---
title: 2D Non-Separable Quadratic Phase Wigner Distribution
url: https://www.emergentmind.com/topics/non-separable-quadratic-phase-wigner-distribution-2d-nsqpwd
type: topic
---

# 2D Non-Separable Quadratic Phase Wigner Distribution

Searching arXiv for the cited papers and closely related work to ground the article.
{"query":"arXiv 2509.19310 \"Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution\" OR 2507.21511 \"Two-Dimensional Nonseparable Fractional Fourier Transform: Theory and Application\"", "max_results": 10}
The **Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution (2D-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 (2D-NSQPFT)**. In the formulation introduced in 2025, the construction is intended to generalize the classical 2D Wigner distribution to signals with coupled, non-separable structure, especially multicomponent two-dimensional linear frequency modulated (2D-LFM) signals [2509.19310]. A closely related antecedent is the **2D nonseparable fractional Fourier transform (NSFRFT)**, whose kernel contains fully coupled quadratic-phase terms and whose action on the 2D Wigner distribution is an explicit 4D rotation; that earlier framework does not define the 2D-NSQPWD by name, but it provides an important transform-theoretic bridge for interpreting nonseparable quadratic-phase Wigner analysis [2507.21511].

## 1. Conceptual placement and motivation

The classical 2D Wigner distribution is written as
\[
\mathcal{W}_f(x_1, x_2, \omega_1, \omega_2) =
\int_{\mathbb{R}^2}
f\left(x_1+\frac{\tau}{2},x_2+\frac{\eta}{2}\right)
\overline{f\left(x_1-\frac{\tau}{2},x_2-\frac{\eta}{2}\right)}
e^{-i(\omega_1\tau+\omega_2\eta)}\,d\tau\,d\eta.
\]
In the 2D-NSQPWD, the Fourier kernel \(e^{-i(\omega_1\tau+\omega_2\eta)}\) is replaced by a non-separable quadratic-phase kernel associated with the 2D-NSQPFT. The stated aim is to preserve the fine localization of Wigner analysis while adapting the representation to coupled 2D signal geometry [2509.19310].

In this setting, **non-separable** means that the kernel is not, in general, a product of two independent 1D kernels. Instead, it may contain mixed terms such as
\[
\omega_1\omega_2,\qquad x_1x_2,\qquad x_1\omega_2,\qquad x_2\omega_1,
\]
introduced through matrix-valued parameters. This matters because many 2D 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 [2509.19310].

The earlier NSFRFT literature identifies a related deficiency in existing 2D FRFT families: the inability to handle 2D non-stationary signals with nonseparable terms and the failure to maintain a consistent 4D rotational relationship with the 2D Wigner distribution. In that work, the proposed NSFRFT is presented as a more general 2D FRFT with four degrees of freedom and a precise Wigner covariance law, thereby supplying a geometric context for later nonseparable quadratic-phase Wigner constructions [2507.21511].

## 2. NSQPFT framework and non-separable kernel structure

The 2D-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 \(\Lambda(B,n)\) is not fully specified in the provided text [2509.19310].

