Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform

Abstract

Uncertainty principle plays an important role in physics, mathematics, signal processing and et al. In this paper, based on the definition and properties of discrete linear canonical transform (DLCT), we introduced the discrete HausdorffYoung inequality. Furthermore, the generalized discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. In addition, the condition of equality via Lagrange optimization was developed, which shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations touch their lowest bounds. On one hand, these new uncertainty relations enrich the ensemble of uncertainty principles, and on the other hand, these derived bounds yield new understanding of discrete signals in new transform domain.

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Zhong, Y. , Wang, X. , Xu, G. , Shao, C. and Ma, Y. (2013) Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform. Journal of Signal and Information Processing, 4, 423-429. doi: 10.4236/jsip.2013.44054.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] R. Ishii and K. Furukawa, “The Uncertainty Principle in Discrete Signals,” IEEE Transactions on Circuits and Systems, Vol. 33, No. 10, 1986, pp. 1032-1034.
[2] L. C. Calvez and P. Vilbe, “On the Uncertainty Principle in Discrete Signals,” IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 39, No. 6, 1992, pp. 394-395. http://dx.doi.org/10.1109/82.145299
[3] S. Shinde and M. G. Vikram, “An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain,” IEEE Transactions of Signal Processing, Vol. 49, No. 11, 2001, pp. 2545-2548. http://dx.doi.org/10.1109/78.960402
[4] G. L. Xu, X. T. Wang and X. G. Xu, “Generalized Entropic Uncertainty Principle on Fractional Fourier Transform,” Signal Processing, Vol. 89, No. 12, 2009, pp. 2692-2697.
http://dx.doi.org/10.1016/j.sigpro.2009.05.014
[5] R. Tao, B. Deng and Y. Wang, “Theory and Application of the Fractional Fourier Transform,” Tsinghua University Press, Beijing, 2009.
[6] S. C. Pei and J. J. Ding, “Eigenfunctions of Fourier and Fractional Fourier Transforms with Complex Offsets and Parameters,” IEEE Trans Circuits and Systems-I: Regular Papers, Vol. 54, No. 7, 2007, pp. 1599-1611.
[7] T. M. Cover and J. A. Thomas, “Elements of Information Theory,” 2nd Edition, John Wiley &Sons, Inc., 2006.
[8] H. Maassen, “A Discrete Entropic Uncertainty Relation,” Quantum Probability and Applications, Springer-Verlag, New York, 1988, pp. 263-266.
[9] C. E. Shannon, “A Mathematical Theory of Communication,” The Bell System Technical Journal, Vol. 27, 1948, pp. 379-656.
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x
[10] A. Rényi, “On Measures of Information and Entropy,” Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, 1960, p. 547.
[11] G. Hardy, J. E. Littlewood and G. Pólya, “Inequalities,” 2nd Edition, Press of University of Cambridge, Cambridge, 1951.
[12] D. Amir, T. M. Cover and J. A. Thomas, “Information Theoretic Inequalities,” IEEE Transactions on Information Theory, Vol. 37, No. 6, 2001, pp. 1501-1508.
[13] A. Dembo, T. M. Cover and J. A. Thomas, “Information Theoretic Inequalities,” IEEE Transactions on Information Theory, Vol. 37, No. 6, 1991, pp. 1501-1518. http://dx.doi.org/10.1109/18.104312
[14] G. L. Xu, X. T. Wang and X. G. Xu, “The Logarithmic, Heisenberg’s and Short-Time Uncertainty Principles Associated with Fractional Fourier Transform,” Signal Processing, Vol. 89, No. 3, 2009, pp. 339-343.
http://dx.doi.org/10.1016/j.sigpro.2008.09.002
[15] G. L. Xu, X. T. Wang and X. G. Xu, “Generalized Uncertainty Principles Associated with Hilbert Transform Signal,” Image and Video Processing, 2013.
[16] G. L. Xu, X. T. Wang and X. G. Xu, “The Entropic Uncertainty Principle in Fractional Fourier Transform Domains,” Signal Processing, Vol. 89, No. 12, 2009, pp. 2692-2697.
http://dx.doi.org/10.1016/j.sigpro.2009.05.014
[17] A. Stern, “Sampling of Linear Canonical Transformed Signals,” Signal Processing, Vol. 86, No. 7, 2006, pp. 1421-1425. http://dx.doi.org/10.1016/j.sigpro.2005.07.031
[18] G. L. Xu, X. T. Wang and X. G. Xu, “Three Cases of Uncertainty Principle for Real Signals in Linear Canonical Transform Domain,” IET Signal Processing, Vol. 3, No. 1, 2009, pp. 85-92.
http://dx.doi.org/10.1049/iet-spr:20080019
[19] A. Stern, “Uncertainty Principles in Linear Canonical Transform Domains and Some of Their Implications in Optics,” Journal of the Optical Society of America, Vol. 25, No. 3, 2008, pp. 647-652.
http://dx.doi.org/10.1364/JOSAA.25.000647
[20] G. L. Xu, X. T. Wang and X. G. Xu, “Uncertainty Inequalities for Linear Canonical Transform,” IET Signal Processing, Vol. 3, No. 5, 2009, pp. 392-402,.
http://dx.doi.org/10.1049/iet-spr.2008.0102

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