Synchrosqueezed ST-FrFT: Instantaneous Analysis

• The paper introduces SS-STFrFT, a method that reassigns energy from short-time FrFT coefficients to the IF axis, enabling improved ridge tracking under the AM–FM model.
• It leverages a unified instantaneous atom framework with quadratic chirplet atoms and adjustable FrFT angles to optimize signal analysis across diverse components.
• The approach enhances concentration for chirped signals compared to SS-STFT, while highlighting practical tradeoffs in window selection and parameter tuning.

Searching arXiv for the specified paper and closely related work on the instantaneous time-frequency atom framework. Synchrosqueezed Short-Time Fractional Fourier Transform (SS-STFrFT) is an instantaneous time-frequency analysis obtained by computing short-time fractional Fourier transform coefficients and then reassigning their energy from the fractional-domain coordinate onto the instantaneous-frequency (IF) axis. Within the unified instantaneous time-frequency atom framework, it is treated as one member of a family that also includes time domain analysis, frequency domain analysis, the fractional Fourier transform (FrFT), and the synchrosqueezed short-time Fourier transform (SS-STFT). The unifying perspective models signals as sums of AM–FM components and interprets each analysis as using a specialized, or limiting, form of a quadratic chirplet template; the resulting instantaneous spectra are organized in a two-parameter continuum indexed by linear frequency and quadratic chirp-rate parameters (Sandoval et al., 7 Aug 2025). Because the excerpted formulation does not include the paper’s equation numbers or exact normalization choices, the presentation conventionally adopts a standard FrFT/STFrFT kernel while preserving the framework’s structural relationships.

1. Unified instantaneous atom formulation

The framework starts from an AM–FM decomposition

$x(t)=\sum_k A_k(t)e^{j\phi_k(t)},$

with instantaneous frequency

$\omega_k(t)=\dot{\phi}_k(t).$

The standard assumptions are that $A_k(t)$ and $\omega_k(t)$ are sufficiently smooth, the components are well separated in instantaneous frequency, and the amplitudes vary slowly relative to their phases.

The analysis template is a two-parameter quadratic chirplet localized by a window $w(t)$ centered at $t=0$ and shifted to $\tau$:

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$

In this atom, $w(\cdot)$ supplies localization, $\omega$ is the linear-frequency parameter, and $\omega_k(t)=\dot{\phi}_k(t).$0 is the quadratic chirp-rate parameter.

Several familiar analyses appear as specializations of this template. Setting $\omega_k(t)=\dot{\phi}_k(t).$1 yields the standard windowed complex sinusoid atom of the STFT. FrFT-based analysis corresponds to a rotation in the time-frequency plane, with the quadratic coefficient aligned with $\omega_k(t)=\dot{\phi}_k(t).$2, so that $\omega_k(t)=\dot{\phi}_k(t).$3. SS-STFT uses the $\omega_k(t)=\dot{\phi}_k(t).$4 atom followed by synchrosqueezing, whereas SS-STFrFT uses a short-time FrFT atom whose phase contains $\omega_k(t)=\dot{\phi}_k(t).$5 and $\omega_k(t)=\dot{\phi}_k(t).$6, followed by a squeezing map from the $\omega_k(t)=\dot{\phi}_k(t).$7 coordinates to the IF axis. This construction places SS-STFrFT within the same atom family rather than outside it.

2. Short-time fractional Fourier structure

The short-time fractional Fourier transform is written using the standard FrFT kernel of angle $\omega_k(t)=\dot{\phi}_k(t).$8:

$\omega_k(t)=\dot{\phi}_k(t).$9

with $A_k(t)$0. Different normalizations exist; this kernel is the one used in the supplied derivation because it is common and analytically convenient.

The STFrFT coefficients are

$A_k(t)$1

Here $A_k(t)$2 is the window centered at time $A_k(t)$3, $A_k(t)$4 is the fractional-domain variable, and the phase contains both the quadratic term $A_k(t)$5 and the bilinear term $A_k(t)$6.

The relation to the quadratic chirplet becomes explicit by expanding the FrFT phase around $A_k(t)$7 and matching coefficients. Locally,

$A_k(t)$8

and the linear term is identified with

$A_k(t)$9

The special case $\omega_k(t)$0 gives $\omega_k(t)$1, hence $\omega_k(t)$2, and $\omega_k(t)$3 reduces, modulo constants, to $\omega_k(t)$4; in that limit the STFrFT becomes the STFT. This makes the FrFT angle a direct control on the quadratic phase of the analyzing atom.

3. Synchrosqueezing map and IF reassignment

Synchrosqueezing reassigns coefficient energy from the analysis coordinate to the IF axis. For a generic analysis with coefficients $\omega_k(t)$5, the instantaneous spectrum is expressed as

$\omega_k(t)$6

In the STFrFT case, the phase of the integrand is

$\omega_k(t)$7

Stationary-phase analysis gives the stationarity condition

$\omega_k(t)$8

Because the window localizes $\omega_k(t)$9 near $w(t)$0, the dominant ridge relation at $w(t)$1 is

$w(t)$2

This yields the IF estimator

$w(t)$3

The synchrosqueezed spectrum is then

$w(t)$4

Under the AM–FM model, well-separated components, and slowly varying amplitudes, this reassigns energy from the $w(t)$5 plane onto the true IF ridges, so that concentration occurs along $w(t)$6 (Sandoval et al., 7 Aug 2025).