The expanded kernel form makes the coupling structure explicit:
\[
\mathcal{K}_{\Omega}(x_1,x_2,\omega_1,\omega_2)
=
\frac{i\,\sqrt{\det(B)}}{2\pi}
\exp\Big\{ i \big( a_{11} \omega_1^2 + (a_{12}+a_{21}) \omega_1 \omega_2 + a_{22} \omega_2^2 + (d_{11}+d_{21}) \omega_1 + (d_{12}+d_{22}) \omega_2 \big) \Big\}
\]
\[
\times
\exp\Big\{ i \big( c_{11} x_1^2 + (c_{12}+c_{21}) x_1 x_2 + c_{22} x_2^2 + (e_{11}+e_{21}) x_1 + (e_{12}+e_{22}) x_2 \big) \Big\}
\]
\[
\times
\exp\Big\{ i\big( x_1(\omega_1 b_{11}+\omega_2 b_{12}) + x_2(b_{12}\omega_1+b_{22}\omega_2) \big)\Big\}.
\]
The paper further introduces
\[
\begin{bmatrix} k_1&k_2&k_3&k_4&k_5 \end{bmatrix}
=
\begin{bmatrix} a_{11}&a_{12}+a_{21}&a_{22}&d_{11}+d_{21}&d_{12}+d_{22} \end{bmatrix},
\]
\[
\begin{bmatrix} m_1&m_2&m_3&m_4&m_5 \end{bmatrix}
=
\begin{bmatrix} c_{11}&c_{12}+c_{21}&c_{22}&e_{11}+e_{21}&e_{12}+e_{22} \end{bmatrix},
\]
with chirp factors
\[
\mathcal{C}_{\mathbf{k}}(\omega_1,\omega_2)
=
\exp\Big\{i\big(k_1\omega_1^2+k_2\omega_1\omega_2+k_3\omega_2^2+k_4\omega_1+k_5\omega_2\big)\Big\},
\]
\[
\mathcal{C}_{\mathbf{m}}(x_1,x_2)
=
\exp\Big\{i\big(m_1x_1^2+m_2x_1x_2+m_3x_2^2+m_4x_1+m_5x_2\big)\Big\}.
\]

This formalism shows that non-separability arises whenever off-diagonal entries are present. In particular, \(b_{12}\neq 0\) introduces cross-coupling in the mixed term \(\boldsymbol{\omega}^T B \mathbf{x}\), while \(k_2\neq 0\) and \(m_2\neq 0\) introduce quadratic cross-terms \(\omega_1\omega_2\) and \(x_1x_2\), respectively [2509.19310].

## 3. Formal definition of the 2D-NSQPWD

The central definition is
\[
\mathcal{W}_f^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\frac{|\det B|\,\mathcal{C}_{\mathbf{k-m}}(\boldsymbol{\omega})}{(2 \pi)^2}
\int_{\mathbb{R}^2}
f_\mathbf{m}\left(\mathbf{x} +\tfrac{1}{2} \boldsymbol{\xi} \right)
\overline{f_\mathbf{k}\left( \mathbf{x} - \tfrac{1}{2} \boldsymbol{\xi} \right)}
\exp\big\{i\, \boldsymbol{\omega}^T B \boldsymbol{\xi} \big\}\, d\boldsymbol{\xi},
\]
where
\[
f_\mathbf{m}(\mathbf{t})=f(\mathbf{t})\,\mathcal{C}_{\mathbf{m}}(\mathbf{t}),\qquad
f_\mathbf{k}(\mathbf{t})=f(\mathbf{t})\,\mathcal{C}_{\mathbf{k}}(\mathbf{t}),
\]
\(\mathbf{x}=(x_1,x_2)^T\), \(\boldsymbol{\xi}=(\tau,\eta)^T\), and \(\boldsymbol{\omega}=(\omega_1,\omega_2)^T\) [2509.19310].

The same object is also written in a more direct kernel form:
\[
\mathcal{W}_f^{\Omega}(x_1,x_2,\omega_1,\omega_2)
=
\int_{\mathbb{R}^2}
f\left(x_1+\frac{\tau}{2},x_2+\frac{\eta}{2}\right)
\overline{f\left(x_1-\frac{\tau}{2},x_2-\frac{\eta}{2}\right)}
\]
\[
\times
\mathcal{K}_{\Omega}\left(x_1+\frac{\tau}{2},x_2+\frac{\eta}{2},\omega_1,\omega_2\right)
\overline{\mathcal{K}_{\Omega}\left(\omega_1,\omega_2,x_1-\frac{\tau}{2},x_2-\frac{\eta}{2}\right)}
\,d\tau\,d\eta.
\]
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 2D Wigner distribution is explicit. If
\[
A=\mathbf{0}_{2\times 2},\quad C=\mathbf{0}_{2\times 2},\quad D=\mathbf{0}_{2\times 2},\quad E=\mathbf{0}_{2\times 2},\quad B=I_{2\times 2},
\]
then, after applying a scaling factor \(4\pi^2\), the definition reduces to the classical 2D-Wigner distribution. Thus the standard 2D-WD appears as a special case corresponding to the absence of additional chirp adaptation and cross-dimensional coupling [2509.19310].

