Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform (original) (raw)
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Weighted Heisenberg-Pauli-Weyl uncertainty principles for the linear canonical transform
The classical uncertainty principle plays an important role in quantum mechanics, signal processing and applied mathematics. With the development of novel signal processing methods, the research of the related uncertainty principles has gradually been one of the most hottest research topics in modern signal processing community. In this paper, the weighted Heisenberg-Pauli-Weyl uncertainty principles for the linear canonical transform (LCT) have been investigated in detail. Firstly, the Plancherel-Parseval-Rayleigh identities associated with the LCT are derived. Secondly, the weighted Heisenberg-Pauli-Weyl uncertainty principles in the LCT domain are investigated based on the derived identities. The signals that can achieve the lower bound of the uncertainty principle are also obtained. The classical Heisenberg uncertainty principles in the Fourier transform (FT) domain are shown to be special cases of our achieved results. Thirdly, examples are provided to show that our weighted Heisenberg-Pauli-Weyl uncertainty principles are sharper than those in the existing literature. Finally, applications of the derived results in time frequency resolution analysis and signal energy concentrations are also analyzed and discussed in detail.
Journal of the Optical Society of America A, 2008
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.
Complex Variables and Elliptic Equations, 2021
The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization of the quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact, we prove the Plancherel formula, and some properties related to the QOLCT. Then, we generalize some different uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, and logarithmic UP to the QOLCT domain in a broader sense.
Uncertainty Principles for the Two-Sided Quaternion Linear Canonical Transform
Circuits, Systems, and Signal Processing, 2020
The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy's uncertainty principle, Beurling's uncertainty principle and Donoho-Stark's uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb's uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff-Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.
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Annals of Physics, 2017
Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. As the corresponding entropies differ, known entropic uncertainty relations are changed. Using differential Shannon entropies, we give a state-dependent formulation with correction term. State-independent uncertainty relations are obtained in terms of the Rényi entropies and the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution.
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The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known "qualitative" uncertainty principles into more quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform.
Uncertainty principles associated with quaternion linear canonical transform and their estimates
Mathematical Methods in The Applied Sciences, 2022
Communicated by Q. Chen In the present paper, we generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic LCT (QLCT). Using the properties of the LCT, we establish an uncertainty principle for the two-sided QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternionic signal minimizes the uncertainty. Copyright
The Linear Canonical Transform (LCT) (Alieva and
2016
The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.
A transform of complementary aspects with applications to entropic uncertainty relations
Journal of Mathematical Physics, 2010
Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d = 2 n which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set of MUBs, and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension d = 4, which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space, and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.
Uncertainty Principles for the Jacobi Transform
Tokyo Journal of Mathematics, 2008
We obtain some uncertainty inequalities for the Jacobi transformf α,β (λ), where we suppose α, β ∈ R and ρ = α + β + 1 ≥ 0. As in the Euclidean case, analogues of the local and global uncertainty principles hold for f α,β. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When β ≤ 0 or β ≤ α, the L 2-norm off α,β (λ)λ is estimated below by the L 2-norm of ρf (x)(cosh x) −1. Otherwise, a similar inequality holds. Especially, when β > α + 1, the discrete part of f appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on SU(1, 1). Then the corresponding uncertainty principle depends, not uniformly on the K-types of f .