JZUS - Journal of Zhejiang University SCIENCE

Frontiers of Information Technology & Electronic Engineering 2022 Vol.23 No.3 P.463-478

http://doi.org/10.1631/FITEE.2000499

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

Author(s): Xiaoxiao HU, Dong CHENG, Kit Ian KOU
Affiliation(s): The First Affiliated Hospital of Wenzhou Medical University, Wenzhou Medical University, Wenzhou 325000, China; more
Corresponding email(s): huxiaoxiao3650@163.com
Key Words: Quaternion Fourier transforms, Quaternion linear canonical transforms, Sampling theorem, Quaternion partial and total Hilbert transforms, Generalized quaternion partial and total Hilbert transforms, Truncation errors

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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.

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

胡晓晓¹，程冬²，高洁欣³
¹温州医科大学第一临床医学院（信息与工程学院），中国温州市，325000
²北京师范大学珠海分校数学与数学教育研究中心，中国珠海市，519087
³澳门大学科技学院数学系，中国澳门
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。

关键词：四元数傅里叶变换；四元数线性正则变换；采样定理；部分和总体四元数希尔伯特变换；部分和总体广义四元数希尔伯特变换；截断误差

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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

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