---
title: 2D Non-Separable Quadratic Phase Fourier Transform
url: https://www.emergentmind.com/topics/non-separable-quadratic-phase-fourier-transform-2d-nsqpft
type: topic
---

# 2D Non-Separable Quadratic Phase Fourier Transform

The **Two-Dimensional Non-Separable Quadratic Phase Fourier Transform (2D-NSQPFT)** is a multidimensional quadratic-phase integral transform for signals in $\mathbb{R}^2$ whose kernel contains genuine cross-dimensional quadratic coupling rather than a tensor product of one-dimensional kernels. In the current literature, it is presented as a two-dimensional extension of the quadratic phase Fourier transform (QPFT), itself a generalization of the Fourier transform (FT), fractional Fourier transform (FrFT), and linear canonical transform (LCT). Its defining feature is the admission of off-diagonal and mixed terms such as $x_1x_2$, $x_1\omega_2$, and $\omega_1\omega_2$, which make it suitable for signals with coupled, sheared, rotated, or chirp-like structure in two dimensions [2402.06645][2505.02526][2509.19310].

## 1. Formal definition and kernel structure

The literature gives the 2D-NSQPFT in a matrix-quadratic form. In the survey formulation, a general non-separable transform is written as
$$
(Q_{\Lambda} f)(\boldsymbol{\omega})=\iint_{\mathbb{R}^2}K_{\Lambda}(\boldsymbol{x},\boldsymbol{\omega})\,f(\boldsymbol{x})\,d\boldsymbol{x},
$$
with kernel
$$
K_{\Lambda}(\boldsymbol{x},\boldsymbol{\omega})=
\frac{1}{2\pi}
\exp\left(
-i\left[
\boldsymbol{x}^{\top}\mathbf{A}\boldsymbol{x}
+2\boldsymbol{x}^{\top}\mathbf{B}\boldsymbol{\omega}
+\boldsymbol{\omega}^{\top}\mathbf{C}\boldsymbol{\omega}
+\mathbf{d}^{\top}\boldsymbol{x}
+\mathbf{e}^{\top}\boldsymbol{\omega}
\right]
\right),
$$
where $\boldsymbol{x},\boldsymbol{\omega}\in\mathbb{R}^2$, $\mathbf{A},\mathbf{B},\mathbf{C}$ are $2\times2$ real symmetric matrices, and $\mathbf{d},\mathbf{e}\in\mathbb{R}^2$ [2402.06645].

A more explicit parameterization is given in the later 2D-NSQPWD work through a tuple $\Omega$ of five real $2\times2$ matrices,
$$
\Omega=(A,B,C,D,E), \qquad \det(B)\neq 0,
$$
and
$$
\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega})
=
\int_{\mathbb{R}^2}
f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},
$$
with
$$
\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})
=
\frac{i\sqrt{\det(B)}}{2\pi}
\exp\Big\{
i\big[
\boldsymbol{\omega}^{T}A\boldsymbol{\omega}
+\boldsymbol{\omega}^{T}B\mathbf{x}
+\mathbf{x}^{T}C\mathbf{x}
+\vec{1}^{T}D\boldsymbol{\omega}
+\vec{1}^{T}E\mathbf{x}
\big]
\Big\},
$$
where $\vec{1}=(1,1)^T$, $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$, and $\mathbf{x}=(x_1,x_2)^T$ [2509.19310].

In both descriptions, non-separability is carried by off-diagonal coefficients. The mixed terms $a_{12},a_{21},b_{12},c_{12},c_{21}$ in the expanded kernel are permitted to be nonzero, so the phase is not restricted to axis-aligned quadratic modulation. This is the essential structural distinction between 2D-NSQPFT and separable 2D QPFT [2509.19310].

## 2. Relation to separable QPFT and classical transforms

The separable two-dimensional QPFT is a tensor product of one-dimensional QPFTs, typically written with one parameter tuple for each coordinate. By contrast, the 2D-NSQPFT uses a true two-dimensional quadratic form and therefore supports cross-dimensional interactions that do not factor into independent transforms along $x_1$ and $x_2$ [2402.06645].

| Aspect | Separable QPFT | 2D-NSQPFT |
|---|---|---|
| Kernel | Product of 1D QPFT kernels | General quadratic form in $\mathbf{x},\mathbf{u}$ |
| Cross-terms | None | Yes, e.g. $x_1\omega_2$ |
| Typical use | Signals/images separable in axes | Oriented features, chirp-anisotropic signals, time-angle analysis |

Special cases are obtained by parameter restriction. When the relevant matrices are diagonal, the transform reduces to the separable QPFT. When all quadratic and linear terms vanish and $B=I$, the 2D-NSQPFT recovers the classical 2D Fourier transform. The survey further notes that specific matrix choices recover coupled FrFT and coupled LCT cases, and the later NSQPWD paper states that the framework encompasses the 2D FT, 2D fractional FT, 2D LCT, Gyrator transform, and other affine/invariant transforms as special cases [2402.06645][2509.19310].

