2D Non-Separable Quadratic Phase Wigner Distribution

• The 2D non-separable quadratic phase Wigner distribution is a bilinear representation that replaces the classical Fourier kernel with a coupled quadratic-phase kernel.
• Its construction via the 2D-NSQPFT framework incorporates matrix-valued parameters to handle cross-dimensional coupling in complex 2D signals.
• The framework demonstrates improved signal localization and cross-term suppression, particularly for multi-component 2D linear frequency modulated signals.

Searching arXiv for the cited papers and closely related work to ground the article. {"^^^^^^^^3query3 (Chauhan et al., 8 Sep 2025) \3^^^^^^^^"Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution\"^^^^^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^^^^^"Two-Dimensional Nonseparable Fractional Fourier Transform: Theory and Application\"", "max_results": ^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^} The Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD) is a bilinear time–frequency distribution defined by replacing the classical Fourier kernel in the two-dimensional Wigner distribution with the kernel of the Two-Dimensional Non-Separable Quadratic Phase Fourier Transform (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPFT). In the formulation introduced in ^^^^3 OR (Li et al., 29 Jul 2025) \3query3 OR (Li et al., 29 Jul 2025) \3^^^^5, the construction is intended to generalize the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution to signals with coupled, non-separable structure, especially multicomponent two-dimensional linear frequency modulated (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-LFM) signals (&&&^^^^3query3^^^^&&&). A closely related antecedent is the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D nonseparable fractional Fourier transform (NSFRFT), whose kernel contains fully coupled quadratic-phase terms and whose action on the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution is an explicit 4D rotation; that earlier framework does not define the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD by name, but it provides an important transform-theoretic bridge for interpreting nonseparable quadratic-phase Wigner analysis (&&&^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^&&&).

^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^. Conceptual placement and motivation

The classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution is written as

PRESERVED_PLACEHOLDER_^^^^3query3^^^^

In the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD, the Fourier kernel PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ is replaced by a non-separable quadratic-phase kernel associated with the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPFT. The stated aim is to preserve the fine localization of Wigner analysis while adapting the representation to coupled ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D signal geometry (&&&^^^^3query3^^^^&&&).

In this setting, non-separable means that the kernel is not, in general, a product of two independent ^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^D kernels. Instead, it may contain mixed terms such as

PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^

introduced through matrix-valued parameters. This matters because many ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D signals are not naturally decomposable into independent horizontal and vertical components. The paper explicitly motivates the framework by reference to coupled modulations, rotated chirps, affine features, and other geometrically structured data (&&&^^^^3query3^^^^&&&).

The earlier NSFRFT literature identifies a related deficiency in existing ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D FRFT families: the inability to handle ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D non-stationary signals with nonseparable terms and the failure to maintain a consistent 4D rotational relationship with the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution. In that work, the proposed NSFRFT is presented as a more general ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D FRFT with four degrees of freedom and a precise Wigner covariance law, thereby supplying a geometric context for later nonseparable quadratic-phase Wigner constructions (&&&^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^&&&).

^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^. NSQPFT framework and non-separable kernel structure

The ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPFT is parameterized by

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$

and, for $f\in L^2(\mathbb{R}^2)$,

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$

where $\mathbf{x}=(x_1,x_2)^T$, $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$, $\vec{1}=(1,1)^T$, and the kernel is

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$

The inverse formula is given in the source paper, although the extracted normalization PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^ is not fully specified in the provided text (&&&^^^^3query3^^^^&&&).

The expanded kernel form makes the coupling structure explicit: PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \33^^^^

The paper further introduces

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^4

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^5

with chirp factors

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^6

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^7

This formalism shows that non-separability arises whenever off-diagonal entries are present. In particular, PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^8 introduces cross-coupling in the mixed term PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^9, while PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3query3^^^^ and PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ introduce quadratic cross-terms PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^ and PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \33^^^^, respectively (&&&^^^^3query3^^^^&&&).

3. Formal definition of the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD

The central definition is

PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^4

where

PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^5

PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^6, PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^7, and PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^8 (&&&^^^^3query3^^^^&&&).

