K. K. Sharma, “New inequalities for signal spreads in linear canonical transform domains,” Signal Process. 90, 880–884 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]
[PubMed]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,”J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[CrossRef]

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

[CrossRef]

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Uncertainty principle for real signals in the linear canonical transform domains,” IEEE Trans. Signal Process. 56, 2677–2683 (2008).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

[CrossRef]
[PubMed]

A. Stern, “Uncertainty principles in linear canonical transform domains and some of their implications in optics,” J. Opt. Soc. Am. A 25, 647–652 (2008).

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

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]

M. J. Bastiaans and T. Alieva, “Classification of lossless first-order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).

[CrossRef]

R. Solimene and R. Pierri, “Number of degrees of freedom of the radiated field over multiple bounded domains,” Opt. Lett. 32, 3113–3115 (2007).

[CrossRef]
[PubMed]

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

A. Stern, “Sampling of compact signals in offset linear canonical transform domains,” Signal Image Video Process. 1, 359–367 (2007).

[CrossRef]

R. Torres, P. Pellat-Finet, and Y. Torres, “Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography,” IEEE Signal Process. Lett. 13, 676–679 (2006).

[CrossRef]

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

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Opt. Commun. 265, 454–460 (2006).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal reconstruction from the undersampled signal samples,” Opt. Commun. 268, 245–252 (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]

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

[CrossRef]
[PubMed]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23, 2494–2500 (2006).

[CrossRef]

B. M. Hennelly and 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 and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).

[CrossRef]

T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).

[CrossRef]

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).

[CrossRef]

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231–1242 (2003).

[CrossRef]

C. Candan and H. M. Ozaktas, “Sampling and series expansion theorems for fractional Fourier and other transforms,” Signal Process. 83, 1455–1457 (2003).

[CrossRef]

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

[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).

[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).

[CrossRef]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).

[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).

[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]

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

[CrossRef]

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).

[CrossRef]

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).

[CrossRef]

A. Zayed, “On the relationship between the Fourier and fractional Fourier transforms,” IEEE Signal Process. Lett. 3, 310–311 (1996).

[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]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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]

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]

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).

[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).

[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).

[CrossRef]

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).

[CrossRef]

M. J. Bastiaans and T. Alieva, “Classification of lossless first-order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).

[CrossRef]

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

[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23, 2494–2500 (2006).

[CrossRef]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).

[CrossRef]

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231–1242 (2003).

[CrossRef]

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

[CrossRef]

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).

[CrossRef]

O. Aytur and H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).

[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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]

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 and T. Alieva, “Classification of lossless first-order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).

[CrossRef]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).

[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, “Transport-equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).

[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).

[CrossRef]

M. J. Bastiaans, “Applications of the Wigner distribution function in optics,” in The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

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

[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]

C. Candan and H. M. Ozaktas, “Sampling and series expansion theorems for fractional Fourier and other transforms,” Signal Process. 83, 1455–1457 (2003).

[CrossRef]

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

[CrossRef]

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).

[CrossRef]

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).

[CrossRef]

L. Cohen, Integral Time-Frequency Analysis (Prentice-Hall, 1995).

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[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]

J. J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan Univ., Taipei, Taiwan, 2001).

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

[CrossRef]

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231–1242 (2003).

[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).

[CrossRef]

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).

[CrossRef]

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

[CrossRef]

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E.Wolf, ed. (Elsevier, 1961), Chap. 4, pp. 109–153.

[CrossRef]

L. Ronchi and F. Gori, “Degrees of freedom for spherical scatterers,” Opt. Lett. 6, 478–480 (1981).

[CrossRef]
[PubMed]

F. Gori and L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).

[CrossRef]

F. Gori, S. Paolucci, and L. Ronchi, “Degrees of freedom of an optical image in coherent illumination, in the presence of aberrations,” J. Opt. Soc. Am. 65, 495–501 (1975).

[CrossRef]

F. Gori and G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).

[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).

[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).

[CrossRef]

F. Gori, “Sampling in optics,” in Advanced Topics in Shannon Sampling and Interpolation Theory (Springer, 1993), Chap. 2, pp. 37–83.

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]
[PubMed]

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

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

[CrossRef]
[PubMed]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

B. M. Hennelly and 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 and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).

[CrossRef]

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Uncertainty principle for real signals in the linear canonical transform domains,” IEEE Trans. Signal Process. 56, 2677–2683 (2008).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal reconstruction from the undersampled signal samples,” Opt. Commun. 268, 245–252 (2006).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Opt. Commun. 265, 454–460 (2006).

[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,”J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[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. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).

[CrossRef]
[PubMed]

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

[CrossRef]

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

[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. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).

[CrossRef]
[PubMed]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).

[CrossRef]

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

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).

