Short-Time Linear Canonical Transform

• STLCT is a localized version of the linear canonical transform that generalizes the STFT by applying a shifted window and chirp modulation.
• It employs a unimodular parameter matrix to control time-frequency localization, supporting rigorous uncertainty principle analyses.
• The framework enables advanced inverse problems such as phase retrieval and sampling reconstruction in non-Fourier canonical domains.

Searching arXiv for recent and foundational STLCT papers to ground the article. The short-time linear canonical transform (STLCT) is a windowed, localized version of the linear canonical transform (LCT), introduced as a time-frequency analysis tool that generalizes the short-time Fourier transform (STFT) by replacing the Fourier kernel with the more general LCT kernel (Gao et al., 2019). In this formulation, a signal is first localized by a shifted window and then analyzed in an LCT domain determined by a unimodular parameter matrix $A=(a,b,c,d)$ with $ad-bc=1$. The resulting representation extends the STFT and short-time fractional Fourier transform settings, supports spectrogram-based analysis, and has been studied in connection with uncertainty principles, phase retrieval, sampling, and generalizations such as the short-time offset linear canonical transform (Gao et al., 2019, Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026, Huo et al., 2018).

1. Definition and canonical-transform setting

The LCT used in STLCT theory is parameterized by

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$

For $b\neq 0$, the LCT of a signal $f(t)$ is

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$

with kernel

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$

and inverse matrix

$A^{-1}=(d,-b,-c,a).$

The main STLCT developments focus on the case $b\neq 0$, since when $b=0$ the LCT degenerates to a chirp multiplication (Gao et al., 2019).

Given a window $ad-bc=1$0 and a complex signal

$ad-bc=1$1

the STLCT is defined by

$ad-bc=1$2

This is the localized LCT analogue of the STFT: $ad-bc=1$3 localizes the signal around time $ad-bc=1$4, and the LCT kernel analyzes that localized component in the $ad-bc=1$5-domain (Gao et al., 2019).

An equivalent notation used in later work is

$ad-bc=1$6

where $ad-bc=1$7 (Dong et al., 26 Aug 2025). A closely related formulation writes

$ad-bc=1$8

with

$ad-bc=1$9

under the standing assumption $$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$0 (Cheng et al., 8 May 2026). These are notational variants of the same windowed LCT principle.

Two special cases organize much of the theory. If

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$1

the LCT becomes the Fourier transform and STLCT reduces to the STFT (Gao et al., 2019). If

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$2

the transform becomes a short-time fractional Fourier transform-type setting (Gao et al., 2019).

2. Structural identities and relation to the STFT

A central structural fact is that STLCT is reducible to an STFT after chirp modulation. One formulation states that

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$3

so the LCT differs from the Fourier transform by chirp factors and a rescaling (Dong et al., 26 Aug 2025). Correspondingly, STLCT magnitudes can be re-expressed through an STFT-like object after chirp modulation and an LCT change of variables: $$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$4 with appropriate matrices $$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$5 (Dong et al., 26 Aug 2025).

Another formulation gives the identity

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$6

where

$$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$7

and $$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$8 denotes the STFT (Cheng et al., 8 May 2026). This identifies STLCT as an STFT of a chirp-modulated signal, up to a known chirp factor and the rescaling $$A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1.$$9.

The 2019 uncertainty paper also establishes a frequency-domain representation: $b\neq 0$0 where

$b\neq 0$1

This identity expresses the STLCT through an interaction between the signal’s LCT and the window’s LCT (Gao et al., 2019).

These formulas explain why STLCT inherits much of the analytic structure of Gabor analysis while extending it to broader canonical domains. This suggests that many STFT techniques can be transported to the STLCT setting after chirp compensation and coordinate rescaling, a point made explicit in later phase-retrieval work (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026).

3. Local signal models, spectrograms, and moment decompositions

The STLCT literature introduces localized signal and spectral quantities to quantify concentration. For a fixed time $b\neq 0$2, the local signal is

$b\neq 0$3

with normalization factor

$b\neq 0$4

and normalized local signal

$b\neq 0$5

The corresponding local spectrum is defined by

$b\neq 0$6

with normalization

$b\neq 0$7

and normalized local spectrum

$b\neq 0$8

These quantities formalize localization in both time and LCT-frequency variables (Gao et al., 2019).

The paper further defines

$b\neq 0$9

and proves the identity

$f(t)$0

Accordingly, the magnitude squared of the STLCT acts like a spectrogram (Gao et al., 2019).

Moment quantities are then defined from $f(t)$1. The mean time and mean frequency are

$f(t)$2

and the time and frequency spreads are

$f(t)$3

$f(t)$4

With the auxiliary matrix

$f(t)$5

the corresponding means and spreads of the window $f(t)$6 are denoted $f(t)$7 (Gao et al., 2019).

