广义测不准原理理论研究
Research on Generalized Uncertainty Principles

作者: 王孝通 , 徐晓刚 :海军大连舰艇学院航海系,辽宁 大连; 徐冠雷 * , 周立佳 , 邵利民 :海军大连舰艇学院军事海洋系,辽宁 大连;

关键词: 分数Fourier变换线性正则变换测不准原理Fractional Fourier Transform (FRFT) Linear Canonical Transform (LCT) Uncertainty Principle

摘要: 测不准原理不仅是物理学中的一个基本问题,也是数学中的一个基本问题,同时对于信息学等多种学科均有较大的影响。从信号处理的角度,本文对分数阶域以及广义分数阶域(线性完整变换域或线性正则变换域)的Heisenberg测不准原理、加窗测不准原理、对数测不准原理、熵测不准原理等理论及其扩展进行了全面分析和综述,与传统测不准原理作了深入分析比较,剖析了广义测不准原理的发展和应用现状,并给出了广义测不准原理尚需进一步研究的问题。

Abstract: The uncertainty principle plays an important role in both physics and mathematics, and it also plays an important role in information science. This paper reviewed the Heisenberg’s uncertainty principles, windowed uncertainty principles, logarithmic uncertainty principles, entropic uncer-tainty principles and so on in the fractional Fourier transform domains and linear canonical transform domains in great details. The development and applications of generalized uncertainty principles were analyzed, and the trend and direction were given as well.

文章引用: 王孝通 , 徐冠雷 , 周立佳 , 邵利民 , 徐晓刚 (2016) 广义测不准原理理论研究。 应用数学进展, 5, 421-434. doi: 10.12677/AAM.2016.53053

参考文献

[1] Chilukuri, M.V. and Dash, P.K. (2004) Multiresolution S-Transform-Based Fuzzy Recognition System for Power Quality Events. IEEE Transactions on Power Delivery, 19, 323-330.
http://dx.doi.org/10.1109/TPWRD.2003.820180

[2] Richard, C. (2005) Time-Frequency Analysis of Visual Evoked Potentials Using Chirplet Transform. IEE Electronics Letters, 41, 217-218.
http://dx.doi.org/10.1049/el:20056712

[3] Wim, S. (1995) The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions. SPIE Proceedings of Wavelet Applications in Signal and Image Processing III, 2569, 68-79.
http://dx.doi.org/10.1117/12.217619

[4] Emmanuel, J. (2003) Ridgelets: Estimating with Ridge Functions. Annals of Statistics, 31, 1561-1599.
http://dx.doi.org/10.1214/aos/1065705119

[5] Donoho, D.L. (2000) Orthonormal Ridgelets and Linear Singularities. SIAM Journal on Mathematical Analysis, 31, 1062-1099.
http://dx.doi.org/10.1137/S0036141098344403

[6] Stockwell, R.G., Man-sinha, L. and Lowe, R.P. (1996) Localization of the Complex Spectrum: The S Transform. IEEE Transactions on Signal Processing, 44, 998-1001.
http://dx.doi.org/10.1109/78.492555

[7] Assous, S., Humeau, A., Tartas, M., Abraham, P. and L’Huillier, J. (2006) S-Transform Applied to Laser Doppler Flowmetry Reactive Hyperemia Signals. IEEE Transaction on Biomedical Engineering, 53, 1032-1037.
http://dx.doi.org/10.1109/TBME.2005.863843

[8] Pinnegar, C.R. and Mansinha, L. (2003) The S-Transform with Windows of Arbitrary and Varying Shape. Geophysics, 68, 381-385.
http://dx.doi.org/10.1190/1.1543223

[9] Tao, R., Li, Y. and Wang, Y. (2009) Short-Time Fractional Fourier Transform and Its Applications. IEEE Transaction on Signal Processing, 58, 2568-2580.
http://dx.doi.org/10.1109/TSP.2009.2028095

[10] 陶然, 邓兵, 王越. 分数阶Fourier 变换及其应用[M]. 北京: 北京清华大学出版社, 2009.