The paper also gives a separable restriction. When all parameter matrices are diagonal,
\[
A=\begin{bmatrix}a_{11}&0\\0&a_{22}\end{bmatrix},\quad
B=\begin{bmatrix}b_{11}&0\\0&b_{22}\end{bmatrix},\quad
C=\begin{bmatrix}c_{11}&0\\0&c_{22}\end{bmatrix},\quad
D=\begin{bmatrix}d_{11}&0\\0&d_{22}\end{bmatrix},\quad
E=\begin{bmatrix}e_{11}&0\\0&e_{22}\end{bmatrix},
\]
the NSQPFT becomes a **2D separable quadratic phase Fourier transform (2D-SQPFT)**, and the associated Wigner construction correspondingly reduces to a separable quadratic-phase family [2509.19310].

## 4. Core analytical properties

The 2D-NSQPWD is equipped with a substantial list of structural identities. The paper states the **conjugate-covariance property**
\[
\overline{\mathcal{W}_f^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)}
=
\mathcal{W}_f^{\Omega^\prime}\left(\mathbf{x}, \boldsymbol{\omega}\right),
\qquad
\Omega^\prime=\{C,B,A,E,D\}.
\]
This replaces the plain real-valuedness often expected in classical Wigner theory by a parameter-swapping conjugation law [2509.19310].

For time reversal, with \(\check{f}(\mathbf{x})=f(-\mathbf{x})\),
\[
\mathcal{W}_{\check{f}}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\mathcal{W}_{f}^{\Omega^{\prime\prime}}(-\mathbf{x}, \boldsymbol{\omega}),
\qquad
\Omega^{\prime\prime}=\{A,-B,C,D,E\}.
\]
This is a covariance relation under reversal that modifies the sign of the coupling matrix \(B\).

The paper gives two marginal relations. The global \(x\)-marginal is
\[
\int_{\mathbb{R}^2}\mathcal{W}_{f}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)\,d\mathbf{x}
=
\mathcal{Q}_{\Omega}[f](\boldsymbol{\omega})\, \mathcal{Q}^{-1}_{\Omega}\big[\overline{f}\big](\boldsymbol{\omega}).
\]
A second relation, obtained from the reconstruction formula at \(\boldsymbol{\xi}=0\), yields a position marginal recovering \(|f(\mathbf{x})|^2\) up to chirp factors and \(B\)-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:
\[
\int_{\mathbb{R}^4}
\mathcal{W}_{f}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)\,
\overline{\mathcal{W}_{g}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)}
\,d\mathbf{x}\,d\boldsymbol{\omega}
=
\frac{|\det B|}{(2 \pi)^2}\, |\langle f,g\rangle|^2.
\]
For \(f=g\), this yields the energy relation
\[
\int_{\mathbb{R}^4}\big|\mathcal{W}_{f}^{\Omega}(\mathbf{x},\boldsymbol{\omega})\big|^2
\,d\mathbf{x}\,d\boldsymbol{\omega}
=
\frac{|\det B|}{(2\pi)^2}\,\|f\|_{L^2(\mathbb{R}^2)}^4.
\]
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 \(\check{f}(\mathbf{x})=f(\mathbf{x}-\mathbf{x}_0)\),
\[
\mathcal{W}_{\check{f}}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\nabla_1(\mathbf{x},\boldsymbol{\omega})\,
\mathcal{W}_{f}^{\Omega}(\mathbf{x}-\mathbf{x}_0, \boldsymbol{\omega}+\boldsymbol{\rho}),
\]
where \(\boldsymbol{\rho}=P\mathbf{x}_0\) and the matrix \(P\), phase factor \(\nabla_1\), matrix \(Q\), and vector \(\lambda\) are given explicitly in the paper. Likewise, for a modulation
\[
\hat f(\mathbf{x})=f(\mathbf{x})\,\exp\{i\,\boldsymbol{\omega}_0^T\mathbf{x}\},
\]
the paper gives
\[
\mathcal{W}_{\hat{f}}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\mathcal{C}_{\mathbf{m-k}}(\mathbf{t})\,
\exp\left\{ -i\,(\boldsymbol{\omega}^T Q \mathbf{t})\right\}
\,\mathcal{W}_f^{\Omega}(\mathbf{x},\boldsymbol{\omega}+ \mathbf{t}),
\]
where \(B\mathbf{t}=\boldsymbol{\omega}_0\). In explicit coordinates,
\[
t_1 = \tilde {b}_{22}\,\omega_{1,0}-\tilde{b}_{12}\,\omega_{2,0},
\qquad
t_2 = -\tilde{b}_{12}\,\omega_{1,0} +\tilde{b}_{11}\,\omega_{2,0}.
\]