The same object is also written in a more direct kernel form: PRESERVED_PLACEHOLDER_^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^9

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$^^^^3query3^^^^

This expression makes clear that the distribution is bilinear in the signal and inherits the quadratic-phase structure of the underlying transform.

The relation to the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution is explicit. If

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^

then, after applying a scaling factor $$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^, the definition reduces to the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-Wigner distribution. Thus the standard ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD appears as a special case corresponding to the absence of additional chirp adaptation and cross-dimensional coupling (&&&^^^^3query3^^^^&&&).

The paper also gives a separable restriction. When all parameter matrices are diagonal,

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$3

the NSQPFT becomes a ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D separable quadratic phase Fourier transform (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-SQPFT), and the associated Wigner construction correspondingly reduces to a separable quadratic-phase family (&&&^^^^3query3^^^^&&&).

4. Core analytical properties

The ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD is equipped with a substantial list of structural identities. The paper states the conjugate-covariance property

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$4

This replaces the plain real-valuedness often expected in classical Wigner theory by a parameter-swapping conjugation law (&&&^^^^3query3^^^^&&&).

For time reversal, with $$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$5,

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$6

This is a covariance relation under reversal that modifies the sign of the coupling matrix $$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$7.

The paper gives two marginal relations. The global $$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$8-marginal is

$$\Omega = \Bigg\{ A=\begin{bmatrix} a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix}, \, B=\begin{bmatrix} b_{11} & b_{12}\ b_{12} & b_{22}\end{bmatrix}, \, \det(B)\ne 0, \, C=\begin{bmatrix} c_{11} & c_{12}\ c_{21} & c_{22}\end{bmatrix}, \, D=\begin{bmatrix} d_{11} & d_{12}\ d_{21} & d_{22}\end{bmatrix}, \, E=\begin{bmatrix} e_{11} & e_{12}\ e_{21} & e_{22}\end{bmatrix} \Bigg\},$$9

A second relation, obtained from the reconstruction formula at $f\in L^2(\mathbb{R}^2)$^^^^3query3^^^^, yields a position marginal recovering $f\in L^2(\mathbb{R}^2)$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ up to chirp factors and $f\in L^2(\mathbb{R}^2)$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^-normalization. The source text notes typographic corruption in this expression, but the intended meaning is that the NSQPWD retains both signal-domain and transform-domain marginal content.

A generalized Moyal identity is proved: $f\in L^2(\mathbb{R}^2)$3 For $f\in L^2(\mathbb{R}^2)$4, this yields the energy relation

$f\in L^2(\mathbb{R}^2)$5

These formulas show that the representation preserves inner-product information in the Wigner-theoretic sense.

Shift covariance is more intricate than in the classical case because the quadratic-phase parameters induce additional phase and coordinate corrections. For a translated signal $f\in L^2(\mathbb{R}^2)$6,

$f\in L^2(\mathbb{R}^2)$7

where $f\in L^2(\mathbb{R}^2)$8 and the matrix $f\in L^2(\mathbb{R}^2)$9, phase factor $$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$^^^^3query3^^^^, matrix $$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^, and vector $$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^ are given explicitly in the paper. Likewise, for a modulation

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$3

the paper gives

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$4

where $$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$5. In explicit coordinates,

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$6

The paper also proves a dilation law,

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$7

and a convolution property,

$$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$8

Together, these identities position the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD as a mathematically structured bilinear representation rather than merely an ad hoc chirp-weighted display (&&&^^^^3query3^^^^&&&).

5. Relation to other representations and geometric context

The ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD is related to several established bilinear and transform-domain constructions. First, the paper identifies it with a chirp-modulated cross-Wigner-type object: $$\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega}) = \int_{\mathbb{R}^2} f(\mathbf{x})\,\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega})\,d\mathbf{x},$$9 This expresses the distribution in terms of chirp-modified signals $\mathbf{x}=(x_1,x_2)^T$^^^^3query3^^^^ and $\mathbf{x}=(x_1,x_2)^T$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^, thereby embedding it in the broader family of cross-Wigner constructions (&&&^^^^3query3^^^^&&&).

