Nonuniform sampling for random signals bandlimited in the linear canonical transform domain

Self Cite

In this paper, we study the inversion of the windowed linear canonical transform. We show that for certain window functions, a function can be recovered from its windowed linear canonical transform with a discrete series. We give necessary and sufficient conditions for the convergence of the series. Both the convergence in norm and the pointwise convergence are studied. Moreover, we get an inversion formula for integrable functions with the Cesàro summability.

Section: Introduction and Main Resultsmentioning

confidence: 99%

Inversion of the windowed linear canonical transform with Riemann sums

Han

Sun

2022

Self Cite

“…Since a signal which is bandlimited in the SAFT domain is not necessarily bandlimited in the classical FT domain (see Example 2.7), many sampling results about FT-bandlimited signals are generalized to the SAFT-bandlimited setting for including more signal models. For example, uniform and nonuniform sampling of deterministic bandlimited signals in the SAFT domain [2,16,21,22,25], sampling and reconstruction of random signals which are bandlimited in the LCT or SAFT domain [6,7,23,24,26]. However, almost all the above results are in the matter of one-dimensional cases.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, one hasR M x,x (𝝉) = R M x,x (𝜏 1 , 𝜏 2 ) = R T x 0 ,x 0 (𝜏 1 )R T x 0 ,x 0 (𝜏 2 ),where T stands for the six parameters (a, b, c, d, p, q) = (3, 1∕𝜋, 𝜋, 2∕3, 0, 0) in (1.2). Moreover, we know from Huo and Sun[7] that x 0 (t) is approximately bandlimited and the bandwidth is 10 in the one-dimensional SAFT domain. Finally, it follows from the item (iii) of Remark 2.2 that x(t) is approximately bandlimited to [−10,10) × [−10,10).…”

mentioning

confidence: 99%

Sampling and reconstruction of multi‐dimensional bandlimited signals in the special affine Fourier transform domain

Gao,

Jiang

2024

The paper is concerned with the sampling and reconstruction of bandlimited signals in the multi‐dimensional special affine Fourier transform domain. First, we define the multi‐dimensional special affine Fourier transform and establish the convolution theorem. Then, sampling and reconstruction of the multi‐dimensional deterministic bandlimited signals in the special affine Fourier transform domain is studied. Finally, the definition of multi‐dimensional random bandlimited signals in the special affine Fourier transform domain is given and a sampling theorem is demonstrated.

“…Thus, the LCT has the potential to provide a more general and flexible analysis tool in signal processing, and it has found numerous applications to real‐world problems in recent years. It is worth noting that a variety of sampling theorems for bandlimited signals in the LCT domain have been extensively investigated 10,11 . Let

{boldB}_{normalΩ}^{2} false(ℝ false) : = false{f \in L^{2} false(ℝ false) : L_{f}^{A} false(u false) = 0 for all false| u false| > normalΩ false}

, and the Shannon sampling theorem in the LCT domain 12,13 states that every

f \in {boldB}_{normalΩ}^{2} false(ℝ false)

can be reconstructed by its samples as

f false(x false) = e^{- j \frac{a}{2 b} x^{2}} true \sum_{n = - \infty}^{\infty} f false(n T false) e^{j \frac{a}{2 b} {false(n T false)}^{2}} sinc ((), \frac{normalΩ false(x - n T false)}{π b}) .

where

T = π b false/ normalΩ

, and the sinc function is given by

sinc (x) = \{\begin{array}{ll} \frac{\sin false(π x false)}{π x}, & x \neq 0, \\ 1, & x = 0 . \end{array}

The sampling series in () converges absolutely and uniformly on

ℝ

.…”

Section: Introductionmentioning

confidence: 99%

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

Huo

2021

Self Cite

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.