Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mustard, D.: Uncertainty principle invariant under fractional Fourier transform. J. Austral. Math. Soc. Ser. B 33, 180–191 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Press of University of Cambridge, Cambridge (1951)

    MATH  Google Scholar 

  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. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhang, D.X.: Modern Signal Processing, 2nd edn, p. 362. Tsinghua University Press, Beijing (2002)

    Google Scholar 

  7. Loughlin, P.J., Cohen, L.: The uncertainty principle: global, local, or both? IEEE Trans. Signal Proc. 52(5), 1218–1227 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, L.: The uncertainty principles of windowed wave functions. Opt. Commun. 179, 221–229 (2000)

    Article  Google Scholar 

  9. Nayak, T.K.: Rao-Cramer type inequalities for mean squared error of prediction. Am. Stat. 56(2), 102–106 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  13. Brunel, N., Nadal, J.-P.: Mutual information, fisher information, and population coding. Neural Comput. 10, 1731–1757 (1998)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  16. Ozaktas, H.M., Aytur, O.: Fractional Fourier domains. Signal Process. 46, 119–124 (1995)

    Article  MATH  Google Scholar 

  17. Tao, R., Qi, L., Wang, Y.: Theory and Application of the Fractional Fourier Transform. Tsinghua University Press, Beingjing (2004)

    Google Scholar 

  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. Mendlovic, D., Ozaktas, H.M.: Fractional Fourier transforms and their optical implementation (I). J. Opt. Soc. Am. A 10(10), 1875–1881 (1993)

    Article  Google Scholar 

  20. Ozaktas, H.M., Mendlovic, D.: Fractional Fourier transforms and their optical implementation (II). J. Opt. Soc. Am. A 10(10), 2522–2531 (1993)

    Article  Google Scholar 

  21. Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pei, S.C., Ding, J.J.: Two-dimensional affine generalized fractional Fourier transform. IEEE Trans. Signal Process. 49(4), 878–897 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wódkiewicz, K.: Operational approach to phase-space measurements in quantum mechanics. Phys. Rev. Lett. 52(13), 1064–1067 (1984)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  40. Zhiqiang, L., Xiao-Yuan, J., Xiaoke, Z.: Progress on approaches to software defect prediction. IET Softw. 12(3), 161–175 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. College of Computer and Information Engineering, Zhejiang Gongshang University, Hangzhou, 310018, China

    Guanlei Xu, Xiaogang Xu & Xun Wang

  2. Dalian Navy Academy, Dalian, 610018, China

    Xiaotong Wang

Authors
  1. Guanlei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaogang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaotong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guanlei Xu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, G., Xu, X., Wang, X. et al. Generalized Cramér–Rao inequality and uncertainty relation for fisher information on FrFT. SIViP 14, 499–507 (2020). https://doi.org/10.1007/s11760-019-01571-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-019-01571-9

Keywords

Search

Navigation