[11] 冉启文, 谭立英. 分数傅里叶光学导论[M]. 北京: 北京科学出版社, 2004.

[12] 张贤达, 保铮. 非平稳信号分析与处理[M]. 北京: 北京国防工业出版社, 1998.

[13] Pei, S.C., Yeh, M.H. and Luo, T.L. (1999) Fractional Fourier Series Expansion for Finite Signals and Dual Extension to Discrete-Time Fractional Fourier Transform. IEEE Transaction on Signal Processing, 47, 2883-2888.
http://dx.doi.org/10.1109/78.790671

[14] Heisenberg, W. (1927) Uber den anschaulichen inhalt der quanten theoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43, 172-198.
http://dx.doi.org/10.1007/BF01397280

[15] Cariolaro, G., Erseghe, T., Kraniauskas, P. and Laurenti, N. (1998) A Unified Framework for the Fractional Fourier Transform. IEEE Transaction on Signal Processing, 46, 3206-3219.
http://dx.doi.org/10.1109/78.735297

[16] Barshan, B., Kutay, M.A. and Ozaktas, H.M. (1997) Optimal Filters with Linear Ca-nonical Transformations. Optics Communications, 135, 32-36.
http://dx.doi.org/10.1016/S0030-4018(96)00598-6

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

[18] Pei, S.C. and Ding, J.J. (2001) Two-Dimensional Affine Generalized Fractional Fourier Transform. IEEE Transaction on Signal Processing, 49, 878-897.
http://dx.doi.org/10.1109/78.912931

[19] Tao, R., Li, B. and Wang, Y. (2007) Spectral Analysis and Reconstruction for Periodic No Uniformly Sampled Signals in Fractional Fourier Domain. IEEE Transactions on Signal Processing, 55, 3541-3547.
http://dx.doi.org/10.1109/TSP.2007.893931

[20] Tao, R., Deng, B., Zhang, W. and Wang, Y. (2008) Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain. IEEE Transactions on Signal Processing, 56, 158-171.
http://dx.doi.org/10.1109/TSP.2007.901666

[21] Xu, G., Wang, X. and Xu, X. (2009) The Logarithmic, Heisenberg’s and Windowed Uncertainty Principles in Fractional Fourier Transform Domains. Signal Processing, 89, 339-343.
http://dx.doi.org/10.1016/j.sigpro.2008.09.002

[22] Xu, G., Wang, X. and Xu, X. (2009) The Entropic Uncertainty Principle in Fractional Fourier Transform Domains. Signal Processing, 89, 2692-2697.
http://dx.doi.org/10.1016/j.sigpro.2009.05.014

[23] Xu, G., Wang, X. and Xu, X. (2010) Novel Uncertainty Relations in Frac-tional Fourier Transform Domain for Real Signals. Chinese Physics B, 19, 294-302.

[24] Xu, G., Wang, X. and Xu, X. (2009) Three Cases of Uncertainty Principle for Real Signals in Linear Canonical Transform Domain. IET Signal Processing, 3, 85-92.
http://dx.doi.org/10.1049/iet-spr:20080019

[25] Zhao, J., Tao, R., Li, Y. and Wang, Y. (2009) Uncertainty Principles for Linear Canonical Transform. IEEE Transactions on Signal Processing, 57, 2856-2858.
http://dx.doi.org/10.1109/TSP.2009.2020039

[26] Xu, G., Wang, X. and Xu, X. (2009) Uncertainty Inequalities for Linear Ca-nonical Transform. IET Signal Processing, 3, 392-402.
http://dx.doi.org/10.1049/iet-spr.2008.0102

[27] Xu, G., Wang, X. and Xu, X. (2009) New Inequalities and Uncertainty Relations on Linear Canonical Transform Revisit. EURASIP Journal on Advances in Signal Processing, 1-17.

