Signal, Image and Video Processing

, Volume 8, Issue 1, pp 85–93 | Cite as

On uncertainty principles for linear canonical transform of complex signals via operator methods

  • Jun ShiEmail author
  • Xiaoping Liu
  • Naitong Zhang
Original Paper


The linear canonical transform (LCT) has been shown to be a useful and powerful analyzing tool in optics and signal processing. Many results of this transform are already known, including its uncertainty principles (UPs). The existing UPs of the LCT for complex signals can only provide sharp bounds with LCT parameters satisfying \(a_1/b_1\ne a_2/b_2\). However, in most cases, we strive to find a lower bound, but not a sharper bound, since a lower bound often leads to optimization problems in signal processing applications. In this paper, we first present a much briefer and more transparent derivation to obtain a general uncertainty principle of the LCT for arbitrary signals via operator methods. Then, we derive lower bounds of three UPs of the LCT for complex signals, which are tighter lower bounds than the existing ones. We also prove that the derived results hold for arbitrary LCT parameters.


Linear canonical transform Complex signal Hermitian operator Time-frequency analysis Uncertainty principle 



This work was completed in parts while Shi J. was visiting the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA. The work was supported in part by the National Basic Research Program of China (Grant No. 2013CB329003), the National Natural Science Foundation of China (Grant No. 61171110), and the Short-Term Overseas Visiting Scholar Program of Harbin Institute of Technology.


  1. 1.
    Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2000)Google Scholar
  2. 2.
    Pei, S.C., Ding, J.J.: Relations between fractional operations and time-frequency distributions and their applications. IEEE Trans. Signal Process. 49, 1638–1655 (2001)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Erseghe, T., Laurenti, N., Cellini, V.: A multicarrier architecture based upon the affine Fourier transform. IEEE Trans. Commun. 53, 853–862 (2005)CrossRefGoogle Scholar
  4. 4.
    Alieva, T., Bastiaans, M.J.: Properties of the linear canonical integral transformation. J. Opt. Soc. Am. A 24, 3658–3665 (2007)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Healy, J.J., Sheridan, J.T.: Fast linear canonical transforms. J. Opt. Soc. Am. A 27, 21–30 (2010)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Liu, Y.L., Kou, K.I., Ho, I.T.: New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms. Signal Process. 90, 933–945 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Shinde, S.: Two channel paraunitary filter banks based on linear canonical transform. IEEE Trans. Signal Process. 59, 832–836 (2011)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Healy, J.J., Sheridan, J.T.: The space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms. J. Opt. Soc. Am. A 28, 786–790 (2011)CrossRefGoogle Scholar
  9. 9.
    Shi, J., Sha, X., Zhang, Q., Zhang, N.: Extrapolation of bandlimited signals in linear canonical transform domain. IEEE Trans. Signal Process. 60, 1502–1508 (2012)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Shi, J., Liu, X., Sha, X., Zhang, N.: Sampling and reconstruction of signals in function spaces associated with the linear canonical transform. IEEE Trans. Signal Process. 60, 6041–6047 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Shi, J., Liu, X., Zhang, N.: Generalized convolution and product theorems associated whit linear canonical transform. Signal Image Video Process. (2012). doi: 10.1007/s11760-012-0348-7
  12. 12.
    Stern, A.: Uncertainty principles in linear canonical transform domains and some of their implications in optics. J. Opt. Soc. Am. A 25, 647–652 (2008)CrossRefGoogle Scholar
  13. 13.
    Sharma, K.K., Joshi, S.D.: Uncertainty principle for real signals in the linear canonical transform domains. IEEE Trans. Signal Process. 56, 2677–2683 (2008)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Zhao, J., Tao, R., Li, Y.L., Wang, Y.: Uncertainty principles for linear canonical transform. IEEE Trans. Signal Process. 57, 2856–2858 (2009)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Zhao, J., Tao, R., Wang, Y.: On signal moments and uncertainty relations associated with linear canonical transform. Signal Process. 90, 2686–2689 (2010)Google Scholar
  16. 16.
    Xu, G., Wang, X., Xu, X.: Three uncertainty relations for real signals associated with linear canonical transform. IET Signal Process. 3, 85–92 (2009) Google Scholar
  17. 17.
    Xu, G., Wang, X., Xu, X.: On uncertainty principle for the linear canonical transform of complex signals. IEEE Trans. Signal Process. 58, 4916–4918 (2010)Google Scholar
  18. 18.
    Shi, J., Chi, Y., Zhang, N.: Multichannel sampling and reconstruction of bandlimited signals in fractional Fourier domain. IEEE Signal Process. Lett. 17, 909–912 (2010)CrossRefGoogle Scholar
  19. 19.
    Gabor, D.: Theory of communication. Inst. Electr. Eng. J. Commun. Eng. 93, 429–457 (1946)Google Scholar
  20. 20.
    Shi, J., Zhang, N., Liu, X.: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 55, 1270–1279 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Korn, P.: Some uncertainty principles for time-frequency transforms of the Cohen class. IEEE Trans. Signal Process. 53, 523–527 (2005)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Durak, L., Arikan, O.: Short-time Fourier transform: two fundamental properties and an optimal implementation. IEEE Trans. Signal Process. 51, 1231–1242 (2003)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Wei, L., Kennedy, R.A., Lamahewa, T.A.: An optimal basis of bandlimited functions for signal analysis and design. IEEE Trans. Signal Process. 58, 5744–5755 (2010)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Nuttall, A., Amoroso, F.: Minimum Gabor bandwidth of M orthogonal signals. IEEE Trans. Inf. Theory 14, 440–444 (1965)CrossRefGoogle Scholar
  25. 25.
    Bajwa, W.U., Haupt, J., Sayeed, A.M., Nowak, R.: Compressed channel sensing: a new approach to estimating sparse multipath channels. Proc. IEEE 98, 1058–1076 (2010)Google Scholar
  26. 26.
    Skolnik, M.I.: Introduction to Radar Systems, 3rd edn. McGraw-Hill, New York, NY (2001)Google Scholar
  27. 27.
    Larson, A.M., Yeh, A.T.: Delivery of sub-10-fs pulses for nonlinear optical microscopy by polarization-maintaining single mode optical fiber. Opt. Express 16, 14723–14730 (2008)CrossRefGoogle Scholar
  28. 28.
    Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207–238 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Cohen, L.: Time-Frequency Analysis. Prentice-Hall, Englewood Cliffs, NJ (1995)Google Scholar
  30. 30.
    Sayeed, A.M., Jones, D.L.: Integral transforms covariant to unitary operators and their implications for joint signal representations. IEEE Trans. Signal Process. 44, 1365–1376 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Communication Research CenterHarbin Institute of TechnologyHarbinChina
  2. 2.Shenzhen Graduate SchoolHarbin Institute of TechnologyShenzhenChina

Personalised recommendations