Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms

Xiaoxiao HU; Dong CHENG; Kit Ian KOU

doi:10.1631/FITEE.2000499

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Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms

Regular Papers | Updated：2022-03-28

- Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms
  Enhanced Publication
- 基于四元数傅里叶变换和线性正则变换的二维四元数信号采样定理
- Frontiers of Information Technology & Electronic Engineering Vol. 23, Issue 3, Pages: 463-478(2022)
- Affiliations：
  
  1.The First Affiliated Hospital of Wenzhou Medical University, Wenzhou Medical University, Wenzhou325000, China
  2.Research Center for Mathematics and Mathematics Education, Beijing Normal University (Zhuhai), Zhuhai519087, China
  3.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China
- Author bio：
  
  ‡Corresponding author
- Funds：
- DOI：10.1631/FITEE.2000499
  CLC： O236
- Received：25 September 2020，
  
  Accepted：18 January 2021，
  
  Published：0 March 2022
- Accepted：
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Xiaoxiao HU, Dong CHENG, Kit Ian KOU. Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms[J]. Frontiers of Information Technology & Electronic Engineering, 2022, 23(3): 463-478.
DOI：

Xiaoxiao HU, Dong CHENG, Kit Ian KOU. Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms[J]. Frontiers of Information Technology & Electronic Engineering, 2022, 23(3): 463-478. DOI： 10.1631/FITEE.2000499.

摘要

本文主要研究在不同形式四元数傅里叶变换和线性正则变换下有限带宽四元数函数的采样定理。证明了有限带宽四元数函数可通过它们的直接采样或经过微分和希尔伯特变换后的采样重构。此外，讨论了不同形式变换下不同类型采样公式之间的关系。首先，如果四元数函数有限带宽区域是关于原点对称的矩形区域，则不同形式四元数傅里叶变换下四元数采样公式具有相同形式；否则，采样公式是不同的。其次，利用双边四元数傅里叶变换和线性正则变换的关系，得到不同形式四元数线性正则变换下有限带宽四元数函数采样定理。再次，分析了采样公式的截断误差。最后，通过仿真展示采样公式的应用。

Abstract

The main purpose of this paper is to study different types of sampling formulas of quaternionic functions

which are bandlimited under various quaternion Fourier and linear canonical transforms. We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms. In addition

the relationships among different types of sampling formulas under various transforms are discussed. First

if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin

then the sampling formulas under various quaternion Fourier transforms are identical. If this rectangle is not symmetric about the origin

then the sampling formulas under various quaternion Fourier transforms are different from each other. Second

using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform

we derive sampling formulas under various quaternion linear canonical transforms. Third

truncation errors of these sampling formulas are estimated. Finally

some simulations are provided to show how the sampling formulas can be used in applications.

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references

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