Matrix-Wigner Distribution

Abstract

In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known $\tau$-Wigner distribution ($\tau$-WD) with only one parameter $\tau$ to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix $\mathcal{M}$. According to operator theory, we construct Heisenberg’s inequalities on the uncertainty product in M-WD domains and formulate two kinds of attainable lower bounds dependent on $\mathcal{M}$. We solve the problem of lower bound minimization and obtain the optimality condition of $\mathcal{M}$, under which the M-WD achieves superior time–frequency resolution. It turns out that the M-WD breaks through the limitation of the $\tau$-WD and gives birth to some novel distributions other than the WD that could generate the highest time–frequency resolution. As an example, the two-dimensional linear frequency-modulated signal is carried out to demonstrate the time–frequency concentration superiority of the M-WD over the short-time Fourier transform and wavelet transform.

1. Introduction

Time–frequency analysis is a crucial aspect of signal processing, enabling a detailed examination of signals in the joint time–frequency domain. Among the various techniques developed, the Wigner Distribution (WD) stands out for its ability to achieve high-resolution analysis. The concept of WD was first introduced by E. Wigner in 1932 in the context of quantum mechanics [1]; approximately fifteen years later, it was employed by J. Ville [2] in the study of non-stationary signals, and then explored by L. Cohen [3,4,5,6], B. Boashash [7,8,9], L. Stanković [10,11,12], G. Folland [13], and P. Korn [14] from the perspective of time–frequency analysis. As one of the most important time–frequency analysis techniques, the WD has found many applications in communication, seismic signal analysis, radar, sonar, image processing, and optical signals processing [15,16,17,18,19,20]. For a function $f\in L^2\left({\mathbb{R}}^N\right)$, its WD is defined by

$$\mathrm{W}f(\mathbf{x},\mathbf{w})=\mathcal{F}{\mathfrak{T}}_{\frac{1}{2}}(f\otimes \overline{f})(\mathbf{x},\mathbf{w}),$$

where the tensor product is $(f\otimes \overline{g})(\mathbf{x},\mathbf{y}):=f\left(\mathbf{x}\right)\overline{g\left(\mathbf{y}\right)}$, the coordinate transformation is ${\mathfrak{T}}_{\frac{1}{2}}h(\mathbf{x},\mathbf{y}):=h\left(\mathbf{x}+\frac{\mathbf{y}}{2},\mathbf{x}-\frac{\mathbf{y}}{2}\right)$, and the partial Fourier transform (FT) on the second variables is $\mathcal{F}h(\mathbf{x},\mathbf{w}):=〈h(\mathbf{x},\mathbf{y}),{\mathrm{e}}^{-2\pi \mathrm{i}\mathbf{y}{\mathbf{w}}^{\mathrm{T}}}〉$ for $h(\mathbf{x},·)\in L^1\left({\mathbb{R}}^N\right)\cap L^2\left({\mathbb{R}}^N\right)$ for almost every $\mathbf{x}\in {\mathbb{R}}^N$. Here, the superscripts $\mathrm{T}$ and — represent the transpose operator and complex conjugate operator, respectively, and $〈·,·〉$ is the distributional bracket.

The WD can offer an accurate representation of a signal’s energy distribution in the time–frequency plane, providing in both time and frequency. This means that the signal’s time and frequency information can be precisely described simultaneously. Although the WD provides a perfect concentration in the time–frequency plane, it is constrained by the classical Heisenberg uncertainty principle, which limits its resolution [21]. Typically, when the lower bound of the uncertainty inequality is reduced, the time–frequency resolution improves. However, the existing lower bounds, proposed by D. Gabor [22] and G. Folland [13] for all functions, then extended by L. Cohen [6] to the complex-valued case, and more recently enhanced by P. Dang [23] and Z. Zhang [24], turn out to be constants dependent only on Heisenberg’s constant (i.e., Planck constant). Consequently, the WD lacks the flexibility for the enhancement of time–frequency resolution, which forces many scholars to consider a parametric approach.

An innovative parameterized Wigner distribution was proposed by P. Boggiatto [25] about fourteen years ago. According to ${\mathfrak{T}}_{\tau }h(\mathbf{x},\mathbf{y}):=h(\mathbf{x}+\tau \mathbf{y},\mathbf{x}-(1-\tau )\mathbf{y})$, $\tau \in [0,1]$, P. Boggiatto extended the change of fixed coordinates ${\mathfrak{T}}_{\frac{1}{2}}$ in the WD to the change of parameterized coordinates ${\mathfrak{T}}_{\tau }$. This parametric formulation thus reads

$$\mathrm{W}\left(\tau \right)f(\mathbf{x},\mathbf{w})=\mathcal{F}{\mathfrak{T}}_{\tau }(f\otimes \overline{f})(\mathbf{x},\mathbf{w}),$$

which is known as the $\tau$-Wigner distribution ($\tau$-WD). It can be seen that the ordinary WD is a special case of the $\tau$-WD, namely, $\mathrm{W}\left(\frac{1}{2}\right)f=\mathrm{W}f$. In addition, the cases $\tau =0$ and $\tau =1$ correspond to the Rihaczek transform and conjugate Rihaczek transform, respectively.

