分数阶Fourier域的二维广义Hilbert变换及Bedrosian定理

数学物理学报 ›› 2011, Vol. 31 ›› Issue (3): 814-828.

分数阶Fourier域的二维广义Hilbert变换及Bedrosian定理

徐冠雷^1,3, 王孝通, 徐晓刚^2,3

1.海军大连舰艇学院 |航海系辽宁大连 116018|2.海军大连舰艇学院 |自动化系辽宁大连 116018|3.海军大连舰艇学院 |光电技术研究所辽宁大连 116018)

收稿日期:2008-05-19 修回日期:2010-06-11 出版日期:2011-06-25 发布日期:2011-06-25
基金资助:
国家自然科学基金(60473141)和辽宁省自然科学基金(20062191)资助

2D Hilbert Transform and Bedrosian’s Principle Associated with Fractional Fourier Transform

XU Guan-Lei^1,3, WANG Xiao-Tong^1,3, XU Xiao-Gang^2,3

1.Department of Navigation, Dalian Naval Academy, Liaoning Dalian 116018;
2.Department of Automation, Dalian Naval Academy, Liaoning Dalian 116018;
3.Institute of Photoelectric Technology, Dalian Naval Academy, Liaoning Dalian 116018

Received:2008-05-19 Revised:2010-06-11 Online:2011-06-25 Published:2011-06-25
Supported by:
国家自然科学基金(60473141)和辽宁省自然科学基金(20062191)资助

摘要/Abstract

摘要：

多维解析信号分析和Hilbert变换在现代信号处理理论和工程应用中有着重要的意义. 该文根据解析信号在广义频域内的能量分布,提出了二维信号的方向广义Hilbert变换、全向广义Hilbert变换以及单象广义Hilbert变换等定义, 并证明了时域和分数阶Fourier域内它们之间的对应关系以及推导出类似于传统Hilbert变换中的性质. 同时,给出了解析信号中具有重要意义的
二维广义Bedrosian定理和相应的理论证明.

关键词: 分数阶Fourier变换, 广义Hilbert变换, 解析信号, Bedrosian定理

Abstract:

Hilbert transform (HT) plays an important role in signal processing. From the energy distributing of analytic signals in the FRFT domain and a few classical 2D elemen-tary Hilbert transform, the definitions of half-planed Hilbert transform, cross-orthant Hilbert transform and single-orthant Hilbert transform are yielded. Meanwhile, the expressions and the mapping in the time domain and the transformed domain are discussed. Moreover, some
important properties and conclusions are obtained as well. Finally, we define and derive 2D Bedrosian’s principle in the FRFT domain, an important property of Hilbert transform.

Key words: Fractional Fourier transform, Generalized Hilbert transform, Analytic signal, Bedrosian’s principle.

中图分类号:

42B05

徐冠雷, 王孝通, 徐晓刚. 分数阶Fourier域的二维广义Hilbert变换及Bedrosian定理[J]. 数学物理学报, 2011, 31(3): 814-828.

XU Guan-Lei, WANG Xiao-Tong, XU Xiao-Gang. 2D Hilbert Transform and Bedrosian’s Principle Associated with Fractional Fourier Transform[J]. Acta mathematica scientia,Series A, 2011, 31(3): 814-828.

参考文献

[1] Namias V. The fractinal order Fourier transform and its application to quantum mechanics. J Inst Math Appl, 1980, 25: 241--265

[2] McBride A C, Kerr F H. On Namias' fractional Fourier transform. IMA J Appl Math, 1987, 39: 159--175

[3] Mendlovic D, Ozaktas H M. Fractional Fourier transforms and their optical implementation (I). J Opt Sco AM A, 1993, 10: 1875--1881

[4] Ozaktas H M, Mendlovic D. Fractional Fourier transforms and their optical implementation (II). J Opt Sco AM A, 1993, 10: 2522--2531

[5] 陶然, 邓兵, 王越. 分数阶Fourier变换在信号处理领域的研究进展.中国科学E辑, 信息科学, 2006, 36: 113--136

[6] 陶然, 邓兵, 王越. 分数阶Fourier 变换的原理与应用. 北京: 清华大学出版社, 2004

[7] Ozaktas H M, Kutay M A, Zalevsky Z. The Fractional Fourier Transform with Applications in Optics and Signal Processing. New York: John Wiley & Sons, 2000