[28] Xia, X.G. (1996) On Bandlimited Signals with Fractional Fourier Transform. IEEE Signal Processing Letter, 3, 72-74.
http://dx.doi.org/10.1109/97.481159

[29] Donoho, D.L. and Huo, X. (2001) Uncertainty Principles and Ideal Atomic Decom-position. IEEE Transactions on Information Theory, 47, 2845-2862.
http://dx.doi.org/10.1109/18.959265

[30] Calvez, L.C. and Vilbe, P. (1992) On the Uncertainty Principle in Discrete Signals. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 39, 394-395.
http://dx.doi.org/10.1109/82.145299

[31] Ishii, R. and Furukawa, K. (1986) The Uncertainty Principle in Discrete Signals. IEEE Transactions on Circuits and Systems, 33, 1032-1034.
http://dx.doi.org/10.1109/TCS.1986.1085842

[32] Doroslovacki, M.I. (1998) Product of Second Moments in Time and Fre-quency for Discrete-Time Signals and the Uncertainty Limit. Signal Processing, 67, 59-76.
http://dx.doi.org/10.1016/S0165-1684(98)00022-X

[33] Venkatesh, Y.V., Kumar, S. and Vidyasagar, G. (2006) On the Uncer-tainty Inequality as Applied to Discrete Signals. International Journal of Mathematics and Mathematical Sciences, 2006, Article ID: 48185.
http://dx.doi.org/10.1155/ijmms/2006/48185

[34] Wilk, G. and Włodarczyk, Z. (2009) Uncertainty Relations in Terms of the Tsallis Entropy. Physical Review A, 79, Article ID: 062108.
http://dx.doi.org/10.1103/PhysRevA.79.062108

[35] Bahri, M., Hitzer, E.S.M., Hayashi, A. and Ashino, R. (2008) An Uncertainty Principle for Quaternion Fourier Transform. Computers and Ma-thematics with Applications, 56, 2398-2410.
http://dx.doi.org/10.1016/j.camwa.2008.05.032

[36] Stark, H. (1971) An Extension of the Hilbert Transform Product Theorem. Proceedings of the IEEE, 59, 1359-1360.
http://dx.doi.org/10.1109/PROC.1971.8420

[37] Havlicek, J.P., Havlicek, J.W., Mamuya, N.D. and Bovik, A.C. (1998) Skewed 2D Hilbert Transforms and Computed AM-FM Models. Proceedings of 1998 International Conference on Image Processing, 1, 602-606.
http://dx.doi.org/10.1109/icip.1998.723573

[38] Thomas, B. and Gerald, S. (2001) Hypercomplex Signals: A Novel Extension of the Analytic Signal to the Multidimensional Case. IEEE Transaction on Signal Processing, 49, 2844-2852.
http://dx.doi.org/10.1109/78.960432

[39] Sangwine, S.J. and Ell, T.A. (2001) Hypercomplex Fourier Transforms of Color Im-ages. IEEE International Conference on Image Processing, 1, 137-140.
http://dx.doi.org/10.1109/icip.2001.958972

[40] Bedrosian, E. (1963) A Product Theorem for Hilbert Transform. Proceedings of the IEEE, 51, 868-869.
http://dx.doi.org/10.1109/PROC.1963.2308

[41] 徐冠雷, 王孝通, 徐晓刚. 二象Hilbert变换[J]. 自然科学进展, 2007, 17(8): 1120-1129.

[42] Fu, Y.X. and Li, L.Q. (2006) A Generalized Bedrosian Theorem in Fractional Fourier Domain. 2006 International Conference on Computational Intelligence and Security, 2, 1785-1788.
http://dx.doi.org/10.1109/ICCIAS.2006.295369

[43] Xu, G., Wang, X. and Xu, X. (2009) Generalized Hilbert Transform and Its Properties in 2D LCT Domain. Signal Processing, 89, 1395-1402.
http://dx.doi.org/10.1016/j.sigpro.2009.01.009

[44] Tao, R., Wang, X. and Wang, Y. (2009) Generalization of Fractional Hilbert Transform. IEEE Signal Processing Letter, 15, 365-368.

