E-mail Alert
RSS
-
摘要:
Hartley变换是傅里叶变换的推广, 它的一个非常好的性质就是把实信号变换成实信号, 从而减少计算量。近些年, 随着分数阶傅里叶变换在信号处理中被广泛的应用, 线性正则变换也逐渐被应用到信号处理, 所以把Hartley变换推广到正则域是一个有研究价值的问题。本文首先通过变化傅里叶变换域Hartley变换的核函数, 得到了一个具有共轭性的核函数, 之后, 通过把该核函数替换成线性正则变换的核函数, 从而得到了正则域的Hartley变换, 在这个定义的基础上, 得到了正则域Hartley变换满足实数性质和奇偶不变性, 之后再利用线性正则变换的Heisenberg不确定性原理, 得到了正则域Hartley变换的Heisenberg不确定性原理。
Abstract:Hartley transform is a generalization of Fourier transform and it transforms the real signal into real signal thereby reducing the amount of computation. In recent years, with the wide applications of fractional Fourier transform in signal processing, linear canonical transform has gradually been applied to signal processing. Hence, it is a valuable problem to generalize Hartley transform in linear canonical transform domain. In this paper, a kernel function with conjugate property is obtained by changing kernel function of Hartley transform in Fourier transform domain. After that, we obtain Hartley transform in linear canonical transform domain by using kernel function of linear canonical transform. Then, Hartley transform in linear canonical transform domain has the properties of real number and odd-even invariance. Finally, by using Heisenberg uncertainty principle in linear canonical transform domain, we obtain Heisenberg uncertainty principle of Hartley transform in linear canonical transform domain.
-
Overview:Fourier transform is a basic tool in the field of signal processing, and with the in-depth research and the rapid development of the computer technology, researchers have managed more and more better results on signal processing. While more and more mathematical tools have been introduced into signal processing. Linear canonical transform is a generalization of Fourier transform and fractional Fourier transform. When researchers deal with the charp signal, they can obtain a very good effect by using the linear canonical transform. Based on the above reasons, more and more researchers begin to pay attention to linear canonical transform.
In this context, many transformations related to Fourier transform have been extended to fractional Fourier transform domains and linear canonical transform domains, such as the classical Wigner-ville distributions and cosine transformations. In Fourier transform domain, Hartley transform, which is the generalization of cosine transform, has a very significant advantage in the ability of transforming one real signal to another real signal, and it can delete the calculation of complex number hence it can cut down the calculation time. Because linear canonical transform kernel is complex to Fourier transform kernel, it is worthy to obtain Hartley transformation in linear canonical transform domain which transforms one real function to another real function. Based on the above issue, combined with linear canonical transform kernel, we define a Hartley transformation kernel, and then we obtain Hartley transformation in linear canonical transform domain. By simple calculations, Hartley transformation in linear canonical transform domain has two properties, which are transformed real function into real function and maintained parity invariant.
We know that the time resolution and the frequency resolution in the Fourier transform cannot be too small at the same time, which is the so-called Heisenberg uncertainty principle in Fourier transform domain. Based on the Heisenberg uncertainty principle in Fourier domain, one can also get the Heisenberg uncertainty principle for Hartley transform by some simple calculations. Since we have obtained the Hartley transformation in linear canonical transform domain, Hence, we guess that the Hartley transformation in linear canonical transform domain should also have the Heisenberg uncertainty principle. In this manuscripts, the Heisenberg uncertainty principle in linear canonical transform domain has been obtained for the real value function, while we simply discusses the entropy uncertainty principle of the Hartley transformation in linear canonical transform domain.
-
-
[1] Hartley R V L. A more symmetrical Fourier analysis applied to transmission problems[J]. Proceedings of the IRE, 1942, 30(3):144-150. doi: 10.1109/JRPROC.1942.234333
[2] Bailey D H, Swarztrauber P N. A fast method for the numerical evaluation of continuous Fourier and Laplace transforms[J]. SIAM Journal on Scientific Computing, 1994, 15(5):1105-1110. doi: 10.1137/0915067
[3] Duffieux P M. The Fourier Transform and Its Applications to Optics[M]. 2nd ed. New York, USA:John Wiley & Sons, 1983.
[4] Ozaktas H M, Kutay M A, Zalevsky Z. The Fractional Fourier Transform with Applications in Optics and Signal Processing[M]. New York:Wiley, 2000.
[5] Wolf K B. Integral Transforms in Science and Engineering[M]. New York:Plenum Press, 1979.
[6] Liu Y L, Kou K I, Ho I T, et al. New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms[J]. Signal Processing, 2010, 90(3):933-945. doi: 10.1016/j.sigpro.2009.09.030
[7] 许天周, 李炳照.线性正则变换及其应用[M].北京:科学出版社, 2013.
[8] 李炳照, 陶然, 王越.线性正则变换域的框架理论研究[J].电子学报, 2007, 35(7):1387-1390. https://www.cnki.com.cn/qikan-DZXU200707030.html
[9] Bai R F, Li B Z, Cheng Q Y. Wigner-Ville distribution associated with the linear canonical transform[J]. Journal of Applied Mathematics, 2012, 2012:740161 http://cn.bing.com/academic/profile?id=0a1819c3daa60ea4248fd6b4288a1a78&encoded=0&v=paper_preview&mkt=zh-cn
[10] Xu G L, Wang X T, Xu X G. Three uncertainty relations for real signals associated with linear canonical transform[J]. IET Signal Processing, 2009, 3(1):85-92. doi: 10.1049/iet-spr:20080019
[11] Folland G B, Sitaram A. The uncertainty principle:a mathematical survey[J]. Journal of Fourier Analysis and Applications, 1997, 3(3):207-238. doi: 10.1007/BF02649110
[12] Shannon C E. Communication theory of secrecy systems[J]. The Bell System Technical Journal, 1949, 28(4):656-715. doi: 10.1002/bltj.1949.28.issue-4
[13] Leipnik R. Entropy and the uncertainty principle[J]. Information and Control, 1959, 2(1):64-79. doi: 10.1016/S0019-9958(59)90082-8
[14] Xu G L, Wang X T, Xu X G. Uncertainty inequalities for linear canonical transform[J]. IET Signal Processing, 2009, 3(5):392-402. doi: 10.1049/iet-spr.2008.0102
[15] Moshinsky M, Quesne C. Linear canonical transformations and their unitary representations[J]. Journal of Mathematical Physics, 1971, 12(8):1772-1780. doi: 10.1063/1.1665805
[16] Collins S A. Lens-system diffraction integral written in terms of matrix optics[J]. Journal of the Optical Society of America, 1970, 60(9):1168-1177. doi: 10.1364/JOSA.60.001168
[17] Zhang Z C. New convolution and product theorem for the linear canonical transform and its applications[J]. Optik-International Journal for Light and Electron Optics, 2016, 127(11):4894-4902. doi: 10.1016/j.ijleo.2016.02.030
[18] Azoug S E, Bouguezel S. A non-linear preprocessing for opto-digital image encryption using multiple-parameter discrete fractional Fourier transform[J]. Optics Communications, 2016, 359:85-94. doi: 10.1016/j.optcom.2015.09.054
-
点击扫一扫
计量
- 文章访问数:
- PDF下载数:
- 施引文献: 0
下载:
