Quadratic Phase Fourier Transforms

• Quadratic Phase Fourier Transforms (QPFT) are a family of linear integral transforms defined by a five-parameter quadratic-phase kernel that generalizes the Fourier, FrFT, and LCT.
• QPFTs enable simultaneous modulation of time and frequency through quadratic, linear, and cross terms, making them ideal for analyzing non-stationary, chirped signals.
• The framework extends to multidimensional, discrete, quaternionic, and windowed variants, providing a unified basis for rigorous uncertainty principles and quantum applications.

The quadratic phase Fourier transform (QPFT) is a parametric family of linear integral transforms that subsumes the classical Fourier transform, fractional Fourier transform (FrFT), linear canonical transform (LCT), and related chirp-based operations as special cases. QPFTs are distinguished by their five-parameter quadratic-phase kernel, enabling the simultaneous modulation of time and frequency variables by quadratic, linear, and cross terms. This increased flexibility allows QPFTs to represent and diagonalize a broad class of physical, signal-processing, and harmonic analysis problems involving non-stationary, chirped, and variable-phase signals. The QPFT framework also generalizes naturally to the multidimensional, discrete, windowed, quaternion-valued, Bessel-related, and Dunkl analytic domains, providing a unified setting for rigorous uncertainty principles, advanced time-frequency representations, numerical algorithms, and quantum applications.

1. General Formulation and Fundamental Properties

The QPFT is formulated as an oscillatory integral operator: $$(Tf)(x) = \int_{\mathbb{R}^n} e^{i\varphi(x, y)}\, a(x, y)\, f(y)\,dy,$$ where $\varphi(x, y)$ is a real-valued quadratic form on $(x, y)$, and $a(x, y)$ is a Shubin amplitude, allowing precise phase-space decay and smoothness control (Cappiello et al., 2018).

For the canonical 1D case, the QPFT of $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ with parameter vector $\Lambda = (A, B, C, D, E)$, $B \neq 0$, is given by: $$(Q_\Lambda f)(u) = \int_{-\infty}^\infty K_\Lambda(x, u)\, f(x) \, dx, \quad K_\Lambda(x, u) = \frac{1}{\sqrt{2\pi}} \exp\big[-i(A x^2 + B x u + C u^2 + D x + E u)\big].$$ This kernel specializes to the classical Fourier transform by setting all parameters except $B=1$ to zero, and to the FrFT or LCT by particular nonzero patterns (Gupta et al., 2024).

Key Operator Properties:

• Unitarity and Parseval: The QPFT is unitary on $L^2$, with $\varphi(x, y)$0 up to normalization (Saoudi, 26 Dec 2025, Saoudi, 21 Jan 2026).
• Inversion: The inverse QPFT is composed with sign-inverted parameters: $\varphi(x, y)$1 (Gupta et al., 2024, Saoudi, 26 Dec 2025).
• Linearity: QPFT is linear.
• Covariance/Shift-Modulation: The transform admits closed-form algebraic rules for time/frequency shift and modulation (phase factors arising from quadratic and linear coefficients) (Gupta et al., 2024).
• Semigroup Structure: The set of all QPFTs is closed under composition, corresponding to the symplectic group (with explicit parameter composition laws) (Cappiello et al., 2018).
• Convolution: There are multiple QPFT-convolution structures, with spectral domain multiplication up to explicit phase corrections (Varghese et al., 5 May 2025, Gupta et al., 2024, Bhat et al., 2022).

2. Geometric and Operator-Theoretic Perspective

QPFTs are realized as Fourier integral operators (FIOs) associated with quadratic phases. The associated symplectic geometry encodes their canonical transformations:

• Quadratic phase functions $\varphi(x, y)$2 parametrize twisted graph Lagrangians in $\varphi(x, y)$3, associated with symplectic matrices $\varphi(x, y)$4.
• Any QPFT can be factorized as the composition of a Weyl pseudo-differential operator (with Shubin-class symbol) and a metaplectic operator covering the symplectic action (Cappiello et al., 2018).
• Schwartz kernels of QPFTs are precisely Lagrangian distributions of order $\varphi(x, y)$5 associated with these twisted graphs.
• This structure yields closedness under composition, inversion, and adjoints, and action on Shubin–Sobolev spaces $\varphi(x, y)$6 (Cappiello et al., 2018).

Canonical Examples:

• FrFT ($\varphi(x, y)$7 a rotation): $\varphi(x, y)$8
• Harmonic oscillator: kernel with phase $\varphi(x, y)$9 (Cappiello et al., 2018).

