Circular Fractional-Order Fourier Transforms

• Circular fractional-order Fourier transformations are defined via quadratic phase kernels with circular (trigonometric) dependence on a fractional parameter, ensuring unitarity.
• They leverage Weyl pseudo-differential calculus and group composition properties to model rotations in time–frequency and phase spaces.
• Applications include diffraction theory, coherent imaging, and signal propagation in complex media such as fractal or non-Euclidean spaces.

A circular fractional-order Fourier transformation is a parameterized family of unitary integral and pseudo-differential operators that generalize the classical Fourier transform by enabling continuous “rotations” in time–frequency or phase space, often realized through kernels with quadratic phase and circular functional dependence on a fractional parameter. These transformations find rigorous formulation in harmonic analysis, pseudo-differential operator theory, and mathematical physics, and are fundamental for modeling signal evolution, diffraction, and imaging in non-Euclidean or curved geometries, including fractal and fractional spaces. The “circular” aspect reflects the periodic or rotational structure of the group of transformations under composition, often governed by trigonometric functions of the fractional parameter.

1. Mathematical Formulation and Unitary Structure

Circular fractional-order Fourier transformations (abbreviated here as CFrFTs, Editor's term) are defined via integral operators with quadratic phase kernels whose dependence on the fractional parameter is governed by circular (trigonometric) functions. In two dimensions:

$$F_a[f](x) = \int_{\mathbb{R}^2} K_a(x, y)\, f(y)\, dy$$

$$K_a(x, y) = C_a\, \exp\left[ -\frac{i}{2}\cot(a)(x^2 + y^2) + \frac{i}{\sin(a)}\, x \cdot y \right],$$

where $a$ is the fractional “angle,” and the normalization constant $C_a$ ensures unitarity and reduction to the classical Fourier transform at $a = \pi/2$. The kernel and symbol structure is compatible with the Weyl pseudo-differential calculus, allowing expression of the transformation as $F_a = a^w(x, D)$ with Weyl symbol

$$a(x, \xi) = e^{-ia}\left(1 + \tan^2\frac{a}{2}\right)\exp\left[2i\pi(x^2 + \xi^2)\tan\frac{a}{2}\right].$$

These operators constitute a one-parameter group with the crucial composition property:

$F_{a_2}\circ F_{a_1} = F_{a_1+a_2} \mod 2\pi,$

mirroring geometric rotations in phase space and underlining the circular topology of the parameter set (Pellat-Finet, 1 Oct 2025, Dong et al., 2016, Zhou, 17 Sep 2024).

2. Relation to Pseudo-Differential Operators and Weyl Calculus

CFrFTs are a distinguished subclass of Weyl pseudo-differential operators with symbols explicitly dependent on the fractional parameter through trigonometric functions. The Weyl calculus provides a systematic approach to their manipulation, ensuring that the composition of two CFrFTs is again a CFrFT, with the order given by the sum of the fractional angles. The phase space representation established by the Weyl symbol encapsulates both the quadratic structure of the kernel and the underlying symplectic symmetry, offering a robust framework for generalizing propagation and imaging laws to arbitrary orders (Pellat-Finet, 1 Oct 2025).

3. Applications in Diffraction Theory and Coherent Imaging

Circular fractional-order Fourier transformations serve as the mathematical underpinning for diffraction and imaging models in scalar wave optics. The diffraction integral for field transfer from a spherical emitter to a receiver, after suitable change of variables (reduced coordinates), is expressed as a CFrFT:

$U_B(p') = e^{ia} F_a[U_A](p'),$

where the fractional order $a$ is determined by an optical parameter $J$ (related to the geometry—curvatures, distances, and refractive indices) via $\cot^2 a = J$. This formulation unifies transfer phenomena—including propagation, focusing, and imaging—within the CFrFT framework. The composition law for CFrFTs enforces the Huygens–Fresnel principle, as the impulse response of sequential propagations is given by the sum of the underlying fractional orders. This composition yields, for instance, the classical imaging laws (e.g., conjugation of vertices and centers of curvature) as algebraic consequences of CFrFT group properties (Pellat-Finet, 1 Oct 2025).