Some essential theories of the $\tau$-WD were established, including its positivity problems (addressed by A. Janssen [26]) and its relation to pseudo-differential operators (studied by M. Wong [27] and M. Shubin [28]). Since its inception, the $\tau$-WD has found widespread applications across signal processing, radar, time–frequency analysis, remote sensing, quantum mechanics, and so on, available in the works of P. Boggiatto [25,29,30], E. Cordero [31,32,33,34,35], F. Luef [36,37], L. D’Elia [38], W. Guo [39], V. Vuojamo [40], Z. Zhang [20], and their collaborators. Particularly, uncertainty principles of the $\tau$-WD are currently derived, including a weak Heisenberg-type version and some local-type versions given formerly by P. Boggiatto, E. Carypis, and A. Oliaro [29,30], and the standard Heisenberg-type version formulated more recently by the authors [20]. In our latest work [20], the tightest universal lower bound $\frac{N^2}{16{\pi }^2}{(2{\tau }^2-2\tau +1)}^2$ on the uncertainty product $\Delta {\mathbf{x}}_{\tau }^2\Delta {\mathbf{w}}_{\tau }^2$ (see Section 2 below) in time-$\tau$-WD and frequency-$\tau$-WD domains for all functions was obtained, as well as two versions of attainable lower bounds $\left(\frac{N^2}{16{\pi }^2}+{\mathrm{Cov}}_{\mathbf{x},\mathbf{w}}^2\right){(2{\tau }^2-2\tau +1)}^2,\left(\frac{N^2}{16{\pi }^2}+{\mathrm{COV}}_{\mathbf{x},\mathbf{w}}^2\right){(2{\tau }^2-2\tau +1)}^2$ for complex-valued functions. Here ${\mathrm{Cov}}_{\mathbf{x},\mathbf{w}},{\mathrm{COV}}_{\mathbf{x},\mathbf{w}}$ (see Section 2 below) denote the covariance and absolute covariance, respectively. These lower bounds are some parametric constants that can be regarded as the functions of $\tau$. By establishing and solving the minimum lower bound problems, it turns out that the optimal parameter for achieving the highest time–frequency resolution is $\tau =\frac{1}{2}$. In other words, the optimal $\tau$-WD is none other than the conventional WD, which indicates that the $\tau$-WD fails to achieve superresolution in the joint time–frequency distribution as compared with the conventional WD.

To sum up, the $\tau$-WD was derived from the idea of the change of parameterized coordinates ${\mathfrak{T}}_{\tau }$. However, with only one parameter $\tau$, the change of single scale coordinates ${\mathfrak{T}}_{\tau }$ technique makes no contribution to the enhancement of time–frequency resolution. It therefore becomes possible to address this problem through the change of multiscale coordinates with multiple parameters.

In this paper we further generalize the notion of $\tau$-WD through the change of multiscale coordinates ${\mathfrak{T}}_{\mathcal{M}}$ with a parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$ according to ${\mathfrak{T}}_{\mathcal{M}}h(\mathbf{x},\mathbf{y}):=h(\mathbf{x}+\mathbf{y}\mathcal{M},\mathbf{x}-\mathbf{y}({\mathbf{I}}_N-\mathcal{M}))$, where ${\mathbf{I}}_N$ denotes the $N\times N$ identity matrix and the class $S^{0,1}\left(N\right)$ consists of all $N\times N$ matrices $\mathcal{M}=\left(\begin{array}{c}{a}_{nm}\end{array}\right)$ satisfying $0\le {\displaystyle \sum _{n=1}^N}{a}_{nm}\le 1\mathrm{for}\mathrm{all}$ $m=1,\cdots ,N$. Namely, for a fixed $N\times N$ parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$, we define the matrix-Wigner distribution (M-WD) of $f\in L^2\left({\mathbb{R}}^N\right)$ by

$$\mathrm{W}\left(\mathcal{M}\right)f(\mathbf{x},\mathbf{w})=\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})(\mathbf{x},\mathbf{w}).$$

(1)

Note that ${\mathbf{I}}_N-\mathcal{M}\in S^{0,1}\left(N\right)$ because of $\mathcal{M}\in S^{0,1}\left(N\right)$, and vice versa. It should also be emphasized here that the M-WD is a special case of the linear perturbations for the WD; see Equation (3) in [41] or Definition 1 in [42].

The $\tau$-WD is a special case of the M-WD corresponding to $\mathcal{M}=\tau {\mathbf{I}}_N$, i.e., $\mathrm{W}\left(\tau {\mathbf{I}}_N\right)f=\mathrm{W}\left(\tau \right)f$. In this case, the conditions $0\le {\displaystyle \sum _{n=1}^N}{a}_{nm}\le 1\mathrm{for}\mathrm{all}m=1,\cdots ,N$ reduce to $\tau \in [0,1]$, which is an essential constraint on the parameter $\tau$ of the $\tau$-WD. This is the main reason why we impose the constraint $\mathcal{M}\in S^{0,1}\left(N\right)$. For $\mathcal{M}={\mathbf{0}}_N$ (i.e., $\tau =0$; here ${\mathbf{0}}_N$ denotes the $N\times N$ null matrix), it corresponds to the Rihaczek transform, while for $\mathcal{M}={\mathbf{I}}_N$ (i.e., $\tau =1$), it corresponds to the conjugate Rihaczek transform. In the cases of $\mathcal{M}\ne \tau {\mathbf{I}}_N$, the M-WD provides a set of new time–frequency distributions that fundamentally vary from the $\tau$-WD. Therefore, the parameter matrix $\mathcal{M}$ may enable the M-WD to have the capacity to enhance time–frequency resolution. To verify this statement, we will need to demonstrate that the optimal M-WDs achieving the highest time–frequency resolution are not unique; they can be either the conventional WD or not. In other words, there are other M-WDs besides the conventional WD that can generate the highest time–frequency resolution.

Motivated by recent work [20], we explore the standard Heisenberg’s uncertainty principles for the M-WD, upon which we establish and solve the minimum lower bound problem to yield the optimality condition that the parameter matrix $\mathcal{M}$ should satisfy.

2. Spreads in M-WD Domains

Heisenberg’s uncertainty principles of the M-WD encompass a series of Heisenberg’s inequalities concerning the uncertainty product in time-M-WD and frequency-M-WD domains. The uncertainty product is determined by the spreads in time-M-WD and frequency-M-WD domains, necessitating a foundational understanding of their background and notation.

