Abstract

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an NlogN algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

© 2005 Optical Society of America

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    [CrossRef]
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2005 (1)

2004 (2)

A. Stern, B. Javidi and , “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004);A. Stern, B. Javidi and , Errata, 21, 2038 (2004).
[CrossRef]

S. C. Pei, J. J. Ding, “Generalized eigenvectors and Fractionalization of offset DFTs and DCTs,” IEEE Trans. Signal Process. 52, 2032–2046 (2004).
[CrossRef]

2003 (4)

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

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

B. M. Hennelly, J. T. Sheridan, “Fractional Fourier transform based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

T. J. Naughton, J. B. Mc Donald, B. Javidi, “Efficient compression of Fresnel fields for internet transmission of three-dimensional images,” Appl. Opt. 42, 4758–4764 (2003).
[CrossRef] [PubMed]

2000 (3)

1999 (3)

O. Matoba, B. Javidi, “Encrypted optical memory using multi-dimensional keys,” Opt. Lett. 24, 762–765 (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]

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

1998 (1)

1997 (1)

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1996 (3)

1995 (2)

1994 (2)

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);S. Abe, J. T. Sheridan and , corrigenda, J. Phys. A 27, 7937–7938.
[CrossRef]

S. Abe, 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]

1981 (1)

R. E. Crochiere, L. R. Rabiner, “Interpolation anddecimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

1979 (1)

1965 (1)

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

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abe, S.

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);S. Abe, J. T. Sheridan and , corrigenda, J. Phys. A 27, 7937–7938.
[CrossRef]

S. Abe, 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]

Arikan, O.

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

Bastians, M. J.

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

M. J. Bastians, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, Amsterdam, 1997).

Bernardo, L. M.

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]

Bihari, B.

Bitran, Y.

Bozdagi, G.

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

Candan, C.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Cariolaro, G.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Chen, R. T.

Cooley, J. W.

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

Crochiere, R. E.

R. E. Crochiere, L. R. Rabiner, “Interpolation anddecimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

Deng, X.

Ding, J. J.

S. C. Pei, J. J. Ding, “Generalized eigenvectors and Fractionalization of offset DFTs and DCTs,” IEEE Trans. Signal Process. 52, 2032–2046 (2004).
[CrossRef]

Dorsch, R.

Dorsch, R. G.

Erseghe, T.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Ferreira, C.

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]

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

Gang, J.

Garcia, J.

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]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grant, P.

B. Mulgrew, P. Grant, J. Thompson, Digital Signal Processing, Concepts and Applications (Macmillan, London, 1999).

Hennelly, B. M.

B. M. Hennelly, J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917–927 (2005).
[CrossRef]

B. M. Hennelly, J. T. Sheridan, “Fractional Fourier transform based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

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

B. M. Hennelly, J. T. Sheridan, “The fast linear canonical transform,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 71–82 (2004).

B. M. Hennelly, J. T. Sheridan, “Efficient algorithms for the linear canonical transform,” in Optical Information Systems II, B. Javidi and D. Psaltis, eds., Proc. SPIE5557, 191–199 (2004).

Hernandez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Ifeachor, E. C.

E. C. Ifeachor, B. W. Jervis, Digital Signal Processing, A Practical Approach (Prentice Hall, Upper Saddle River, N.J. 1999).

Illueca, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Javidi, B.

Jervis, B. W.

E. C. Ifeachor, B. W. Jervis, Digital Signal Processing, A Practical Approach (Prentice Hall, Upper Saddle River, N.J. 1999).

Konforti, N.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kraniauskas, P.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Kutay, M. A.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

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

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

Lohmann, A.

Lohmann, A. W.

Marinho, F.

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]

Marinho, F. J.

Mas, D.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[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]

Matoba, O.

Mc Donald, J. B.

Mendlovic, D.

Miret, J. J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Mulgrew, B.

B. Mulgrew, P. Grant, J. Thompson, Digital Signal Processing, Concepts and Applications (Macmillan, London, 1999).

Mullis, C. T.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Pearson Addison-Wesley, Boston, Mass., 1987).

Naughton, T. J.

Ozaktas, H. M.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

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

Y. Bitran, D. Mendlovic, R. Dorsch, A. Lohmann, H. M. Ozaktas, “Fractional Fourier transform: simulations and experimental results,” Appl. Opt. 34, 1329–1332 (1995).
[CrossRef] [PubMed]

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

Pei, S. C.