[45] Pei, S.C. and Ding, J.J. (2001) Two-Dimensional Affine Generalized Fractional Fourier Transform. IEEE Transaction on Signal Processing, 49, 1638-1655.

[46] Xu, G., Wang, X. and Xu, X. (2009) Improved Bi-Dimensional EMD and Hilbert Spectrum for the Analysis of Textures. Pattern Recognition, 42, 718-734.
http://dx.doi.org/10.1016/j.patcog.2008.09.017

[47] Zheng, S., Shi, W., Liu, J. and Tian, J. (2008) Remote Sensing Image Fusion Using Multiscale Mapped LS-SVM. IEEE Transactions on Geoscience and Remote Sensing, 46, 1313-1322.
http://dx.doi.org/10.1109/TGRS.2007.912737

[48] Zheng, S., Shi, W., Liu, J. and Tian, J. (2007) Multi Source Image Fusion Method Using Support Value Transform. IEEE Transactions on Image Processing, 16, 1831-1839.
http://dx.doi.org/10.1109/TIP.2007.896687

[49] Aanæs, H., Sveinsson, J.R., Nielsen, A., Thomas, B. and Benediktsson, J. (2008) Model-Based Satellite Image Fusion. IEEE Transactions on Geoscience and Remote Sensing, 46, 1336-1346.
http://dx.doi.org/10.1109/TGRS.2008.916475

[50] Thomas, C., Ranchin, T., Wald, L. and Chanussot, J. (2008) Synthesis of Multispectral Images to High Spatial Resolution: A Critical Review of Fusion Methods Based on Remote Sensing Physics. IEEE Transactions on Geoscience and Remote Sensing, 46, 1301-1312.
http://dx.doi.org/10.1109/TGRS.2007.912448

[51] Liu, Z., Forsyth, D.S., Safizadeh, M.S. and Fahr, A. (2008) A Data-Fusion Scheme for Quantitative Image Analysis by Using Locally Weighted Regression and Dempster-Shafer Theory. IEEE Transactions on Instrumentation and Measurement, 57, 2554-2560.
http://dx.doi.org/10.1109/TIM.2008.924933

[52] 覃征, 鲍复民, 等. 数字图象融合[M]. 西安: 西安交通大学出版社, 2004: 7.

[53] 那彦, 焦李成. 基于多分辨分析理论的图象融合方法[M]. 西安: 西安电子科技大学出版社, 2007: 5.

[54] Xu, G., Wang, X. and Xu, X. (2010) On Uncertainty Principle for the Linear Canonical Transform of Complex Signals. IEEE Transactions on Signal Processing, 58, 4916-4918.
http://dx.doi.org/10.1109/TSP.2010.2050201

[55] 徐冠雷. 分量分解的信号变换及信号分辨率分析研究[D]: [博士学位论文]. 大连: 海军大连舰艇学院, 2009: 7.

[56] Ding, J.J. and Pei, S.C. (2013) Heisenberg’s Uncertainty Principles for the 2-D Nonseparable Linear Canonical Transforms. Signal Processing, 93, 1027-1043.
http://dx.doi.org/10.1016/j.sigpro.2012.11.023

[57] Shi, J., Liu, X. and Zhang, N. (2012) On Uncertainty Principle for Signal Concentrations with Fractional Fourier Transform. Signal Processing, 92, 2830-2836.
http://dx.doi.org/10.1016/j.sigpro.2012.04.008

[58] Zhao, J., Tao, R. and Wang, Y. (2010) On Signal Moments and Uncertainty Relations Associated with Linear Canonical Transform. Signal Processing, 90, 2686-2689.
http://dx.doi.org/10.1016/j.sigpro.2010.03.017

[59] Xu, G., Wang, X., Zhou, L. and Xu, X. (2013) New Inequalities on Sparse Representation in Pairs of Bases. IET Signal Processing, 7, 674-683.
http://dx.doi.org/10.1049/iet-spr.2012.0365