The spread is characterized by the second moment normalized using $L^2$-norm, induced by the inner product. First, simplify the $L^2$-norm ${\parallel \mathrm{W}\left(\mathcal{M}\right)f\parallel }_2$ within the spreads in M-WD domains. This simplification can be achieved according to the Parseval’s theorem of the FT and the invariance of $L^2$-norm under the coordinate transformation ${\mathfrak{T}}_{\mathcal{M}}$, reducing ${\parallel \mathrm{W}\left(\mathcal{M}\right)f\parallel }_2$ to ${\parallel f\parallel }_2^2$. Then the spreads in time-M-WD and frequency-M-WD domains are given below as follows:

Definition 1.

Let $\mathrm{W}\left(\mathcal{M}\right)f$ represent the M-WD of $f\in L^2\left({\mathbb{R}}^N\right)$ with respect to the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$. It is then given by the following: (i) Spread in the time-M-WD domain: $\Delta {\mathbf{x}}_{\mathcal{M}}^2=\frac{{∥\left(\mathbf{x}-{\mathbf{x}}_{\mathcal{M}}^0\right)\mathrm{W}\left(\mathcal{M}\right)f∥}_2^2}{{\parallel f\parallel }_2^4}$, where the moment vector in the time-M-WD domain: ${\mathbf{x}}_{\mathcal{M}}^0=\frac{\langle \mathbf{x}\mathrm{W}\left(\mathcal{M}\right)f,\mathrm{W}\left(\mathcal{M}\right)f\rangle }{{\parallel f\parallel }_2^4}$; (ii) Spread in the frequency-M-WD domain: $\Delta {\mathbf{w}}_{\mathcal{M}}^2=\frac{{∥\left(\mathbf{w}-{\mathbf{w}}_{\mathcal{M}}^0\right)\mathrm{W}\left(\mathcal{M}\right)f∥}_2^2}{{\parallel f\parallel }_2^4}$, where the moment vector in the frequency-M-WD domain: ${\mathbf{w}}_{\mathcal{M}}^0=\frac{\langle \mathbf{w}\mathrm{W}\left(\mathcal{M}\right)f,\mathrm{W}\left(\mathcal{M}\right)f\rangle }{{\parallel f\parallel }_2^4}$. Here $\langle \circ (\mathbf{x},\mathbf{w}),\diamond (\mathbf{x},\mathbf{w})\rangle ={\int }_{{\mathbb{R}}^{2N}}\circ (\mathbf{x},\mathbf{w})\overline{\diamond (\mathbf{x},\mathbf{w})}\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{w}$ denotes the inner product for functions defined on ${\mathbb{R}}^{2N}$.

Remark 1.

In the case of $\mathcal{M}=\tau {\mathbf{I}}_N$, the spread $\Delta {\mathbf{x}}_{\mathcal{M}}^2$ in the time-M-WD domain and the spread $\Delta {\mathbf{w}}_{\mathcal{M}}^2$ in the frequency-M-WD domain reduce to the spread $\Delta {\mathbf{x}}_{\tau }^2=\frac{{∥\left(\mathbf{x}-{\mathbf{x}}_{\tau }^0\right)\mathrm{W}\left(\tau \right)f∥}_2^2}{{\parallel f\parallel }_2^4}$ in the time-τ-WD domain and the spread $\Delta {\mathbf{w}}_{\tau }^2=\frac{{∥\left(\mathbf{w}-{\mathbf{w}}_{\tau }^0\right)\mathrm{W}\left(\tau \right)f∥}_2^2}{{\parallel f\parallel }_2^4}$ in the frequency-τ-WD domain, respectively. Here, ${\mathbf{x}}_{\tau }^0=\frac{\langle \mathbf{x}\mathrm{W}\left(\tau \right)f,\mathrm{W}\left(\tau \right)f\rangle }{{\parallel f\parallel }_2^4}$ and ${\mathbf{w}}_{\tau }^0=\frac{\langle \mathbf{w}\mathrm{W}\left(\tau \right)f,\mathrm{W}\left(\tau \right)f\rangle }{{\parallel f\parallel }_2^4}$ denote the moment vectors in time-τ-WD and frequency-τ-WD domains, respectively.

To establish Heisenberg’s inequalities on the uncertainty product $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ in time-M-WD and frequency-M-WD domains, it is imperative to provide an explanation of the background and symbols.

Definition 2

(See Definitions 1.3 and 1.4 in [43]). Let $\mathcal{F}f$ and $\mathrm{tr}(·)$ represent the FT of f and the trace operator, respectively. For any $f,\mathbf{x}f,\mathbf{w}\mathcal{F}f\in L^2\left({\mathbb{R}}^N\right)$, it is given by the following: (i) Spread matrix and spread in the time domain: $\mathbf{X}=\left(\begin{array}{c}\Delta {\mathbf{x}}_{n,m}^2\end{array}\right)=\frac{〈{\left(\mathbf{x}-{\mathbf{x}}^0\right)}^{\mathrm{T}}f,\left(\mathbf{x}-{\mathbf{x}}^0\right)f〉}{{\parallel f\parallel }_2^2}$ and $\Delta {\mathbf{x}}^2=\mathrm{tr}\left(\mathbf{X}\right)=\frac{{∥\left(\mathbf{x}-{\mathbf{x}}^0\right)f∥}_2^2}{{\parallel f\parallel }_2^2}$, where the moment vector in the time domain: ${\mathbf{x}}^0=\frac{\langle \mathbf{x}f,f\rangle }{{\parallel f\parallel }_2^2}$; (ii) Spread matrix and spread in the frequency domain: $\mathbf{W}=\left(\begin{array}{c}\Delta {\mathbf{w}}_{n,m}^2\end{array}\right)=\frac{〈{\left(\mathbf{w}-{\mathbf{w}}^0\right)}^{\mathrm{T}}\mathcal{F}f,\left(\mathbf{w}-{\mathbf{w}}^0\right)\mathcal{F}f〉}{{\parallel f\parallel }_2^2}$ and $\Delta {\mathbf{w}}^2=\mathrm{tr}\left(\mathbf{W}\right)=\frac{{∥\left(\mathbf{w}-{\mathbf{w}}^0\right)\mathcal{F}f∥}_2^2}{{\parallel f\parallel }_2^2}$, where the moment vector in the frequency domain: ${\mathbf{w}}^0=\frac{\langle \mathbf{w}\mathcal{F}f,\mathcal{F}f\rangle }{{\parallel f\parallel }_2^2}$.