S. C. Pei, J. J. Ding, “Generalized eigenvectors and Fractionalization of offset DFTs and DCTs,” IEEE Trans. Signal Process. 52, 2032–2046 (2004).
[CrossRef]

Perez, J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Rabiner, L. R.

R. E. Crochiere, L. R. Rabiner, “Interpolation anddecimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

Rhodes, W. T.

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, Bellingham, Wash., 2002), pp. 343–356.

W. T. Rhodes, “Numerical simulation of Fresnel-regime wave propagation: the light tube model,” in Wave-Optical Systems Engineering, F. Wyrowski, ed., 4436, 21–26 (2001).

Roberts, R. A.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Pearson Addison-Wesley, Boston, Mass., 1987).

Sheridan, J. T.

B. M. Hennelly, J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917–927 (2005).
[CrossRef]

B. M. Hennelly, J. T. Sheridan, “Fractional Fourier transform based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

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

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);S. Abe, J. T. Sheridan and , corrigenda, J. Phys. A 27, 7937–7938.
[CrossRef]

S. Abe, 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]

B. M. Hennelly, J. T. Sheridan, “Efficient algorithms for the linear canonical transform,” in Optical Information Systems II, B. Javidi and D. Psaltis, eds., Proc. SPIE5557, 191–199 (2004).

B. M. Hennelly, J. T. Sheridan, “The fast linear canonical transform,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 71–82 (2004).

Stern, A.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Tajahuerce, E.

Thompson, J.

B. Mulgrew, P. Grant, J. Thompson, Digital Signal Processing, Concepts and Applications (Macmillan, London, 1999).

Tukey, J. W.

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

Vazquez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

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

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

Zhao, F.

Appl. Opt. (4)

IEEE Trans. Signal Process. (4)

S. C. Pei, J. J. Ding, “Generalized eigenvectors and Fractionalization of offset DFTs and DCTs,” IEEE Trans. Signal Process. 52, 2032–2046 (2004).
[CrossRef]

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

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

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Phys. A (1)

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);S. Abe, J. T. Sheridan and , corrigenda, J. Phys. A 27, 7937–7938.
[CrossRef]

Math. Comput. (1)

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

Opt. Commun. (4)

B. M. Hennelly, J. T. Sheridan, “Fractional Fourier transform based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[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. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Opt. Lett. (2)

Optik (1)

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

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Proc. IEEE (1)

R. E. Crochiere, L. R. Rabiner, “Interpolation anddecimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
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W. T. Rhodes, “Numerical simulation of Fresnel-regime wave propagation: the light tube model,” in Wave-Optical Systems Engineering, F. Wyrowski, ed., 4436, 21–26 (2001).

M. J. Bastians, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, Amsterdam, 1997).

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

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

B. M. Hennelly, J. T. Sheridan, “Efficient algorithms for the linear canonical transform,” in Optical Information Systems II, B. Javidi and D. Psaltis, eds., Proc. SPIE5557, 191–199 (2004).

B. M. Hennelly, J. T. Sheridan, “The fast linear canonical transform,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 71–82 (2004).

B. Mulgrew, P. Grant, J. Thompson, Digital Signal Processing, Concepts and Applications (Macmillan, London, 1999).

E. C. Ifeachor, B. W. Jervis, Digital Signal Processing, A Practical Approach (Prentice Hall, Upper Saddle River, N.J. 1999).

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Figures (5)

Fig. 1
Fig. 1

Magnitude of discrete Fourier transform calculated with FLCT algorithm setting α = γ = 0 , β = 1 .

Fig. 2
Fig. 2

(a) Magnitude and (b) phase of discrete (normalized) fractional Fourier transform (angle θ = π 4 ) calculated with FLCT algorithm setting α = γ = 1 tan ( π 4 ) , β = 1 sin ( π 4 ) .

Fig. 3
Fig. 3

(a) Magnitude and (b) phase of discrete Fresnel transform calculated with FLCT algorithm setting α = γ = β = 1 λ z .

Fig. 4
Fig. 4

QPS with two lenses and three sections of free space.

Fig. 5
Fig. 5

Magnitude of DLCT of Rect ( x 2 ) for the two-lens system shown in Fig. 4 calculated with FLCT algorithm.

