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

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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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.

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.

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Authors

Contributions

Xiaoxiao HU designed the research. Xiaoxiao HU and Dong CHENG processed the data. Xiaoxiao HU drafted the paper. Kit Ian KOU helped organize the paper. Xiaoxiao HU and Dong CHENG revised and finalized the paper.

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Correspondence to Xiaoxiao Hu  (胡晓晓).

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Xiaoxiao HU, Dong CHENG, and Kit Ian KOU declare that they have no conflict of interest.

Additional information

Project supported by the Research Development Foundation of Wenzhou Medical University, China (No. QTJ18012), the Wenzhou Science and Technology Bureau of China (No. G2020031), the Guangdong Basic and Applied Basic Research Foundation of China (No. 2019A1515111185), the Science and Technology Development Fund, Macau Special Administrative Region, China (No. FDCT/085/2018/A2), and the University of Macau, China (No. MYRG2019-00039-FST)

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

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Hu, X., Cheng, D. & Kou, K.I. Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms. Front Inform Technol Electron Eng 23, 463–478 (2022). https://doi.org/10.1631/FITEE.2000499

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  • DOI: https://doi.org/10.1631/FITEE.2000499

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