Definition 3

(See Definition 1.3 in [43]). Let $\mathcal{F}f$, $|·|$ and ${\nabla }_{\mathbf{x}}(·)=\left(\frac{\partial (·)}{\partial x_1},\cdots ,\frac{\partial (·)}{\partial x_N}\right)$ denote the FT of f, the absolute operator that applies element-wise absolute value to vectors, and the gradient operator of functions defined on $\mathbf{x}\in {\mathbb{R}}^N$, respectively. For a complex-valued function $f=\lambda {\mathrm{e}}^{2\pi \mathrm{i}\phi }$, if $f,\mathbf{x}f,\mathbf{w}\mathcal{F}f\in L^2\left({\mathbb{R}}^N\right)$, ${\nabla }_{\mathbf{x}}\lambda ,{\nabla }_{\mathbf{x}}\phi ,{\nabla }_{\mathbf{x}}f$ exist at any point $\mathbf{x}\in {\mathbb{R}}^N$, it is then given by the following: (i) Covariance in the time–frequency domain: ${\mathrm{Cov}}_{\mathbf{x},\mathbf{w}}=\frac{〈\left(\mathbf{x}-{\mathbf{x}}^0\right)f,{\left({\nabla }_{\mathbf{x}}\phi -{\mathbf{w}}^0\right)}^{\mathrm{T}}f〉}{{\parallel f\parallel }_2^2}$; (ii) Absolute covariance in the time–frequency domain: ${\mathrm{COV}}_{\mathbf{x},\mathbf{w}}=\frac{〈\left|\mathbf{x}-{\mathbf{x}}^0\right|f,{\left|{\nabla }_{\mathbf{x}}\phi -{\mathbf{w}}^0\right|}^{\mathrm{T}}f〉}{{\parallel f\parallel }_2^2}$.

3. Uncertainty Product in M-WD Domains

Section 2 has provided some preliminary knowledge and works. Building on this foundation, the primary focus of this section is to transform the uncertainty product $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ in time-M-WD and frequency-M-WD domains into the uncertainty product $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2$ in time and frequency domains. This conversion represents an essential step toward formulating Heisenberg’s inequalities concerning $\Delta \mathbf{x}{\mathcal{M}}^2\Delta \mathbf{w}{\mathcal{M}}^2$.

3.1. Spread in the Time-M-WD Domain Revisited

This section converts the spread $\Delta {\mathbf{x}}_{\mathcal{M}}^2$ in the time-M-WD domain to the spread $\Delta {\mathbf{x}}^2$ in the time domain. The original definition $\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})$ of the M-WD proves effective for this purpose, as the tensor product $f\otimes \overline{f}$ explicitly contains only time variables.

We first disclose a trace relation between $\Delta {\mathbf{x}}_{\mathcal{M}}^2$ and the spread matrix $\mathbf{X}$ in the time domain.

Lemma 1.

Let $\mathrm{W}\left(\mathcal{M}\right)f$ denote the M-WD of f with respect to the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$. If $f,\mathbf{x}f\in L^2\left({\mathbb{R}}^N\right)$, there is a trace relation

$$\Delta {\mathbf{x}}_{\mathcal{M}}^2=\mathrm{tr}\left(P_{\mathcal{M}}\mathbf{X}\right),$$

(2)

where $P_{\mathcal{M}}=2\mathcal{M}{\mathcal{M}}^{\mathrm{T}}-\mathcal{M}-{\mathcal{M}}^{\mathrm{T}}+{\mathbf{I}}_N$.

Proof. 

According to the definition of the M-WD, i.e., $\mathrm{W}\left(\mathcal{M}\right)f=\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})$, the moment vector ${\mathbf{x}}_{\mathcal{M}}^0$ in the time-M-WD domain can be expressed as

$$\begin{array}{cc}\hfill {\mathbf{x}}_{\mathcal{M}}^0=& \frac{〈\mathbf{x}\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f}),\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})〉}{{\parallel f\parallel }_2^4}\hfill \\ \hfill =& \frac{〈\mathbf{x}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f}),{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})〉}{{\parallel f\parallel }_2^4}.\hfill \end{array}$$

By introducing the variables substitution $(\mathbf{x},\mathbf{y})\to (\mathbf{x},\mathbf{y})$$\left(\begin{array}{cc}{\mathbf{I}}_N-\mathcal{M}& {\mathbf{I}}_N\\ \mathcal{M}& -{\mathbf{I}}_N\end{array}\right)$ yields

$$\begin{array}{cc}\hfill {\mathbf{x}}_{\mathcal{M}}^0=& \frac{〈[\mathbf{x}({\mathbf{I}}_N-\mathcal{M})+\mathbf{y}\mathcal{M}]f\otimes \overline{f},f\otimes \overline{f}〉}{{\parallel f\parallel }_2^4}\hfill \\ \hfill =& \frac{〈\mathbf{x}f,f〉}{{\parallel f\parallel }_2^2}({\mathbf{I}}_N-\mathcal{M})+\frac{〈\mathbf{y}\overline{f},\overline{f}〉}{{\parallel f\parallel }_2^2}\mathcal{M}\hfill \\ \hfill =& {\mathbf{x}}^0.\hfill \end{array}$$