The paper also proves a dilation law,
\[
\mathcal{W}_{D_\lambda f}^\Omega\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\frac{1}{\lambda}\,
\mathcal{W}_f^\Omega\left(\lambda \mathbf{x}, \frac{\boldsymbol{\omega}}{\lambda}\right),
\qquad
D_\lambda f(\mathbf{x})=\sqrt{\lambda}\,f(\lambda \mathbf{x}),
\]
and a convolution property,
\[
\mathcal{W}_{f*g}^{\Omega}\left(\mathbf{x},\boldsymbol{\omega}\right)
=
\frac{ (2\pi)^2\,\mathcal{C}_{\mathbf{m-k}}(\boldsymbol{\omega})}{|\det B|}
\int_{\mathbb{R}^2}
\mathcal{W}_f^{\Omega}\left(\mathbf{u},\boldsymbol{\omega}\right)
\mathcal{W}_g^{\Omega}\left(\mathbf{x}-\mathbf{u},\boldsymbol{\omega}\right)\,d\mathbf{u}.
\]
Together, these identities position the 2D-NSQPWD as a mathematically structured bilinear representation rather than merely an ad hoc chirp-weighted display [2509.19310].

## 5. Relation to other representations and geometric context

The 2D-NSQPWD is related to several established bilinear and transform-domain constructions. First, the paper identifies it with a chirp-modulated cross-Wigner-type object:
\[
\mathcal{W}_f^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\frac{|\det B|\,\mathcal{C}_{\mathbf{k-m}}(\boldsymbol{\omega})}{(2 \pi)^2}\,
\mathcal{W}_{f_\mathbf{m},f_\mathbf{k}}^{\Omega}(\mathbf{x}, -B\boldsymbol{\omega}).
\]
This expresses the distribution in terms of chirp-modified signals \(f_\mathbf{m}\) and \(f_\mathbf{k}\), thereby embedding it in the broader family of cross-Wigner constructions [2509.19310].

Second, the paper derives a direct relation to the **2D short-time Fourier transform (2D-STFT)**:
\[
S_{f,g}\left(\mathbf{x},\boldsymbol{\omega}\right)
=
\int_{\mathbb{R}^2}
f(\boldsymbol{\xi})\, g(\boldsymbol{\xi}-\mathbf{x})\,
\exp\left\{-i\, \boldsymbol{\omega}^T\boldsymbol{\xi}\right\}\,d\boldsymbol{\xi},
\]
and
\[
\mathcal{W}_{f}^{\Omega}\left(\frac{\mathbf x}{2},\tilde{B}\boldsymbol{\omega}\right)
=
\nabla_2(\mathbf x, \boldsymbol{\omega})\,S_{f,g}\left(\mathbf x,\boldsymbol{\omega} \right),
\]
with
\[
\tilde{B}= \begin{bmatrix} -\tilde{b}_{22} & \tilde{b}_{12} \\ \tilde{b}_{12} & -\tilde{b}_{11} \end{bmatrix},
\]
and a window \(g\) determined by \(f_\mathbf{k}\), \(\mathcal{C}_{\mathbf m}\), and the quadratic form in \(m_1,m_2,m_3\). This gives a windowed representation-theoretic interpretation of the NSQPWD.

