Uncertainty principles associated with the offset linear canonical transform

Huo, Haiye; Sun, Wenchang; Xiao, Li

doi:10.1002/mma.5353

Cited by 25 publications

(13 citation statements)

References 44 publications

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“…In this regard we establish the Donoho-Stark's and the Lieb's UP for the LCWT, which in turn provide a lower bound for the measure of essential support of the LCWT. See also [13], [31], for similar results in case of other integral transforms. (iv) to study the Shapiro's mean dispersion theorem for the LCWT.…”

Section: List Of Abbreviations Ft -Fourier Transformmentioning

confidence: 63%

Linear canonical wavelet transform and the associated uncertainty principles

Gupta¹,

Verma²,

Cattani³

2022

Preprint

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We define a novel time-frequency analyzing tool, namely linear canonical wavelet transform (LCWT) and study some of its important properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark's and Lieb's uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give Shapiro's mean dispersion theorem for the proposed LCWT.

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Section: List Of Abbreviations Ft -Fourier Transformmentioning

confidence: 63%

Linear canonical wavelet transform and the associated uncertainty principles

Gupta¹,

Verma²,

Cattani³

2022

Preprint

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“…) , the two-sided OQLCT leads to the two-sided QLCT (quaternion linear canonical transform). [23][24][25] From the above definition, it is noted that for b k = 0, k = 1, 2, the two-sided OQLCT is of no particular interest for our objective in this work. Hence, without loss of generality, we set b k > 0 in the following.…”

Section: Lemma 22 ( 12 )mentioning

confidence: 99%

Uncertainty principles for the two‐sided offset quaternion linear canonical transform

Zhu

Zheng

2021

Math Methods in App Sciences

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The offset quaternion linear canonical transform (OQLCT) provides a more general framework for a number of linear integral transforms in signal processing and optics, such as quaternion Fourier transform (QFT), fractional quaternion Fourier transform (FrQFT), and linear canonical transform (QLCT). We devote this paper to various different of uncertainty principles (UPs) for the two-sided OQLCT, which including logarithmic UP, Heisenberg-type UP, Hardy's UP, Beurling's UP, Entropic UP, Donoho-Stark's UP, and Local UP. Moreover, we also prove Lieb's UP for the two-sided short-time offset quaternion linear canonical transform (SOQLCT). KEYWORDS offset quaternion linear canonical transform (OQLCT), quaternion Fourier transform (QFT), uncertainty principle (UP), short-time offset quaternion linear canonical transform (SOQLCT) MSC CLASSIFICATION 42B10; 11R52; 94A12

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“…The linear canonical transform (LCT) 1–9 is a four‐parameter ( a , b , c , d ) class of linear integral transforms, which plays an important role in optics and digital signal processing. Let

A = false(a, b; c, d false), a, b, c, d \in ℝ

, and

a d - b c = 1

, then the LCT of a signal

f false(x false) \in L^{2} false(ℝ false)

associated with parameter A is defined by

L_{f}^{A} false(u false) = {scriptL}_{A} false{f false(x false) false} false(u false) = ({, \begin{matrix} left left \\ array \int_{ℝ} f (x) K_{A} (u, x) d x, & array b \neq 0, \\ array \sqrt{d} e^{j \frac{c d}{2} u^{2}} f (d u), & array b = 0, \end{matrix})

where

K_{A} (u, x) = \sqrt{\frac{1}{j 2 π b}} e^{j \frac{a}{2 b} x^{2} - j \frac{1}{b} u x + j \frac{d}{2 b} u^{2}} .

By choosing specific values for parameter A , several well‐known linear transforms turn out to be special cases of the LCT in (), for example, Fourier transform, fractional Fourier transform (FrFT), and Fresnel transform.…”

Section: Introductionmentioning

confidence: 99%

Truncation error estimates for two‐dimensional sampling series associated with the linear canonical transform

Huo

2021

Math Methods in App Sciences

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The linear canonical transform (LCT) provides a more general framework for many well‐known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi‐dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two‐dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples.

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Uncertainty principles associated with the offset linear canonical transform

Cited by 25 publications

References 44 publications

Linear canonical wavelet transform and the associated uncertainty principles

Linear canonical wavelet transform and the associated uncertainty principles

Uncertainty principles for the two‐sided offset quaternion linear canonical transform

Truncation error estimates for two‐dimensional sampling series associated with the linear canonical transform

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