[CrossRef]

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

[CrossRef]

G.-X. Xie, B.-Z. Li, and Z. Wang, “Identical relation of interpolation and decimation in the linear canonical transform domain,” in Proceedings of the International Conference on Signal Processing, ICSP 2008 (IEEE, 2008), pp. 72–75.

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]

A. W. Lohmann, Optical Information Processing, lecture notes (Optik+Info, Post Office Box 51, Uttenreuth, Germany, 1986).

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).

R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1966).

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[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]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).

[CrossRef]

F. S. Oktem, “Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints,” Master’s thesis, (Bilkent Univ., Turkey, 2009).

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

[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,”J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[CrossRef]

F. S. 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. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).

[CrossRef]
[PubMed]

C. Candan and H. M. Ozaktas, “Sampling and series expansion theorems for fractional Fourier and other transforms,” Signal Process. 83, 1455–1457 (2003).

[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).

[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).

[CrossRef]

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).

[CrossRef]

O. Aytur and H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).

[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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]

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

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

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

[CrossRef]

R. Torres, P. Pellat-Finet, and Y. Torres, “Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography,” IEEE Signal Process. Lett. 13, 676–679 (2006).

[CrossRef]

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[CrossRef]

F. Gori and L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).

[CrossRef]

L. Ronchi and F. Gori, “Degrees of freedom for spherical scatterers,” Opt. Lett. 6, 478–480 (1981).

[CrossRef]
[PubMed]

F. Gori, S. Paolucci, and L. Ronchi, “Degrees of freedom of an optical image in coherent illumination, in the presence of aberrations,” J. Opt. Soc. Am. 65, 495–501 (1975).

[CrossRef]

K. K. Sharma, “New inequalities for signal spreads in linear canonical transform domains,” Signal Process. 90, 880–884 (2010).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Uncertainty principle for real signals in the linear canonical transform domains,” IEEE Trans. Signal Process. 56, 2677–2683 (2008).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal reconstruction from the undersampled signal samples,” Opt. Commun. 268, 245–252 (2006).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Opt. Commun. 265, 454–460 (2006).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]
[PubMed]

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

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

[CrossRef]
[PubMed]

B. M. Hennelly and 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 and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).

[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]

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]

A. E. Siegman, Lasers (University Science Books, 1986).

A. Stern, “Uncertainty principles in linear canonical transform domains and some of their implications in optics,” J. Opt. Soc. Am. A 25, 647–652 (2008).

[CrossRef]

A. Stern, “Sampling of compact signals in offset linear canonical transform domains,” Signal Image Video Process. 1, 359–367 (2007).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

A. Stern, “Why is the linear canonical transform so little known?” in 5th International Workshop on Information Optics, Toledo, Spain, 5–7 June 2006, pp. 225–234.

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[CrossRef]

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[CrossRef]

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

[CrossRef]

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

[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]

R. Torres, P. Pellat-Finet, and Y. Torres, “Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography,” IEEE Signal Process. Lett. 13, 676–679 (2006).

[CrossRef]

R. Torres, P. Pellat-Finet, and Y. Torres, “Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography,” IEEE Signal Process. Lett. 13, 676–679 (2006).

[CrossRef]

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[CrossRef]

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

[CrossRef]

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

[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]

G.-X. Xie, B.-Z. Li, and Z. Wang, “Identical relation of interpolation and decimation in the linear canonical transform domain,” in Proceedings of the International Conference on Signal Processing, ICSP 2008 (IEEE, 2008), pp. 72–75.

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).

[CrossRef]

G.-X. Xie, B.-Z. Li, and Z. Wang, “Identical relation of interpolation and decimation in the linear canonical transform domain,” in Proceedings of the International Conference on Signal Processing, ICSP 2008 (IEEE, 2008), pp. 72–75.

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

A. Zayed, “On the relationship between the Fourier and fractional Fourier transforms,” IEEE Signal Process. Lett. 3, 310–311 (1996).

[CrossRef]

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[CrossRef]

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[CrossRef]

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

[CrossRef]

H. Zhao, Q.-W. Ran, J. Ma, and L.-Y. Tan, “On bandlimited signals associated with linear canonical transform,” IEEE Signal Process. Lett. 16, 343–345 (2009).

[CrossRef]

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

[CrossRef]

X.-G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).

[CrossRef]

A. Zayed, “On the relationship between the Fourier and fractional Fourier transforms,” IEEE Signal Process. Lett. 3, 310–311 (1996).

[CrossRef]

R. Torres, P. Pellat-Finet, and Y. Torres, “Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography,” IEEE Signal Process. Lett. 13, 676–679 (2006).

[CrossRef]

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).

[CrossRef]

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

[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]

L. Durak and O. Arikan, “Short-time Fourier transform: two fundamental properties and an optimal implementation,” IEEE Trans. Signal Process. 51, 1231–1242 (2003).

