Xu Guanlei Dalian Navy Academy

Transcription

1 Xu Guanlei Dalian Navy Academy

2 Werner Heisenberg x p 2 So, it is called as Heisenberg's uncertainty principle

3 2 2 t u 1/4 2 2 ( ) 2 u u u F u du where t t t f ( t) dt

4 What will happen for UP? Three directions: (and others such as mathematical cases) (1)The lower bound or sharper bound; (2)The new forms of UP; (3)The new applications of UP.

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6 In 2001, Sudarshan Shinde et al. introduced the sharper bound for UP in An uncertainty principle for real signals in the fractional Fourier transform domain, in IEEE TSP: Only for real signals

7 K. K. Sharma and S. D. Joshi, Uncertainty principle for real signals in the linear canonical transform domains, IEEE Trans. Signal Process., vol. 56, no. 7, pp , Only for real signals

8 What about the complex signals?

9 G. L. Xu, X. T. Wang, and X. G. Xu, On uncertainty principle for the linear canonical transform of complex signals, IEEE Trans. Signal Process., vol. 58, no. 9, pp , For the complex signals.

10 Pei Dang, Guan-Tie Deng, and Tao Qian, A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase Derivative,2013,IEEE TSP: The complex signals

11 There are also other papers published in all kinds of Journels, however, here we only review a few papers. The other papers such as follows: K. K. Sharma, New inequalities for signal spreads in linear canonical Transform domains, Signal Process., vol. 90, no. 3, pp , A.Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Amer. A, vol. 25, pp , J. Zhao, R. Tao, Y. L. Li, and Y. Wang, Uncertainty principles for linear canonical transform, IEEE Trans. Signal Process., vol. 57, no. 7, pp , J. Zhao, R. Tao, and Y. Wang, On signal moments and uncertainty relations associated with linear canonical transform, Signal Process., vol. 90, no. 9, pp , P. Dang, G. T. Deng, and T. Qian, A sharper uncertainty principle, J. Functional Anal., vol. 265, pp , Jun Shi,Xiaoping Liu,Naitong Zhang, On uncertainty principle for signal concentrations with fractional Fourier transform, Signal Processing 92 (2012) Soo-Chang Pei, Jian-Jiun Ding, Uncertainty Principle of the 2-D Affine Generalized Fractional Fourier Transform, Proceedings of 2009 APSIPA Annual Summit and Conference, Sapporo, Japan, October 4-7, 2009

12 What about the New Forms?

13 A. Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Amer. A, vol. 25, pp , The all parameters of a,b,c,d in LCT are employed in these forms.

14 G. L. Xu, X. T.Wang, and X. G. Xu, Three uncertainty relations for real signals associated with linear canonical transform, IET Signal Process., vol. 3, no. 1, pp , Real Signals The all parameters of a,b,c,d in LCT are employed in these forms.

15 What about the complex signals?

16 G. L. Xu, X. T. Wang, and X. G. Xu, On uncertainty principle for the linear canonical transform of complex signals, IEEE Trans. Signal Process., vol. 58, no. 9, pp , Complex Signals The all parameters of a,b,c,d in LCT are employed in these forms.

17 Yan Yang, KitIan Kou, Uncertainty principles for hyper complex signals in the linear canonical transform domains, Signal Processing,95(2014) For the hyper complex signals or quaternion signals

18 Xu Guanlei, Wang Xiaotong, Xu Xiaogang [J]. Generalized Uncertainty Principles associated with Hilbert Transform [J]. Signal, Image and Video Processing, 2014, 8(2): For the complex signals by Hilbert transform out of the real signals

19 Xu Guanlei, Wang Xiaotong, Xu Xiaogang. The Logarithmic, Heisenberg s and Windowed Uncertainty Principles in Fractional Fourier Transform Domains[J]. Signal Processing, 2009, 89(3) : For Logairthmic and Windowed UP

20 For Entropic UP Xu Guanlei, Wang Xiaotong, Xu Xiaogang. Generalized entropic uncertainty principle on fractional Fourier transform [J]. Signal Processing, 2009, 89(12):

21

22 in

23 In practice, the signals are digital or discrete, we must give the discrete UP at first. Xu Guanlei, Wang Xiaotong, Zhou Lijia, Shao Limin, Xu Xiaogang. Discrete Entropic Uncertainty Relations Associated with FRFT [J]. Journal of Signal and Information Processing,2013,3B: Xu Guanlei, Wang Xiaotong, Xu Xiaogang, Hu jiang, Li Binyu. Discrete Inequalities on LCT[J]. Journal of Signal and Information Processing, 2015,6(2):

24 [1]D. Donoho, P. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 1989, 49: [2]D. L. Donoho, X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 2001,47(7): [3]D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 2006, 52: [6]M. Elad, A. M. Bruckstein, A generalized uncertainty principle and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 2002, 48(9): [7]A. Feuer, A. Nemirovski, On sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 2003, 49(6): [8]R. Gribonval, M. Nielsen, Sparse representations in unions of bases, IEEE Trans. Inf. Theory, 2003, 49(12): [9]Y. Li, S. Amari, Two conditions for equivalence of 0-norm solution and 1-norm solution in sparse representation, IEEE Trans. Neural Netw., 2010, 21 (7): [10]J.J. Fuchs, On sparse representations in arbitrary redundant bases, IEEE Trans. Inf. Theory, 2004, 50 (6): [12]K. Patrick, D. Giuseppe, B. Helmut, Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets, IEEE Trans. Inf. Theory, 2012, 58(1):

25 [19]E.J. Candès, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 2005, 59: [23]E. J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 2006, 52 (2): Xu Guanlei, Wang Xiaotong, Zhou Lijia, Xu Xiaogang. New Inequalities on Sparse Representation in Pairs of Bases [J]. IET Signal Processing, 2013, 7(8), pp Xu Guanlei, Wang Xiaotong, Xu Xiaogang, Zhou Lijia. Entropic Inequalities on Sparse Representation[J]. IET Signal Processing,2016.

26 Xu Guanlei, Wang Xiaotong, Zhou Lijia, Xu Xiaogang. New Inequalities on Sparse Representation in Pairs of Bases [J]. IET Signal Processing, 2013, 7(8), pp Xu Guanlei, Wang Xiaotong, Xu Xiaogang, Zhou Lijia. Entropic Inequalities on Sparse Representation[J]. IET Signal Processing,2016. These relations tell us how the signal is sparse and how to select bases.

27 (1)The lower bound or sharper bound; (2)The new forms of UP; (3)The new applications of UP. They are the potential Fundationals or Bases of CS and Sparse Representation.

28 The Classical/Traditional UP FRFT and LCT : extensions Heisenberg s UP Entropic UP extension Windowed UP Logarithmic UP Sharper Ones New Forms with More Parameters Application Generalized Extensions Applicatio n Sparse Representation?

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