X-ray Wavelet-Chirplet Transform (XWCT)

• XWCT is a transform that integrates a localized X-ray step to enhance decay along the chirprate axis, yielding clearer ridge extraction and improved mode retrieval.
• The method uses third-order synchrosqueezing to accurately estimate instantaneous frequency and chirprate, even in signals with crossing IF curves.
• Experimental results demonstrate reduced RMSE in IF, chirprate, and mode recovery, confirming XWCT’s superiority over traditional wavelet-chirplet transforms.

The X-ray Wavelet-Chirplet Transform (XWCT) is an X-ray transform-based wavelet-chirprate transform introduced for the analysis of multicomponent non-stationary signals with crossover instantaneous-frequency (IF) curves. It starts from the wavelet-chirplet transform (WCT), then applies a localized X-ray integral to the magnitude of the WCT along prescribed lines in a time-scale-chirprate parametrization. In the formulation of Jiang et al., the purpose of this construction is to obtain superior decay along the chirprate direction, because the WCT decays rapidly in the frequency-related variable but only algebraically in chirprate. The associated third-order synchrosqueezed XWCT produces a sharp three-dimensional time-frequency-chirprate representation and is used for accurate IF estimation, chirprate estimation, and mode retrieval without requiring multiple synchrosqueezing operations (Jiang et al., 25 Aug 2025).

1. Analytical setting and motivation

Recent work on the chirplet transform and wavelet-chirplet transform established that time-frequency-chirprate methods can estimate IFs and chirprates and can retrieve modes from multicomponent signals with crossover IF curves. In the formulation underlying XWCT, the central difficulty is that chirprate estimation remains less accurate than IF estimation because the chirplet transform or WCT decays slowly along the chirprate direction. Synchrosqueezed chirplet-transform variants and multiple synchrosqueezing strategies improve concentration, but only moderately, so the XWCT is introduced specifically to enhance decay along the chirprate axis (Jiang et al., 25 Aug 2025).

A common misunderstanding is to treat XWCT as merely a synchrosqueezed WCT. In the cited construction, the XWCT itself is the localized X-ray transform of WCT magnitudes; synchrosqueezing is an additional reassignment step that produces the synchrosqueezed XWCT (SXWCT). The distinction is structural: the X-ray stage modifies decay and leakage behavior, whereas the synchrosqueezing stage reassigns energy toward IF-chirprate ridges.

2. Continuous WCT and the definition of XWCT

The continuous WCT is defined for a signal $x(t)$, carrier frequency $\mu>0$, chirp-rate parameter $\lambda\in\mathbb R$, and a sufficiently smooth window $g\in L^1\cap L^2$ by

$$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$

If $\lambda=0$ and $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$ is a Morlet-like wavelet, one recovers the usual continuous wavelet transform (Jiang et al., 25 Aug 2025).

For a single locally quadratic chirp mode

$$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$

the local stationary-phase approximation gives

$$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$

with

$$G(\eta,\nu) = \int_{-\infty}^\infty g(u)\,e^{-\,i2\pi\,\eta\,u}\,e^{-\,i\pi\,\nu\,u^2}\,du.$$

This identifies the WCT as a representation on time, scale, and chirprate coordinates in which IF and chirprate enter through the arguments of a second-order polynomial Fourier transform of the window.

The XWCT is defined after the change of variables

$\mu>0$0

With a nonnegative X-ray window $\mu>0$1 satisfying $\mu>0$2, the XWCT is

$\mu>0$3

Equivalently, in $\mu>0$4-coordinates,

$\mu>0$5

Thus the XWCT is a localized X-ray transform of WCT magnitudes along lines of direction $\mu>0$6 in $\mu>0$7-space.

3. Chirprate decay, leakage, and geometric effect of the X-ray step

The motivation for the X-ray construction is explicit in the decay law of the WCT. In the summary of the method, $\mu>0$8 decays rapidly in

$\mu>0$9

but only algebraically in

$\lambda\in\mathbb R$0

For a pure linear chirp $\lambda\in\mathbb R$1, one obtains

$\lambda\in\mathbb R$2

so the factor

$\lambda\in\mathbb R$3

implies $\lambda\in\mathbb R$4 decay at large $\lambda\in\mathbb R$5, which limits chirprate resolution (Jiang et al., 25 Aug 2025).