Second, the paper derives a direct relation to the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D short-time Fourier transform (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-STFT): $\mathbf{x}=(x_1,x_2)^T$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^ and

$\mathbf{x}=(x_1,x_2)^T$3

with

$\mathbf{x}=(x_1,x_2)^T$4

and a window $\mathbf{x}=(x_1,x_2)^T$5 determined by $\mathbf{x}=(x_1,x_2)^T$6, $\mathbf{x}=(x_1,x_2)^T$7, and the quadratic form in $\mathbf{x}=(x_1,x_2)^T$8. This gives a windowed representation-theoretic interpretation of the NSQPWD.

Third, the source paper explicitly lists special families included in the framework: the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-Wigner distribution, the gyrator-Wigner distribution, the fractional Fourier-associated ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD, and, through diagonal restrictions, separable families linked to the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-FT, Fresnel transform, FrFT, LCT, and scaling transforms. The following summary organizes the named specializations.

Special case	Parameter choice	Result
Classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD	$\mathbf{x}=(x_1,x_2)^T$9	Reduces to classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD up to $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$^^^^3query3^^^^ scaling
Separable quadratic-phase case	All parameter matrices diagonal	^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-SQPFT / separable quadratic-phase family
Gyrator case	$\boldsymbol{\omega}=(\omega_1,\omega_2)^T$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ as in Eq. (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^.^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^)	Gyrator-associated Wigner distribution
Fractional Fourier case	Diagonal $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^ with $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$3, $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$4 entries	FrFT-associated ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD

For the gyrator case,

$\boldsymbol{\omega}=(\omega_1,\omega_2)^T$5

the distribution becomes

$\boldsymbol{\omega}=(\omega_1,\omega_2)^T$6

with

$\boldsymbol{\omega}=(\omega_1,\omega_2)^T$7

For $\boldsymbol{\omega}=(\omega_1,\omega_2)^T$8, this reduces to the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD (&&&^^^^3query3^^^^&&&).

The geometric background supplied by the NSFRFT literature is relevant here. That work proves that a nonseparable quadratic-phase transform can act on the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution by an explicit 4D rotation

$\boldsymbol{\omega}=(\omega_1,\omega_2)^T$9

or, in matrix form,

$\vec{1}=(1,1)^T$^^^^3query3^^^^

The paper is explicit that $\vec{1}=(1,1)^T$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ is a specific 4D rotation matrix. This does not define the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD itself, but it suggests a geometric interpretation: nonseparable quadratic-phase Wigner analysis can be viewed as Wigner analysis in a phase-space coordinate system adapted to coupled chirp structure (&&&^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^&&&).

6. Signal classes, localization behavior, and cross-term suppression

The principal application domain in the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD paper is the analysis of ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D linear frequency modulated (^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-LFM) signals. For the single-component case,

$\vec{1}=(1,1)^T$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^

When $\vec{1}=(1,1)^T$3, the paper derives

$\vec{1}=(1,1)^T$4

$\vec{1}=(1,1)^T$5

This formula is central because it shows how a chirp can be mapped to a concentrated sinc-like structure in the NSQPWD domain by suitable parameter choice (&&&^^^^3query3^^^^&&&).

The numerical single-component example is

$\vec{1}=(1,1)^T$6

with transform parameters

$\vec{1}=(1,1)^T$7

The paper notes the strong non-separable structure in the off-diagonal entries of $\vec{1}=(1,1)^T$8, $\vec{1}=(1,1)^T$9, and $$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$^^^^3query3^^^^.

For a bi-component signal,

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^

with

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$3

the NSQPWD decomposes into two auto-terms and two cross-terms: $$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$4 Under $$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$5, $$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$6, and $$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$7, the cross-term reduces to a phase factor multiplied by the same sinc factors that govern the component localization.