[CrossRef]

R. Tao, B. Deng, W.-Q. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 56, 158–171 (2008).

[CrossRef]

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

[CrossRef]

K. K. Sharma and S. D. Joshi, “Uncertainty principle for real signals in the linear canonical transform domains,” IEEE Trans. Signal Process. 56, 2677–2683 (2008).

[CrossRef]

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

[CrossRef]

G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–499 (1955).

[CrossRef]

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–803 (1969).

[CrossRef]
[PubMed]

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).

[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).

[CrossRef]

F. Gori and G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).

[CrossRef]

F. Gori, S. Paolucci, and L. Ronchi, “Degrees of freedom of an optical image in coherent illumination, in the presence of aberrations,” J. Opt. Soc. Am. 65, 495–501 (1975).

[CrossRef]

F. Gori and L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).

[CrossRef]

M. Nazarathy and J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).

[CrossRef]

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

B. M. Hennelly and 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 and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).

[CrossRef]

R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).

[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).

[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).

[CrossRef]

G. Newsam and R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985).

[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]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).

[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23, 2494–2500 (2006).

[CrossRef]

M. J. Bastiaans and T. Alieva, “Classification of lossless first-order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).

[CrossRef]

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

[CrossRef]

A. Stern, “Uncertainty principles in linear canonical transform domains and some of their implications in optics,” J. Opt. Soc. Am. A 25, 647–652 (2008).

[CrossRef]

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).

[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,”J. Opt. Soc. Am. A 27, 1288–1302 (2010).

[CrossRef]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).

[CrossRef]

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]

M. J. Bastiaans, “Transport-equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).

[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).

[CrossRef]

H. M. Ozaktas and M. F. Erden, “Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Opt. Commun. 265, 454–460 (2006).

[CrossRef]

K. K. Sharma and S. D. Joshi, “Signal reconstruction from the undersampled signal samples,” Opt. Commun. 268, 245–252 (2006).

[CrossRef]

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).

[CrossRef]

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).

[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).

[CrossRef]

O. Aytur and H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).

[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett. 33, 228–230 (2008).

[CrossRef]
[PubMed]

R. Solimene and R. Pierri, “Number of degrees of freedom of the radiated field over multiple bounded domains,” Opt. Lett. 32, 3113–3115 (2007).

[CrossRef]
[PubMed]

L. Ronchi and F. Gori, “Degrees of freedom for spherical scatterers,” Opt. Lett. 6, 478–480 (1981).

[CrossRef]
[PubMed]

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]

T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).

[CrossRef]

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

[CrossRef]
[PubMed]

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).

[CrossRef]
[PubMed]

J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).

[CrossRef]
[PubMed]

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).

[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]

A. Stern, “Sampling of compact signals in offset linear canonical transform domains,” Signal Image Video Process. 1, 359–367 (2007).

[CrossRef]

H. M. Ozaktas and O. Aytur, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).

[CrossRef]

C. Candan and H. M. Ozaktas, “Sampling and series expansion theorems for fractional Fourier and other transforms,” Signal Process. 83, 1455–1457 (2003).

[CrossRef]

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

[CrossRef]

K. K. Sharma, “New inequalities for signal spreads in linear canonical transform domains,” Signal Process. 90, 880–884 (2010).

[CrossRef]

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]

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

[CrossRef]

L. Cohen, Integral Time-Frequency Analysis (Prentice-Hall, 1995).

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).

G.-X. Xie, B.-Z. Li, and Z. Wang, “Identical relation of interpolation and decimation in the linear canonical transform domain,” in Proceedings of the International Conference on Signal Processing, ICSP 2008 (IEEE, 2008), pp. 72–75.

F. S. Oktem, “Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints,” Master’s thesis, (Bilkent Univ., Turkey, 2009).

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E.Wolf, ed. (Elsevier, 1961), Chap. 4, pp. 109–153.

[CrossRef]

A. W. Lohmann, Optical Information Processing, lecture notes (Optik+Info, Post Office Box 51, Uttenreuth, Germany, 1986).

F. Gori, “Sampling in optics,” in Advanced Topics in Shannon Sampling and Interpolation Theory (Springer, 1993), Chap. 2, pp. 37–83.

[CrossRef]

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

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9: Construction and properties of canonical transforms.

R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1966).

A. E. Siegman, Lasers (University Science Books, 1986).

J. J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan Univ., Taipei, Taiwan, 2001).

M. J. Bastiaans, “Applications of the Wigner distribution function in optics,” in The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

A. Stern, “Why is the linear canonical transform so little known?” in 5th International Workshop on Information Optics, Toledo, Spain, 5–7 June 2006, pp. 225–234.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).