Third, the source paper explicitly lists special families included in the framework: the classical 2D-Wigner distribution, the gyrator-Wigner distribution, the fractional Fourier-associated 2D-WD, and, through diagonal restrictions, separable families linked to the 2D-FT, Fresnel transform, FrFT, LCT, and scaling transforms. The following summary organizes the named specializations.

| Special case | Parameter choice | Result |
|---|---|---|
| Classical 2D-WD | \(A=C=D=E=0,\; B=I\) | Reduces to classical 2D-WD up to \(4\pi^2\) scaling |
| Separable quadratic-phase case | All parameter matrices diagonal | 2D-SQPFT / separable quadratic-phase family |
| Gyrator case | \(\Omega_\theta\) as in Eq. (2.10) | Gyrator-associated Wigner distribution |
| Fractional Fourier case | Diagonal \(A,B,C\) with \(\cot\theta_i\), \(-\csc\theta_i\) entries | FrFT-associated 2D-WD |

For the gyrator case,
\[
\Omega_\theta = \Bigg\{
A = \begin{bmatrix} 0 & \frac{1}{2}\cot\theta \\ \frac{1}{2}\cot\theta & 0 \end{bmatrix},
\,
B = \begin{bmatrix} 0 & -\csc\theta\\ -\csc\theta &0 \end{bmatrix},
\,
C = \begin{bmatrix} 0 & \frac{1}{2}\cot\theta \\ \frac{1}{2}\cot\theta & 0 \end{bmatrix},
\,
D=0,\,
E=0
\Bigg\},
\]
the distribution becomes
\[
\mathcal{W}_f^{\Omega_\theta}(\mathbf{x}, \boldsymbol{\omega})
=
\frac{|\csc^2\theta|}{(2\pi)^2}
\int_{\mathbb{R}^2}
f\left( \mathbf{x} + \tfrac{1}{2} \boldsymbol{\xi} \right)
\overline{f\left( \mathbf{x} - \tfrac{1}{2} \boldsymbol{\xi} \right)}
\, e^{i\, \mathbf{x}^T J \boldsymbol{\xi} \cot\theta - i\, \boldsymbol{\omega}^T J \boldsymbol{\xi} \csc\theta}
\, d\boldsymbol{\xi},
\]
with
\[
J= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.
\]
For \(\theta=\pi/2\), this reduces to the classical 2D-WD [2509.19310].

The geometric background supplied by the NSFRFT literature is relevant here. That work proves that a nonseparable quadratic-phase transform can act on the 2D Wigner distribution by an explicit 4D rotation
\[
W_{F_P}(x, y, u, v) = W_f(\widetilde{x}, \widetilde{y}, \widetilde{u}, \widetilde{v}),
\]
or, in matrix form,
\[
W_{F_P}(\mathbf z)=W_f(\mathbf R_{4D}^{\mathrm{NS}}\mathbf z).
\]
The paper is explicit that \(\mathbf R_{4D}^{\mathrm{NS}}\) is a specific 4D rotation matrix. This does not define the 2D-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 [2507.21511].

