Discrete Inequalities on LCT

Guanlei Xu¹, Xiaotong Wang², Xiaogang Xu²

¹Ocean Department of Dalian Naval Academy, Dalian, China.
²Navgation Department of Dalian Naval Academy, Dalian, China.

DOI: 10.4236/jsip.2015.62014                

Abstract

Linear canonical transform (LCT) is widely used in physical optics, mathematics and information processing. This paper investigates the generalized uncertainty principles, which plays an important role in physics, of LCT for concentrated data in limited supports. The discrete generalized uncertainty relation, whose bounds are related to LCT parameters and data lengths, is derived in theory. The uncertainty principle discloses that the data in LCT domains may have much higher concentration than that in traditional domains.

Keywords

Linear Canonical Transform (LCT), Uncertainty Inequality

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Zhang, X.D. (2002) Modern Signal Processing. 2nd Edition, Tsinghua University Press, Beijing, 362.
[2]	Selig, K.K. (2002) Uncertainty Principles Revisited. Electronic Transactions on Numerical Analysis, 14, 165-177.
[3]	Dembo, A., Cover, T.M. and Thomas, J.A. (2001) Information Theoretic Inequalities. IEEE Transactions on Information Theory, 37, 1501-1508.
[4]	Loughlin, P.J. and Cohen, L. (2004) The Uncertainty Principle: Global, Local, or Both? IEEE Transac-tions on Signal Processing, 52, 1218-1227.
[5]	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
[6]	Tao, R., Deng, B. and Wang, Y. (2009) Theory and Application of the Fractional Fourier Transform. Tsinghua University Press, Beijing.
[7]	Maassen, H. (1988) A Discrete Entropic Uncertainty Relation. Quantum Probability and Applications, Springer-Verlag, New York, 263-266.
[8]	Stern, A. (2007) Sampling of Compact Signals in Offset Linear Canonical Transform Domains. Signal, Image and Video Processing, 1, 359-367.
[9]	Shinde, S. and Vikram, M.G. (2001) An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain. IEEE Transactions on Signal Processing, 49, 2545-2548.
[10]	Mustard, D. (1991) Uncertainty Principle Invariant under Fractional Fourier Transform. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 33, 180-191. http://dx.doi.org/10.1017/S0334270000006986
[11]	Bialynicki-Birula, I. (1985) Entropic Uncertainty Relations in Quantum Mechanics. In: Accardi, L. and von Waldenfels, W., Eds., Quantum Probability and Applications II, Lecture Notes in Mathematics, Volume 1136, Springer, Berlin 90.
[12]	Aytür, O. and Ozaktas, H.M. (1995) Non-Orthogonal Domains in Phase Space of Quantum Optics and Their Relation to Fractional Fourier Transforms. Optics Communications, 120, 166-170. http://dx.doi.org/10.1016/0030-4018(95)00452-E
[13]	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
[14]	Sharma, K.K. and Joshi, S.D. (2008) Uncertainty Principle for Real Signals in the Linear Canonical Transform Domains. IEEE Transactions on Signal Processing, 56, 2677-2683. http://dx.doi.org/10.1109/TSP.2008.917384
[15]	Zhao, J., Tao, R., Li, Y.L. 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
[16]	Xu, G.L., Wang, X.T. and Xu, X.G. (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
[17]	Xu, G.L., Wang, X.T. and Xu, X.G. (2009) New Inequalities and Uncertainty Relations on Linear Canonical Transform Revisit. EURASIP Journal on Advances in Signal Processing, 2009, Article ID: 563265.http://dx.doi.org/10.1155/2009/563265
[18]	Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Generalized Entropic Uncertainty Principle on Fractional Fourier Transform. Signal Processing, 89, 2692-2697. http://dx.doi.org/10.1016/j.sigpro.2009.05.014
[19]	Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Uncertainty Inequalities for Linear Canonical Transform. IET Signal Processing, 3, 392-402. http://dx.doi.org/10.1049/iet-spr.2008.0102
[20]	Xu, G.L., Wang, X.T. and Xu, X.G. (2009) The Logarithmic, Heisenberg’s and Short-Time Uncertainty Principles Associated with Fractional Fourier Transform. Signal Processing, 89, 339-343. http://dx.doi.org/10.1016/j.sigpro.2008.09.002
[21]	Xu, G.L., Wang, X.T. and Xu, X.G. (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
[22]	Somaraju, R. and Hanlen, L.W. (2006) Uncertainty Principles for Signal Concentrations. Proceedings of the 7th Australian Communications Theory Workshop, Perth, 1-3 February 2006, 38-42. http://dx.doi.org/10.1109/AUSCTW.2006.1625252
[23]	Donoho, D.L. and Huo, X. (2001) Uncertainty Principles and Ideal Atomic Decomposition. IEEE Transactions on Information Theory, 47, 2845-2862. http://dx.doi.org/10.1109/18.959265
[24]	Donoho, D.L. and Stark, P.B. (1989) Uncertainty Principles and Signal Recovery. SIAM Journal on Applied Mathematics, 49, 906-930. http://dx.doi.org/10.1137/0149053
[25]	Elad, M. and Bruckstein, A.M. (2002) A Generalized Uncertainty Principle and Sparse Representation in Pairs of Bases. IEEE Transactions on Information Theory, 48, 2558-2567. http://dx.doi.org/10.1109/TIT.2002.801410
[26]	Xu, G.L., Wang, X.T. and Xu, X.G. (2010) Novel Uncertainty Relations in Fractional Fourier Transform Domain for Real Signals. Chinese Physics B, 19, Article ID: 014203. http://dx.doi.org/10.1088/1674-1056/19/1/014203
[27]	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 Transactions on Circuits and System II: Analog and Digital Signal Processing, 47, 2883-2888.
[28]	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. http://dx.doi.org/10.1364/JOSAA.20.000522
[29]	Qi, L., Tao, R., Zhou, S. and Wang, Y. (2004) Detection and Parameter Estimation of Multicomponent LFM Signal Based on the Fractional Fourier Transform. Science in China Series F, 47, 184-198. http://dx.doi.org/10.1360/02yf0456