A new convolution theorem associated with the linear canonical transform

Huo, Haiye

doi:10.1007/s11760-018-1337-2

Cited by 13 publications

(11 citation statements)

References 28 publications

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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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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}} .

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 four parameters in the LCT give extra degrees of freedom, which makes this transformation a very flexible one, and a powerful tool in the fields of signal processing, filter design, radar system analysis, pattern recognition, optics, solvability of integral and differential equations, as well as in many other areas of applied sciences (Anh et al, 2017, 2019, Barshan et al, 1997, Deng et al, 2006, Goel and Singh, 2013, Sharma and Joshi, 2006, Shi et al, 2012, 2014, Zhang, 2016a. Many properties of the LCT are currently well known (Sharma andJoshi, 2006, Xu andLi, 2013).…”

Section: Introductionmentioning

confidence: 99%

A new convolution operator for the linear canonical transform with applications

2021

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The linear canonical transform plays an important role in engineering and many applied fields, as it is the case of optics and signal processing. In this paper, a new convolution for the linear canonical transform is proposed and a corresponding product theorem is deduced. It is also proved a generalized Young's inequality for the introduced convolution operator. Moreover, necessary and sufficient conditions are obtained for the solvability of a class of convolution type integral equations associated with the linear canonical transform. Finally, the obtained results are implemented in multiplicative filters design, through the product in both the linear canonical transform domain and the time domain, where specific computations and comparisons are exposed.

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“…The linear canonical transform (LCT) is a general class of linear integral transformations with three free parameters, which includes many well-known linear transforms as its special cases, such as Fourier transform (FT), fractional FT, scaling operation, and Fresnel transform [1][2][3][4][5][6][7]. Therefore, there has been growing interest in studying the LCT and its properties pertaining to applications across the fields of signal processing and optics [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Multivariate Dynamical Sampling in $l^2(\mathbb{Z}^2)$ and Shift-Invariant Spaces Associated with Linear Canonical Transform

Huo¹,

Xiao²

2021

Preprint

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In this paper, we investigate the multivariate dynamical sampling problem in l 2 (Z 2 ) associated with the two-dimensional discrete time non-separable linear canonical transform (2D-DT-NS-LCT) and shift-invariant spaces associated with the two-dimensional non-separable linear canonical transform (2D-NS-LCT), respectively. Specifically, we derive a sufficient and necessary condition under which a sequence in l 2 (Z 2 ) (or a function in a shift-invariant space) can be stably recovered from its dynamical sampling measurements associated with the 2D-DT-NS-LCT (or the 2D-NS-LCT). We also present a simple example to elucidate our main results.

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A new convolution theorem associated with the linear canonical transform

Cited by 13 publications

References 28 publications

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

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

A new convolution operator for the linear canonical transform with applications

Multivariate Dynamical Sampling in $l^2(\mathbb{Z}^2)$ and Shift-Invariant Spaces Associated with Linear Canonical Transform

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