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

The principal application domain in the 2D-NSQPWD paper is the analysis of **2D linear frequency modulated (2D-LFM)** signals. For the single-component case,
\[
\mathcal{S}(\mathbf{x}) = \kappa_0 \exp\big\{i \left((\alpha_0 x_1 + \beta_0 x_1^2) + (\mu_0 x_2 + \lambda_0 x_2^2) \right)\big\},
\quad
\mathbf{x} \in \left[ -\frac{T}{2}, \frac{T}{2} \right] \times \left[ -\frac{T}{2}, \frac{T}{2} \right].
\]
When \(\mathbf{m}=\mathbf{k}\), the paper derives
\[
\mathcal{W}_\mathcal{S}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\frac{|\kappa_0|^2\,T^2\,|\det B| }{(2 \pi)^2}
\,
\text{sinc}\left\{ \frac{T}{2} \left( m_4 + m_2 x_2 + \alpha_0 + 2 x_1 (m_1 + \beta_0) + b_{11} \omega_1 + b_{12} \omega_2 \right)\right\}
\]
\[
\times
\text{sinc}\left\{ \frac{T}{2} \left( m_5 + m_2 x_1 + \mu_0 + 2 x_2 (m_3 + \lambda_0) + b_{12} \omega_1 + b_{22} \omega_2 \right)\right\}.
\]
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 [2509.19310].

The numerical single-component example is
\[
\mathcal{S}(x_1,x_2)
=
\exp \left\{i\, ( (0.3 x_1 + 0.2 x_1^2) + (0.1 x_2 + 0.5x_2^2) )\right\},
\qquad
T=40,
\]
with transform parameters
\[
\Omega_0 = \Bigg\{
A = \begin{bmatrix} 1 & -5 \\ 5 & 1 \end{bmatrix},
\,
B = \begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix},
\,
C = \begin{bmatrix} 1 & \frac{-13}{7} \\ \frac{13}{7} & 1 \end{bmatrix},
\,
D = \begin{bmatrix} 2 & 1 \\ 2 & 5 \end{bmatrix},
\,
E = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
\Bigg\}.
\]
The paper notes the strong non-separable structure in the off-diagonal entries of \(A\), \(B\), and \(C\).

For a bi-component signal,
\[
\mathcal{U}(\mathbf{x})=\mathcal{U}_1(\mathbf{x})+\mathcal{U}_2(\mathbf{x}),
\]
with
\[
\mathcal{U}_1(\mathbf{x}) = \kappa_1 \exp\left\{i\, \big( (\alpha_1 x_1 + \beta_1 x_1^{2}) + (\mu_1 x_2 + \lambda_1 x_2^{2})\big) \right\},
\]
\[
\mathcal{U}_2(\mathbf{x})=\kappa_2 \exp\left\{i\,\big( (\alpha_2 x_1 + \beta_2 x_1^{2}) + (\mu_2 x_2 + \lambda_2 x_2^{2}) \big)\right\},
\]
the NSQPWD decomposes into two auto-terms and two cross-terms:
\[
\mathcal{W}_\mathcal{U}^{\Omega}\left(\mathbf{x}, \boldsymbol{\omega}\right)
=
\mathcal{W}^{\Omega}_{\mathcal{U}_{1}}\left(\mathbf{x}, \boldsymbol{\omega}\right)
+
\mathcal{W}^{\Omega}_{\mathcal{U}_{2}}\left(\mathbf{x}, \boldsymbol{\omega}\right)
+
\mathcal{W}^{\Omega}_{\mathcal{U}_{1}, \mathcal{U}_{2}}\left(\mathbf{x}, \boldsymbol{\omega}\right)
+
\mathcal{W}^{\Omega}_{\mathcal{U}_{2}, \mathcal{U}_{1}}\left(\mathbf{x}, \boldsymbol{\omega}\right).
\]
Under \(\mathbf{m}=\mathbf{k}\), \(\beta_1=\beta_2\), and \(\lambda_1=\lambda_2\), 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
\[
\mathcal{U}(\mathbf{x})
=
4\,e^{i \left(0.2x_1 + 0.05x_1^2 + 0.15x_2 + 0.04x_2^2\right)}
+
e^{i \left(0.4x_1 + 0.05 x_1^2 + 0.2x_2 + 0.04x_2^2\right)},
\qquad
T=40.
\]

