Discrete Fractional Transforms

• Discrete fractional transforms are finite-dimensional, unitary operators that interpolate continuously between the identity and standard transforms.
• They enable advanced time–frequency–space analysis and chirp-based representations, with applications in DSP, optics, communications, and quantum computing.
• Key methodologies include eigendecomposition and Vandermonde interpolation, ensuring properties such as unitarity, additivity, and efficient computation.

A discrete fractional transform is any finite-dimensional (typically unitary or invertible) linear operator that extends a classical discrete transform (such as the discrete Fourier, Hadamard, or Radon transform), parameterized continuously (or at least densely) by a fractional index or rotation angle. Such transforms interpolate analytically between identity (order zero), the standard transform (integral order), and parity/reversal, while preserving core algebraic properties like unitarity, additivity of orders, and invertibility. They are essential for time–frequency–space analysis in finite settings, enable chirp-based signal representations, and underpin numerous applications in digital signal processing, optics, communications, and quantum information. The field encompasses one-dimensional, multidimensional, real, complex, quaternionic, and fractal settings, as well as both analytic (diagonalization/interpolation-based) and group-theoretic (representation-theoretic) definitions.

1. Core Theory and Construction Principles

The canonical construction of a discrete fractional transform (DFT-type) begins with an orthogonally/unitarily diagonalizable base matrix $T$ (e.g., the DFT, Hadamard, or Kravchuk matrix) with eigenvalues $\{\lambda_k\}$. The fractional power $T^{\alpha}$ for arbitrary real/complex $\alpha$ is defined via

$$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$

For the DFT matrix, eigenvalues $\lambda_k = e^{-2\pi i k/N}$. The "Vandermonde approach" expresses $T^\alpha$ as a polynomial in $T$ via Lagrange interpolation,

$T^\alpha = \sum_{j=0}^{N-1} c_j T^j,$

with coefficients $c_j$ chosen such that $\{\lambda_k\}$0 for all $\{\lambda_k\}$1; the system $\{\lambda_k\}$2 is solved for $\{\lambda_k\}$3 with $\{\lambda_k\}$4 (Moya-Cessa et al., 2016). This approach, or equivalent diagonalization, underlies the Discrete Fractional Fourier Transform (DFrFT), the Discrete Fractional Hadamard Transform (DFRHT), and many related constructs.

In group-theoretic constructions, as in the arithmetic finite fractional Fourier transform (AFrFT), the transform arises as the unitary representation of a rotation generator in the symplectic group $\{\lambda_k\}$5 acting on the discrete toroidal phase space $\{\lambda_k\}$6. This guarantees preservation of discrete "rotational" structure and Euclidean distance modulo $\{\lambda_k\}$7, and enables quantum circuit implementations with bounded complexity scaling (Floratos et al., 2024).

2. Families and Specializations

Discrete fractional transforms comprise several major subfamilies, each with unique mathematical structure and preferred operational context.

• Fractional Discrete Fourier Transform (DFrFT): Defined either by analytic interpolation (Vandermonde/eigendecomposition), or via a cascaded chirp–DFT–chirp pipeline; unitarity, additivity, and invertibility follow directly (Moya-Cessa et al., 2016, Zayas, 2019, Weimann et al., 2015, Oswald et al., 8 May 2025).
• Affine DFrFT: Alters the kernel to retain an exact circular-convolution property, important for communication systems relying on OFDM, at the cost of replacing the DFT’s chirp-exponential kernel by a more general angle-dependent chirp (Nafchi et al., 2020).
• Discrete Fractional Hadamard Transform (DFRHT): Fractionalized Hadamard matrices for $\{\lambda_k\}$8, constructed by spectral decomposition and recursively organized in radix-2 structures to achieve $\{\lambda_k\}$9 complexity (Cariow et al., 2015).
• Finite/Number-Theoretic Fractional Transforms: The AFrFT defines the transform as a unitary matrix from the action of $T^{\alpha}$0 on discrete phase space, admitting explicit quantum implementations and enabling generalization to all linear canonical transforms (Floratos et al., 2024).
• Multidimensional and Nonseparable Transforms: The multidimensional generalization includes the 2D nonseparable fractional Fourier transform (NSFRFT), the discrete gyrator transform (DGT), and the quaternion quadratic phase transform (DQQPFT), all allowing for flexible rotation and scaling in high-dimensional time–frequency–space and for multicomponent (e.g., color image or vector field) processing (Li et al., 29 Jul 2025, Pei et al., 2017, Dar, 2024).
• Fractal/Local Fractional Transforms: The Discrete Yang–Fourier transform (DYFT) replaces the exponential kernel by the Mittag–Leffler function, rendering the transform suitable for discrete fractal signals (Yang, 2011).

