Linear Canonical Transforms’ Discretization Formula Based on the Frequency-Domain Convolution Theory

  • Qiwei LiuEmail author
  • Yanheng Ma
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 463)


The Linear Canonical Transforms’ (LCT) domain not only contains the informations of time and frequency, but also includes the informations of varying frequency with time. Because of these characteristics which makes it have a unique advantage in dealing with nonstationary signals, it attracts more and more scientists and the engineers’ attention. Fourier Transform (FT), Fractional Fourier Transform (FrFT), Fresnel Transform and Scale Transforms can be seen as a special LCT’s form. And in the combination of time-frequency analysis method (Wigner distribution, short-time Fourier transform (STFT), Fuzzy function) [2] on the basis of Linear Canonical Transforms will have more development space. However, the discretization analysis of Linear Canonical Transforms is not clear enough, especially the discretization formula is not suitable for computer’s calculation. Therefore, based on the sampling theorem of bandpass signal, the discretization formula is deduced based on the frequency-domain discretization formula and the convolution formula, and the correctness and reversibility are verified on the computer’s calculation.


Linear Canonical Transforms (LCT) Fourier Transforms (FT) Fractional Fourier Transforms (FrFT) Discretization Frequency-domain discretization formula 


  1. 1.
    Sharma, K.K., Joshi, S.D.: Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265(2), 454–460 (2006)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(8), 1638–1655 (2001)Google Scholar
  3. 3.
    Hlawatsch, F., Bartels, G.F.B.: Linear and quadratic time-frequency signal representation. IEEE Signal Process. Mag. 9(2), 21–67 (1992)Google Scholar
  4. 4.
    Robinson, E.A.: A historical perspective of spectrum estimation. Proc. IEEE 70(9), 885–907 (1982)Google Scholar
  5. 5.
    Torres, R., Pellat, F.P., Torres, Y.: Fractional convolution, fractional correlation and their translation invariance properties. Signal Process. 90(6), 1976–1984 (2010)Google Scholar
  6. 6.
    Hennelly, B.M., Sheridan, J.T.: Fast numerical algorithm for the linear canonical transform. J. Opt. Soc. Am. A 22(5), 928–937 (2005)Google Scholar
  7. 7.
    Healy, J.J., Sheridan, J.T.: Cases where the linear canonical transform of a signal has compact support or is band-limited. Opt. Letters 33(3), 228–230 (2008)Google Scholar
  8. 8.
    Xiang, Q., Qin, K.Y., Zhang, C.W.: Sampling theorems of band-limited signals in the linear canonical transform domain. In: 2009 International Workshop on Information Security and Application, Qingdao, pp. 126–129 (2009)Google Scholar
  9. 9.
    Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)Google Scholar
  10. 10.
    Tao, R., Li, B.Z., Wang, Y.: Spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional Fourier domain. IEEE Trans. Signal Process. 55(7), 3541–3547 (2007)Google Scholar
  11. 11.
    Li, Y., Zhang, F., Li, Y., Tao, R.: An application of the linear canonical transform correlation for detection of LFM signals. IET Signal Process. (2015)Google Scholar
  12. 12.
    Zhang, F., Hu, Y., Tao, R., et al.: Signal reconstruction from partial information of discrete linear canonical transform. Optical Eng. 53(3), 1709–1717 (2014)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Shijiazhuang Engineering UniversityShijiazhuangChina

Personalised recommendations