Advances in Applied Mathematics
Vol.3 No.03(2014), Article ID:13996,5 pages
DOI:10.12677/AAM.2014.33020

Uncertainty Principle for a Kind of Quaternionic Linear Canonical Transform

Yingxiong Fu, Zhen Xiong*

School of Mathematics and Statistics, Hubei University, Wuhan

Email: *aaxiongzhen@126.com

Copyright © 2014 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received: May 20th, 2014; revised: Jun. 18th, 2014; accepted: Jun. 27th, 2014

ABSTRACT

In this paper, based on the properties of the left-sided quaternionic linear canonical transform (QLCT), an uncertainty principle is established for the left-sided QLCT. It states that the product of the variances of quaternion-valued signals in the spatial and frequency domains has a lower bound and only a 2D Gaussian signal minimizes the uncertainty principle.

Keywords:Quaternion, Left-Sided Quaternionic Linear Canonical Transform, Uncertainty Principle

Email: *aaxiongzhen@126.com

1. 引言

Hamilton在1843年提出四元数的概念[1] [2] ，作为最简单的超复数，人们不断地对其进行探讨，结合经典的傅立叶变换，学者们提出了四元数傅立叶变换[3] -[7] 。由于四元数乘法的不可交换性，人们可以定义三种不同类型的四元数傅立叶变换，即左边四元数傅立叶变换，右边四元数傅立叶变换和双边四元数傅立叶变换。目前，众多文献在四元数域内运用右边四元数傅立叶变换取得了大量的研究成果。基于右边四元数傅立叶变换的性质，文献[7] 解决了右边四元数傅立叶变换的不确定性原理。

2. 预备知识

(1)

(2)

3. 主要结论

1. [1]   Hamilton, W.R. (1866) Elements of Quaternions. Longmans, Green and Co., London.

2. [2]   Kantor, I.L. and Solodovnikov, A.S. (1989) Hypercomplex number: An elementary introduction to algebras. Springer-Verlag, New York.

3. [3]   Pei, S.C. and Cheng, C.M. (1997) A novel block truncation coding of color image using a quaternion-moment-preserving principle. IEEE Transactions on Communications, 45, 583-595.

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6. [6]   Evans, C.J., Sangwine, S.J. and Ell, T.A. (2000) Hypercomplex color-sensitive smoothing filters. IEEE International Conference on Image Processing, 1, 541-544.

7. [7]   Mawardi, B., Hitzer, E., Hayashi, A. and Ashino, R. (2008) An uncertainty principle for quaternion Fourier transform. Computers and Mathematics with Applications, 56, 2411-2417.

8. [8]   Moshinsky, M. and Quesne, C. (1971) Linear canonical transformations and their unitary representations. Journal of Mathematical Physics, 12, 1772-1783.

9. [9]   Collins, S.A. (1970) Lens-system diffraction integral written in terms of matrix optics. Journal of the Optical Society of America, 60, 1168-1177.

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11. [11]   Ozaktas, H.M., Kutay, M.A. and Zalevsky, Z. (2001) The fractional Fourier transform with applications in optics and signal processing. Wiley, New York, 93-95.

12. [12]   Tao, R., Qi, L. and Wang, Y. (2004) Theory and applications of the fractional Fourier transform. Tsinghua University Press, Beijing, 73-76.

13. [13]   Barshan, B., Kutay, M.A. and Ozaktas, H.M. (1997) Optimal filters with linear canonical transformations. Optics Communications, 135, 32-36.

14. [14]   Pei, S.C. and Ding, J.J. (2003) Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. Journal of the Optical Society of America A, 20, 522-532.

15. [15]   Sharma, K.K. and Joshi, S.D. (2006) Signal separation using linear canonical and fractional Fourier transforms. Optics Communications, 256, 454-460.

16. [16]   Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Generalized Hilbert transform and its properties in 2D LCT domain. Signal Process, 89, 1395-1402.

17. [17]   Tao, R., Li, B.Z., Wang, Y. and Aggrey, G.K. (2008) On sampling of band-limited signals associated with the linear canonical transform. IEEE Transactions on Signal Processing, 56, 5454-5464.

18. [18]   Stern, A. (2008) Uncertainty principles in linear canonical transform domains and some their implications in optics. Journal of the Optical Society of America A, 25, 647-652.

NOTES

*通讯作者。