Equations (69)

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δ T ( x ) = n = + δ ( x n T ) ,
f T ( x ) = f ( x ) δ T ( x ) = n = f ( n T ) δ ( x n T ) ,
f T ( x ) = 1 T f ( x ) k = + exp ( j 2 π k f S x ) .
L α β γ ( y ) = Θ α β γ { f ( x ) } ( y ) = + K ( x , y ) f ( x ) d x ,
K ( x , y ) = A exp [ j π ( α x 2 2 β x y + γ y 2 ) ] .
Θ α β γ [ exp ( j 2 π ξ x ) f ( x ) ] ( y ) = exp ( j π γ ξ 2 β 2 ) exp ( j 2 π y ξ γ β ) × Θ α β γ { f ( x ) } ( y ξ β ) .
Θ α β γ [ f ( x ξ ) ] ( y ) = exp [ j π ξ 2 ( α γ α 2 β 2 ) ] × exp [ j 2 π y ξ ( γ α β β ) ] Θ α β γ { f ( x ) } ( y ξ α β ) .
Θ α β γ { f T ( x ) } ( y ) = 1 T k = + { exp [ j π ( k T ) 2 γ β 2 ] × exp [ j 2 π ( k T ) γ y β ] Θ α β γ { f ( x ) } ( y k T β ) } .
Θ α β γ { f T ( x ) } ( y ) = + [ n = + f ( n ) δ ( x n T ) ] exp ( j 2 π β x y ) exp [ j π ( α x 2 + γ y 2 ) ] d x = exp ( j π γ y 2 ) n = + f ( n T ) exp [ j π α ( n T ) 2 ] exp ( j 2 π β y n T ) .
Θ α β γ { f [ ( n l ) T ] } ( y ) = exp ( j π γ y 2 ) n = + f [ ( n l ) T ] × exp [ j π α ( n T ) 2 ] exp ( j 2 π β y n T ) .
Θ α β γ { f [ ( n l ) T ] } ( y ) = exp ( j π γ y 2 ) i = + f ( i T ) exp { j π α [ ( i + l ) T ] 2 } exp [ j 2 π β y ( i + l ) T ] = exp [ j π ( l T ) 2 ( α γ α 2 β 2 ) ] exp [ j 2 π y l T ( α γ β β ) ] × i = + f ( i T ) exp { j π [ α ( i T ) 2 2 β i T ( y α l T β ) + γ ( y α l T β ) 2 ] } .
Θ α β γ { f [ ( n l ) T ] } ( y ) = exp [ j π l 2 T 2 ( α γ α 2 β 2 ) ] × exp [ j 2 π y l T ( γ α β β ) ] Θ α β γ { f ( x ) } ( y l T α β ) ,
exp ( j π γ y 2 ) n = N 2 N 2 1 f ( n T ) exp [ j π α ( n T ) 2 ] exp ( j 2 π β y n T ) .
1 2 T β y 1 2 T β 1 N T β
exp [ j π γ ( m T y ) 2 ] n = N 2 N 2 1 f ( n T ) exp [ j π α ( n T ) 2 ] exp [ j 2 π β ( n T ) ( m T y ) ] = exp [ j π γ ( m N T β ) 2 ] n = N 2 N 2 1 f ( n T ) exp [ j π α ( n T ) 2 ] exp ( j 2 π n m N ) ,
N 2 m N 2 1 .
L α β γ T y ( y ) = 1 T y L α β γ ( y ) k = + exp [ j 2 π k ( 1 T y ) y ] .
Θ γ β α { L α β γ T y ( y ) } ( x ) = Θ γ β α { 1 T y L α β γ ( y ) k = + exp [ j 2 π k ( 1 T y ) y ] } ( x ) .
N T β k = + { exp [ j π γ ( k N T ) 2 ] exp [ j 2 π x γ ( k N T ) ] f ( x k N T ) } .
exp [ j π γ ( m N T β ) 2 ] n = [ f ( n T ) exp ( j 2 π ξ n T ) ] exp [ j π α ( n T ) 2 ] exp ( j 2 π n m N ) = exp [ j π γ ( ξ β ) 2 ] exp ( j 2 π γ m ξ N T β 2 ) exp [ j π γ ( m ξ N T N T β ) 2 ] n = f ( n T ) exp [ j π α ( n T ) 2 ] exp [ j 2 π n ( m ξ N T ) N ] .