Then, the spread $\Delta {\mathbf{x}}_{\mathcal{M}}^2$ in the time-M-WD domain becomes

$$\begin{array}{cc}\hfill \Delta {\mathbf{x}}_{\mathcal{M}}^2=& \frac{{∥\left[\left(\mathbf{x}-{\mathbf{x}}^0\right)({\mathbf{I}}_N-\mathcal{M})+\left(\mathbf{y}-{\mathbf{x}}^0\right)\mathcal{M}\right]f\otimes \overline{f}∥}_2^2}{{\parallel f\parallel }_2^4}\hfill \\ \hfill =& \frac{{∥\left(\mathbf{x}-{\mathbf{x}}^0\right)({\mathbf{I}}_N-\mathcal{M})f∥}_2^2}{{\parallel f\parallel }_2^2}+\frac{{∥\left(\mathbf{y}-{\mathbf{x}}^0\right)\mathcal{M}\overline{f}∥}_2^2}{{\parallel f\parallel }_2^2}\hfill \\ \hfill =& \mathrm{tr}\left({({\mathbf{I}}_N-\mathcal{M})}^{\mathrm{T}}\mathbf{X}({\mathbf{I}}_N-\mathcal{M})\right)+\mathrm{tr}\left({\mathcal{M}}^{\mathrm{T}}\mathbf{X}\mathcal{M}\right).\hfill \end{array}$$

Due to the cyclic permutation invariance of trace, we arrive at the conclusion. □

Motivated by the observations of $\Delta {\mathbf{x}}^2=\mathrm{tr}\left(\mathbf{X}\right)$, the symmetry of $P_{\mathcal{M}}$ ($P_{\mathcal{M}}$ is indeed positive definite) and the positive semidefiniteness of $\mathbf{X}$ (see Proposition 2.3 in [44]), we further formulate an inequality relation between $\Delta {\mathbf{x}}_{\mathcal{M}}^2$ and $\Delta {\mathbf{x}}^2$.

Corollary 1.

Let $\mathrm{W}\left(\mathcal{M}\right)f$ represent the M-WD of f with respect to the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$. If $f,\mathbf{x}f\in L^2\left({\mathbb{R}}^N\right)$, there is an inequality relation

$$\Delta {\mathbf{x}}_{\mathcal{M}}^2\ge {\lambda }_{min}\left(P_{\mathcal{M}}\right)\Delta {\mathbf{x}}^2,$$

(3)

where ${\lambda }_{min}(·)$ is the operator for the minimum eigenvalue of a matrix. The equality holds if $P_{\mathcal{M}}$ is a scalar matrix.

Proof. 

By employing the well-known trace inequality, i.e., $\mathrm{tr}\left(\mathbf{U}\mathbf{V}\right)\ge {\lambda }_{min}\left(\mathbf{U}\right)\mathrm{tr}\left(\mathbf{V}\right)$, where $\mathbf{U}$ is a symmetric matrix and $\mathbf{V}$ is a positive semidefinite matrix, referring to Lemma 1 in [45] or the inequality (1) in [46] concerning the right-hand side of (2), we arrive at the conclusion. □

3.2. Spread in the Frequency-M-WD Domain Revisited

This section converts the spread $\Delta {\mathbf{w}}_{\mathcal{M}}^2$ in the frequency-M-WD domain to the spread $\Delta {\mathbf{w}}^2$ in the frequency domain. However, due to the tensor product $f\otimes \overline{f}$ being a function of time variables, the original definition of the M-WD, $\mathcal{F}{\mathfrak{T}}_{\mathcal{M}}(f\otimes \overline{f})$, is no longer an effective technology for this task. Hence, the M-WD must be rewritten to produce a tensor product solely involving frequency variables.

Let ${\mathcal{F}}^{-1}$, $\delta$, $T_{\mathbf{w},}$ and $M_{\mathbf{x}}$ denote the inverse Fourier, Dirac delta, translation, and modulation operators, respectively. Then, the M-WD can be rewritten in terms of the FT as

$$\begin{array}{cc}\hfill \mathrm{W}\left(\mathcal{M}\right)f(\mathbf{x},\mathbf{w})=& \mathcal{F}{\mathfrak{T}}_{\mathcal{M}}\left({\mathcal{F}}^{-1}\left(\mathcal{F}f\right)\otimes \overline{{\mathcal{F}}^{-1}\left(\mathcal{F}f\right)}\right)(\mathbf{x},\mathbf{w})\hfill \\ \hfill =& 〈M_{\mathbf{x}}\mathcal{F}f\otimes \overline{M_{\mathbf{x}}\mathcal{F}f},T_{\mathbf{w}}\delta \left(\mathbf{u}\mathcal{M}+\mathbf{v}({\mathbf{I}}_N-\mathcal{M})\right)〉.\hfill \end{array}$$

(4)

By first introducing the variables substitution $(\mathbf{u},\mathbf{v})\to (\mathbf{u},\mathbf{v})\left(\begin{array}{cc}{\mathbf{I}}_N& {\mathbf{I}}_N\\ -({\mathbf{I}}_N-\mathcal{M})& \mathcal{M}\end{array}\right)$ and then using the sifting property of Dirac delta functions, Equation (4) becomes

$$\begin{array}{cc}\hfill \mathrm{W}\left(\mathcal{M}\right)f(\mathbf{x},\mathbf{w})=& 〈{\mathfrak{T}}_{\mathcal{M}}\left(\overline{M_{\mathbf{x}}\mathcal{F}f}\otimes M_{\mathbf{x}}\mathcal{F}f\right),T_{\mathbf{w}}\delta \left(\mathbf{u}\right)〉\hfill \\ \hfill =& \mathcal{F}{\mathfrak{T}}_{\mathcal{M}}\left(\overline{\mathcal{F}f}\otimes \mathcal{F}f\right)(\mathbf{w},\mathbf{x}),\hfill \end{array}$$

