×

Sampling and discretization of the linear canonical transform. (English) Zbl 1157.94332

Summary: The numerical approximation of the linear canonical transform (LCT) is of importance in modeling first order optical systems and many signal processing applications. We have considered an approach based on discretizing the continuous LCT, making careful consideration of the consequences for the range and resolution of the output.

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
65T60 Numerical methods for wavelets

Keywords:

LCT; sampling; DLCT; FLCT; FFT

Software:

FFTW
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Koc, A.; Ozaktas, H. M.; Candan, C.; Kutay, M. A.: Digital computation of linear canonical transforms, IEEE trans. Signal process. 56, No. 6, 2383-2394 (2008) · Zbl 1390.94247
[2] Abe, S.; Sheridan, J. T.: Optical operations on wave functions as the abelian subgroups of the special affine Fourier transformation, Opt. lett. 19, 1801-1803 (1994)
[3] Hennelly, B. M.; Sheridan, J. T.: Optical encryption and the spacebandwidth product, Opt. comm. 247, No. 4 – 6, 291-305 (March 2005)
[4] Hennelly, B. M.; Kelly, D. P.; Ward, J. E.; Patten, R.; Gopinathan, U.; O’neill, F. T.; Sheridan, J. T.: Metrology and the linear canonical transform, J. mod. Opt. 53, No. 15, 2167-2186 (October 2006) · Zbl 1133.78004 · doi:10.1080/09500340600810473
[5] Mas, D.; Perez, J.; Illueca, C.; Espinosa, J.; Hernandez, C.; Vazquez, C.; Miret, J. J.: Determination of chromatic aberration in the human eye by means of fresnel propagation theory, SPIE med. Imag. 5959, No. 1, 595911 (September 2005)
[6] Ozaktas, H. M.; Zalevsky, Z.; Kutay, M. A.: The fractional Fourier transform with applications in optics and signal processing, (2001)
[7] M.J. Bastiaans, K.B. Wolf, Phase reconstruction from intensity measurements in one-parameter canonical-transform systems, in: Proceedings of the Seventh International Symposium on Signal Processing and its Applications, vol. 1, 2003, July 2003, pp. 589 – 592.
[8] Gopinathan, U.; Situ, G.; Naughton, T. J.; Sheridan, J. T.: Noninterferrometric phase retrieval using a fractional Fourier system, J. opt. Soc. amer. A 25, 108-115 (2008)
[9] Barshan, B.; Kutay, M. A.; Ozaktas, H. M.: Optimal filtering with linear canonical transformations, Opt. comm. 135, No. 1 – 3, 32-36 (1997)
[10] Pei, S. -C.; Ding, J. -J.: Simplified fractional Fourier transforms, J. opt. Soc. amer. A 17, No. 12, 2355-2367 (2000)
[11] Goodman, J. W.: Speckle phenomena in optics: theory and applications, (2007)
[12] Goodman, J. W.: Introduction to Fourier optics, (2005)
[13] Jr., S. A. Collins: Lens-system diffraction integral written in terms of matrix optics, J. opt. Soc. amer. 60, 1168-1177 (1970)
[14] Bastiaans, M. J.: Wigner distribution function and its application to first-order optics, J. opt. Soc. amer. 69, 1710-1716 (1979)
[15] O’neill, J. C.; Flandrin, P.; Williams, W. J.: On the existence of discrete Wigner distributions, IEEE signal process. Lett. 6, No. 12, 304-306 (December 1999)
[16] A.W. Lohmann, The space-bandwidth product, applied to spatial filtering and holography, Research Paper RJ-438, IBM San Jose Research Laboratory, San Jose, CA, 1967, pp. 1 – 23.
[17] Healy, J. J.; Sheridan, J. T.: Cases where the linear canonical transform of a signal has compact support or is band-limited, Opt. lett. 33, No. 3, 228-230 (February 2008)
