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DEStech Transactions on Engineering and Technology Research

Bi-dimensional Generalized Uncertainty Relations Associated with Hilbert Transform in LCT Domain

Guanlei Xu, Shao Limin, Xiaotong Wang, Xiaogang Xu

Published Dec 21, 2016• 1 citation
Publisher

Abstract

This paper discusses the Heisenberg’s uncertainty relations associated with two types of 2D LCT (linear canonical transform). These uncertainty bounds are related with the transform parameters of LCT. In addition, we show that the certainty bounds for the complex signals derived by LCT are different with that for the general complex signals in most cases. The special cases of these principles in traditional domains are provided as well. First, these new uncertainty relations enrich the ensemble of uncertainty principles. Second, these derived bounds yield new viewpoint of complex signals via 2D Hilbert transform.

DOI
10.12783/dtetr/icmite20162016/4580

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This paper discusses the Heisenberg’s uncertainty relations associated with two types of 2D LCT (linear canonical transform). These uncertainty bounds are related with the transform parameters of LCT. In addition, we show that the certainty bounds for the complex signals derived by LCT are different with that for the general complex signals in most cases. The special cases of these principles in traditional domains are provided as well. First, these new uncertainty relations enrich the ensemble of uncertainty principles. Second, these derived bounds yield new viewpoint of complex signals via 2D Hilbert transform.
The uncertainty principle plays an important role in mathematics, physics, signal processing, and so on. Firstly, based on definition of the linear canonical transform (LCT) and the traditional Pitt's inequality, one novel Pitt's inequality in the LCT domains is obtained, which is connected with the LCT parameters and Then one novel logarithmic uncertainty principle is derived from this novel Pitt's inequality in the LCT domains, which is associated with parameters of the two LCTs. Secondly, from the relation between the original function and LCT, one entropic uncertainty principle and one Heisenberg's uncertainty principle in the LCT domains are derived, which are associated with the LCT parameters and The reason why the three lower bounds are only associated with LCT parameters and and independent of and is presented. The results show it is possible that the bounds tend to zeros.
Uncertainty principle plays an important role in multiple fields such as physics, mathematics, signal processing, etc. The linear canonical transform (LCT) has been used widely in optics and information processing and so on. In this paper, a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced. These newly deduced uncertainty relations not only introduce new physical interpretation in signal processing, but also build the relations between the uncertainty lower bounds and the LCT transform parameters a, b, c and d for the first time, which give us the new ideas for the analysis and potential applications. In addition, these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts. Furthermore, some numeric examples are given to demonstrate the efficiency of these newly deduced uncertainty inequalities.
The linear canonical transform (LCT) is the name of a parameterized continuum of transforms that include, as particular cases, many widely used transforms in optics such as the Fourier transform, fractional Fourier transform, and Fresnel transform. It provides a generalized mathematical tool for representing the response of any first-order optical system in a simple and insightful way. In this work we present four uncertainty relations between LCT pairs and discuss their implications in some common optical systems.
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
In this paper, the generic uncertainty relation (UR) for kernel-based transformations (KT) of signals is derived. Instead of using the statistics approach as shown in the literature before, here we employ quantum mechanical operator approach for directly deriving the UR for KT's. We are able to do this because we have found the quantum operator realization of KT. Our new method is concise and applicable to any kinds of KT's with continuous and discrete parameters and variables. An explicit result of UR for a family of KT's including FrFT, generalized fractional transformation (GFrT) and linear canonical transformation (LCT) is provided as an application of our new method.