For the XWCT, the corresponding approximation is

$\lambda\in\mathbb R$6

The cited analysis states that, by choosing $\lambda\in\mathbb R$7 sharply concentrated, $\lambda\in\mathbb R$8 becomes exponentially small whenever $\lambda\in\mathbb R$9. The resulting interpretation is geometric as well as analytic: the weighted line integral is designed to suppress chirprate leakage by aggregating magnitude along directions aligned with the transform’s intrinsic ridge geometry.

This faster decay is the main reason that the XWCT is introduced. The paper’s comparison is not merely qualitative; it explicitly contrasts algebraic WCT decay in chirprate with exponential decay for the XWCT after the localized line integral. The practical significance is that ridge extraction in the time-frequency-chirprate volume becomes substantially less ambiguous near IF crossings.

4. Third-order synchrosqueezing and the SXWCT

To sharpen concentration and recover IF and chirprate from WCT or XWCT coefficients, the construction uses third-order reassignment operators. The third-order IF reference operator and chirprate reference operator are

$g\in L^1\cap L^2$0

$g\in L^1\cap L^2$1

where $g\in L^1\cap L^2$2 are explicit determinants built from $g\in L^1\cap L^2$3 (Jiang et al., 25 Aug 2025).

For a cubic phase

$g\in L^1\cap L^2$4

the method checks that these operators recover exactly $g\in L^1\cap L^2$5 and $g\in L^1\cap L^2$6. In this sense, the third-order construction is tailored to higher-order phase variation rather than only locally linear or quadratic behavior.

The third-order synchrosqueezed WCT (SWCT) is defined by

$g\in L^1\cap L^2$7

The third-order synchrosqueezed XWCT (SXWCT) replaces $g\in L^1\cap L^2$8 by $g\in L^1\cap L^2$9: $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$0 The result is a three-dimensional time-frequency-chirprate representation in which the X-ray step sharpens chirprate decay and the synchrosqueezing step concentrates energy onto the estimated IF-chirprate ridges.

5. Mode retrieval and computational workflow

The mode-retrieval procedure is stated in algorithmic form. The inputs are a multicomponent signal $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$1, carrier $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$2, window $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$3 (Gaussian of width $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$4), X-ray window $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$5, and the number of modes $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$6 (Jiang et al., 25 Aug 2025).

The parameter-tuning stage chooses $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$7 to minimize the Rényi entropy of $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$8. After that, discrete WCT samples $$U^g_x(a,b,\lambda) = \int_{-\infty}^\infty x(t)\,\frac1a\,g\!\Big(\frac{t-b}a\Big)\, \exp\bigl[ -\,i2\pi\mu\frac{t-b}a -\,i\pi\,\lambda(t-b)^2 \bigr] \,dt, \qquad a>0,\; b,\lambda\in\mathbb R.$$9 are computed via FFT/IFFT. The XWCT is then formed by the discrete line integral

$\lambda=0$0

along the line

$\lambda=0$1

The third-order reassignment quantities $\lambda=0$2 and $\lambda=0$3 are evaluated at each $\lambda=0$4, and synchrosqueezing bins the coefficients into a discrete $\lambda=0$5 grid: $\lambda=0$6

Ridge extraction proceeds by finding, for each fixed $\lambda=0$7, the $\lambda=0$8 largest distinct peaks of $\lambda=0$9. Their coordinates are recorded as

$G(u)=\overline{g(u)}e^{+i2\pi\mu u}$0

Mode recovery then uses a small linear system,

$G(u)=\overline{g(u)}e^{+i2\pi\mu u}$1

with

$G(u)=\overline{g(u)}e^{+i2\pi\mu u}$2

and

$G(u)=\overline{g(u)}e^{+i2\pi\mu u}$3

In practice the algorithm inverts $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$4 or uses its pseudo-inverse to obtain the mode estimates $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$5. The cited presentation emphasizes that the recovery is based on ridge localization in the synchrosqueezed volume rather than direct component-wise filtering in the original time-frequency plane.