[18] Jerri, A. J.: The Shannon sampling theorem — its various extensions and applications: a tutorial review, IEEE proc. 65, 1565-1596 (November 1977) · Zbl 0442.94002 · doi:10.1109/PROC.1977.10771
[19] Hennelly, B. M.; Sheridan, J. T.: Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, fresnel, and linear canonical transforms, J. opt. Soc. amer. A 22, No. 5, 917-927 (May 2005)
[20] Kelly, D. P.; Hennelly, B. M.; Rhodes, W. T.; Sheridan, J. T.: Analytical and numerical analysis of linear optical systems, Opt. eng. 45, No. 8, 088201 (August 2006)
[21] J.-J. Ding, Research of fractional Fourier transform and linear canonical transform, Ph.D. Thesis, National Taiwan University, Taipei, Taiwan, ROC, 2001.
[22] Stern, A.: Sampling of linear canonical transformed signals, Signal processing 86, No. 7, 917-927 (July 2006) · Zbl 1172.94344 · doi:10.1016/j.sigpro.2005.07.031
[23] Deng, B.; Tao, R.; Wang, Y.: Convolution theorems for the linear canonical transform and their applications, Sci. China (Ser. F inform. Sci.) 49, No. 5, 592-603 (September 2006)
[24] Bing-Zhao, L.; Tao, R.; Wang, Y.: New sampling formulae related to linear canonical transform, Signal processing 87, No. 5, 983-990 (May 2007) · Zbl 1186.94201 · doi:10.1016/j.sigpro.2006.09.008
[25] Ozaktas, H. M.; Ko, A.; Sari, I.; Kutay, M. A.: Efficient computation of quadratic-phase integrals in optics, Opt. lett. 31, No. 1, 35-37 (2006)
[26] Zhao, J.; Tao, R.; Wang, Y.: Sampling rate conversion for linear canonical transform, Signal processing 88, No. 11, 2825-2832 (November 2008) · Zbl 1151.94440 · doi:10.1016/j.sigpro.2008.06.008
[27] Pei, S. -C.; Ding, J. -J.: Closed-form discrete fractional and affine Fourier transforms, IEEE trans. Signal process. 48, No. 5, 1338-1353 (May 2000) · Zbl 1018.94002 · doi:10.1109/78.839981
[28] Hennelly, B. M.; Sheridan, J. T.: Fast numerical algorithm for the linear canonical transform, J. opt. Soc. amer. A 22, No. 5, 928-937 (May 2005)
[29] Frigo, M.; Johnson, S. G.: The design and implementation of FFTW3, Proc. IEEE 93, No. 2, 216-231 (2005)
[30] J.J. Healy, J.T. Sheridan, Applications of fast algorithms for the numerical calculation of optical signal transforms, Proc. SPIE 6187 (April 2006).
[31] J.J. Healy, J.T. Sheridan, New fast algorithm for the numerical computation of quadratic-phase integrals, Proc. SPIE 6313 (August 2006).
[32] Cooley, J. W.; Tukey, J. W.: An algorithm for the machine computation of complex Fourier series, Math. comput. 19, 297-301 (1965) · Zbl 0127.09002 · doi:10.2307/2003354
[33] W.M. Gentleman, G. Sande, Fast Fourier transforms — for fun and profit, in: Proceedings of the AFIPS Fall Joint Computer Conference, vol. 29, 1966, pp. 563 – 578.
[34] Deng, X.; Bihari, B.; Gang, J.; Zhao, F.; Chen, R. T.: Fast algorithm for chirp transforms with zooming-in ability and its applications, J. opt. Soc. amer. A 17, No. 4, 762-771 (April 2000)
[35] Ozaktas, H. M.; Arikan, O.; Kutay, M. A.; Bozdagt, G.: Digital computation of the fractional Fourier transform, IEEE trans. Signal process. 44, No. 9, 2141-2150 (1996)
[36] Graham, R. L.; Knuth, D. E.; Patashnik, O.: Concrete mathematics: A foundation for computer science, (1990) · Zbl 0836.00001
[37] J.J. Healy, B.M. Hennelly, J.T. Sheridan, An additional sampling criterion for the linear canonical transform, Opt. Lett. 33 (22) (2008) 2599 – 2601.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.