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Generalized Cramér–Rao inequality and uncertainty relation for fisher information on FrFT

  • Guanlei XuEmail author
  • Xiaogang Xu
  • Xun Wang
  • Xiaotong Wang
Original Paper

Abstract

Uncertainty principle plays an important role in signal processing, physics and mathematics and so on. In this paper, four novel uncertainty inequalities including the new generalized Cramér–Rao inequalities and the new uncertainty relations on Fisher information associated with fractional Fourier transform (FrFT) are deduced for the first time. These novel uncertainty inequalities extend the traditional Cramér–Rao inequality and the uncertainty relation on Fisher information to the generalized cases. Compared with the traditional Cramér–Rao inequality, the generalized Cramér–Rao inequalities’ bounds are sharper and tighter. In addition, the generalized Cramér–Rao inequalities build the relation between the Cramér–Rao bounds and the FrFT transform angles, which seem to be quaint compared with the traditional counterparts. Furthermore, the generalized Cramér–Rao inequalities give the relation between the FrFT’s variance and FrFT’s gradient’s integral in only one single transform domain, which is fully novel. On the other hand, compared with the traditional uncertainty relation on Fisher information, the newly deduced uncertainty relations on Fisher information yield the sharper and tighter bounds. These deduced inequalities are novel, and they will yield the potential advantage in the parameter estimation in the FrFT domain. Finally, examples are given to show the efficiency of these newly deduced inequalities.

Keywords

Fractional Fourier transform (FrFT) Uncertainty principle Cramér–Rao inequality Fisher information 

Notes

Acknowledgements

This work is fully supported by NSFCs (6197050275, 61471412, 61771020, 6197011044) and LZ15F020001.