(5)

which is a valid technique for converting $\Delta {\mathbf{w}}_{\mathcal{M}}^2$ to $\Delta {\mathbf{w}}^2$, since only frequency variables are explicitly present in the tensor product $\overline{\mathcal{F}f}\otimes \mathcal{F}f$. Indeed, combining (1) and (5) gives $\mathrm{W}\left(\mathcal{M}\right)f(\mathbf{x},\mathbf{w})=\mathrm{W}\left(\mathcal{M}\right)\overline{\mathcal{F}f}(\mathbf{w},\mathbf{x})$, which implies that the M-WD of f and the M-WD of its FT’s conjugate $\overline{\mathcal{F}f}$ are symmetric with respect to $\mathbf{x}=\mathbf{w}$. With Lemma 1 and this symmetry, there is a trace relation between $\Delta {\mathbf{w}}_{\mathcal{M}}^2$ and the spread matrix $\mathbf{W}$ in the frequency domain

$$\Delta {\mathbf{w}}_{\mathcal{M}}^2=\mathrm{tr}\left(P_{\mathcal{M}}\mathbf{W}\right),$$

and then similar to Corollary 1, there is an inequality relation between $\Delta {\mathbf{w}}_{\mathcal{M}}^2$ and $\Delta {\mathbf{w}}^2$:

$$\Delta {\mathbf{w}}_{\mathcal{M}}^2\ge {\lambda }_{min}\left(P_{\mathcal{M}}\right)\Delta {\mathbf{w}}^2.$$

(6)

3.3. The Product of Spreads

By multiplying (3) and (6), $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ can be converted to $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2$.

Corollary 2.

Let $\mathcal{F}f$ and $\mathrm{W}\left(\mathcal{M}\right)f$ denote the FT of f and the M-WD of f with the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$, respectively. If $f,\mathbf{x}f,\mathbf{w}\mathcal{F}f\in L^2\left({\mathbb{R}}^N\right)$, there is an inequality relation between $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ and $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2$:

$$\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2\ge {\lambda }_{min}^2\left(P_{\mathcal{M}}\right)\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2.$$

(7)

The equality holds if $P_{\mathcal{M}}$ is a scalar matrix.

4. Heisenberg’s Inequalities in the M-WD Setting

The estimation of $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ has been translated to that of $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2$, as implied by Corollary 2. Then, the existing lower bounds for all functions and complex-valued functions on $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2$ can be used to obtain the corresponding lower bounds on $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$.

The first type is the tightest universal lower bound for all functions given below.

Theorem 1.

Let $\mathcal{F}f$ be the FT of f and $\mathrm{W}\left(\mathcal{M}\right)f$ be the M-WD of f associated with the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$, and let $f,\mathbf{x}f,\mathbf{w}\mathcal{F}f\in L^2\left({\mathbb{R}}^N\right)$. If ${\nabla }_{\mathbf{x}}f$ exists at any point $\mathbf{x}\in {\mathbb{R}}^N$, an inequality exists as follows:

$$\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2\ge \frac{N^2}{16{\pi }^2}{\lambda }_{min}^2\left(P_{\mathcal{M}}\right).$$

(8)

When f is nearly nonzero everywhere, if $P_{\mathcal{M}}$ is a scalar matrix and f is the optimal Gaussian-enveloped complex exponential function that is ${\mathrm{e}}^{-\frac{1}{2\zeta }{∥\mathbf{x}-{\mathbf{x}}^0∥}^2+\sigma }{\mathrm{e}}^{2\pi \mathrm{i}\left({\mathbf{w}}^0{\mathbf{x}}^{\mathrm{T}}+\varsigma \right)}$, where $\zeta >0$ and $\sigma ,\varsigma \in \mathbb{R}$, the equation holds.

Proof. 

By combining (7) with a well-known inequality $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2\ge \frac{N^2}{16{\pi }^2}$ (see Corollary 2.8 in [13]), we arrive at the conclusion. □

The other category is the lower bounds for complex-valued functions.

Theorem 2.

For a complex-valued function $f=\lambda {\mathrm{e}}^{2\pi \mathrm{i}\phi }$, let $\mathcal{F}f$ and $\mathrm{W}\left(\mathcal{M}\right)f$ represent the FT of f and the M-WD of f with respect to the parameter matrix $\mathcal{M}\in S^{0,1}\left(N\right)$, respectively. If $f,\mathbf{x}f,\mathbf{w}\mathcal{F}f\in L^2\left({\mathbb{R}}^N\right)$, ${\nabla }_{\mathbf{x}}\lambda ,{\nabla }_{\mathbf{x}}\phi ,{\nabla }_{\mathbf{x}}f$ exist at any point $\mathbf{x}\in {\mathbb{R}}^N$, an inequality chain exists as follows:

$$\begin{array}{cc}\hfill \Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2& \ge \left(\frac{N^2}{16{\pi }^2}+{\mathrm{COV}}_{\mathbf{x},\mathbf{w}}^2\right){\lambda }_{min}^2\left(P_{\mathcal{M}}\right)\hfill \\ \hfill & \ge \left(\frac{N^2}{16{\pi }^2}+{\mathrm{Cov}}_{\mathbf{x},\mathbf{w}}^2\right){\lambda }_{min}^2\left(P_{\mathcal{M}}\right).\hfill \end{array}$$

(9)

When ${\nabla }_{\mathbf{x}}\phi$ is continuous and λ is nearly nonzero everywhere, if $P_{\mathcal{M}}$ is a scalar matrix and f is the optimal Gaussian-enveloped chirp function, i.e., Equation (4) in [24], the first equality holds; if $P_{\mathcal{M}}$ is a scalar matrix and f is the optimal Gaussian-enveloped chirp function with ${\mathbf{k}}_{j_3}={\mathbf{k}}_{j_4}=\varnothing$, where ${\mathbf{k}}_{j_3}$ and ${\mathbf{k}}_{j_4}$ are given respectively by Equations (8) and (9) in [24], the second equality holds.

