Fractional Fourier Transform (FrFT)

• Fractional Fourier transform (FrFT) is a unitary linear integral transform that generalizes the Fourier transform by introducing a continuous rotation parameter in the time–frequency plane.
• It employs chirp-based decomposition and fast algorithms to interpolate between time and frequency representations, enabling efficient signal processing and optical implementations.
• FrFT's well-defined group structure and quantum-mechanical basis provide deep insights for phase-space analysis, phase retrieval, and advanced applications in communications and imaging.

The fractional Fourier transform (FrFT) is a unitary linear integral transform that generalizes the conventional Fourier transform by introducing a continuous order parameter, corresponding to an arbitrary-angle rotation in the time–frequency (phase) space. The FrFT has become a foundational tool across physics, signal processing, optics, quantum information, and applied mathematics, enabling analysis and synthesis techniques that interpolate smoothly between time and frequency representations.

1. Mathematical Definition and Core Properties

Let $f(t) \in L^2(\mathbb{R})$. The one-dimensional FrFT of real order (or rotation angle) $\alpha$ is given by: $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$ where the kernel is: $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$ for $\alpha \notin \pi\mathbb{Z}$. The FrFT of order $\alpha$ is the unitary operator $F_\alpha = \exp(i\alpha \hat{H})$, where $\hat{H} = (\hat{p}^2/2 + \hat{q}^2/2 - 1/2)$ is the dimensionless harmonic-oscillator Hamiltonian and $(\hat{q}, \hat{p})$ are suitably scaled conjugate operators (Niewelt et al., 2023, Gutiérrez et al., 10 Jun 2025).

Special cases:

• $\alpha = 0$: $\alpha$0 (identity transformation)
• $\alpha$1: $\alpha$2 coincides with the conventional Fourier transform
• For integer multiples: $\alpha$3, where $\alpha$4 is the standard Fourier transform
• The inverse: $\alpha$5
• Linearity and unitarity: $\alpha$6 is linear and preserves $\alpha$7 norm

Group structure: The FrFTs obey $\alpha$8 for all real $\alpha$9, endowing them with a continuous group structure (Gutiérrez et al., 10 Jun 2025, Li et al., 29 Jul 2025).

Geometric interpretation: FrFT implements a rotation by angle $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$0 in the time–frequency plane, which is most rigorously described using the Wigner distribution. For Wigner distribution $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$1,

$$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$2

i.e., a rigid rotation in the joint phase space (Gutiérrez et al., 10 Jun 2025, Niewelt et al., 2023, Yang et al., 2023).

2. Operational and Algorithmic Realizations

Chirp–Fourier Decomposition: The FrFT can be constructed by three steps: (1) input chirp pre-multiplication, (2) ordinary Fourier transform, (3) output chirp post-multiplication. For a function $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$3: $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$4 where $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$5 is a normalization constant, $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$6 is the classical Fourier transform (Chen et al., 2020, Gutiérrez et al., 10 Jun 2025).

Discrete/fast algorithms: FrFT of a discrete signal $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$7 can be computed via chirp multiplication–FFT–chirp multiplication sequence, achieving $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$8 complexity. Enhanced schemes incorporate Newton–Cotes quadrature weighting for improved accuracy (Nzokem, 2023), as well as closed-form matrix kernels for exact discrete versions (e.g., affine DFrFT) with properties paralleling those of the DFT, such as circular convolution (Nafchi et al., 2020).

3. Phase Space and Quantum-Mechanical Perspective

Quantum origins: The FrFT emerges naturally as a change-of-basis operation in quantum mechanics: applying a fractional power of the harmonic oscillator’s time-evolution operator rotates between the position and momentum representations. Specifically, the generator is the number operator $$F_\alpha[f](u) = \int_{-\infty}^{\infty} K_\alpha(t,u) f(t) \, dt$$9, so $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$0, and the eigenfunctions are the Hermite-Gaussian modes, with $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$1 (Chen et al., 2013).

Generalized fractional transforms: This structure can be generalized: given a family of mutually commuting Hermitian operators, one defines a multi-parameter group of generalized fractional transforms, each with corresponding kernel and eigenfunction decomposition (Chen et al., 2013).

4. Extensions: Multidimensional and Variant FrFTs

Higher-dimensional generalizations:

• Separable and nonseparable 2D FrFTs: For $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$2, the nonseparable 2D FrFT is parameterized by four degrees of freedom—parameters $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$3 satisfying $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$4, with a kernel that unifies the separable 2D FRFT, gyrator transform, and coupled FRFT as special cases (Li et al., 29 Jul 2025). This NSFRFT preserves a full $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$5 phase-space rotation, enabling advanced joint time-frequency processing in two dimensions.
• Holomorphic FrFT (HFrFT): In this variant, the output is an entire function of a complexified time-frequency variable, intertwining the standard FrFT with a complex-domain heat kernel, and the transform is unitary between $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$6 and a weighted holomorphic function space (Kirwin et al., 2019).

