides a rigorous transform-based route to IS. Starting from the continuous wavelet transform

$k$ 9

one defines a local frequency estimate wherever $\phi_k'(t)$ 0:

$\phi_k'(t)$ 1

The synchrosqueezing transform then reallocates each wavelet coefficient in the frequency direction toward $\phi_k'(t)$ 2, yielding

$\phi_k'(t)$ 3

This IS is sparse, invertible, and stable: ridge-tracking in $\phi_k'(t)$ 4 permits recovery of the constituent modes, and the paper reports robustness under small deterministic perturbations and Gaussian white noise (Thakur, 2014).

Sparse Frequency Analysis (SFA) approaches IS from a convex variational standpoint. A discrete-time real signal is modeled as

$\phi_k'(t)$ 5

with narrow-band sinusoids whose amplitudes vary in time. The instantaneous parameters are

$\phi_k'(t)$ 6

The amplitudes are estimated by minimizing total variation in $\phi_k'(t)$ 7 and $\phi_k'(t)$ 8 together with a spectral sparsity term, either under perfect reconstruction or under a quadratic data-fidelity penalty in noise. The optimization is solved by ADMM with variable splitting and a nested MM step for the coupled $\phi_k'(t)$ 9 sparsity term. Once the amplitudes are estimated, the instantaneous spectrum is defined on the discrete grid by

$t$ 0

which preserves abrupt amplitude and phase changes more sharply than classical linear filtering (Ding et al., 2013).

A different line of work replaces the periodic boundary condition (PBC) underlying STFT implementations with the Linear eXtrapolation Condition (LXC). In that formulation, a real signal is locally modeled as a sum of complex AM–FM modes

$t$ 1

near each analysis time $t$ 2, where the poles

$t$ 3

are extracted via a small Toeplitz system under the no-wrap-around LXC constraint. Amplitudes are then obtained by least squares, and each mode contributes a Lorentzian-shaped local spectrum

$t$ 4

Summing over modes yields a framewise instantaneous spectrum suitable for pulse-series signals (Ishiyama, 3 Feb 2026).

The quadratic-chirplet framework of instantaneous time-frequency atoms places several of these constructions into a single two-parameter continuum. With

$t$ 5

the IS is $t$ 6, and time-domain analysis, frequency-domain analysis, fractional Fourier analysis, synchrosqueezed STFT, and synchrosqueezed STFrFT arise as limiting or specialized cases in the $t$ 7 plane. This framework explicitly treats those analyses as decompositions into AM–FM components using specialized or limiting forms of a quadratic chirplet (Sandoval et al., 7 Aug 2025).

4. Multivariate instantaneous spectrum and physical moments

In multivariate analysis, IS is generalized from scalar amplitude–frequency traces to matrix-valued instantaneous moments. For a two- or three-component real-valued time series $t$ 8 or $t$ 9, the analytic signal is

$S_f(t,\omega)$ 0

The instantaneous amplitude and instantaneous frequency are defined by

$S_f(t,\omega)$ 1

One then introduces the one-sided spectral matrix and its global moments, together with instantaneous moment matrices

$S_f(t,\omega)$ 2

Their traces recover instantaneous power, instantaneous frequency, and bandwidth (Lilly, 2012).

The distinctive result in this line of work is that these instantaneous spectral moments coincide exactly with the physical moments of a canonical time-varying ellipse traced by the signal. The ellipse is parameterized by semi-axes, orbital phase, and, in three dimensions, a rotation. Scalar, vector, and tensor moments such as variance, normal vector, moment of inertia, circulation, and angular momentum are defined for the corresponding ring of particles. The paper proves exact identities linking the matrix-valued spectral moments to those physical moments, and, taking the trace of $S_f(t,\omega)$ 3, obtains the formula

$S_f(t,\omega)$ 4

Thus circulation equals instantaneous power times instantaneous frequency (Lilly, 2012).

This correspondence gives IS a geometric and mechanical interpretation. The amplitude becomes the root-mean-square radius of the ellipse, while the frequency decomposes into orbital and precessional contributions. In this setting, the instantaneous spectrum is not merely a sharper spectrogram; it is a local moment decomposition of an evolving geometric object. The paper further notes that multivariate wavelet-ridge analysis can be used in noisy data to estimate $S_f(t,\omega)$ 5 and hence the instantaneous moment structure (Lilly, 2012).