Short-Time OLCT: A Time-Frequency Transform

• Short-Time OLCT is a localized time–frequency transform that replaces the standard LCT kernel with an offset version for enhanced flexibility.
• It extends uncertainty principles by incorporating time shifts and modulation parameters, enabling robust analysis of chirped and shifted signals.
• STOLCT supports phase retrieval, convolution, and filtering applications, unifying classical transforms within a framework for adaptive signal recovery.

A short-time offset linear canonical transform (STOLCT) is a localized time–frequency methodology that generalizes the short-time linear canonical transform (STLCT) to the broader offset linear canonical framework, enabling the analysis of signals under arbitrary time–shift and frequency modulation parameters. This extension is predicated on substituting the standard LCT kernel with an OLCT kernel, thus introducing extra degrees of flexibility—time shift and modulation—that are critical for both theoretical signal localization bounds and diverse applications in optics, signal processing, and phase retrieval.

1. Kernel Generalization and Formal Definition

The transition from STLCT to STOLCT entails replacing the LCT kernel with the offset linear canonical kernel in the windowed transform framework. Whereas the STLCT is defined as

$$V_g^A f (x, u) = \int_{-\infty}^{+\infty} f(t) g(t - x) K_A(t, u) dt,$$

where $K_A$ is the LCT kernel, the OLCT kernel is specified as: $$K_A(t, u) = \sqrt{\frac{1}{j2\pi b}} \exp\left\{ j \left[ \frac{a}{2b} t^2 - \frac{1}{b} t(u-T) + \frac{d}{2b}(u-T)^2 - n u + \varphi \right] \right\}$$ for matrix parameters $$A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$ (with $ad - bc = 1$), real shifts $T$ and $n$, and explicit phase $\varphi$.

Thus, the STOLCT is defined by

$$V_{g, A} f (x, u) = \int_{-\infty}^{+\infty} f(t) g(t - x) K_A(t, u) dt,$$

where $g$ is a windowing function and $K_A$ includes the offset parameters, so the transform adapts analytically to chirped and shifted signals, and encompasses all classical cases (FT, FrFT, LCT) as special cases (Huo et al., 2018).

2. Mathematical Structure and Localization

The transformation realized by STOLCT windows the input signal with $g(t-x)$, thus providing time localization, then applies the OLCT kernel to the localized signal. Explicitly, the kernel’s offset terms $(T, n)$ impart the ability to analyze signals that are nonstationary, modulated, or spatially shifted.

When $T = n = 0$, the transform reduces to the STLCT, preserving compatibility with existing time–frequency analyses. This structure is also naturally generalized to higher dimensions, as in the octonion and quaternion settings, or to polar coordinates in 2D applications for radially symmetric or angularly periodic signals (Bhat et al., 2021).

3. Uncertainty Principles: Localization Bounds

Uncertainty principles for STOLCT quantify the trade-offs between time and frequency localization. In particular, Lieb’s uncertainty principle is extended for the STOLCT (Theorem 4.3 in (Huo et al., 2018)): $$\| V_{g,A} f \|_{L^p(\mathbb{R}^2)} \leq (2\pi b)^{-1/p} \| f \|_{L^2} \| g \|_{L^2}$$ for $f, g \in L^2(\mathbb{R})$ and $2 < p < \infty$, showing that excessive concentration in both domains is prohibited.

Additionally, lower bounds on the essential support $\Omega$ of STOLCT representations are established: $$| \Omega | \geq 2 \pi b (1-\epsilon)^{p-\text{dependent factor}}$$ with $\epsilon$ representing allowable error energy (Huo et al., 2018).

Generalizations of Nazarov's, Hardy's, Beurling's, logarithmic Pitt, and entropic uncertainty principles are available for the OLCT domain, and their implications extend by analogy to short-time analyses (Huo, 2018), ensuring that a signal and its localized OLCT representation cannot be simultaneously arbitrarily concentrated or rapidly decaying.

