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Generalized Uncertainty Principles for the Offset Linear Canonical Transform

Haiye Huo

Published Feb 11, 2018• 0 citations
OpenAlex

Abstract

The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we generalize some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.

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The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we generalize some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.
The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.
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.
The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application 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 (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domain
The uncertainty principle plays an important role in both physics and mathematics, and it also plays an important role in information science. This paper reviewed the Heisenberg’s uncertainty principles, windowed uncertainty principles, logarithmic uncertainty principles, entropic uncertainty principles and so on in the fractional Fourier transform domains and linear canonical transform domains in great details. The development and applications of generalized uncertainty principles were analyzed, and the trend and direction were given as well.
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.