A common simplification is to treat all 2D quadratic-phase transforms as tensor-product extensions of the one-dimensional theory. The non-separable literature rejects that identification explicitly: the decisive issue is whether the kernel admits off-diagonal coupling. In that sense, 2D-NSQPFT is not merely a higher-dimensional QPFT, but a coupled quadratic-phase model.

## 3. Core analytic properties

For the multidimensional QPFT, the theoretical framework includes inversion theorems and Parseval-type identities. In the $N$-dimensional formulation, the transform is written as
$$
(Q_A f)(\mathbf{w})=\int_{\mathbb{R}^N}K_A(\mathbf{w},\mathbf{x})\,f(\mathbf{x})\,d\mathbf{x},
$$
and, under mild regularity,
$$
f(\mathbf{x})=\int_{\mathbb{R}^N}K_{-A}(\mathbf{w},\mathbf{x})\,(Q_A f)(\mathbf{w})\,d\mathbf{w}.
$$
The same work states Parseval’s identity in the form
$$
\langle f,g\rangle=\langle Q_A f,Q_A g\rangle,
\qquad
\|f\|_{L^2}^2=\|Q_A f\|_{L^2}^2,
$$
so the multidimensional QPFT acts as a unitary, energy-preserving operator [2505.02526].

The survey likewise records inversion, Parseval/Plancherel, and norm preservation for the one-dimensional QPFT and states that inversion, Parseval, and sharp Hausdorff–Young carry to the separable two-dimensional setting in tensor-product form. For the non-separable case, it emphasizes that the kernel requires block-matrix analysis rather than coordinatewise factorization [2402.06645].

The same survey reviews uncertainty principles in QPFT settings, including Heisenberg-type, logarithmic, Donoho–Stark concentration, and Renyi and Shannon-type inequalities. For the non-separable setting summarized there, the uncertainty behavior is described as dimension-dependent, with constants and bounds tied to the quadratic kernel parameters rather than to the Euclidean axes alone [2402.06645].

The multidimensional theory also includes a Boas-type theorem for spectral characterization in the QPFT domain. In the formulation summarized for the 2025 multidimensional paper, this yields necessary and sufficient conditions for a function’s QPFT to vanish on a region of transform space, with implications for reconstruction, band-limiting, and sampling [2505.02526].

## 4. Convolution, correlation, and operator structure

A substantial part of QPFT theory concerns convolution adapted to the quadratic-phase kernel. The survey reports several QPFT convolution definitions in one dimension and states that their two-dimensional extension is direct in the separable case but more involved in the non-separable case. A generalized 2D QPFT convolution is given by
$$
(f\otimes_{\Lambda} g)(\mathbf{x})
=
\iint_{\mathbb{R}^2}
f(\mathbf{t})\,g(\mathbf{x}-\mathbf{t})\,
e^{2i\,\mathbf{t}^{\top}\mathbf{A}(\mathbf{x}-\mathbf{t})}
\,d\mathbf{t},
$$
with transform-domain multiplication up to a phase factor. The same source notes that chirp-modified convolutions may lose commutativity and associativity unless special symmetries hold, whereas chirp-free convolutions are sometimes constructed to recover commutative and associative behavior [2402.06645].

The multidimensional QPFT paper systematizes this structure by introducing three types of generalized convolutions and a corresponding correlation framework. Its summary states that these generalizations extend conventional convolution for multiple variables and are then used for multiplicative filter design and for solving integral equations in the QPFT setting [2505.02526].

The practical significance is direct. In the multiplicative filtering viewpoint, the filter transfer function is the transform of the kernel or filter in the QPFT domain, and signal components localized by the quadratic-phase geometry can be retained or suppressed through transform-domain multiplication. In the integral-equation setting, the QPFT converts a convolution equation into an algebraic equation in transform space, followed by inversion [2505.02526].

## 5. Time-frequency distributions built on the 2D-NSQPFT

A major development around 2D-NSQPFT is the construction of Wigner-type distributions by replacing the classical Fourier kernel with the non-separable quadratic-phase kernel. The 2025 paper "A Novel Two-Dimensional Wigner Distribution Framework via the Quadratic Phase Fourier Transform with a Non-Separable Kernel" defines the Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution (2D-NSQPWD) as
$$
\mathcal{W}_f^{\Omega}(\mathbf{x},\boldsymbol{\omega})
=
\frac{|\det B|\,\mathcal{C}_{\mathbf{k}-\mathbf{m}}(\boldsymbol{\omega})}{(2\pi)^2}
\int_{\mathbb{R}^2}
f_{\mathbf{m}}\!\left(\mathbf{x}+\frac{1}{2}\boldsymbol{\xi}\right)
\overline{f_{\mathbf{k}}\!\left(\mathbf{x}-\frac{1}{2}\boldsymbol{\xi}\right)}
\exp\!\left(i\,\boldsymbol{\omega}^{T}B\boldsymbol{\xi}\right)
\,d\boldsymbol{\xi},
$$
where the modulation factors are determined by the NSQPFT parameters [2509.19310].