[8] Zayed A I. Hilbert transform associated with the fractional Fourier transform. IEEE Signal Processing Letters, 1998, 5: 206--208

[9] Fu Y X, Li L Q. A generalized Bedrosian theorem in fractional Fourier Domain. IEEE Proc, 2006: 1785--1788

[10] Lohmann A W, Mendlovic D, Zalevsky Z. Fractional Hilbert transform. Opt Lett, 1996, 21: 281--283

[11] Pei S C, Yeh M H. Discrete fractional Hilbert transform. IEEE Trans Circuits and Systems-II: Analog and Digital Signal Processing, 2000, 11: 1307--1311

[12] Tseng C C, Pei S C. Design of discrete-time fractional Hilbert transformer. IEEE Proc, ISCAS, 2000, V: 525--528

[13] Pei S C, Ding J J. The generalized Hilbert transform and its applications to 2-D edge detection. IEEE Proc, ICASSP, 2003, III: 357--360

[14] Stark H. An extension of the Hilbert transform product theorm. IEEE Proc, 1971, 59: 1359--1360

[15] Havlicek J P, Havlicek J W, Ngao D, et al. Skewed 2D Hilbert transforms and computed AM-FM models. IEEE Proc, 1998, 59 : 602--606

[16] Hahn L S. Multidimensional complex signals with single-orthant spectra. IEEE Proc, 1992, 80: 1287--1300

[17] Thomas B, Gerald S. Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case. IEEE Trans Signal Processing, 2001, 49: 2844--2852

[18] Chang J H, Pei S C, Ding J J. 2D quaternion Fourier spectral analysis and its applications.IEEE Proc, 2004, 3: 241--244

[19] Sangwine S J, Ell T A. Hypercomplex Fourier transforms of color images. IEEE Proc, 2001, 1: 137--140

[20] 徐冠雷, 王孝通, 徐晓刚. 二象Hilbert变换. 自然科学进展, 2007, 17: 168--178

[21} Pei S C, Yeh M H. Two dimensional discrete fractional Fourier transform. Signal Processing, 1998, 67: 99--108

[22] Pei S C, Ding J J. Two-Dimensional Affine generalized fractional Fourier transform. IEEE Trans Signal Processing, 2001, 49: 878--897

[23] Gabor D. Theory of communication. IEEE, 1946, 93: 429--457

[24] Ville J. Theorie et applications de la notion de signal analytique. Cables et Transmission, 1948, 2A: 61--74

[25] Boashash B. Estimating and interpreting the instantaneous frequency of a signal-part 1: fundamentals. IEEE Proc, 1992, 80: 520--539

[26] Boashash B. Estimating and interpreting the instantaneous frequency of a signal-part 2: algorithms and applications. IEEE Proc, 1992, 80: 540--568

[27] Cohen L, Lee C. Standard deviation of instantaneous frequency. International Conference on Acoustics, Speech, and Signal Processing, 1989, 4: 2238--2241

[28] Joseph M F, Friedlander B. Parameter estimation of 2-D random amplitude polynomial-phase signals. IEEE Trans Signal Processing, 1999, 47: 1795--1810

[29] Maragos P, Bovik A C. Demodulation of images modeled by amplitude-frequency modulationsusing multidimensional energy separation. IEEE Proc, ICIP, 1994, 3: 421--425

[30] Spagnolini U. 2-D phase unwrapping and instantaneous frequency estimation. IEEE Trans Geoscience and Remote Sensing, 1995, 33: 579--589

[31] Cohen L. What is a multicomponent signal. IEEE Proc, ICASSP'92, 1992, 5: 113--116

[32] 徐冠雷, 王孝通, 徐晓刚等. 多分量到单分量可用EMD分解的条件及判据. 自然科学进展, 2006, 16: 1356--1360

[33] Bedrosian E. A product theorem for Hilbert transform. IEEE, 1963, 51: 868--869

[34] 黄晋, 吕涛, 朱瑞. 解带有Hilbert核的奇异积分方程的高精度组合算法. 数学物理学报, 2009, 29A(1): 103--113

[35] 唐立, 杨文胜. 边值问题的概率数值方法.数学物理学报, 2008, 28A(1): 81--85