6. Experimental behavior and comparative results

The reported experiments use three synthetic crossover-chirp examples with sampling $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$6 Hz and a Gaussian window. In Example 1, with two linear chirps

$G(u)=\overline{g(u)}e^{+i2\pi\mu u}$7

the IF crossover occurs at $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$8. The reported chirprate RMSE near crossover is approximately $G(u)=\overline{g(u)}e^{+i2\pi\mu u}$9 for WCT+SWCT and approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$0 for $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$1MSWCT, whereas the SXWCT gives chirprate RMSE approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$2. The mode-retrieval RMSE is approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$3 for SXWCT versus approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$4 for MSWCT. In Example 3, with two sinusoidal-chirp modes

$$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$5

the reported IF-RMSE is approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$6, CR-RMSE approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$7, and mode-RMSE approximately $$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$8–$$x_k(t)=A_k(t)e^{\,i2\pi\,\phi_k(t)}, \qquad \phi_k'(t)\approx\phi_k'(b)+\tfrac12\phi_k''(b)(t-b),$$9 for MSWCT, while SXWCT gives IF-RMSE approximately $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$0, CR-RMSE approximately $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$1, and mode-RMSE approximately $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$2–$$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$3 (Jiang et al., 25 Aug 2025).

For Example 2, involving two cubic chirps

$$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$4

the published RMSE table is:

Method	IF1, IF2	CR1, CR2, Mode1, Mode2
SWCT	0.2555, 0.1403	2.2492, 0.4831, 0.1191, 0.0759
5×MSWCT	0.1021, 0.1094	0.8319, 0.8643, 0.0581, 0.0546
MSCT	0.1438, 0.1438	0.5512, 0.5512, 0.0472, 0.0471
SXWCT	0.0357, 0.0357	0.0727, 0.0727, 0.0276, 0.0229

These examples are presented as evidence for two separate effects. First, the X-ray stage improves decay along the chirprate axis. Second, the third-order synchrosqueezing stage provides sharply localized IF-chirprate ridges and supports mode retrieval from crossover configurations.

7. Scope, limitations, and relation to X-ray WLCT

The advantages stated for the XWCT framework are exponential-scale decay in chirprate, clean 3D ridge extraction, the sufficiency of a single synchrosqueezing step, and exact recovery for cubic phases under the third-order reassignment model. The limitations stated in the same summary are equally specific: the extra X-ray integral step adds $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$5 workload in naive form; the choice of X-ray window $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$6 and its support trades off decay versus bias; and edge and boundary artifacts require local averaging. The potential applications listed are radar micro-Doppler analysis, machinery vibration diagnosis, biomedical signals such as EEG/ECG with intersecting rhythms, and more generally multicomponent frequency-modulated signal separation where IF curves cross (Jiang et al., 25 Aug 2025).

A later and broader development places the X-ray idea inside a windowed linear canonical transform (WLCT) framework. That work develops a novel WLCT, discusses four types of WLCTs, uses a special X-ray transform to sharpen the time-frequency-chirprate representation, and derives a corresponding three-dimensional synchrosqueezed transform for signal separation (Li et al., 12 Oct 2025). In that setting, Type 2 WLCT coincides with the chirplet transform up to $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$7, and the X-ray transform integrates $$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$8 along line directions

$$U^g_{x_k}(a,b,\lambda)\approx x_k(b)\, G\!\bigl(\mu-a\,\phi_k'(b),\,a^2(-\phi_k''(b))\bigr),$$9

yielding the X-ray WLCT (XWLCT). A plausible implication is that XWCT and XWLCT instantiate the same analytical strategy—localized X-ray sharpening of a three-parameter time-frequency-chirprate representation—within different front-end transforms (Li et al., 12 Oct 2025).