exp ( j 2 π ξ m N T β ) exp [ j π γ ( m N T β ) 2 ] n = f ( n T ) exp [ j π α ( n T ) 2 ] exp ( j 2 π n m N ) = exp [ j π γ ( m N T β ) 2 ] n = f ( n T ) exp ( j 2 π α n ξ T β ) exp ( j π α ξ 2 β 2 ) exp { j π α [ ( n ξ T β ) T ] 2 } exp [ j 2 π m ( n ξ T β ) N ] .
D L α β γ ( m T y ) = D Θ α β γ T , N [ f ( n T ) ] ( m T y ) = n = N 2 N 2 1 f ( n T ) exp [ j π α ( n T ) 2 ] exp ( j 2 π n m N ) × exp [ j π γ ( m N T β ) 2 ] ,
T = h N ,
D Θ α β γ T , N { f ( n T ) } ( m T y ) = n = N 2 N 2 1 f ( n T ) exp [ j π α ( n h ) 2 N ] exp ( j 2 π n m N ) exp [ j π γ N ( m h β ) 2 ] .
W N , h n , m = exp [ j π α ( n h ) 2 N ] exp ( j 2 π n m N ) exp [ j π γ N ( m h β ) 2 ] .
n = + f ( n T ) exp ( j 2 π ξ n h N ) W N , h n , m = exp [ j π γ ( ξ β ) 2 ] exp ( j 2 π γ m ξ h N β 2 ) n = + f ( n T ) W N , h n , ( m ξ h N ) .
n = + f ( n T ) W N , h n , m exp ( j 2 π ξ m h β N ) = exp ( j π α ξ 2 β 2 ) n = + f ( n T ) exp ( j 2 π α n ξ h N β ) W N , h ( n ξ N h β ) , m .
f ( n T ) , N 2 n N 2 1 ,
a ( n ) = f ( n T ) ,
b ( n ) = f [ ( n + N 2 ) T ] , N 2 n 1 .
D Θ α β γ T , N { f ( n T ) } ( m T y ) = n = N 2 N 2 1 f ( n T ) W N , h n , m = n = N 2 1 [ a ( n ) W N , h n , m + b ( n ) W N , h ( n + N 2 ) , m ] ,
D Θ α β γ T , N { f ( n T ) } ( m T y ) = n = N 2 1 { a ( n ) W N , h n , m + b ( n ) exp [ j π α h 2 ( n + N 4 ) ] exp ( j π m ) W N , h n , m } .
c ( n ) = b ( n ) μ 0 ( n ) , μ 0 ( n ) = exp [ j π α h 2 ( n + N 4 ) ] ,
n = N 2 1 [ a ( n ) + c ( n ) exp ( j π m ) ] W N , h n , m .
D Θ α β γ T , N [ f ( n T ) ] ( 2 m T y ) = n = N 2 1 [ a ( n ) + c ( n ) ] W N , h n , 2 m ,
D Θ α β γ T , N [ f ( n T ) ] [ ( 2 m + 1 ) T y ] = n = N 2 1 [ a ( n ) c ( n ) ] W N , h n , ( 2 m + 1 ) ,
p ( n ) = a ( 2 n ) + c ( 2 n ) ,
q ( n ) = a ( 2 n + 1 ) + c ( 2 n + 1 ) ,
r ( n ) = a ( 2 n ) c ( 2 n ) ,
s ( n ) = a ( 2 n + 1 ) c ( 2 n + 1 ) ,
D Θ α β γ T , N { f ( n T ) } ( 2 m T y ) = n = N 4 1 [ p ( n ) W N , h 2 n , 2 m + q ( n ) W N , h ( 2 n + 1 ) , 2 m ] ,
D Θ α β γ T , N { f ( n T ) } [ ( 2 m + 1 ) T y ] = n = N 4 1 [ r ( n ) W N , h 2 n , ( 2 m + 1 ) + s ( n ) W N , h ( 2 n + 1 ) , ( 2 m + 1 ) ] .
W N , h 2 n , 2 m = W ( N 4 ) , h n , m .
D Θ α β γ T , N { f ( n T ) } ( 2 m T y ) = n = N 4 1 [ p ( n ) W ( N 4 ) , h n , m + q ( n ) W ( N 4 ) , h ( n + 1 2 ) , m ] ,
D Θ α β γ T , N { f ( n T ) } [ ( 2 m + 1 ) T y ] = n = N 4 1 [ r ( n ) W ( N 4 ) , h n , ( m + 1 2 ) + s ( n ) W ( N 4 ) , h ( n + 1 2 ) , ( m + 1 2 ) ] .