The specific simulated bi-component signal is

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$8

For the tri-component case,

$$\mathcal{K}_{\Omega}(\mathbf{x},\boldsymbol{\omega}) = \frac{i\,\sqrt{\det(B)}}{2\pi} \exp\Big\{ i\big( \boldsymbol{\omega}^{T}A\boldsymbol{\omega} +\boldsymbol{\omega}^{T}B\mathbf{x} +\mathbf{x}^{T}C\mathbf{x} +\vec{1}D\boldsymbol{\omega} +\vec{1}E\mathbf{x} \big) \Big\}.$$9

the distribution contains three auto-terms and six cross-terms. The parameter values used are

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3query3^^^^

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^

with

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3 OR (Li et al., 29 Jul 2025) \3^^^^

The paper’s principal practical claim is that the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD provides better cross-term suppression and better localization than the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-Wigner distribution for the studied ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-LFM examples. The reported evidence is partly analytical and partly visual. For the single-component case, the paper states that the representation produces sharp peaks and clear contour ridges. For the bi-component case, it reports that at PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query33^^^^ the energy is sharply localized with clear auto-terms and minimal cross-term interference; at PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^4 the energy remains concentrated but is slightly broadened along the frequency axes, with faint cross-term artifacts; and at PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^5 the concentration decreases moderately and cross-terms become more pronounced. For the tri-component example, the paper states that the LFM components form clear diagonal ridges, that contour maps show sharp high-energy bands aligned with these ridges, that cross-term suppression remains effective, and that even at PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^6 dB the method localizes and separates overlapping components reliably (&&&^^^^3query3^^^^&&&).

An important caveat stated in the source text is that the excerpt does not provide explicit side-by-side numerical metrics against the classical ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-WD; the comparison is primarily analytical and visual. A plausible implication is that the method’s empirical advantage is presently documented more by qualitative concentration behavior than by a standardized benchmark suite.

7. Assumptions, constraints, limitations, and transform-theoretic context

The main theoretical setup assumes

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^7

The examples use finite-support chirp signals over

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^8

The key admissibility condition on the transform parameters is

PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^9

Thus PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3query3^^^^ must be symmetric and invertible (&&&^^^^3query3^^^^&&&).

The work is described in the source as theoretical with application-oriented demonstrations. It develops the transform, proves core identities, and illustrates the construction on synthetic LFM signals. It does not provide a detailed computational complexity analysis, a discrete algorithm, or an implementation-accuracy study. The text also notes notational and typesetting corruption in some formulas, especially in the inverse normalization and parts of the marginal/reconstruction expressions.

This limitation contrasts with the earlier NSFRFT paper, which is algorithmically more explicit. That work presents three discrete algorithms, including two fast algorithms with computational complexity PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^, and reports representative timings of about PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^–PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \33^^^^ s for the direct method on PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^4 examples, about PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^5–PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^6 s for Algorithm ^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^, and about PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^7–PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^8 s for Algorithm ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^. It also reports NMSE values around PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^9 to PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3query3^^^^ for Algorithm ^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^, around PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^ for Algorithm ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^, and reversibility NMSE PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^ for Algorithm ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^ (&&&^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^&&&). This does not directly supply a numerical scheme for the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD, but it indicates that nonseparable quadratic-phase transform frameworks can admit efficient discretizations.

The transform-theoretic connection is conceptually significant. The NSFRFT paper does not define the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD, and it does not derive an ambiguity-function relation, Cohen-class kernel formula, or a direct NSFRFT-associated Wigner integral kernel under that name. However, it supplies three ingredients that closely parallel the later NSQPWD construction: a nonseparable quadratic-phase transform kernel, a matrix/canonical representation, and an exact Wigner covariance law under a 4D rotation (&&&^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3^^^^&&&). This suggests a broader research program in which nonseparable quadratic-phase transforms and adapted Wigner distributions are viewed as members of a unified phase-space framework.

Within that broader context, the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D-NSQPWD can be characterized as a matrix-parameterized, chirp-adapted, non-separable generalization of the ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D Wigner distribution. Its defining feature is not merely the addition of quadratic terms, but the controlled introduction of cross-dimensional couplings through PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \33^^^^, PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^4, PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^5, PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^6, and PRESERVED_PLACEHOLDER_^^^^3arXiv (Chauhan et al., 8 Sep 2025) \3 OR (Li et al., 29 Jul 2025) \3^^^^7. The intended analytical payoff is a representation aligned with coupled ^^^^3 OR (Li et al., 29 Jul 2025) \3^^^^D chirp geometry rather than with independent axis-wise oscillations (&&&^^^^3query3^^^^&&&).