A key decomposition lemma yields

$f(t)$8

$f(t)$9

These relations separate signal and window contributions to the STLCT centers and spreads (Gao et al., 2019). A plausible implication is that window design directly controls the attainable localization geometry in the STLCT plane, not merely the smoothness of the analysis operator.

4. Uncertainty principles in STLCT domains

The foundational uncertainty paper generalizes several classical principles from Fourier and LCT analysis to STLCT for complex signals (Gao et al., 2019). The starting point is the known LCT uncertainty principle

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$0

Using the moment decomposition above, the paper proves the STLCT time-frequency uncertainty relation

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$1

This is Theorem 1 and is presented as the STLCT analogue of the classical time-frequency uncertainty relation (Gao et al., 2019).

The same paper establishes a two-domain uncertainty relation for two different LCT parameter matrices

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$2

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$3

This is Theorem 2 and quantifies the incompatibility of concentration in two distinct STLCT frequency-like domains (Gao et al., 2019).

For the fractional Fourier special case,

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$4

the bound becomes

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$5

The dependence on $$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$6 shows that separation of fractional angles strengthens the lower bound (Gao et al., 2019).

The spectrogram itself also satisfies a conditional uncertainty principle. For fixed $$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$7, the conditional mean of $$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$8 is

$$L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt,$$9

and for fixed $$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$0, the conditional mean of $$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$1 is

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$2

The conditional standard deviations are

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$3

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$4

Theorem 3 states

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$5

where $$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$6 are Hermitian operators, specifically

$$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$7

This is an operator-based uncertainty principle for the shape of the STLCT spectrogram itself (Gao et al., 2019).

The broader short-time canonical-transform literature extends this perspective. The short-time OLCT, which reduces to STLCT when $$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$8 and $$K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right),$$9, satisfies a Lieb-type uncertainty principle: $A^{-1}=(d,-b,-c,a).$0 and an essential support lower bound

$A^{-1}=(d,-b,-c,a).$1

under the stated normalization and concentration hypotheses (Huo et al., 2018). Since short-time OLCT is explicitly described as a generalization of STLCT, these results situate STLCT within a wider uncertainty-theoretic hierarchy (Huo et al., 2018).

5. Phase retrieval, sampling geometry, and reconstruction theory

Recent work has shifted STLCT research from concentration inequalities toward inverse problems. A 2025 study considers phase retrieval from intensity measurements

$A^{-1}=(d,-b,-c,a).$2

with equivalence relation

$A^{-1}=(d,-b,-c,a).$3

The central question is whether

$A^{-1}=(d,-b,-c,a).$4

implies $A^{-1}=(d,-b,-c,a).$5 (Dong et al., 26 Aug 2025).

The main positive result holds on a rectangular square-root lattice

$A^{-1}=(d,-b,-c,a).$6

with

$A^{-1}=(d,-b,-c,a).$7

so that $A^{-1}=(d,-b,-c,a).$8, where

$A^{-1}=(d,-b,-c,a).$9

If the window $b\neq 0$0 belongs to

$b\neq 0$1

and

$b\neq 0$2

then for every $b\neq 0$3,

$b\neq 0$4

if and only if $b\neq 0$5 (Dong et al., 26 Aug 2025).

The same paper gives negative results for uniform lattices in $b\neq 0$6. For a Gaussian window $b\neq 0$7,

$b\neq 0$8

satisfy

$b\neq 0$9

but

$b=0$0

on a suitable lattice strip/line arrangement, including failure on sets of the form

$b=0$1

as well as on uniform lattices (Dong et al., 26 Aug 2025). The paper attributes this to an “ambiguity by interference” phenomenon.

Uniform sampling becomes viable again under model restrictions. For $b=0$2, $b=0$3, if

$b=0$4

then

$b=0$5

if and only if

$b=0$6

This identifies a sampling-density condition under which phase retrieval on a uniform lattice is possible for band-limited functions (Dong et al., 26 Aug 2025).

A 2026 paper develops a more explicit reconstruction and stability theory in the complex Gaussian shift-invariant space

$b=0$7

It proves that every signal in $b=0$8 is uniquely determined, up to a global unimodular constant, by phaseless STLCT measurements on

$b=0$9

provided $ad-bc=1$00 for some $ad-bc=1$01 (Cheng et al., 8 May 2026). The paper also derives an explicit reconstruction formula and a local stability bound on an interval $ad-bc=1$02: $ad-bc=1$03 with the constants and functions given explicitly in the paper (Cheng et al., 8 May 2026).

