Research ArticleOpen AccessGoogle Scholar indexed
Discrete Entropic Uncertainty Relations Associated with FRFT
Ocean department of Dalian Naval Academy, Dalian, China, 116018
Navgation department of Dalian Naval Academy, Dalian, China, 116018.
Ocean department of Dalian Naval Academy, Dalian, China, 116018
Ocean department of Dalian Naval Academy, Dalian, China, 116018
Navgation department of Dalian Naval Academy, Dalian, China, 116018.
- 1 Ocean department of Dalian Naval Academy, Dalian, China, 116018
- 2 Navgation department of Dalian Naval Academy, Dalian, China, 116018.
- 3 Ocean department of Dalian Naval Academy, Dalian, China, 116018
- 4 Ocean department of Dalian Naval Academy, Dalian, China, 116018
- 5 Navgation department of Dalian Naval Academy, Dalian, China, 116018.
Journal of Signal and Information Processing·Volume 04 (2013)·Pages 120–124·Published 1 August 2013·DOI10.4236/jsip.2013.43B021
Copy link · social · email
Abstract
Based on the definition and properties of discrete fractional Fourier transform (DFRFT), we introduced the discrete Hausdorff-Young inequality. Furthermore, the discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. Also, the condition of equality via Lagrange optimization was developed, as 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 reach their lowest bounds. In addition, the resolution analysis via the uncertainty is discussed as well.
KeywordsDiscrete Fractional Fourier Transform (DFRFT)Uncertainty PrincipleRényi EntropyShannon Entropy
- R. Ishii and K. Furukawa, “The Uncertainty Principle in Discrete Signals,” IEEE Trans Circuits and Systems, Vol. 33, No. 10, 1986, pp. 1032-1034.
- L. C. Calvez and P. Vilbe, “On the Uncertainty Principle in Discrete Signals,” IEEE Trans Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 39, No. 6, 1992, pp. 394-395. doi:10.1109/82.145299
- S. Shinde and M. G. Vikram, “An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain,” IEEE Transactions Signal Processing, Vol. 49, No. 11, 2001, pp. 2545-2548. doi:10.1109/78.960402
- 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. doi:10.1016/j.sigpro.2009.05.014
- R. Tao, B. Deng and Y. Wang, “Theory and Application of the Fractional Fourier Transform,” Tsinghua University Press, Beijing, 2009
- 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, 2007, Vol. 54, No. 7, pp. 1599-1611.
- T. M. Cover and J. A. Thomas, “Elements of Information Theory, Second Edition,” John Wiley &Sons, Inc.,2006.
- H. Maassen, “A Discrete Entropic Uncertainty Relation,” Quantum Probability and Applications, V, Springer-Verlag, New York, 1988, pp. 263-266.
- C. E. Shannon. “A Mathematical Theory of Communication,” The Bell System Technical Journal, Vol. 27, 1948, pp. 379-656
- A. Rényi, On Measures of Information and Entropy, In: Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, 1960, p. 547.
- G. Hardy, J. E. Littlewood and G. Pólya Inequalities. 2nd edition, Press of University of Cambridge, 1951.
- D. Amir, T. M. Cover and J. A. Thomas, “Information Theoretic Inequalities,” IEEE Trans Information Theory, Vol. 37, No. 6, 2001, pp. 1501-1508.
- D. Victor, O. Murad and P. Tomasz, “Resolution in Time–Frequency,” IEEE Transactions on Signal Processing, Vol. 47, No. 3, 1999, pp. 783-788. doi:10.1109/78.747783
- T. Przebinda, V. DeBrunner and M. Özayd1n, “The Optimal Transform for the Discrete Hirschman Uncertainty Principle,” IEEE Transactions on information theory, Vol. 47, No. 5, 2001, pp. 2086-2090. doi:10.1109/18.930948