D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).

[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).

[CrossRef]
[PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).

[CrossRef]
[PubMed]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution filtering and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).

[CrossRef]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).

[CrossRef]
[PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).

[CrossRef]
[PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).

[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).

[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).

[CrossRef]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).

[CrossRef]
[PubMed]

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).

[CrossRef]
[PubMed]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).

[CrossRef]
[PubMed]

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).

[CrossRef]
[PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).

[CrossRef]
[PubMed]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).

[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994).

[CrossRef]

X. Deng, B. Bihari, J. Gan, F. Zhao, R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000).

[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).

[CrossRef]
[PubMed]

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).

[CrossRef]
[PubMed]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).

[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).

[CrossRef]
[PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).

[CrossRef]
[PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution filtering and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).

[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).

[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).

[CrossRef]

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution filtering and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).

[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).

[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).

[CrossRef]

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).

[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).

[CrossRef]
[PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).

[CrossRef]
[PubMed]

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).

[CrossRef]
[PubMed]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).

[CrossRef]
[PubMed]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994).

[CrossRef]

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).

[CrossRef]

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).

[CrossRef]
[PubMed]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).

[CrossRef]
[PubMed]

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).

[CrossRef]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).

[CrossRef]
[PubMed]

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).

[CrossRef]
[PubMed]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).

[CrossRef]
[PubMed]

S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).

[CrossRef]

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution filtering and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).

[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).

[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).

[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).

[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).

[CrossRef]

X. Deng, B. Bihari, J. Gan, F. Zhao, R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000).

[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994).

[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997).

[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).

[CrossRef]

M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993).

[CrossRef]
[PubMed]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).

[CrossRef]
[PubMed]

B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990).

[CrossRef]
[PubMed]

W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991).

[CrossRef]
[PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).

[CrossRef]
[PubMed]

H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).

A. Bultheel, H. E. Martinez Sulbaran, “Computation of the fractional Fourier transform” (2003); http://www.cs.kuleuven.ac.be/cwis/research/nalag/papers/ade/frftcomp/ccomp.pdf .

H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.

J. O’Neill, “Discrete TFDs (fracft.m)”: a collection of Matlab files for time-frequency analysis (1999); ftp.mathworks.com/pub/contrib/v5/signal/DiscreteTFDs/.

M. A. Kutay, “FracF: Fast computation of the fractional Fourier transform (1996)”; http://www.ee.bilkent.edu.tr/∼haldun/fracF.m .