Proof. 

By combining (7) with the inequality $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2\ge \frac{N^2}{16{\pi }^2}+{\mathrm{COV}}_{\mathbf{x},\mathbf{w}}^2$ (see Corollary 1 in [24] or Theorem 2.2 in [23]), and with the inequality $\Delta {\mathbf{x}}^2\Delta {\mathbf{w}}^2\ge \frac{N^2}{16{\pi }^2}+{\mathrm{Cov}}_{\mathbf{x},\mathbf{w}}^2$ (see Chapter 3 in [6]), we arrive at the conclusion. □

Remark 2.

In the case of $\mathcal{M}=\tau {\mathbf{I}}_N$, the inequality (8) and the inequality chain (9) reduce to $\Delta {\mathbf{x}}_{\tau }^2\Delta {\mathbf{w}}_{\tau }^2\ge \frac{N^2}{16{\pi }^2}{(2{\tau }^2-2\tau +1)}^2$ and $\Delta {\mathbf{x}}_{\tau }^2\Delta {\mathbf{w}}_{\tau }^2\ge \left(\frac{N^2}{16{\pi }^2}+{\mathrm{COV}}_{\mathbf{x},\mathbf{w}}^2\right){(2{\tau }^2-2\tau +1)}^2\ge \left(\frac{N^2}{16{\pi }^2}+{\mathrm{Cov}}_{\mathbf{x},\mathbf{w}}^2\right){(2{\tau }^2-2\tau +1)}^2$, respectively, i.e., Heisenberg’s inequalities in the τ-WD setting introduced in [20].

5. The Optimal M-WDs

Minimizing the lower bounds found in (8) and (9) yields the optimal M-WDs achieving the highest time–frequency resolution. Evidently, this equates to addressing the minimization problem

$$\underset{\mathcal{M}\in S^{0,1}\left(N\right)}{min}{\lambda }_{min}\left(P_{\mathcal{M}}\right),$$

(10)

namely,

$$\underset{\mathcal{M}\in S^{0,1}\left(N\right),\parallel \mathbf{u}\parallel =1}{min}\mathbf{u}{P}_{\mathcal{M}}{\mathbf{u}}^{\mathrm{T}}.$$

(11)

Using the Lagrange multipliers method, we introduce an auxiliary function, commonly referred to as the Lagrange function

$$\mathcal{L}(\mathcal{M},\mathbf{u},\rho )=\mathbf{u}{P}_{\mathcal{M}}{\mathbf{u}}^{\mathrm{T}}+\rho (\mathbf{u}{\mathbf{u}}^{\mathrm{T}}-1),$$

and solve a system of algebraic equations

$$\{\begin{array}{}\mathrm{(12)}& \frac{\partial \mathcal{L}}{\partial \mathcal{M}}=2{\mathbf{u}}^{\mathrm{T}}\mathbf{u}(2\mathcal{M}-{\mathbf{I}}_N)={\mathbf{0}}_N,\mathrm{(13)}& \frac{\partial \mathcal{L}}{\partial \mathbf{u}}=2\left(P_{\mathcal{M}}+\rho {\mathbf{I}}_N\right){\mathbf{u}}^{\mathrm{T}}={\mathbf{0}}^{\mathrm{T}},\mathrm{(14)}& \frac{\partial \mathcal{L}}{\partial \rho }=\mathbf{u}{\mathbf{u}}^{\mathrm{T}}-1=0,\end{array}$$

where $\mathbf{0}$ denotes the null vector in ${\mathbb{R}}^N$. It is clear that $\mathcal{M}=\frac{{\mathbf{I}}_N}{2}$, $\mathbf{u}{\mathbf{u}}^{\mathrm{T}}=1$, and $\rho =-\frac{1}{2}$ satisfy the above equations. Substituting $\rho =-\frac{1}{2}$ into (13) gives $\mathbf{u}(2\mathcal{M}-{\mathbf{I}}_N)=\mathbf{0}$, which implies (12). Therefore, the optimal solutions of (11) are given by $\mathbf{u}(2\mathcal{M}-{\mathbf{I}}_N)=\mathbf{0}$ and $\mathbf{u}{\mathbf{u}}^{\mathrm{T}}=1$, from which we draw the following conclusion.

Theorem 3.

The optimality condition of the parameter matrix $\mathcal{M}$ of the lower bounds optimization problem (i.e., the minimization problem (10)) is that the determinant of $2\mathcal{M}-{\mathbf{I}}_N$ is equal to zero, i.e., $\det (2\mathcal{M}-{\mathbf{I}}_N)=0$.

Proof. 

The homogeneous system of linear equations $\mathbf{u}(2\mathcal{M}-{\mathbf{I}}_N)=\mathbf{0}$ has a nonzero solution ${\mathbf{u}}_0$ whose unitization $\frac{{\mathbf{u}}_0}{\parallel {\mathbf{u}}_0\parallel }$ is also the solution if and only if the rank of the coefficient matrix $(2\mathcal{M}-{\mathbf{I}}_N)$ is less than N, i.e., $\mathrm{Rank}(2\mathcal{M}-{\mathbf{I}}_N)<N$. □

Remark 3.

The class $\left\{\mathrm{W}\left(\mathcal{M}\right)f\right|\mathcal{M}\in S^{0,1}\left(N\right),\det (2\mathcal{M}-{\mathbf{I}}_N)=0\}$ consists of all of the optimal M-WDs achieving the highest time–frequency resolution. It contains not only the conventional WD corresponding to $\mathcal{M}=\frac{{\mathbf{I}}_N}{2}$, but also other distributions for $1\le \mathrm{Rank}(2\mathcal{M}-{\mathbf{I}}_N)<N$ which seem more useful and effective than the WD in time–frequency analysis because of some degrees of freedom.

