Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Sharma, K.K., Joshi, S.D.: Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265(2), 454–460 (2006)
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)
Hlawatsch, F., Bartels, G.F.B.: Linear and quadratic time-frequency signal representation. IEEE Signal Process. Mag. 9(2), 21–67 (1992)
Robinson, E.A.: A historical perspective of spectrum estimation. Proc. IEEE 70(9), 885–907 (1982)
Torres, R., Pellat, F.P., Torres, Y.: Fractional convolution, fractional correlation and their translation invariance properties. Signal Process. 90(6), 1976–1984 (2010)
Hennelly, B.M., Sheridan, J.T.: Fast numerical algorithm for the linear canonical transform. J. Opt. Soc. Am. A 22(5), 928–937 (2005)
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)
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)
Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Liu, Q., Ma, Y. (2019). Linear Canonical Transforms’ Discretization Formula Based on the Frequency-Domain Convolution Theory. In: Liang, Q., Mu, J., Jia, M., Wang, W., Feng, X., Zhang, B. (eds) Communications, Signal Processing, and Systems. CSPS 2017. Lecture Notes in Electrical Engineering, vol 463. Springer, Singapore. https://doi.org/10.1007/978-981-10-6571-2_166
Download citation
DOI: https://doi.org/10.1007/978-981-10-6571-2_166
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6570-5
Online ISBN: 978-981-10-6571-2
eBook Packages: EngineeringEngineering (R0)