4. Closed-form instantaneous spectrum

The target instantaneous spectrum in the time-domain idealization is

$w(t)$7

The unified framework shows that several different analyses lead, after the appropriate IF mapping, to the same ridge-support structure. For SS-STFT,

$w(t)$8

with the standard time-derivative phase estimator

$w(t)$9

and under AM–FM separation

$t=0$0

For SS-STFrFT, the corresponding closed-form concentration is

$t=0$1

where $t=0$2 is the window’s value at zero and $t=0$3 depends on the FrFT normalization. In the supplied formulation this is the SS-STFrFT analogue of the SS-STFT concentration result. It guarantees that, when components are well separated in IF, energy collapses onto the true IF ridges, enabling ridge tracking and modal reconstruction (Sandoval et al., 7 Aug 2025).

A common misconception is to treat the fractional-domain coordinate $t=0$4 itself as an IF coordinate. In this framework it is not: $t=0$5 is an intermediate coordinate whose interpretation depends on $t=0$6, and synchrosqueezing recovers the physically relevant IF axis through $t=0$7.

5. Canonical signal classes and ridge geometry

For a single sinusoid,

$t=0$8

the analytic representation is $t=0$9, hence

$\tau$0

The ridge in the $\tau$1 plane is

$\tau$2

and the synchrosqueezed instantaneous spectrum is

$\tau$3

For a linear chirp,

$\tau$4

the IF is

$\tau$5

The ridge satisfies

$\tau$6

so that

$\tau$7

The corresponding synchrosqueezed spectrum is

$\tau$8

For a quadratic chirp,

$\tau$9

the IF is

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$0

Under the localization assumption around $$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$1,

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$2

and therefore

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$3

The synchrosqueezed spectrum becomes

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$4

For a multicomponent signal

$$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$5

each component contributes an independent ridge $$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$6 determined by $$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$7, and SS-STFrFT concentrates the corresponding energy on $$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$8.

6. Discrete implementation, parameter tradeoffs, and reconstruction

A discrete-time implementation begins by choosing a window $$g_{\tau,\omega,c_2}(t)=w(t-\tau)\exp\left\{j\left[\omega(t-\tau)+\frac{1}{2}c_2(t-\tau)^2\right]\right\}.$$9 such as Gaussian or Hann, selecting frame centers $w(\cdot)$0 on a grid with hop size $w(\cdot)$1, choosing a set of fractional angles $w(\cdot)$2, and defining an appropriate grid of $w(\cdot)$3 values for each angle. With sampling rate $w(\cdot)$4 and $w(\cdot)$5, the discrete STFrFT for a frame centered at $w(\cdot)$6 is

$w(\cdot)$7

Efficient implementations use fast FrFT algorithms; alternatively, the kernel may be precomputed or evaluated through a chirp-Z approach.

The supplied algorithmic guidance uses the analytical ridge map

$w(\cdot)$8

instead of numerically differentiating $w(\cdot)$9. A derivative-based estimator analogous to SS-STFT,

$\omega$0

is also possible, but the explicit mapping is described as simpler and robust.

Parameter selection follows the usual localization tradeoff. Shorter windows improve time localization, longer windows improve frequency resolution, and Gaussian windows are preferred for chirps because they minimize spread. Angles near $\omega$1 are appropriate for approximately sinusoidal content, whereas deviations from $\omega$2 better align linear or quadratic chirps with the kernel phase. If strong chirps are expected, selecting $\omega$3 so that $\omega$4 approximates the chirp curvature enhances pre-alignment before squeezing.

The method’s comparative behavior is also stated explicitly. SS-STFrFT typically concentrates chirped components better than SS-STFT when $\omega$5 is tuned, and its IF ridges are straighter and more concentrated for chirps, simplifying ridge tracking and modal extraction. When components have different chirp rates, choosing $\omega$6 near each component’s effective $\omega$7 makes their ridges more distinct before squeezing. The principal limitations are the need to choose $\omega$8, reduced concentration under mis-specified $\omega$9, numerical instability near $\omega_k(t)=\dot{\phi}_k(t).$00 or $\omega_k(t)=\dot{\phi}_k(t).$01 because $\omega_k(t)=\dot{\phi}_k(t).$02 and $\omega_k(t)=\dot{\phi}_k(t).$03 magnify errors, and the possibility that strongly nonlinear chirps may benefit more from adaptive or chirplet dictionaries than from a fixed angle.

Under well-separated components and sufficient concentration, reconstruction is expressed as

$\omega_k(t)=\dot{\phi}_k(t).$04

up to a constant depending on the window and normalization. For an individual component localized in an IF band $\omega_k(t)=\dot{\phi}_k(t).$05 around $\omega_k(t)=\dot{\phi}_k(t).$06,

$\omega_k(t)=\dot{\phi}_k(t).$07

This places SS-STFrFT in the unified picture as an analysis that first rotates and localizes chirped content through the FrFT kernel and then squeezes the resulting representation back onto the IF axis, producing a concentrated instantaneous spectrum within the two-parameter continuum of quadratic chirplet atoms (Sandoval et al., 7 Aug 2025).