Quaternion Windowed Linear Canonical Transform of Two-Dimensional Quaternionic Signals

We investigate the 2D quaternion windowed linear canonical transform(QWLCT) in this paper. Firstly, we propose the new definition of the QWLCT, and then several important properties of newly defined QWLCT, such as bounded, shift, modulation, orthogonality relation, are derived based on the spectral representation of the quaternionic linear canonical transform(QLCT). Finally, by the Heisenberg uncertainty principle for the QLCT and the orthogonality relation property for the QWLCT, the Heisenberg uncertainty principle for the QWLCT is established.Comment: This article is 14 pages lon

Uncertainty principles for the windowed offset linear canonical transform

The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy's uncertainty principle, Beurling's uncertainty principle, Lieb's uncertainty principle, Donoho-Stark's uncertainty principle, Amrein-Berthier-Benedicks's uncertainty principle, Nazarov's uncertainty principle and Logarithmic uncertainty principle.Comment: 12 page

Uncertainty Principles for Quaternion Windowed Offset Linear Canonical Transform of Two Dimensional Signals

The offset linear canonical transform encompassing the numerous integral transforms, is a promising tool for analyzing non-stationary signals with more degrees of freedom. In this paper, we generalize the windowed offset linear canonical transform for quaternion-valued signals, by introducing a novel time-frequency transform namely the quaternion windowed offset linear canonical transform of 2D quaternion-valued signals. We initiate our investigation by studying some fundamental properties of the proposed transform including inner product relation, energy conservation, and reproducing formula by employing the machinery of quaternion offset linear canonical transforms. Some uncertainty principles such as Heisenberg-Weyl, logarithmic and local uncertainty principle are also derived for quaternion windowed offset linear canonical transform. Finally, we gave an example of quaternion windowed offset linear canonical transform.Comment: 16 pages. arXiv admin note: text overlap with arXiv:2006.0675

Convolution and correlation theorems for the windowed offset linear canonical transform

In this paper, some important properties of the windowed offset linear canonical transform (WOLCT) such as shift, modulation and orthogonality relation are introduced. Based on these properties we derive the convolution and correlation theorems for the WOLCT.Comment: The article consists of six page

Uncertainty Principles For the continuous Gabor quaternion linear canonical transform

Gabor transform is one of the performed tools for time-frequency signal analysis. The principal aim of this paper is to generalize the Gabor Fourier transform to the quaternion linear canonical transform. Actually, this transform gives us more flexibility to studied nonstationary and local signals associated with the quaternion linear canonical transform. Some useful properties are derived, such as Plancherel and inversion formulas. And we prove some uncertainty principles: those including Heisenberg's, Lieb's and logarithmic inequalities. We finish by analogs of concentration and Benedick's type theorems

Short time quaternion quadratic phase Fourier transform and its uncertainty principles

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff-Young inequality for QQPFT. We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner-Ville distribution associated with QQPFT and obtain the Lieb's uncertainty and entropy uncertainty principles for these three transforms

Uncertainty principles for the windowed Hankel transform

The aim of this paper is to prove some new uncertainty principles for the windowed Hankel transform. They include uncertainty principle for orthonormal sequence, local uncertainty principle, logarithmic uncertainty principle and Heisenberg-type uncertainty principle. As a side result, we obtain the Shapiro's dispersion theorem for the windowed Hankel transform.Comment: 15 page

Uncertainty principles for the short-time linear canonical transform of complex signals

The short-time linear canonical transform (STLCT) can be identified as a generalization of the short-time Fourier transform (STFT). It is a novel time-frequency analysis tool. In this paper, we generalize some different uncertainty principles for the STLCT of complex signals. Firstly, one uncertainty principle for the STLCT of complex signals in time and frequency domains is derived. Secondly, the uncertainty principle for the STLCT of complex signals in two STLCT domains is obtained. Finally, the uncertainty principle for the two conditional standard deviations of the spectrogram associated with the STLCT is discussed.Comment: 10 page

Quaternion Linear Canonical Wavelet Transform and The Corresponding Uncertainty Inequalities

The linear canonical wavelet transform has been shown to be a valuable and powerful time-frequency analyzing tool for optics and signal processing. In this article, we propose a novel transform called quaternion linear canonical wavelet transform which is designed to represent two dimensional quaternion-valued signals at different scales, locations and orientations. The proposed transform not only inherits the features of quaternion wavelet transform but also has the capability of signal representation in quaternion linear canonical domain. We investigate the fundamental properties of quaternion linear canonical wavelet transform including Parseval's formula, energy conservation, inversion formula, and characterization of its range using the machinery of quaternion linear canonical transform and its convolution. We conclude our investigation by deriving an analogue of the classical Heisenberg-Pauli-Weyl uncertainty inequality and the associated logarithmic and local versions for the quaternion linear canonical wavelet transform.Comment: 20 page

Local uncertainty principles for the two-sided Gabor quaternion Fourier transform

By the important applications of Gabor transform in time-frequency analysis and signal analysis, in this paper, we consider the Gabor quaternion Fourier transform (GQFT), and we prove of it a version Benedicks-type uncertainty principle for GQFT and some local concentration uncertainty principles