Abstract

The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande–Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions.

© 2010 Optical Society of America

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    [CrossRef]
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  52. T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).
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    [CrossRef]
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    [CrossRef]

2009

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).
[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).
[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).
[CrossRef]

2008

2007

T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658-3665 (2007).
[CrossRef]

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).
[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).
[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).
[CrossRef]

2006

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).
[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).
[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).
[CrossRef]

2005

2004

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. (Bellingham) 43, 2557-2563 (2004).
[CrossRef]

2003

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

2000

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).
[CrossRef]

1997

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

1996

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Trans. Signal Process. 3, 72-74 (March 1996).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).
[CrossRef]

1994

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801-1803 (1994).
[CrossRef] [PubMed]

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach,” J. Phys. A 27, 4179-4187 (1994).
[CrossRef]

1981

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
[CrossRef]

1980

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

1979

1970

1968

C. M. Rader, “Discrete Fourier transforms when the number of data samples is prime,” Proc. IEEE 56, 1107-1108 (1968).
[CrossRef]

1965

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
[CrossRef]

Abe, S.

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach,” J. Phys. A 27, 4179-4187 (1994).
[CrossRef]

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801-1803 (1994).
[CrossRef] [PubMed]

Alieva, T.

Alper Kutay, M.

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).
[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

Averbuch, A.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

Barshan, B.

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral transformation,” J. Opt. Soc. Am. A 24, 3658-3665 (2007).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710-1716 (1979).
[CrossRef]

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in one-parameter canonical-transform systems,” in Proceedings of Seventh International Symposium on Signal Processing and Its Applications, Vol. 1 (IEEE, 2003), pp. 589-592.
[CrossRef]

Bozdagt, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

Brenner, K.-H.

X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47, E88-E98 (2008).
[CrossRef] [PubMed]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).
[CrossRef]

Coifman, R. R.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

Collins, S. A.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
[CrossRef]

Cortes, A.

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).
[CrossRef]

Deng, B.

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).
[CrossRef]

Ding, J.-J.

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).
[CrossRef]

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

Donoho, D. L.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

Dorsch, R. G.

Elad, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

Ferreira, C.

Frigo, M.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Gentleman, W. M.

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Gopinathan, U.

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293-297 (1981).
[CrossRef]

Healy, J. J.

Hennelly, B. M.

Israeli, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmon. Anal. 21, 145-167 (2006).
[CrossRef]

Johnson, S. G.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Joseph, J.

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).
[CrossRef]

Kelly, D. P.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).
[CrossRef]

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).
[CrossRef] [PubMed]

Kernighan, B.

B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

Kloos, G.

G. Kloos, Matrix Methods for Optical Layout (SPIE Press, 2007).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

Kuo, J.-C.

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Li, B.-Z.

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).
[CrossRef]

Liu, X.

Liu, Y.

Lohmann, A. W.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470-473 (1996).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980).
[CrossRef]

Lundy, T. J.

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).
[CrossRef]

Mendlovic, D.

Naughton, T. J.

Nelleri, A.

A. Nelleri, J. Joseph, and K. Singh, “Digital Fresnel field encryption for three-dimensional information security,” Opt. Eng. (Bellingham) 46, 045801 (8 pages) (2007).
[CrossRef]

Oktem, F.

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).
[CrossRef]

O'Neill, F. T.

Onural, L.

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. (Bellingham) 43, 2557-2563 (2004).
[CrossRef]

Ozaktas, H. M.

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383-2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).
[CrossRef]

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Patten, R. F.

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B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

Sande, G.

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.

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H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).
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A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).
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J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).
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J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).
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U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108-115 (2008).
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R. F. Patten, B. M. Hennelly, D. P. Kelly, F. T. O'Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31, 32-34 (2006).
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B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).
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B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

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A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).
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J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).
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T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).
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J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).
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B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).
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B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).
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J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26, 1858-1867 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23, 2861-2870 (2006).
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B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

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J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

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H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).
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H. M. Ozaktas, A. Koç. I. Sari, and M. Alper Kutay, “Efficient computation of quadratic-phase integrals in optics,” Appl. Opt. 31, 35-37 (2006).
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Computing

T. J. Lundy and J. Van Buskirk, “A new matrix approach to real FFTs and convolutions of length 2k,” Computing 80, 23-45 (2007).
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F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727-730 (2009).
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S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338-1353 (2000).
[CrossRef]

A. Cortes, I. Velez, and J. F. Sevillano, “Radix rk FFTs: matricial representation and SDC/SDF pipeline implementation,” IEEE Trans. Signal Process. 57, 2824-2839 (2009).
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H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141-2150 (1996).
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X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Trans. Signal Process. 3, 72-74 (March 1996).
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J. Mod. Opt.

B. M. Hennelly, D. P. Kelly, R. F. Patten, J. E. Ward, U. Gopinathan, F. T. O'Neill, and J. T. Sheridan, “Metrology and the linear canonical transform,” J. Mod. Opt. 53, 2167-2186 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

S. Abe and J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach,” J. Phys. A 27, 4179-4187 (1994).
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J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297-301 (1965).
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B. Barshan, M. Alper Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32-36 (1997).
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Opt. Lett.

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B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Stuttgart) 114, 251-265 (2003).

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M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
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B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Sci. China Ser. F, Inf. Sci. 49, 592-603 (2006).
[CrossRef]

Signal Process.

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983-990 (2007).
[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825-2832 (2008).
[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421-1425 (2006).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641-648 (2009).
[CrossRef]

Other

A. Stern, “Why is the linear canonical transform so little known?” in Proceedings of 5th International Workshop on Information Optics, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds (Springer, 2006), pp. 225-234.