To prevent exponential deterioration with respect to interval size, the same work introduces an anchor-point condition: there exist points

$ad-bc=1$04

and $ad-bc=1$05 such that

$ad-bc=1$06

With

$ad-bc=1$07

the global stability constant depends on the maximal spacing between adjacent anchor points rather than the full interval size (Cheng et al., 8 May 2026). This suggests a phase-propagation mechanism localized by nonvanishing samples.

The same paper further gives a finite-data noisy model

$ad-bc=1$08

and develops an explicit reconstruction algorithm with quantitative robustness guarantees (Cheng et al., 8 May 2026).

6. Generalizations, higher-dimensional variants, and comparative frameworks

STLCT has been embedded into several larger transform families. The most immediate is the short-time offset linear canonical transform. In the OLCT setting, the parameter set is

$ad-bc=1$09

where $ad-bc=1$10 is a time-shift parameter and $ad-bc=1$11 is a frequency-modulation / frequency-offset parameter (Huo et al., 2018). The short-time OLCT is defined by

$ad-bc=1$12

and the paper explicitly states that when

$ad-bc=1$13

the short-time OLCT reduces to STLCT (Huo et al., 2018). This places STLCT as the zero-offset member of a six-parameter localized canonical-transform family.

A 2026 study introduces a convolution-based short time offset linear canonical transform (STOLCT) and treats STLCT as the benchmark transform introduced in earlier work. In that paper, STLCT is described as a short-time analysis transform built from the convolution structure of the LCT, and its reconstruction/sampling formula is quoted as

$ad-bc=1$14

The authors use this only for numerical comparison against their STOLCT sampling theorem and report that reconstruction errors decrease as $ad-bc=1$15 increases for both methods, while STOLCT “produces smaller errors, demonstrating better reconstruction performance” than STLCT (Mahato et al., 12 Jun 2026). Since the paper’s discussion of STLCT is explicitly indirect, this should be read as a comparative statement within the STOLCT framework rather than a redevelopment of STLCT theory itself.

Higher-dimensional and hypercomplex extensions also use STLCT as a reference model. In three dimensions, a paper on octonion analysis defines the 3D-STLCT by

$ad-bc=1$16

with three coordinate-wise LCT parameter matrices $ad-bc=1$17, $ad-bc=1$18 (Bhat et al., 2021). The paper’s main object is a short-time octonion linear canonical transform, but it explicitly presents 3D-STLCT as the classical comparison object and gives a structural relation between them (Bhat et al., 2021).

A more speculative extension appears in work on the linear canonical space-time transform for $ad-bc=1$19-valued signals. That paper does not define a short-time LCST explicitly, but it states that its results form the theoretical backbone for any STLCT or short-time linear canonical space-time transform built in the $ad-bc=1$20 setting (Xu et al., 2024). This suggests a possible path from one-dimensional STLCT to Clifford-valued space-time canonical analysis, though the short-time construction itself is not supplied there.

7. Conceptual significance and recurring themes

Across the literature, STLCT occupies a specific position: it is the short-time, windowed form of the LCT, just as the STFT is the short-time form of the Fourier transform (Gao et al., 2019). Its chief role is to provide localized analysis in canonical transform domains broader than ordinary frequency, including Fourier and fractional Fourier cases (Gao et al., 2019).

Three themes recur.

First, localization is constrained but structured. The uncertainty relations show that STLCT inherits fundamental lower bounds on simultaneous concentration in time and canonical-frequency variables, in two different STLCT domains, and in the conditional geometry of its spectrogram (Gao et al., 2019). This indicates that generalized canonical flexibility does not remove the localization trade-offs already familiar from Fourier analysis.

Second, the transform is tightly linked to Gabor analysis. The chirp-modulation identities reduce STLCT to STFT-like objects, enabling transfer of uniqueness, analyticity, and sampling arguments between the two settings (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026). In the Fourier case

$ad-bc=1$21

the STLCT becomes the Gabor transform exactly (Cheng et al., 8 May 2026).

Third, sampling geometry matters as much as transform geometry. For general $ad-bc=1$22 signals, phaseless STLCT recovery is possible on carefully designed rectangular square-root lattices, fails on uniform lattices in $ad-bc=1$23, and becomes possible again on uniform lattices for band-limited spaces or Gaussian shift-invariant spaces under explicit structural conditions (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026). A plausible implication is that the main obstruction in STLCT phase retrieval is not the canonical kernel itself but the interaction between window analyticity, function class, and sampling set.

In this sense, STLCT has evolved from a generalized time-frequency representation into a platform for inverse problems and sampling theory. The 2019 uncertainty-principle results established its concentration geometry (Gao et al., 2019); subsequent work on phase retrieval, explicit reconstruction, and stability has shown that this geometry can also support rigorous recovery theory in non-Fourier canonical domains (Dong et al., 26 Aug 2025, Cheng et al., 8 May 2026).