[60] Xu, G., Wang, X., Zhou, L., Shao, L. and Xu, X. (2013) Discrete Entropic Uncertainty Relations Associated with FRFT. Journal of Signal and Information Processing, 4, 120-124.
http://dx.doi.org/10.4236/jsip.2013.43B021

[61] Xu, G., Wang, X. and Xu, X. (2014) Generalized Uncertainty Principles Associated with Hilbert Transform. Signal, Image and Video Processing, 8, 279-285.
http://dx.doi.org/10.1007/s11760-013-0547-x

[62] Xu, G., Wang, X., Xu, X., Hu, J. and Li, B. (2015) Discrete Inequalities on LCT. Journal of Signal and Information Processing, 6, 146-152.
http://dx.doi.org/10.4236/jsip.2015.62014

[63] Heinig, H.P. and Smith, M. (1986) Extensions of the Heisenberg-Weyl Inequality. International Journal of Mathematics and Science, 9, 185-192.
http://dx.doi.org/10.1155/S0161171286000212

[64] Selig, K.K. (2002) Uncertainty Principles Revisited. Electronic Transactions on Numerical Analysis, 14, 145-177.

[65] Folland, G.B. and Sitaram, A. (1997) The Uncertainty Principle: A Mathematical Survey. The Journal of Fourier Analysis and Applications, 3, 207-238.
http://dx.doi.org/10.1007/BF02649110

[66] Stankovic, L., Alieva, T. and Bastiaans, M.J. (2003) Time-Frequency Signal Analysis Based on the Windowed Fractional Fourier Transform. Signal Processing, 83, 2459-2468.
http://dx.doi.org/10.1016/S0165-1684(03)00197-X

[67] Cohen, L. (2000) The uncertainty principles of windowed wave functions. Optics Communications, 179, 221–229.
http://dx.doi.org/10.1016/S0030-4018(00)00454-5

[68] Loughlin, P.J. and Cohen, L. (2004) The Uncertainty Principle: Global, Local, or Both? IEEE Transaction on Signal Processing, 52, 1218-1227.
http://dx.doi.org/10.1109/TSP.2004.826160

[69] Beckner, W. (1995) Pitt’s Inequality and the Uncertainty Principle. Proceedings of the American Mathematical Society, 123, 1897-1905.
http://dx.doi.org/10.1090/s0002-9939-1995-1254832-9

[70] Beckner, W. (1975) Inequalities in Fourier analysis. The Annals of Mathematics, 2nd Ser, 102, 159-182.
http://dx.doi.org/10.2307/1970980

[71] Hirschman Jr., I.I. (1957) A Note on Entropy. American Journal of Mathematics, 79, 152-156.
http://dx.doi.org/10.2307/2372390

[72] Majerník, V., Majerníková, E. and Shpyrko, S. (2003) Uncertainty Relations Expressed by Shannon-Like Entropies. CEJP, 3, 393-420.
http://dx.doi.org/10.2478/bf02475852

[73] Iwo, B.B. (1985) Entropic Uncertainty Relations in Quantum Mechanics. Quantum Probability and Applications II. In: Accardi, L. and von Waldenfels, W., Eds., Lecture Notes in Mathematics 1136, Springer, Berlin, 90.

[74] Maassen, H. (1988) A Discrete Entropic Uncertainty Relation. Quantum Probability and Applications V. Springer- Verlag, New York, 263-266.

[75] Maassen, H. and Uffink, J.B.M. (1983) Generalized Entropic Uncertainty Relations. Physical Review Letters, 60, 1103-1106.
http://dx.doi.org/10.1103/PhysRevLett.60.1103

[76] Amir, D., Cover, T.M. and Thomas, J.A. (2001) Information Theoretic Inequalities. IEEE Transactions on Information Theory, 37, 1501-1508.

[77] Iwo, B.B. (2006) Formulation of the Uncertainty Rela-tions in Terms of the Rényi Entropies. Physical Review A, 74, Article ID: 052101.

[78] Iwo, B.B. (2007) Rényi Entropy and the Uncertainty Relations. In, Adenier, G., Fuchs, C.A. and Khrennikov, A.Y., Eds., Foundations of Probability and Physics, American Institute of Physics, Melville, 52-62.

