The generalization of the Poisson sum formula associated with the linear canonical transform. (English) Zbl 1264.42001

Summary: The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.


42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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[1] R. Tao, B. Deng, and Y. Wang, Fractional Fourier Transform and Its Applications, Tsinghua University Press, Beijing, China, 2009.
[2] M. Zhu, B.-Z. Li, and G.-F. Yan, “Aliased polyphase sampling associated with the linear canonical transform,” IET Signal Processing, vol. 6, no. 6, pp. 594-599, 2012.
[3] K. B. Wolf, Integral Transforms in Science and Engineering, vol. 11 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, NY, USA, 1979. · Zbl 0514.42013
[4] C.-P. Li, B.-Z. Li, and T.-Z. Xu, “Approximating bandlimited signals associated with the LCT domain from nonuniform samples at unknown locations,” Signal Processing, vol. 92, no. 7, pp. 1658-1664, 2012.
[5] D. Wei, Q. Ran, and Y. Li, “Reconstruction of band-limited signals from multichannel and periodic nonuniform samples in the linear canonical transform domain,” Optics Communications, vol. 284, no. 19, pp. 4307-4315, 2011.
[6] W. Qiu, B.-Z. Li, and X.-W. Li, “Speech recovery based on the linear canonical transform,” Speech Communication, vol. 50, no. 1, pp. 40-50, 2013.
[7] B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China. Series F. Information Sciences, vol. 49, no. 5, pp. 592-603, 2006.
[8] S.-C. Pei and J.-J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Transactions on Signal Processing, vol. 49, no. 8, pp. 1638-1655, 2001. · Zbl 1369.94258
[9] B.-Z. Li and T.-Z. Xu, “Spectral analysis of sampled signals in the linear canonical transform domain,” Mathematical Problems in Engineering, vol. 2012, Article ID 536464, 19 pages, 2012. · Zbl 1264.94057
[10] B.-Z. Li, R. Tao, T.-Z. Xu, and Y. Wang, “The Poisson sum formulae associated with the fractional Fourier transform,” Signal Processing, vol. 89, no. 5, pp. 851-856, 2009. · Zbl 1161.94339
[11] K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Optics Communications, vol. 265, no. 2, pp. 454-460, 2006.
[12] R. Tao, B.-Z. Li, Y. Wang, and G. K. Aggrey, “On sampling of band-limited signals associated with the linear canonical transform,” IEEE Transactions on Signal Processing, vol. 56, no. 11, pp. 5454-5464, 2008. · Zbl 1390.94661
[13] B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Processing, vol. 87, no. 5, pp. 983-990, 2007. · Zbl 1186.94201
[14] J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Processing, vol. 89, no. 4, pp. 641-648, 2009. · Zbl 1157.94332
[15] H. G. Feichtinger, “Parseval’s relationship for nonuniform samples of signals with several variables,” IEEE Transactions on Signal Processing, vol. 40, no. 5, pp. 1262-1263, 1992. · Zbl 0825.94076
[16] F. Marvasi, “Reflections on the Poisson sum formula and the uniform sampling,” IEICE Transactions, vol. e67, no. 9, pp. 494-501.
[17] A. Córdoba, “La formule sommatoire de Poisson,” Comptes Rendus des Séances de l’Académie des Sciences. Série I. Mathématique, vol. 306, no. 8, pp. 373-376, 1988. · Zbl 0663.42019
[18] A. Córdoba, “Dirac combs,” Letters in Mathematical Physics, vol. 17, no. 3, pp. 191-196, 1989. · Zbl 0681.42013
[19] J.-P. Kahane and P.-G. Lemarié-Rieusset, “Remarques sur la formule sommatoire de Poisson,” Studia Mathematica, vol. 109, no. 3, pp. 303-316, 1994. · Zbl 0820.42004
[20] A. L. Durán, R. Estrada, and R. P. Kanwal, “Extensions of the Poisson summation formula,” Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 581-606, 1998. · Zbl 0940.46019
[21] K. Gröchenig, “An uncertainty principle related to the Poisson summation formula,” Studia Mathematica, vol. 121, no. 1, pp. 87-104, 1996. · Zbl 0866.42005
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