D Θ α β γ T , N { f ( n T ) } ( 2 m T y ) = n = N 4 1 p ( n ) W ( N 4 ) , h n , m + exp ( j π α ξ 2 β 2 ) exp ( j 2 π ξ m h β N 4 ) n = N 4 1 q ( n ) exp ( j 2 π n α ξ h β N 4 ) W ( N 4 ) , h ( n + 1 2 ξ N 4 h β ) , m .
D Θ α β γ T , N { f ( n T ) } ( 2 m T y ) = n = N 4 1 p ( n ) W ( N 4 ) , h n , m + exp ( j π α h 2 N ) exp ( j 4 π m N ) n = N 4 1 q ( n ) exp ( j 4 π α n h 2 β N ) W ( N 4 ) , h n , m = A ( m ) + μ 1 ( m ) B ( m ) ,
μ 1 ( m ) = exp ( j π α h 2 N ) exp ( j 4 π m N ) ,
μ 2 ( n ) = exp ( j 4 π α n h 2 β N ) ,
A ( m ) = n = N 4 1 p ( n ) W ( N 4 ) , h n , m ,
B ( m ) = n = N 4 1 [ q ( n ) μ 2 ( n ) ] W ( N 4 ) , h n , m .
A ( m ) = A ( m + N 4 ) μ 3 ( m ) , B ( m ) = B ( m + N 4 ) μ 3 ( m ) ,
μ 3 ( m ) = exp ( j π N γ 4 h 2 β 2 ) exp ( j 2 π m γ β 2 h 2 ) ,
A ( m ) = A ( m N 4 ) μ 4 ( m ) , B ( m ) = B ( m N 4 ) μ 4 ( m ) ,
μ 4 ( m ) = exp [ j π γ ( 2 m N 4 ) h 2 β 2 ] .
D Θ α β γ T , N { f ( n T ) } [ ( 2 m + 1 ) T y ] = n = N 4 1 r ( n ) W ( N 4 ) , h n , ( m + 1 2 ) + n = N 4 1 s ( n ) W ( N 4 ) , h ( n + 1 2 ) , ( m + 1 2 )
= n = N 4 1 r ( n ) exp ( j 2 π q n h N 4 ) exp ( j 2 π q n h N 4 ) W ( N 4 ) , h n , ( m + 1 2 ) + n = N 4 1 s ( n ) exp ( j 2 π q n h N 4 ) exp ( j 2 π q n h N 4 ) W ( N 4 ) , h ( n + 1 2 ) , ( m + 1 2 ) .
D Θ α β γ T , N { f ( n T ) } [ ( 2 m + 1 ) T y ] = exp ( j π γ β 2 h 2 N ) exp ( j 4 π m γ h 2 β 2 N ) n = N 4 1 r ( n ) exp ( j 4 π n N ) W ( N 4 ) , h n , m + exp ( j π γ β 2 h 2 N ) exp ( j 4 π m γ h 2 β 2 N ) n = N 4 1 s ( n ) exp ( j 4 π n N ) W ( N 4 ) , h ( n + 1 2 ) , m = μ 5 ( m ) n = N 4 1 r ( n ) μ 6 ( n ) W ( N 4 ) , h n , m + μ 5 ( m ) n = N 4 1 s ( n ) μ 6 ( n ) W ( N 4 ) , h ( n + 1 2 ) , m ,
μ 5 ( m ) = exp ( j π γ β 2 h 2 N ) exp ( j 4 π m γ h 2 β 2 N ) ,
μ 6 ( n ) = exp ( j 4 π n N ) .
D Θ α β γ T , N { f ( n T ) } [ ( 2 m + 1 ) T y ]
= μ 5 ( m ) C ( m ) + μ 7 μ 5 ( m ) μ 1 ( m ) D ( m ) ,
μ 7 = exp ( 2 π N ) ,
C ( m ) = n = N 4 1 r ( n ) μ 6 ( n ) W ( N 4 ) , h n , m ,
D ( m ) = n = N 4 1 s ( n ) μ 6 ( n ) μ 2 ( n ) W ( N 4 ) , h n , m .
C ( m ) = C ( m + N 4 ) μ 3 ( m ) , D ( m ) = D ( m + N 4 ) μ 3 ( m ) ,
D ( m ) = D ( m N 4 ) μ 4 ( m ) , D ( m ) = D ( m N 4 ) μ 4 ( m ) .
Θ α 3 β 3 γ 3 { f ( x ) } ( y ) = Θ α 2 β 2 γ 2 { Θ α 1 β 1 γ 1 { f ( x ) } ( y ) } ( y ) ,
[ γ 3 β 3 1 β 3 β 3 + α 3 γ 3 β 3 α 3 β 3 ] = [ γ 2 β 2 1 β 2 β 2 + α 2 γ 2 β 2 α 2 β 2 ] × [ γ 1 β 1 1 β 1 β 1 + α 1 γ 1 β 1 α 1 β 1 ] .

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