[79] Gill, J. (2005) An Entropy Measure of Uncertainty in Vote Choice. Electoral Studies, 24, 371-392.
http://dx.doi.org/10.1016/j.electstud.2004.10.009

[80] Rényi, A. (1960) Some Fundamental Questions of Information Theory. Selected Papers of Alfred Renyi, Vol 2, pp 526-552, Akademia Kiado, Budapest, 1976.

[81] Rényi, A. (1960) On Measures of In-formation and Entropy. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, Berkeley, 20 June-30 July 1960, 547-561.

[82] Shannon, C.E. (1948) A Mathematical Theory of Communication. The Bell System Technical Journal, 27, 379-656.
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x

[83] Wódkiewicz, K. (1984) Operational Approach to Phase-Space Measurements in Quantum Mechanics. Physical Review Letters, 52, 1064-1067.
http://dx.doi.org/10.1103/PhysRevLett.52.1064

[84] Mustard, D. (1991) Uncertainty Principle Invariant under Fractional Fourier Transform. Journal of the Australian Mathematical Society, 33, 180-191.
http://dx.doi.org/10.1017/S0334270000006986

[85] Ozaktas, H.M. and Aytur, O. (1995) Fractional Fourier Domains. Signal Processing, 46, 119-124.
http://dx.doi.org/10.1016/0165-1684(95)00076-P

[86] Shinde, S. and Vikram, M.G. (2001) An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain. IEEE Transaction on Signal Processing, 49, 2545-2548.
http://dx.doi.org/10.1109/78.960402

[87] Stern, A. (2007) Sampling of Compact Signals in Offset Linear Canonical Transform Domains. Signal, Image and Video Processing, 1, 359-367.
http://dx.doi.org/10.1007/s11760-007-0029-0

[88] Stern, A. (2008) Uncertainty Principles in Linear Canonical Transform Domains and Some of Their Implications in Optics. Journal of the Optical Society of America A, 25, 647-652.
http://dx.doi.org/10.1364/JOSAA.25.000647

[89] Aytur, O. and Ozaktas, H.M. (1995) Non-Orthogonal Domains in Phase Space of Quantum Optics and Their Relation to Fractional Fourier Transform. Optics Communi-cations, 120, 166-170.
http://dx.doi.org/10.1016/0030-4018(95)00452-E

[90] Lohmann, A.W. (1994) Relationships between the Radon-Wigner and Fractional Fourier Transfoms. Journal of the Optical Society of America A, 11, 1398-1401.
http://dx.doi.org/10.1364/JOSAA.11.001798

[91] Sharma, K.K. and Joshi, S.D. (2008) Uncertainty Principle for Real Signals in the Linear Canonical Transform Domains. IEEE Transaction on Signal Processing, 56, 2677-2683.
http://dx.doi.org/10.1109/TSP.2008.917384

[92] Adams, M.D., Kossentini, F. and Ward, R.K. (2002) Generalized S Transform. IEEE Transaction on Signal Processing, 50, 2831-2842.
http://dx.doi.org/10.1109/TSP.2002.804085

[93] Schimmel, M. and Gallart, J. (2005) The Inverse S-Transform in Filters with Time-Frequency Localization. IEEE Transaction on Signal Processing, 53, 4417-4422.
http://dx.doi.org/10.1109/TSP.2005.857065

[94] Simon, C., Ventosa, S., Schimmel, M. and Heldring, A. (2007) The S-Transform and Its Inverses: Side Effects of Discretizing and Filtering. IEEE Transaction on Signal Processing, 55, 4928-4937.
http://dx.doi.org/10.1109/TSP.2007.897893

[95] Xu, G., Wang, X., Xu, X. and Zhou, L. (2016) Entropic Inequalities on Sparse Representation. IET Signal Processing, 10, 413-421.
http://dx.doi.org/10.1049/iet-spr.2014.0072

分享
Top