Moyal's Formula Research Articles

Uncertainty principles for coupled fractional Wigner-Ville distribution.

The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.

N‐dimensional wave packet transform and associated uncertainty principles in the free metaplectic transform domain

The free metaplectic transformation (FMT) is an ‐dimensional linear canonical transform. This transform is much useful, especially in multidimensional signal processing and applications. In this paper, our aim is to achieve an efficient time‐frequency representation of higher‐dimensional nonstationary signals by introducing the novel free metaplectic wave packet transform (FM‐WPT) in , based on the elegant convolution structure associated with the free metaplectic transforms. The FM‐WPT preserves the properties of classical wave packet transform (WPT) in and has better mathematical properties. Further, the validity of the proposed transform is demonstrated via a lucid example. The preliminary analysis encompasses the derivation of fundamental properties of the novel FM‐WPT, including boundedness, reconstruction formula, Moyal's formula, and the reproducing kernel. To extend the scope of the study, we formulate several uncertainty inequalities, including Lieb's inequality, Pitt's inequality, logarithmic inequality, Heisenberg's uncertainty inequality, and Nazarov's uncertainty inequality for the proposed transform.

Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles

<abstract><p>The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various applications in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short-time free metaplectic transform (ST-FMT). In this paper, we investigate the ST-FMT. First we propose the definition of the ST-FMT and provide the time-frequency analysis of the proposed transform in the FMT domain. Second we establish the relationship between the ST-FMT and short-time Fourier transform (STFT) in $ L^2(\mathbb R^n) $ and investigate the basic properties of the proposed transform including the reconstruction formula, Moyal's formula. The emergence of the ST-FMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. We extend some different uncertainty principles (UPs) from quantum mechanics including Lieb's inequality, Pitt's inequality, Hausdorff-Young inequality, Heisenberg's UP, Hardy's UP, Beurling's UP, Logarithmic UP and Nazarov's UP. Finally, we give a numerical example and a possible applications of the proposed ST-FMT.</p></abstract>

A sparse Bayesian framework for discovering interpretable nonlinear stochastic dynamical systems with Gaussian white noise

Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equation discovery are mostly applicable to deterministic systems and require both input and state measurements. We here propose a novel data-driven framework for discovering nonlinear stochastic dynamical systems with Gaussian white noise. The proposed framework blends concepts of stochastic calculus, sparse learning algorithms, and Bayesian statistics to learn the governing physics from data. In particular, we combine sparsity–promoting spike and slab prior, Bayes law, and the Kramers–Moyal formula to identify stochastic differential equations from data. The proposed framework is highly efficient and works with sparse, noisy, and incomplete output measurements. The efficacy and robustness of the proposed approach are illustrated in several numerical examples involving both complete and partial state measurements. The results obtained indicate the potential of the proposed approach in discovering nonlinear stochastic dynamical systems subjected to Gaussian white noise excitation.

On the polyanalytic short-time Fourier transform in the quaternionic setting

<p style='text-indent:20px;'>In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.</p>

On the Quaternionic Short-Time Fourier and Segal\u2013Bargmann Transforms

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

Linear canonical Stockwell transform

The linear canonical transform (LCT) has long been recognized as a competent tool for optics and signal processing. However, the LCT cannot reveal the local LCT-frequency contents due to its global kernel and as such makes the LCT incompetent in situations demanding a joint information of time and linear canonical domain-frequency. In this paper, we propose the linear canonical Stockwell transform (LCST) in order to rectify the limitations of the LCT and the Stockwell transform (ST). Firstly, we examine the resolution of the novel transform in the time and LCT domains and then derive some of its basic properties, such as Moyal's formula, inversion formula and provide a characterization of the range. Furthermore, we derive a direct relationship between the well known Wigner distribution and the proposed LCST. In sequel, a discrete version of the linear canonical Stockwell transform is also presented. Finally, we extend the scope of the proposed work by studying the LCST in the realm of almost periodic functions.

Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

Unified Wigner–Ville distribution and ambiguity function in the linear canonical transform domain

In this paper, we propose a novel integral transform by combining the generalized Wigner–Ville distribution (WDL) with the linear canonical transform (LCT). The new integral transform unifies the WDL and the generalized ambiguity function (AFL), and then can be considered as a generalization of the classical Wigner–Ville distribution (WVD) and ambiguity function (AF). Some useful properties of the new integral transform are derived, including conjugation symmetry property, conjugation invariance of LCT, marginal properties, shifting properties, anti-derivative property and Moyal formula. The relationships between the new integral transform and other common time–frequency analysis tools are discussed, such as the LCT, the short-time Fourier transform (STFT) and the short-time linear canonical transform (STLCT). The applications of the newly defined integral transform in the detection of one-component and bi-component linear frequency-modulated (LFM) signals embedded in white Gaussian noise are investigated. The comparisons of the detection performance of the new integral transform with that of the WDL and AFL are also performed to show the preponderance of the proposed techniques. The simulation results indicate that the new integral transform achieves better detection performance than the WDL and AFL.