Simplified and reduced-order variants: The SmFrFT and ROFrFT offer alternative kernels and convolution structures designed to simplify analytical manipulation or to restore "elegance" to algebraic properties such as the convolution theorem, enabling direct pointwise multiplication in the FrFT domain or tractable analytical expressions (Kumar, 2018, Kumar, 2018).

5. Applications Across Disciplines

Signal processing:

• Time–frequency analysis: FrFT enables continuous interpolation between time and frequency domains, yielding optimal representations for chirp-like and nonstationary signals. It has found use in denoising, time–frequency filtering, cepstral analysis (fractional cepstrum), and sound synthesis ("alpha-domain") (Gutiérrez et al., 10 Jun 2025, Miah et al., 2012).
• Digital communications: Affine discrete FrFTs generalize OFDM modulation, allowing circular convolution and efficient fractional/bandwidth equalization for chirped or time-varying channels (Nafchi et al., 2020).

Optics and quantum information:

• Optical systems: FrFT generalizes lens and free-space propagation concepts to arbitrary phase-space rotations, with direct physical implementations via interleaved dispersive (spectral phase) and time-lens (temporal quadratic phase) elements. Both atomic quantum memories and electro-optic modulators realize time-frequency FrFTs for quantum and classical photonic signals (Niewelt et al., 2023, Lipka et al., 2023).
• Quantum information: The phase-space rotation interpretation enables temporal mode sorting, high-dimensional quantum communication, and tomographic phase-space measurements. In atomic quantum memory, programmable FrFT operations are achieved by synthesizing sequences of quadratic spectral and temporal phase modulations (Niewelt et al., 2023).

Numerical inversion and probability: FrFT is a fast and accurate method for inverting characteristic functions to PDFs and their derivatives, used in maximum likelihood estimation of financial models (variance-gamma, generalized tempered stable, etc.), outperforming classical FFT-based schemes in accuracy and flexibility, particularly when tail behavior or flexible output grids are essential (Nzokem, 2022, Nzokem et al., 2022).

Image and diffraction processing: In coherent diffraction imaging, FrFT-based measurement models unify near-field and far-field propagation, enabling single-shot phase retrieval with untrained neural networks and breaking ambiguities present in pure Fourier measurements (Yang et al., 2023).

Wavelet theory: FrFT supports the construction of fractional MRAs and (bi)orthogonal wavelet families, where dilation and translation parameters are accompanied by order-dependent phase factors. Riesz-basis and biorthogonality properties are characterized by the periodized FrFT energy and associated filters (Ahmad et al., 2020).

6. Analytical and Computational Features

Variant / Property	Domain/Kernel	Notable Feature / Use
Standard FrFT	$$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$7, rotation angle $$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$8	General interpolation time$$K_\alpha(t,u) = \sqrt{\frac{1-i\cot\alpha}{2\pi}} \exp\left\{ i \frac{(t^2+u^2)\cot\alpha - 2 t u \csc\alpha}{2} \right\}$$9freq.
Affine DFrFT	Matrix kernel, closed-form	Circular convolution, OFDM, eigenstructure
NSFRFT (2D nonseparable)	4-param. $\alpha \notin \pi\mathbb{Z}$0, $\alpha \notin \pi\mathbb{Z}$1 rot.	Unified multi-DOF for 2D signal/image analysis
HFrFT	Holomorphic extension, heat kernel smoothing	Analysis in holomorphic function spaces
SmFrFT, ROFrFT	Reduced/simpler kernels	Simplified analytical/algorithmic properties

• Numerical implementation: The FrFT admits $\alpha \notin \pi\mathbb{Z}$2 numerical schemes via decomposition into chirps and FFTs, with high-order quadrature (e.g., Newton–Cotes) further improving accuracy for inverse integral calculations (Nzokem, 2023, Nzokem, 2022).
• Analytic tractability: Variant definitions (ROFrFT, SmFrFT) provide explicitly closed-form transforms for prototypical signals (Gaussians, chirps, delta functions) and convolution theorems that reduce to pointwise multiplication in the FrFT domain (Kumar, 2018, Kumar, 2018).

7. Limitations, Open Problems, and Outlook

Error sources in experiments: Implementations are affected by device-limited coherence, finite memory efficiency (atomic FrFT), magnetic field stability, inhomogeneous broadening, and spurious phase contributions (Niewelt et al., 2023, Lipka et al., 2023).

Algorithmic trade-offs: While fast algorithms exist, optimizing performance for windowed, adaptive, or multidimensional scenarios and quantifying discretization/aliasing errors in practical applications remains an active area (Nzokem, 2023, Li et al., 29 Jul 2025).

Open implications in phase retrieval: The use of FrFT domains for support-free and single-shot phase retrieval is promising, but failure modes persist at certain orders (e.g., $\alpha \notin \pi\mathbb{Z}$3 for classical phase retrieval), and further research is needed to generalize uniqueness guarantees and robustness (Yang et al., 2023).

Broader significance: The FrFT unifies continuous phase-space rotation, optimal time–frequency representation, and physically implementable optical/quantum transformations, offering a mathematically rich and practically versatile generalization of Fourier analysis that continues to drive innovation in signal analysis, computing, and photonic information processing.