3. Multidimensional, Non-Separable, and Quantum Extensions

Multidimensional QPFT generalizes by parameterizing each coordinate: $(x, y)$0 with kernel parameters $(x, y)$1 (Varghese et al., 5 May 2025).

Non-Separable Cases: The 2D-NSQPFT uses a non-separable kernel parameterized by five $(x, y)$2 real matrices, enabling full coupling between variables. This supports advanced Wigner distributions (2D-NSQPWD), which maintain all symmetries and marginals while providing superior cross-term suppression and adaptability for multicomponent, chirped signals (Chauhan et al., 8 Sep 2025).

Quantum and Discrete Models:

• Quadratic Discrete Fourier Transform (QDFT): Two-parameter (often $(x, y)$3) unitary matrix generalizing the DFT, with explicit quadratic-phase structure. QDFT families are the algebraic backbone for constructing mutually unbiased bases in finite prime dimensions and quantum information protocols (Kibler, 2010, Atakishiyev et al., 2010).
• Quadratic Quantum Fourier Transform (QQFT): Implementation of QPFT in cold atom lattices as a number-conserving unitary. On single-particle states, QQFT coincides with QFT; in many-particle subspaces it coordinates with bosonic/fermionic statistics and supports programmable engineering of non-local Hamiltonians such as discrete Poincaré crystals and flat Chern bands (Wang et al., 2022).
• Finite Number-Theoretic Constructions: Arithmetic fractional FT (AFrFT) via Weil/metaplectic representations, with explicit efficient quantum circuits for qudits, allowing $(x, y)$4 gate decomposition and $(x, y)$5 depth (Floratos et al., 2024).

4. Windowed, Localized, and Bessel/Dunkl Extensions

Windowed and Localized QPFT:

• Windowed QPFT (WQPFT): Generalizes the STFT to QPFTs by analyzing with a shifted window. Inherits full inversion and reproducing-kernel identities; convolution and norm-preserving results carry through (Varghese et al., 5 Jul 2025, Bhat et al., 2022).
• Quadratic-Phase Wave Packet Transform (QP-WPT): Localizes chirped structures by integrating against phase-modulated and spatially concentrated packets; enables joint time-frequency-quadratic-phase analysis with explicit uncertainty, reconstruction, and norm bounds (Bhat et al., 2022).

Bessel and Dunkl Generalizations:

• QP Fourier–Bessel Transform: Convolution and translation structure adapted to the Hankel/Bessel context; unitarity and Donoho-Stark uncertainty extend with radial modifications. Kernel incorporates $(x, y)$6 (Saoudi, 21 Jan 2026).
• QP Dunkl Transform: Parameterizes over Dunkl’s kernel $(x, y)$7, embedding QPFT as the $(x, y)$8 specialization. All core results (inversion, convolution, Heisenberg bounds) transfer, demonstrating wide generality (Saoudi, 26 Dec 2025).

5. Quaternion, Quaternionic, and Multicomponent QPFTs

Quaternionic QPFT (QQPFT):

• Defined for quaternion-valued functions, both in continuous and discrete 2D (and higher). Uses two-sided kernels (e.g., left with $(x, y)$9, right with $a(x, y)$0), ensuring non-commutativity and enabling analytic manipulation of color/image/multichannel data (Gupta et al., 2022, Dar et al., 2022, Dar, 2024).
• Plancherel, sharp Hausdorff–Young, and explicit entropic uncertainty principles are established with proper normalization ($a(x, y)$1 Jacobians) (Gupta et al., 2022).
• Quaternionic analogues of ambiguity functions and Wigner–Ville distributions are explicitly constructed, retaining the full uncertainty and inner-product structure (Gupta et al., 2022, Dar, 2024).
• Fast computation algorithms exist, reducing the 2D quaternionic QPFT to sums of complex FFTs and post-processing (Dar, 2024).

6. Time-Frequency Analysis, Wigner Distributions, Uncertainty Principles

Advanced Wigner and Ambiguity Distributions:

• Substitution of the Fourier kernel in Wigner/Ambiguity integrals by QPFT kernels yields time–frequency representations in which chirped signals are localized as straight ridges, with full retention of marginal and Moyal identities (Dar et al., 20 Mar 2025, Chauhan et al., 8 Sep 2025).
• Non-separable QPFT Wigner distributions allow décorrelating cross-terms of multi-component linear frequency modulated signals, improving detection and estimation performance (Chauhan et al., 8 Sep 2025, Dar et al., 20 Mar 2025).