Example 1.

For a two-dimensional linear frequency-modulated signal $f(x_1,x_2)=$${\mathrm{e}}^{\pi \mathrm{i}\left(x_1+2{x_1}^2+3x_2+4{x_2}^2\right)}$, the M-WD of f with the matrix $\mathcal{M}=\left(\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{5}\end{array}\right)$ is

$$\begin{array}{cc}\hfill & \mathrm{W}\left(\mathcal{M}\right)f(x_1,x_2,w_1,w_2)=\mathcal{F}{\mathfrak{T}}_{\left(\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{5}\end{array}\right)}(f\otimes \overline{f})(x_1,x_2,w_1,w_2)\hfill \\ \hfill =& {\int \int }_{{\mathbb{R}}^2}f\left(x_1+\frac{1}{2}{y}_1,x_2+\frac{1}{5}{y}_2\right)\overline{f\left(x_1-\frac{1}{2}{y}_1,x_2-\frac{4}{5}{y}_2\right)}{\mathrm{e}}^{-2\pi \mathrm{i}(y_1{w}_1+y_2{w}_2)}\mathrm{d}{y}_1\mathrm{d}{y}_2\hfill \\ \hfill =& {\int }_{\mathbb{R}}{\mathrm{e}}^{\pi \mathrm{i}\left(y_1+4x_1{y}_1-2w_1{y}_1\right)}\mathrm{d}{y}_1{\int }_{\mathbb{R}}{\mathrm{e}}^{-\frac{12}{5}\pi \mathrm{i}{y_2}^2}{\mathrm{e}}^{\pi \mathrm{i}\left(3y_2+8x_2{y}_2-2w_2{y}_2\right)}\mathrm{d}{y}_2.\hfill \end{array}$$

(15)

According to the Gauss integral formula, it follows that

$$\begin{array}{cc}\hfill \left|\mathrm{W}\left(\mathcal{M}\right)f(x_1,x_2,w_1,w_2)\right|=& \left|\delta \left(\frac{1}{2}+2x_1-w_1\right)\sqrt{\frac{5}{12\mathrm{i}}}{\mathrm{e}}^{\frac{5\pi \mathrm{i}{\left(3+8x_2-2w_2\right)}^2}{48}}\right|\hfill \\ \hfill =& \sqrt{\frac{5}{12}}\delta \left(\frac{1}{2}+2x_1-w_1\right).\hfill \end{array}$$

(16)

From the above equation, it is evident that the M-WD of the linear frequency-modulated signal f can generate impulse, achieving excellent time–frequency concentration. The short-time Fourier transform (STFT) and wavelet transform (WT) are also commonly used tools for time–frequency analysis. The STFT the signal f is given by

$$\begin{array}{c}\hfill {\mathrm{STFT}}_f(x_1,x_2,w_1,w_2)={\int \int }_{{\mathbb{R}}^2}{\mathrm{e}}^{\pi \mathrm{i}\left(t_1+2{t_1}^2+3t_2+4{t_2}^2\right)}g(t_1-x_1,t_2-x_2){\mathrm{e}}^{-2\pi \mathrm{i}(t_1{w}_1+t_2{w}_2)}\mathrm{d}{t}_1\mathrm{d}{t}_2,\end{array}$$

(17)

where g is the window function. The WT the signal f is given by

$$\begin{array}{c}\hfill {\mathrm{WT}}_f(x_1,x_2,w_1,w_2)={\int \int }_{{\mathbb{R}}^2}{\mathrm{e}}^{\pi \mathrm{i}\left(t_1+2{t_1}^2+3t_2+4{t_2}^2\right)}\frac{1}{\sqrt{\left|x_1{x}_2\right|}}\psi \left(\frac{t_1-w_1}{x_1},\frac{t_2-w_2}{x_2}\right)\mathrm{d}{t}_1\mathrm{d}{t}_2,\end{array}$$

(18)

where ψ is the wavelet function.

As seen from Equations (16)–(18), the STFT partitions a signal into short, overlapping segments and computes the FT for each segment, but its resolution is limited by the fixed window size, creating a trade-off between time and frequency resolution. The WT, using variable-sized windows for different frequency components, also faces limitations based on the chosen mother wavelet. Despite their adaptive nature, neither STFT nor WT achieves optimal time–frequency concentration. In comparison, the M-WD provides superior time–frequency concentration, accurately representing the signal’s energy distribution with in both time and frequency domains.

6. Conclusions

The single scale $\tau$-WD was extended to the M-WD through the change of multiscale coordinates ${\mathfrak{T}}_{\mathcal{M}}$ with a parameter matrix $\mathcal{M}$. Heisenberg’s uncertainty principles for the M-WD were established, providing two types of attainable lower bounds on the uncertainty product $\Delta {\mathbf{x}}_{\mathcal{M}}^2\Delta {\mathbf{w}}_{\mathcal{M}}^2$ in time-M-WD and frequency-M-WD domains. These lower bounds are attained by the optimal Gaussian-enveloped complex exponential function or the optimal Gaussian-enveloped chirp function. It turns out that they are the functions of $\mathcal{M}$, and the relevant section of $\mathcal{M}$ for both reads ${\lambda }_{min}^2\left(2\mathcal{M}{\mathcal{M}}^{\mathrm{T}}-\mathcal{M}-{\mathcal{M}}^{\mathrm{T}}+{\mathbf{I}}_N\right)$. It reaches its minimum at the optimality condition $\det (2\mathcal{M}-{\mathbf{I}}_N)=0$, based on which the optimal $\tau$-WD achieving the highest time–frequency resolution is just the conventional WD. The optimal M-WDs break through this limitation to generate other distributions with some degrees of freedom besides the WD. A comparison of the M-WD and some classical time–frequency analysis tools, including the STFT and WT, indicates that it exhibits a prominent superiority in time–frequency concentration.