Within that framework, the following properties are established: time and frequency shift invariance, marginal behavior, conjugate symmetry, convolution relations, and Moyal’s identity. In particular, the marginal structure connects the distribution back to the forward and inverse 2D-NSQPFT, and Moyal’s identity gives
$$
\int_{\mathbb{R}^4}
\mathcal{W}_f^{\Omega}(\mathbf{x},\boldsymbol{\omega})\,
\overline{\mathcal{W}_g^{\Omega}(\mathbf{x},\boldsymbol{\omega})}
\,d\mathbf{x}\,d\boldsymbol{\omega}
=
\frac{|\det B|}{(2\pi)^2}|\langle f,g\rangle|^2.
$$
The same paper also derives a relation between the 2D-NSQPWD and the two-dimensional short-time Fourier transform:
$$
\mathcal{W}_{f}^{\Omega}\left(\frac{\mathbf{x}}{2},\tilde{B}\boldsymbol{\omega}\right)
=
\nabla_2(\mathbf{x},\boldsymbol{\omega})\,S_{f,g}(\mathbf{x},\boldsymbol{\omega}),
$$
with $\nabla_2$ determined by the NSQPFT parameters [2509.19310].

Simulation results reported there show the 2D-NSQPWD on single-, bi-, and tri-component 2D linear frequency modulated signals, with greater energy concentration, improved separation of auto-terms, and stronger suppression of cross-terms than the classical Wigner distribution. The same paper attributes this to the adaptability of the non-separable quadratic-phase kernel, which can be tuned through fifteen independent matrix parameters [2509.19310].

The one-dimensional precursor is the advanced quadratic-phase Wigner distribution and ambiguity function in the QPFT domain. That work establishes Moyal, marginal, symmetry, shift, and reconstruction properties for AQWD/AQAF and reports improved effectiveness in linear frequency-modulated signal detection relative to traditional WD/AF and earlier QPFT-based forms. This suggests a coherent research trajectory in which classical time-frequency bilinear forms are systematically re-expressed in quadratic-phase domains and then specialized to richer non-separable settings [2503.16258].

## 6. Applications, extensions, and related non-separable frameworks

The survey identifies image processing, optics, radar/sonar, and orientation-aware analysis as principal application areas for 2D-NSQPFT. Its stated use cases include anisotropic feature detection, beam propagation and quadratic phase systems, 2D signal analysis with time-frequency coupling, and affine- or orientation-sensitive processing. The multidimensional QPFT paper adds multiplicative filter design and the solution of integral equations as explicit applications of the transform-domain algebra [2402.06645][2505.02526].

Quaternionic and multichannel extensions form an additional branch of the theory. For quaternion-valued functions, the survey gives a 2D quaternionic QPFT of the form
$$
(\mathcal{Q}^{\wedge_1,\wedge_2}_{\mathbb{H}}f)(u_1,u_2)
=
\iint_{\mathbb{R}^2}
\mathcal{K}^i_{\wedge_1}(t_1,u_1)\,
f(t_1,t_2)\,
\mathcal{K}^j_{\wedge_2}(t_2,u_2)\,
dt_1dt_2,
$$
and states that, in non-separable cases, the kernel is built from general quadratic forms with cross-terms such as $t_1t_2$ and $t_1u_2$. The same summary connects these extensions to orientation analysis, directional decompositions, and chirp/aniso-frequency analysis for signals and images [2402.06645].

A closely related but distinct development is the Two-Dimensional Nonseparable Fractional Fourier Transform (2D NSFRFT). It is not a 2D-NSQPFT, but it occupies the same non-separable quadratic-phase landscape. That work presents a transform with four degrees of freedom, includes the 2D SFRFT, Gyrator transform, and coupled FRFT as special cases, maintains a general 4D rotational relationship with the 2D Wigner distribution, and provides two fast discrete algorithms with computational complexity $O(N^2\log N)$ alongside a direct $O(N^4)$ method [2507.21511]. A plausible implication is that algorithmic and geometric techniques developed for non-separable fractional transforms may inform future discrete formulations of 2D-NSQPFT itself.

Across these strands, the role of 2D-NSQPFT is stable: it provides a mathematically flexible quadratic-phase representation for two-dimensional data in which cross-dimensional coupling is intrinsic rather than incidental. That is the point at which it departs from tensor-product generalizations and becomes a dedicated framework for non-separable phase geometry.

Source: https://www.emergentmind.com/topics/non-separable-quadratic-phase-fourier-transform-2d-nsqpft