For the tri-component case,
\[
\mathcal{U}(\mathbf{x}) = \sum_{n=1}^{3} \mathcal{U}_n(\mathbf{x}),
\qquad
\mathcal{U}_n(\mathbf{x}) = \kappa_n \exp\left\{ i\left[ (\alpha_n x_1 + \beta_n x_1^2) + (\mu_n x_2 + \lambda_n x_2^2) \right] \right\},
\]
the distribution contains three auto-terms and six cross-terms. The parameter values used are
\[
\kappa_n\in\{5,1,11\},\quad
\alpha_n\in\{2,4,6\},\quad
\beta_n\in\{0.05,0.05,0.05\},
\]
\[
\mu_n\in\{0.15,6,0.25\},\quad
\lambda_n\in\{0.04,0.04,0.04\},\quad
T=40,
\]
with
\[
\Omega_1 = \Bigg\{
A = \begin{bmatrix} 1 & \frac{-1}{7} \\ \frac{1}{7} & 1 \end{bmatrix},
\,
B = \begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix},
\,
C = \begin{bmatrix} 1 & \frac{-19}{5} \\ \frac{19}{5} & 1 \end{bmatrix},
\,
D = \begin{bmatrix} 4 & 5 \\ 0 & 7 \end{bmatrix},
\,
E = \begin{bmatrix} 2 & 7 \\ 2 & 5 \end{bmatrix}
\Bigg\}.
\]

The paper’s principal practical claim is that the 2D-NSQPWD provides **better cross-term suppression and better localization** than the classical 2D-Wigner distribution for the studied 2D-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 \(\mathbf{x}=(0.60,0.10)\) the energy is sharply localized with clear auto-terms and minimal cross-term interference; at \(\mathbf{x}=(0.40,0.30)\) the energy remains concentrated but is slightly broadened along the frequency axes, with faint cross-term artifacts; and at \(\mathbf{x}=(0.20,0.30)\) 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 \( \mathrm{SNR}=10 \) dB the method localizes and separates overlapping components reliably [2509.19310].

An important caveat stated in the source text is that the excerpt does not provide explicit side-by-side numerical metrics against the classical 2D-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
\[
f\in L^2(\mathbb{R}^2).
\]
The examples use finite-support chirp signals over
\[
\left[-\frac{T}{2},\frac{T}{2}\right]^2.
\]
The key admissibility condition on the transform parameters is
\[
B=B^T,\qquad \det(B)\neq 0.
\]
Thus \(B\) must be symmetric and invertible [2509.19310].

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 \(O(N^2\log N)\), and reports representative timings of about \(162\)–\(166\) s for the direct method on \(200\times 200\) examples, about \(0.008\)–\(0.010\) s for Algorithm 1, and about \(0.006\)–\(0.008\) s for Algorithm 2. It also reports NMSE values around \(10^{-3}\) to \(10^{-2}\) for Algorithm 1, around \(10^{-8}\) for Algorithm 2, and reversibility NMSE \(7.989\times 10^{-31}\) for Algorithm 2 [2507.21511]. This does not directly supply a numerical scheme for the 2D-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 2D-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 [2507.21511]. 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 2D-NSQPWD can be characterized as a **matrix-parameterized, chirp-adapted, non-separable generalization of the 2D Wigner distribution**. Its defining feature is not merely the addition of quadratic terms, but the controlled introduction of cross-dimensional couplings through \(A\), \(B\), \(C\), \(D\), and \(E\). The intended analytical payoff is a representation aligned with coupled 2D chirp geometry rather than with independent axis-wise oscillations [2509.19310].

Source: https://www.emergentmind.com/topics/non-separable-quadratic-phase-wigner-distribution-2d-nsqpwd