Heisenberg Uncertainty Inequality Research Articles

N$$ N $$‐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.

Quadratic phase S-Transform: Properties and uncertainty principles

In this paper, a novel quadratic phase S-transform (QPST) is proposed, by generalizing the S-transform (ST) with five parameters a, b, c,d and e. QPST displays the time and quadratic phase domain-frequency information jointly in the time-frequency plane. Firstly, we define the novel QPST and give its relation with quadratic phase Fourier transform (QPFT). Secondly, several important properties of newly defined QPST, such as conjugation, translation, modulation, orthogonality relation and reconstruction formula are derived. Finally, we formulate several classes of uncertainty inequalities, such as the Heisenberg uncertainty inequality and logarithmic uncertainty inequality.

Quadratic-phase wave packet transform

The quadratic-phase Fourier transform (QPFT) has gained much popularity in recent years because of its applications in image and signal processing. However, the QPFT is inadequate for localizing the quadratic-phase spectrum, which is required in some applications. In this paper, the quadratic-phase wave packet transform (QP-WPT) is proposed to address this problem, based on the wave packet transform (WPT) and QPFT. Firstly, we propose the definition of the QP-WPT and give its relation with windowed Fourier transform (WFT). Secondly, several notable inequalities and important properties of newly defined QP-WPT, such as boundedness, reconstruction formula, Moyal’s formula, reproducing kernel are derived. Finally, we formulate several classes of uncertainty inequalities, such as Leib’s uncertainty principle, logarithmic uncertainty inequality and the Heisenberg uncertainty inequality.

Heisenberg uncertainty inequality for Gabor transform on nilpotent Lie groups

Heisenberg uncertainty inequality for Gabor transform on nilpotent Lie groups

Heisenberg uncertainty inequality for certain Lie groups

We establish analogues of Heisenberg uncertainty inequality for some classes of Lie groups, such as connected and simply connected nilpotent Lie groups, diamond Lie groups and Heisenberg motion groups.

Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group Hn. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.

A variation on uncertainty principles for the generalized q-Bessel Fourier transform

The aim of this paper is to prove new uncertainty principles for the generalized q-Fourier Bessel transform studied earlier in [9]. To do so we prove a Nash-type inequality and a Carlson-type inequality for this transformation. From this we deduce a variation on Heisenberg's uncertainty inequality and Faris's local uncertainty principle. We also prove a variation on Donoho–Stark's uncertainty principle. Our results can be applied to the q-Bessel Fourier transform [7].

Heisenberg uncertainty inequality for Gabor transform

We discuss Heisenberg uncertainty inequality for groups of the form K ⋉ R n , K is a separable unimodular locally compact group of type I. This inequality is also proved for Gabor transform for several classes of groups of the form K ⋉ R n .

Generalized analogs of the Heisenberg uncertainty inequality

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

Heisenberg-type inequalities for the Weinstein operator

The aim of this paper is to prove new uncertainty principles for the Weinstein operator. The first of these results is a sharp Heisenberg-type inequality for the Weinstein transform, that is, for and , The second result states that the previous inequality can be refined for an orthonormal basis for , that is, if is an orthonormal basis for , then As a side result, we prove a new version of Heisenberg's uncertainty inequality for the Weinstein–Gabor transform, which states that the Weinstein–Gabor transform of a nonzero function with respect to a nonzero window function cannot be time and frequency concentrated around zero.

Time–frequency concentration of the windowed Hankel transform

The aim of this paper is to prove a version of Heisenberg's uncertainty inequality for the windowed Hankel transform. As a side results, we extend some others uncertainty principles about sets of finite measure to the windowed Hankel transform which are valid for the windowed Fourier transform.

Uncertainty principles involving L 1-norms for the Dunkl transform

The aim of this paper is to prove new uncertainty principles for the Dunkl transform. More precisely, we show a variation on Heisenberg's uncertainty inequality, the local uncertainty principle and Donoho–Stark's uncertainty principle.

H‐methods in applied sciences

AbstractThe author has developed a framework for mathematical modelling within applied sciences. It is characteristic for data from ‘nature and industry’ that they have reduced rank for inference. It means that full rank solutions normally do not give satisfactory solutions. The basic idea of H‐methods is to build up the mathematical model in steps by using weighing schemes. Each weighing scheme produces a score and/or a loading vector that are expected to perform a certain task. Optimisation procedures are used to obtain ‘the best’ solution at each step. At each step, the optimisation is concerned with finding a balance between the estimation task and the prediction task. The name H‐methods has been chosen because of close analogy with the Heisenberg uncertainty inequality. A similar situation is present in modelling data. The mathematical modelling stops, when the prediction aspect of the model cannot be improved. H‐methods have been applied to wide range of fields within applied sciences. In each case, the H‐methods provide with superior solutions compared to the traditional ones. A background for the H‐methods is presented. The H‐principle of mathematical modelling is explained. It is shown how the principle leads to well‐defined optimisation procedures. This is illustrated in the case of linear regression. The H‐methods have been applied in different areas: general linear models, nonlinear models, multi‐block methods, path modelling, multi‐way data analysis, growth models, dynamic models and pattern recognition. Copyright © 2008 John Wiley & Sons, Ltd.

On the uncertainty inequality as applied to discrete signals

Given a continuous‐time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme for exactly reconstructing it from its discrete samples. We analyze the relationship between concentration (or compactness) in the temporal/spectral domains of the (i) continuous‐time and (ii) discrete‐time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product of effective temporal and spectral spreads of the signal. On the other hand, the discrete‐time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem: for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal? In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.

Principal squeezing of vacuum fluctuations

With regard to the squeezing detection process a more substantial definition of squeezing is proposed based on the sharpened Heisenberg uncertainty inequality. This notion involves the quadrature correlation and is rotationally invariant.