References

  1. 1.
    Shinde, S., Vikram, M.G.: An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Sig. Process. 49(11), 2545–2548 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Mustard, D.: Uncertainty principle invariant under fractional Fourier transform. J. Austral. Math. Soc. Ser. B 33, 180–191 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Press of University of Cambridge, Cambridge (1951)zbMATHGoogle Scholar
  4. 4.
    Selig, K.K.: Uncertainty Principles Revisited, Technische Universitat Munchen, Tech. Rep., 2001 (online). http://www-lit.ma.tum.de/veroeff/quel/010.47001.pdf
  5. 5.
    Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Zhang, D.X.: Modern Signal Processing, 2nd edn, p. 362. Tsinghua University Press, Beijing (2002)Google Scholar
  7. 7.
    Loughlin, P.J., Cohen, L.: The uncertainty principle: global, local, or both? IEEE Trans. Signal Proc. 52(5), 1218–1227 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cohen, L.: The uncertainty principles of windowed wave functions. Opt. Commun. 179, 221–229 (2000)Google Scholar
  9. 9.
    Nayak, T.K.: Rao-Cramer type inequalities for mean squared error of prediction. Am. Stat. 56(2), 102–106 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gajek, L., Kałuszka, M.: Nonexponential applications of a global Cramèr–Rao inequality. Stat. J. Theor. Appl. Stat. 26(2), 111–122 (1995).  https://doi.org/10.1080/02331889508802472 zbMATHGoogle Scholar
  11. 11.
    Dehesa, J.S., González-Férez, R., Sánchez-Moreno, P.: The Fisher-information-based uncertainty relation, Cramer–Rao inequality and kinetic energy for the D-dimensional central problem. J. Phys. A Math. Theor. 40, 1845–1856 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dehesa, J.S., Martínez-Finkelshtei, A., Sorokin, V.N.: Information-theoretic measures for Morse and Pöschl–Teller potentials. Mol. Phys. 104(4), 613–622 (2006)Google Scholar
  13. 13.
    Brunel, N., Nadal, J.-P.: Mutual information, fisher information, and population coding. Neural Comput. 10, 1731–1757 (1998)Google Scholar
  14. 14.
    Sánchez-Moreno, P., Plastino, A.R., Dehesa, J.S.: A quantum uncertainty relation based on Fisher’s information. J. Phys. A Math. Theor. 44, 065301:1–065301:9 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Aytur, O., Ozaktas, H.M.: Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform. Opt. Commun. 120, 166–170 (1995)Google Scholar
  16. 16.
    Ozaktas, H.M., Aytur, O.: Fractional Fourier domains. Signal Process. 46, 119–124 (1995)zbMATHGoogle Scholar
  17. 17.
    Tao, R., Qi, L., Wang, Y.: Theory and Application of the Fractional Fourier Transform. Tsinghua University Press, Beingjing (2004)Google Scholar
  18. 18.
    Tao, R., Deng, B., Wang, Y.: Theory and Application of the Fractional Fourier Transform. Beijing Beijing Tsinghua University Press, Beingjing (2009)Google Scholar
  19. 19.
    Mendlovic, D., Ozaktas, H.M.: Fractional Fourier transforms and their optical implementation (I). J. Opt. Soc. Am. A 10(10), 1875–1881 (1993)Google Scholar
  20. 20.
    Ozaktas, H.M., Mendlovic, D.: Fractional Fourier transforms and their optical implementation (II). J. Opt. Soc. Am. A 10(10), 2522–2531 (1993)Google Scholar
  21. 21.
    Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pei, S.C., Ding, J.J.: Relations between fractional operations and time–frequency distributions, and their applications. IEEE Trans. Signal Proc. 49(8), 1638–1655 (2001)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pei, S.C., Ding, J.J.: Two-dimensional affine generalized fractional Fourier transform. IEEE Trans. Signal Process. 49(4), 878–897 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wódkiewicz, K.: Operational approach to phase-space measurements in quantum mechanics. Phys. Rev. Lett. 52(13), 1064–1067 (1984)MathSciNetGoogle Scholar
  25. 25.
    Stankovic, L., Alieva, T., Bastiaans, M.J.: Time–frequency signal analysis based on the windowed fractional Fourier transform. Signal Process. 83, 2459–2468 (2003)zbMATHGoogle Scholar
  26. 26.
    Xu, G., Wang, X., Xu, X.: Three cases of uncertainty principle for real signals in linear canonical transform domain. IET Signal Process. 3(1), 85–92 (2009)MathSciNetGoogle Scholar
  27. 27.
    Zhao, J., Tao, R., Wang, Y.: On signal moments and uncertainty relations associated with linear canonical transform. Signal Process. 90(9), 2686–2689 (2010)zbMATHGoogle Scholar
  28. 28.
    Sharma, K.K., Joshi, S.D.: Uncertainty principle for real signals in the linear canonical transform domains. IEEE Trans. Signal Process. 56(7), 2677–2683 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhao, J., Tao, R., Li, Y.L., Wang, Y.: Uncertainty principles for linear canonical transform. IEEE Trans. Signal Process. 57(7), 2856–2858 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Stern, A.: Uncertainty principles in linear canonical transform domains and some of their implications in optics. J. Opt. Soc. Am. A 25(3), 647–652 (2008)MathSciNetGoogle Scholar
  31. 31.
    Pei, S.C., Ding, J.J.: Uncertainty principle of the 2-D affine generalized fractional Fourier transform. In: Proceedings of APSIPA, pp. 1–4. Sapporo, Japan (2009)Google Scholar
  32. 32.
    Dang, P., Deng, G.T., Qian, T.: A tighter uncertainty principle for linear canonical transform in terms of phase derivative. IEEE Trans. Signal Process. 61(21), 5153–5164 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Yan, Y., Kou, K.I.: Uncertainty principles for hypercomplex signals in the linear canonical transform domains. Signal Process. 95(2), 67–75 (2014)Google Scholar
  34. 34.
    Li, B.Z., Tao, R., Xu, T.Z., et al.: The Poisson sum formulae associated with the fractional Fourier transform. Signal Process. 89(5), 851–856 (2009)zbMATHGoogle Scholar
  35. 35.
    Bing Zhao, L.I., Tao, R., Wang, Y.: Hilbert transform associated with the linear canonical transform. Acta Armamentarii 27(5), 827–830 (2006)Google Scholar
  36. 36.
    Jing, X.Y., Wu, F., Dong, X., et al.: An improved SDA based defect prediction framework for both within-project and cross-project class-imbalance problems. IEEE Trans. Softw. Eng. 43(4), 321–339 (2017)Google Scholar
  37. 37.
    Li, Z., Jing, X.Y., Zhu, X., et al.: On the multiple sources and privacy preservation issues for heterogeneous defect prediction. IEEE Trans. Softw. Eng. 45(4), 391–411 (2019)Google Scholar
  38. 38.
    Jing, X.Y., Zhu, X., Wu, F., et al.: Super-resolution person re-identification with semi-coupled low-rank discriminant dictionary learning. IEEE Trans. Image Process. 26(3), 1363–1378 (2017)MathSciNetGoogle Scholar
  39. 39.
    Jing, X., Zhang, D.: A face and palmprint recognition approach based on discriminant DCT feature extraction. IEEE Trans. Syst. Man Cybern. B 34(6), 2405–2415 (2004)Google Scholar
  40. 40.
    Zhiqiang, L., Xiao-Yuan, J., Xiaoke, Z.: Progress on approaches to software defect prediction. IET Softw. 12(3), 161–175 (2018)Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Guanlei Xu
    • 1
    Email author
  • Xiaogang Xu
    • 1
  • Xun Wang
    • 1
  • Xiaotong Wang
    • 2
  1. 1.College of Computer and Information EngineeringZhejiang Gongshang UniversityHangzhouChina
  2. 2.Dalian Navy AcademyDalianChina

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