3. Algebraic Properties and Computational Aspects

Key algebraic features are nearly universal:

• Unitarity: For real fractional order, transforms are unitary; i.e., $T^{\alpha}$1, spectrum on the unit circle.
• Invertibility and Additivity: $T^{\alpha}$2 and $T^{\alpha}$3.
• Polynomial/Recursive Structure: Via Cayley–Hamilton, any $T^{\alpha}$4 can be written as a polynomial of degree $T^{\alpha}$5 in $T^{\alpha}$6 (Moya-Cessa et al., 2016, Cariow et al., 2015).

Associated computational strategies and relevant complexities vary:

Transform	Approach	Complexity
DFrFT (Vandermonde)	$T^{\alpha}$7 (generic), $T^{\alpha}$8 with DFT structure, reuse for fixed $T^{\alpha}$9	Acceptable for $\alpha$0 (Moya-Cessa et al., 2016)
DFRHT (Sylvester)	Radix-2, sparse expansion	$\alpha$1 (Cariow et al., 2015)
Chirp–FFT Pipeline	Pre/post chirps + DFT	$\alpha$2 (Zayas, 2019, Nafchi et al., 2020)
Multidimensional	Cascade, FFT, eigen-expansion	$\alpha$3 for full orthonormalization (Li et al., 29 Jul 2025, Pei et al., 2017)
Arithmetic/AFrFT	Group representation + FFT	Classical $\alpha$4, quantum $\alpha$5 gate count (Floratos et al., 2024)
Nonseparable FRFT	Chirp multiplications, FFTs, or direct sum	Fast $\alpha$6, Direct $\alpha$7 (Li et al., 29 Jul 2025)

Numerical stability is generally robust for moderate $\alpha$8; for large $\alpha$9 or certain kernel parameters, care is required to avoid ill-conditioning.

4. Extensions: Multidimensional and Generalized Domains

Discrete fractional transforms extend naturally to higher dimensions and exotic algebraic settings.

• Multidimensional Fractional Transforms: The 2D DFrFT, DGT, CFRFT, NSFRFT, and DQQPFT allow for arbitrary rotation and scale mixing among spatial and frequency-space axes. The 2D NSFRFT introduces a 4-parameter nonseparable kernel supporting full 4D rotational symmetry in the Wigner domain, subsuming SFRFT, GT, and CFRFT as special cases (Li et al., 29 Jul 2025). The DQQPFT further supports quaternion-valued signals, offering left/right chirp structures, convolution/multiplication theorems, and fast FFT-based computation (Dar, 2024).
• Gyrator and Canonical Transforms: DGTs operate as discrete group rotations in "twisted" domains, and can be realized in several algorithmic forms (LCC, DFT, CCC, DHGF), with properties and computational constraints varying accordingly. Eigenfunction-based DHGF variants achieve exact algebraic closure, though with $$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$0 cost (Pei et al., 2017).
• Fractal Kernels: The DYFT serves as a discrete analog of local fractional calculus, enabling spectral analysis of fractal or self-similar discrete-time signals where classical derivatives fail (Yang, 2011).
• Number-Theoretic and Quantum Extensions: The AFrFT links to finite group actions and enables efficient quantum circuit architectures for fractional transforms and general LCTs (Floratos et al., 2024).

5. Applications and Implementation Modalities

Discrete fractional transforms have been realized in a wide spectrum of practical and physical settings.

