Offset Linear Canonical Transform

• OLCT is a six-parameter integral transform that generalizes classical transforms by integrating time shift and frequency modulation into its framework.
• The transform extends key uncertainty principles by quantitatively bounding time–frequency localization with explicit measures based on the kernel parameter b.
• Its applications in signal processing and optics enable advanced filtering, reconstruction, and diffraction modeling while preserving L²-norm structures.

The offset linear canonical transform (OLCT) is a six-parameter generalization of the linear canonical transform, incorporating both time shifting and frequency modulation to create a unifying framework for most linear integral transforms commonly used in signal processing and optics. The OLCT parameter set $A = (a, b, c, d, T, n)$, where $ad-bc=1$, enables the recovery of the Fourier, fractional Fourier, Fresnel, and standard linear canonical transforms as special cases. Its mathematical flexibility allows precise control over time-frequency localization and is foundational for rigorous uncertainty analysis.

1. Definition and Transform Structure

Let $f \in L^2(\mathbb{R})$, the OLCT is defined for $b>0$ as

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

where the kernel is

$$K_A(t,u) = \frac{1}{\sqrt{j2\pi b}} \exp \left\{ j \left[ \frac{a}{2b} t^2 - \frac{1}{b} t(u-T) + \frac{d(u-T)^2 + 2(u-T)bn - 2nT - aT^2}{2b} \right] \right\}.$$

This generalizes the four-parameter LCT kernel by introducing the time shift $T$ and frequency modulation $n$. The normalization and phase, dictated by $b$ and the remaining parameters, maintain the symplectic structure underpinning most physically meaningful integral transforms.

2. Uncertainty Principles in the OLCT Domain

Uncertainty principles for the OLCT domain quantify fundamental limitations for the simultaneous localization of a function and its transform.

2.1 Heisenberg’s Uncertainty Principle

The classical Heisenberg uncertainty relation, which prohibits sharp localization in both time and frequency, is parametrically adapted to the OLCT, with the lower bound modulated by the kernel parameter $b$. Explicit derivation and generalization for the OLCT have been established in prior work; the principle remains: neither $f$ nor $O_A f$ can be sharply supported unless $f=0$.

2.2 Donoho–Stark’s Uncertainty Principle

For $f \in L^2(\mathbb{R})$ $\epsilon_Q$-concentrated on $\Omega \subset \mathbb{R}$ and $O_A f$ $\epsilon_R$-concentrated on $T \subset \mathbb{R}$,

$$|\Omega|\,|T| \geq 2\pi b (1-\epsilon_Q-\epsilon_R)^2$$

(see Theorem 3.4). If both $\epsilon_Q$ and $\epsilon_R$ are zero, this reduces to a lower bound on the product of the Lebesgue measures of the concentration sets. Consequently, “too much” simultaneous spatial and transform concentration is forbidden, directly generalizing the classical uncertainty product to the OLCT with $b$ scaling.

2.3 Amrein–Berthier–Benedicks’s Principle

If $f \in L^1(\mathbb{R})$ is supported on $\Omega$ and $O_A f$ is supported on $T$, with $|\Omega|\,|T|<+\infty$, then $f$ must vanish identically (Theorem 3.6). This underpins the impossibility of simultaneously compactly supported functions and OLCTs.

2.4 Lieb’s Uncertainty Principle for Short-Time OLCT

For the windowed, or “short-time” OLCT,

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

and any $2\leq p<\infty$,

$$\|V_{g,A}f\|_{L^p(\mathbb{R}^2)} \leq (2\pi b)^{-1/2+1/p} \|f\|_2 \|g\|_2$$

(see Theorem 4.3). Corollary 4.4 further gives a lower bound for the essential support, i.e., size of the joint time-frequency region with most signal energy.

3. Structure and Properties of the Short-Time OLCT

The short-time OLCT brings time–frequency localization into the OLCT framework, analogous to the short-time Fourier transform. It leverages an $L^2$ window function to analyze nonstationary or locally-structured signals. Parseval’s relation is generalized: $$\iint |V_{g,A}f(x,u)|^2 dx\,du = \|f\|_2^2 \|g\|_2^2,$$ highlighting the preservation of $L^2$ structure under the short-time OLCT. Energy concentration properties and Lieb-type $L^p$ bounds provide quantitative handles on time–OLCT-frequency localization trade-offs.

4. Applications and Implications in Signal Processing and Optics

The OLCT, encompassing the FT, FrFT, LCT, and Fresnel transforms, is exploited in multiple domains due to its enriched parameter space.

• Signal Processing: OLCT enables the development of generalized filtering, sampling, and reconstruction schemes that naturally adapt to chirp-like and nonstationary signal phenomena.
• Optics: The OLCT models quadratic-phase systems, e.g., Fresnel diffraction and graded-index media, capturing phenomena unaddressable by the FT alone.
• Uncertainty Principles: Practically, the derived bounds enforce strict constraints on simultaneous temporal and spectral detail—vital for high-resolution reconstructions in inverse problems, pattern analysis, and information compression.
• Short-Time Analysis: The short-time OLCT’s bounded essential support guides the design of analysis windows for optimally capturing locally transient phenomena.

5. Mathematical and Theoretical Significance

By generalizing integral transform theory’s uncertainty principles—including Heisenberg, Donoho–Stark, Amrein–Berthier–Benedicks, and Lieb—the OLCT framework extends foundational limits inherent to time–frequency analysis. The explicit dependence on the kernel parameter $b$ modulates these bounds, matching the OLCT’s nonseparable parameterization. Theoretical structure, such as the preservation of $L^2$-norms and extensions to localized domains, underpins advanced harmonic and functional-analytic investigations in both signal processing and mathematical physics.

6. Concluding Remarks

The offset linear canonical transform serves as a mathematically rigorous generalization, integrating both shift and modulation into canonical transform frameworks and offering expanded flexibility for real-world systems and theoretical study. Its suite of associated uncertainty principles ensures that any attempt at aggressive signal localization—either in the original or transform domain—is quantitatively limited, with implications for the design, analysis, and optimization of time–frequency methods across engineering and physics. The short-time OLCT, through Lieb-type bounds, ensures that joint time/OLCT-localized representations remain fundamentally constrained in support size and energy concentration, providing practical guidance for windowed time–frequency analysis in advanced applications.

Key formulas and quantitative bounds are as established throughout §§3, 4, and 5 of (Huo et al., 2018).