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in one-parameter canonical-transform systems,” in Proceedings of Seventh International Symposium on Signal Processing and Its Applications, Vol. 1 (IEEE, 2003), pp. 589-592.
[CrossRef]

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[CrossRef]

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T. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods (Wiley-VCH, 2005).

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H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

B. Kernighan, and D. Ritchie, The C Programming Language (Prentice-Hall, 1978).

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S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media, 2003).

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H.J.Caulfield, ed. (SPIE Press, 2002) pp. 343-356 (2002).

J.-C. Kuo, C.-H. Wen, and A.-Y. Wu, “Implementation of a programmable 64~2048-point FFT/IFFT processor for OFDM-based communication systems,” in Proceedings of the 2003 International Symposium on Circuits and Systems, Vol. 2 (IEEE, 2003), pp. 121-124.

W. M. Gentleman and G. Sande, “Fast Fourier transforms--for fun and profit,” in Proceedings of AFIPS Fall Joint Computer Conference, Vol. 29 (ACM, 1966), pp. 563-578.

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Figures (5)

Fig. 1
Fig. 1

Sampling according to [25]. (a) The signal is assumed to be bounded in space, x, and in the Fourier domain, k. (b) The signal is sampled at a rate not less than the Nyquist rate, making it periodic (but not overlapping) along the frequency axis (dashed lines indicate replicas). For convenience, only the two closest replicas are shown. (c) An LCT operation is performed on the discrete signal. (d) The transformed signal is sampled (gray lines indicate replicas created by this process). Note the irregularity of the replicas, which may complicate recovery of the continuous signal from the samples or even introduce aliasing effects for some choices of sampling rates as described in [21]. In this case, it is clear that if we include more replica terms from the first sampling operation, there will be some overlap with the zeroth-order term.

Fig. 2
Fig. 2

Sampling according to [22, 23]. (a) The signal is assumed to be bounded in space and in the output linear canonical domain. (b) The signal is sampled at a rate not less than that given by Ding’s LCT sampling theorem, making it periodic along the frequency axis (dashed lines indicate replicas). For convenience, only the two closest replicas are shown. (c) An LCT operation is performed on the discrete signal. (d) The transformed signal is sampled (gray lines indicate replicas created by this process). Note the regularity of the replicas—a consequence of the bounds chosen for the original signal. These diagrams have been scaled to provide clearer illustration, but this does not alter the argument.

Fig. 3
Fig. 3

(a) Amplitude and (b) wrapped phase of the LCT for parameters α = 636.619249 , β = 636.620034 , and γ = 636.619249 of a 1D rectangular aperture of width w = 9.96 cm , determined analytically. λ = 1 .

Fig. 4
Fig. 4

Comparison between the analytic result of Fig. 4 and the same calculation approximated using the discrete LCT. (a) Magnitude error, expressed as a percentage of the value at zero of the discrete output. (b) Phase error.

Fig. 5
Fig. 5

Comparison between the result of Fig. 4 performed using O ( N 2 ) and O ( N log N ) algorithms ( N = 2048 ) . (a) Phase error. (b) Magnitude error, expressed as a percentage of the value at zero of the discrete output.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

f M ( y ) = L M { f ( x ) } ( y ) = { 1 2 π B exp ( j π 4 ) f ( x ) exp { j 2 B ( A x 2 2 x y + D y 2 ) } d x , B 0 1 A exp ( j 2 C A y 2 ) f ( y A ) B = 0 } ,
f ( x ) = T L M 1 { rect ( y L ) L M { f ( n T ) } ( y ) } ( x ) ,
rect ( x ) = { 1 | x | < 0.5 0.5 | x | = 0.5 0 | x | > 0.5 } .
W { f M ( y ) } ( y , k y ) = W { f ( x ) } ( D x B k x , C x + A k x ) .
D ϴ α β γ T , N { g ( n T ) } ( m T y ) = β 2 π exp ( j π 4 ) n = N 2 ( N 2 ) 1 g ( n T ) W N n , m ,
W N n , m = exp { j π [ α ( n T ) 2 2 n m N + γ ( m N T β ) 2 ] } .
L [ m ] = β 2 π exp ( j π 4 ) n = 0 N 1 f [ n ] W N n , m .
f i [ n ] = f [ n + i N r ] .
L [ m ] = n = 0 ( N r ) 1 i = 1 r f i [ n ] W N n + i N r , m .
W N n + i N r , m W N n , m = μ 1 μ 2 ,
μ 1 = exp [ j 2 π i m r ] ,
μ 2 = exp { j π α ( N r ) i T 2 [ 2 n + i N r ] } .
L [ m ] = n = 0 ( N r ) 1 i = 0 r 1 μ 1 μ 2 f i [ n ] W N n , m .
L c [ m ] = L [ r m + c ] .
W N n , m = W p N n , p m ,
L [ m ] = n = 0 ( N r ) 1 i = 0 r 1 μ 1 μ 2 f i [ n ] W N r n , m r .
k = ( m c ) r ,
μ 3 μ 4 = W N n , k + c r W N n , k ,
μ 3 = exp ( j 2 π n c r N ) ,
μ 4 = exp [ j π γ c ( 2 k r + c ) ( N T β r ) 2 ] .
L c [ k ] = μ 4 n = 0 ( N r ) 1 i = 0 r 1 μ 1 μ 2 μ 3 f i [ n ] W N r n , k .
μ 1 = exp [ j 2 π c r ] .
f c [ n ] = i = 0 r 1 f i [ n ] μ 1 μ 2 μ 3 .
L c [ m ] = μ 4 n = 0 ( N r ) 1 f c [ n ] W N r n , k .
L M ( y ) = k exp ( j l y 2 ) { erfi [ ( 1 + j ) ( m n y ) ] + erfi [ ( 1 + j ) ( m + n y ) ] } ,

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