The geometric ordering of key points (vertices, centers of curvature) along the optical axis is preserved up to a circular permutation under imaging described by CFrFTs. This preservation is a direct manifestation of the circular nature of the transformation parameter and the symplectic symmetries intrinsic to the Weyl symbol (Pellat-Finet, 1 Oct 2025).

4. Analytic Properties, Kernel Structure, and Function Space Regularity

CFrFTs are unitary on $L^2(\mathbb{R}^n)$ and their kernels and symbols are analytic in the fractional parameter away from singularities (e.g., at integer multiples of $\pi$). The mapping $\alpha \mapsto F_\alpha f$ is strongly continuous in $L^2$ provided the phase coefficients are continuous functions of $\alpha$. Sufficient regularity (e.g., $f$ in a Sobolev space with order exceeding critical indices) assures pointwise and almost-everywhere convergence with respect to the parameter (Zhou, 17 Sep 2024). The spectral properties are dictated by the kernel structure: for example, eigenfunctions are Hermite–Gaussian functions (or related complete systems), and eigenvalues are explicit phase factors $e^{-ina}$, reflecting the circular spectral structure.

In certain generalized settings, the quadratic-phase parameterization may be replaced by linear or hyperbolic forms—yet only the circular (trigonometric) dependence yields CFrFTs with genuine rotational group structure (Zhou, 17 Sep 2024).

5. Extensions to Discrete, Fractal, and Clifford Analysis Frameworks

Discrete analogues, such as the discrete fractional Fourier transform (DFrFT) and affine DFrFT, maintain the rotational and circular convolution properties by adapting the kernel and incorporating phase corrections to ensure unitarity and group composition. The convolution theorem for affine DFrFTs establishes that the transform of a chirp-weighted circular convolution corresponds to pointwise multiplication plus a phase correction, enabling signal processing schemes (e.g., in communication systems) structurally analogous to those in the Euclidean setting (Nafchi et al., 2020).

CFrFTs are also natural extensions to non-integer (fractal) dimensional spaces and Clifford algebras. In fractional spaces with anomalous measures, an infinite class of transforms—unitary under their respective measures—generalizes the classical Fourier transform and diagonalizes corresponding Laplacians, facilitating spectral analysis in fractal or quantum gravity models (Calcagni et al., 2012). In Clifford analysis, two-parameter generalizations (fractional Clifford–Fourier transforms) introduce both a radial (fractional/rotational) and an angular degree of freedom, providing covariant transforms for multivariate harmonic analysis (Bie et al., 2012).

6. Impact, Applications, and Theoretical Implications

CFrFTs bridge Fourier analysis, quantum mechanics, optical physics, and signal processing by providing a continuous deformation between time and frequency representations. Their circular group property is critical in describing the dynamical evolution of signals under quadratic Hamiltonians (e.g., in quantum mechanics), time–frequency localization (e.g., in audio and seismic processing), and wave propagation through general optical systems (Gutiérrez et al., 10 Jun 2025, Miah et al., 2012).

The pseudo-differential operator interpretation establishes robust algebraic and analytic machinery for composing, inverting, and analyzing these operators, facilitating advances in high-resolution imaging, generalizing the mathematical structure of field transfers, and providing connections to geometric optics via preservation of canonical point arrangements (Pellat-Finet, 1 Oct 2025). The analytical structure, spectral decomposition, and group-theoretical underpinnings make CFrFTs indispensable in the rigorous treatment of fractional and non-integer signal transforms, as well as in the propagation of singularities for dispersive PDEs and in the paper of quantum field theories on non-trivial geometries.

In summary, circular fractional-order Fourier transformations encapsulate a robust class of unitary, rotationally-parameterized operators with deep connections to harmonic analysis, geometry, signal processing, and physics, anchored by their integral kernel structure, Weyl symbolic calculus, group property, and essential role in modeling wave propagation in complex media (Pellat-Finet, 1 Oct 2025, Zhou, 17 Sep 2024, Calcagni et al., 2012, Bie et al., 2012).