4. Convolution, Correlation, and Filter Theorems

Novel convolution and correlation theorems specific to OLCT extend classical Fourier relationships. For signals $f, g \in L^1(\mathbb{R})$: $$(f * g)(x) = \int_{-\infty}^{\infty} f(t) g(x-t) \exp\{ i \theta(t,x) \} dt,$$ where $\theta$ encodes the OLCT structure (Mahato et al., 2 May 2025).

Correspondingly, the OLCT domain supports a product theorem: $OM(fg)(u) = T(u) OM[f](2u) OM[g](2u),$ with $T(u)$ a modulation factor analytically determined by OLCT parameters. This enables the design of multiplicative filters acting in the OLCT domain: $$f_{\text{out}}(x) = OM^{-1} \{ OM[f_{\text{in}}](2u) T(u) GM(2u) \}(x),$$ where $GM(2u)$ encodes the filter’s frequency response. The extension to short-time domains (STOLCT) implies enhanced adaptability for localized signal recovery, robust to spectral time–frequency shifts (Mahato et al., 2 May 2025).

5. Phase Retrieval with STOLCT

The STOLCT provides redundancy crucial for phase retrieval: recovering a signal from magnitude-only local time–frequency measurements. For $f \in L^2(\mathbb{R})$, window $\varphi$, and kernel $K_A$, the STOLCT is

$$\mathcal{V}_{\varphi}^A f(v,u) = \int_{-\infty}^{+\infty} f(t) \overline{\varphi(t-v)} K_A(t,u) dt.$$

It is established that if the ambiguity function $\mathcal{A}_\varphi$ of the window does not vanish on a set with positive measure, then $f$ can be uniquely reconstructed (up to a global phase) from measurements $|\mathcal{V}_{\varphi}^A f(v,u)|$, both for nonseparable general functions and for bandlimited signals in FT or OLCT domains, provided appropriate sampling and mild regularity conditions on $\mathcal{A}_\varphi$ (Liu et al., 4 Jun 2025).

6. Generalizations in Polar, Quaternion, and Octonion Domains

In 2D contexts, the OLCT is extended to polar coordinates, facilitating the analysis of angularly periodic and radially symmetric functions via the offset linear canonical Hankel transform (OLCHT). Angular harmonics are mapped through OLCHT to decompose the transform in terms of Bessel function components. Spatial shift and convolution theorems are generalized, with direct implications for localized analysis techniques, such as STOLCT in computed tomography and optics (Zhao et al., 2021).

Quaternion and octonion extensions leverage orthogonal plane split to apply OLCT componentwise to multi-channel signals, preserving uncertainty principles and computational efficiency. For three-dimensional and multichannel data, the STOLCT is formulated by combining windowed OLCTs across dimensions and supporting reconstruction, convolution, and norm-preserving properties relevant for color image and vector field analysis (Bhat et al., 2021, Mahato et al., 11 Sep 2025).

7. Applications and Implications in Signal Processing and Optics

STOLCT’s flexible parametrization enables tailored analysis of signals subject to arbitrary shifts, chirps, and modulation—critical in radar, communications, imaging, and adaptive optics. Uncertainty bounds ensure robust feature extraction and prevent instability due to over-concentration in time–frequency representations. Multiplicative filter design is facilitated by convolution theorems in the OLCT, and phase retrieval is enabled with uniqueness guarantees for both continuous and bandlimited signals.

The unification under OLCT generalizes FT, FrFT, and LCT frameworks into a flexible family compatible with modern signal processing demands—including discrete-time extensions and optimal windowing under uncertainty constraints. STOLCT provides a theoretical and practical foundation for local, adaptive, and robust time–frequency transformations, signal recovery, and system identification in challenging environments (Huo et al., 2018, Huo, 2018, Mahato et al., 2 May 2025, Liu et al., 4 Jun 2025, Zhao et al., 2021, Bhat et al., 2021, Mahato et al., 11 Sep 2025).