Ambiguity function based on the linear canonical transform

The ambiguity function (AF) as an important tool for time-frequency analysis, has widely been used in radar signal processing, sonar technology and so on. It does well in analysing chirp signals. However, it fails in estimating the cubic phase (CP) signal, which is required in many applications. As the generalisation of the Fourier transform (FT), the linear canonical transform (LCT) has received much attention, and has found many applications in filter design, pattern recognition, optics and so on. In this study, the authors define a new kind of AF - the AF based on the LCT (LCTAF), which is proposed to estimate the CP signal. Some important properties of LCTAF are discussed, such as symmetry and conjugation property, shifting property and Moyal formula. The relationships between the LCTAF and other time-frequency analysis distributions are derived, including the classical AF, the Wigner distribution function (WDF) based on LCT, the short-FT and the wavelet transform. The linear canonical AF (LCAF) is another kind of AF. The authors also discuss its relation with AF and WDF and give some new properties of the LCAF. At last, the LCTAF is applied for estimating the CP signal. The simulation indicates that the LCTAF is useful and effective.

On deformation of associative algebras and graph homology

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an “almost contraction” leading to a canonical solution of the corresponding Maurer–Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as “initiator.” Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the “merger operation” provides a canonical solution at least in the case of linear Poisson structures.

A unified framework for the Sussman, Moyal, and Janssen formulas

This paper derives three fundamental identities in the radar and sonar literature, namely, Sussman' identity for ambiguity functions, Moyal's formula which establishes the value of the inner product between two scattering functions, and Janseen's formula which establishes identities for mixed inner products between waveforms and Gabor wavelets. Starting from the fundamental convolution identity, we derive Sussman's identity. Following from an initial value theorem of Fourier analysis, we obtained Moyal's formula. Following from Poisson's sum formula and an initial value theorem, we also obtained Janssen's equality. The relationship between these three identities is as follows: Janssen's formula is a sampled-data version of Moyal's formula, and both follow from Sussman's identity. In turn, Sussman's identity is a consequence of the fundamental convolution identity.

Bohr-Sommerfeld Rules to All Orders

H on the real line. The formula gives the highorder terms in the expansion in powers of h of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we address here is only about ways to compute these corrections. Existence of corrections to any order to Bohr-Sommerfeld rules is well known and can be found for example in [8] and [15] Section 4.5. Our way to get these high-order corrections is inspired by A. Voros’s thesis (1977) [13], [14]. The reference [1], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of

Discrete time and frequency Wigner-Ville distribution: Moyal's formula and aliasing

In this letter, we propose a new definition of the discrete time and frequency Wigner-Ville distribution. The proposed distribution not only displays a readable representation (small aliasing) but also exhibits unitarity and is easy to compute. We compare the time-frequency representation associated with this proposed definition with other existing ones.

Signal detection using time-frequency distributions with nonunity kernels

A new technique is proposed to solve the simple binary signal-detection problem using a nonunity kernel time-frequency signal detector (GNKD). The GNKD is based on a Cohen time-frequency power spectrum, employing nonunity kernels only. This class of signal detectors includes the Choi-Williams detector (CWWD) and the recently proposed hyperbolic detector (HyD). This work extends the work done by Kumar and Carroll, who investigated the cross unity-kernel Wigner-Ville detector (CWD), which is a special case of the GNKD class. The discrete Moyal's formula for the nonunity kernel time-frequency distribution is derived. The performance of the GNKD is then compared to that of the CWD and the cross-correlator (CORR) detectors by calculating the signal-to-noise ratio (SNR) and the loss factor Q. The GNKD is shown to be better than both the CWD and the CORR with improvement in the SNR by a factor of &. The HyD can improve the SNR by about 18% compared to the CWWD. Detection of some practical nonstationary signals is also inves- tigated to exemplify the proposed method. © 2001 Society of Photo-Optical

Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding centre motion

We derive a product rule for gauge invariant Weyl symbols which provides a generalization of the well known Moyal formula to the case of non-vanishing electromagnetic fields. Applying our result to the guiding centre problem we expand the guiding centre Hamiltonian into an asymptotic power series with respect to both Planck's constant and an adiabaticity parameter already present in the classical theory. This expansion is used to determine the influence of quantum-mechanical effects on guiding centre motion.

Bilinear signal synthesis

The authors discuss the signal synthesis problem in the general framework of bilinear signal representations (BSRs), thereby obtaining a unified treatment which encompasses, e.g., the Wigner distribution and ambiguity function as special cases. The inclusion of a signal space constraint serves to impart flexibility to the signal synthesis process and to relax mathematical requirements. The characterization of signal spaces either by orthogonal projection operators or by orthonormal bases leads to two different signal synthesis methods. Both methods assume the BSR to be unitary (i.e., satisfy Moyal's formula) on the signal space on which signal synthesis is performed. As an application of the general signal synthesis methods, band-limited signal synthesis in the case of the Wigner distribution is considered.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Regularity and unitarity of bilinear time-frequency signal representations

Two structural properties of bilinear time-frequency representations (BTFRs) of signals are introduced and studied. The definition of these properties is based on a linear-operator description of BTFRs. The first property, termed regularity, has important implications with respect to the recovery of signals from the BTFR outcome, the derivation of other bilinear signal representations from a BTFR, the BTFRs reaction to linear signal transformations, and the construction of bases of induced BTFR-domain spaces. The second property, called unitarity, is equivalent to validity of Moyal's formula (1949). Unitarity is thus necessary and sufficient for a closed-form solution of optimal signal synthesis and for a BTFR formulation or optimal detection/estimation methods. Unitarity also allows the systematic construction of BTFR product relations like the Wigner distribution's interference formula and the ambiguity function's self-transformation property. Unitarity permits the construction of induced orthogonal projection operators and guarantees the orthonormality of induced basis functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>