• Time–Frequency and Space–Frequency Analysis: DFrFT and its variants realize continuous-domain rotation in discrete settings, essential for representing chirped and time-varying spectra, and for optimizing the representation of nonstationary or multicomponent signals (Zayas, 2019, Weimann et al., 2015).
• Digital Communications: The affine DFrFT enables circular-convolution theorems, supporting fractional-domain OFDM and advanced equalization in channels with nontrivial time-frequency scattering (Nafchi et al., 2020).
• Optics and Quantum Systems: DFrFTs, especially in J$$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$1-lattices or coupled-waveguide arrays, physically realize fractional transforms in classical and quantum optical networks—including path-entangled biphoton transformations—and precisely demonstrate algebraic properties such as the shift theorem (Weimann et al., 2015).
• Microwave Engineering: The analog DFrFT via a metamaterial coupled lines network (MCLN) performs real-time DFrFT at microwave and mm-wave frequencies, leveraging the equivalence of tridiagonal coupling evolution with DFrFT eigendecomposition, facilitating low-latency, power-free time–frequency processing (Keshavarz et al., 2024).
• Image and Signal Processing: Multidimensional transforms (NSFRFT, DGT, DQQPFT) are foundational in image encryption, watermarking, denoising, mode conversion, and aliasing-resistant filtering for both scalar and vector (quaternion) data (Li et al., 29 Jul 2025, Pei et al., 2017, Dar, 2024).
• Quantum Information and Cryptography: AFrFT and its generalizations provide both group-theory-grounded mixing operators and efficient, local gate quantum-circuit realizations, powering quantum cryptography and tomography tasks (Floratos et al., 2024).

6. Comparative Table of Representative Transforms

The following summarizes the variety of discrete fractional transforms (DFTs and variants), highlighting kernel form, key properties, and principal domains of application.

Transform	Kernel Structure	Algebraic Properties	Typical Application Domains
DFrFT (Eigen/Chirp/FFT)	$$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$2 or chirp–FFT–chirp	Unitary, Additive, Invertible	DSP, Optics, Communications
Affine DFrFT	Chirp-normalized, circular	True circular-convolution, Pointwise mult.	OFDM, Radar, Chirp-based Filtering
DFRHT (Hadamard)	Sylvester, radix-2	Unitary, $$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$3-periodic, $$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$4	Imaging, Codes, Beamforming
AFrFT (Arithmetic, SO$$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$5[Z$$T^\alpha = V \Lambda^\alpha V^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_0, ..., \lambda_{N-1}),\ \Lambda^\alpha = \operatorname{diag}(\lambda_0^\alpha, ..., \lambda_{N-1}^\alpha).$$6])	Finite-rotational/unitary	Group representation, Quantum circuit	Quantum algorithms, Cryptography
NSFRFT (2D Nonseparable)	4-parameter, nonseparable	4D Wigner rotation, Unitary, Additive	Image denoising, Encryption, Filtering
DGT (Gyrator)	LCC/DFT/CCC/HG eigenbasis	Additive (approx/exact), Unitary	Twisted optics, Image processing
DQQPFT (Quaternion)	Left/right quaternion chirps	Unitary, Modulation, Plancherel, Convolution	2D LTV filtering, RGB image processing
DYFT (Yang-Fractal)	Mittag–Leffler (fractal)	Local-fractal, Invertible, Parseval	Fractal signal analysis, Texture

7. Research Trends and Future Directions

Recent work has focused on increasing generality (nonseparable, quaternionic, finite-geometry, fractal bases), robustness of algebraic properties (especially additivity, tight invertibility, and symmetry), and practical hardware-compatible implementations (FFT-based fast algorithms, analog networks, and quantum circuits). Notably, novel multidimensional forms (NSFRFT, DGT, DQQPFT), arithmetic/quantum-circuit constructs (AFrFT), and microwave analog realizations (MCLN) have been introduced and experimentally validated.

Outstanding open challenges include fast algorithms for more exotic kernels (e.g., fractal/Mittag–Leffler), further group-theoretic canonical forms, large-scale high-precision numerical stability, and integration with machine-learning-driven systems for real-time adaptation of fractional parameters. The field continues to expand into quantum information, compressive sensing, ultrafast optical/microwave hardware, and multimodal high-dimensional signal analysis.