Uncertainty Principles:

• Heisenberg-type: $a(x, y)$2 (Gupta et al., 2024, Dar et al., 2022, Saoudi, 26 Dec 2025).
• Logarithmic, Rényi, Shannon: Entropy and log-integral inequalities hold with explicit $a(x, y)$3-dependence (Gupta et al., 2024, Gupta et al., 2022, Dar et al., 2022).
• Donoho–Stark: Joint support concentration yields $a(x, y)$4 for any $a(x, y)$5 $a(x, y)$6–concentrated in time/frequency domains (Gupta et al., 2024, Dar et al., 2022, Saoudi, 26 Dec 2025, Saoudi, 21 Jan 2026).
• Quantum, Bessel, and Dunkl analogues are proven, always reflecting the structural normalization constants of the kernel (Gupta et al., 2022, Saoudi, 26 Dec 2025, Saoudi, 21 Jan 2026).

7. Discrete, Finite, and Algebraic Structures

• Discrete QPFTs parameterized by quadratic phases in the summation variable serve dual roles: providing orthogonal/unitary bases, especially important in quantum information (maximal sets of mutually unbiased bases in prime dimensions) (Atakishiyev et al., 2010, Kibler, 2010).
• Group–theoretic approaches (SU(2), number-theoretic $a(x, y)$7 representations) yield explicit constructions of QPFT kernels as generalized Hadamard matrices, linked to quadratic Gauss sums and Weyl/Pauli operators (Atakishiyev et al., 2010, Floratos et al., 2024).
• Quantum circuit methods for implementing general QPFTs with quadratic phase diagonal gates, modular multipliers, and efficient decompositions exist for both qubits and qudits. Resource requirements match the best known $a(x, y)$8 gate complexity and $a(x, y)$9 depth under nearest-neighbor constraints (Floratos et al., 2024).

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References:

• (Cappiello et al., 2018): General FIO and symplectic structure of QPFT.
• (Gupta et al., 2024): Mathematical survey, continuous QPFT, and uncertainty/analysis offshoots.
• (Varghese et al., 5 May 2025): Multidimensional QPFT theory/applications.
• (Dar et al., 20 Mar 2025): Advanced Wigner/Ambiguity QPFT-based time–frequency analysis.
• (Tuan et al., 2024): Hausdorff–Young and Lᵖ–boundedness of QPFT.
• (Chauhan et al., 8 Sep 2025): 2D non-separable QPFT and associated Wigner formalism.
• (Varghese et al., 5 Jul 2025): Windowed QPFT, structure, convolution, and kernel theory.
• (Bhat et al., 2022): QPFT wave packet transforms and uncertainty principles.
• (Gupta et al., 2022, Dar et al., 2022, Dar, 2024): Quaternionic and 2D QQPFT, uncertainty, and fast algorithms.
• (Saoudi, 21 Jan 2026): Quadratic-phase Fourier–Bessel transform.
• (Saoudi, 26 Dec 2025): Quadratic-phase Dunkl transform, Bessel/Dunkl generalization.
• (Atakishiyev et al., 2010, Kibler, 2010): Discrete QPFT, mutually unbiased bases.
• (Wang et al., 2022): Quadratic quantum Fourier transform in optical lattice platforms.
• (Floratos et al., 2024): Weil/metaplectic approaches, finite QPFTs, and efficient quantum circuits.

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Summary Table: QPFT Domains & Key Features

Setting	Kernel Structure	Unitarity/Invertibility
Classical Continuous	$f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$0	Yes ($f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$1 and $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$2)
Multidimensional	Product or non-separable quadratic forms	Yes
Discrete / Finite	$f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$3	Yes (unitary matrix)
Quaternionic (continuous)	Two-sided yield: $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$4	Parseval/Plancherel
Windowed / Localized	$f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$5	Full inversion/reproducing kernel
Bessel/Dunkl	$f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$6, or Dunkl kernel	Yes (weighted $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$7)
Quantum Lattice (QQFT)	Exponentials of quadratic forms in occupation operators	Yes (number-conserving unitary)

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The QPFT family provides an analytically rigorous and extremely versatile integral transform framework with broad utility in analytic, physical, operator-theoretic, numerical, and quantum-information settings. It acts as a technical and conceptual unifier for the analysis and implementation of quadratic-phase, chirped, and non-stationary signal structures.