Abstract

Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space–frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space–frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space–bandwidth product. The bicanonical width product, which is a generalization of the space–bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.

© 2010 Optical Society of America

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2010 (4)

2009 (3)

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]

2008 (7)

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]

2007 (5)

2006 (8)

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]

2005 (4)

2004 (1)

2003 (2)

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]

2000 (3)

1999 (1)

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

1997 (4)

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]

1996 (5)

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]

1995 (3)

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]

1994 (3)

1989 (1)

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

1985 (1)

1982 (2)

1981 (2)

1979 (2)

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]

1978 (1)

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

1975 (1)

1974 (1)

1973 (1)

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

1971 (1)

1970 (1)

1969 (1)

1955 (1)

Abe, S.

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]

Agarwal, G. S.

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

Alieva, T.

Arikan, O.

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]

Aytur, O.

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]

Bagini, V.

Barakat, R.

Barshan, B.

Bastiaans, M. J.

Bozdagi, G.

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]

Calvo, M. L.

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]

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

Carioraro, G.

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

Cellini, V.

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

Cohen, L.

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

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

Collins, S. A.

Deng, B.

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]

Ding, J. J.

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

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]

Dorsch, R. G.

Durak, L.

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]

Erden, M. F.

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]

Erseghe, T.

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]

Ferreira, C.

Gabor, D.

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

Gori, F.

Guattari, G.

Healy, J. J.

Hennelly, B. M.

Hesselink, L.

Hua, J.

James, D. F. V.

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

Javidi, B.

Joshi, S. D.

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]

Koç, A.

Kraniauskas, P.

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

Kutay, M.

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]

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, 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).

Laurenti, N.

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

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]

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.

Li, G.

Liu, L.

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]

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).

Luneburg, R. K.

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

Ma, J.

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]

Mendlovic, D.

Miller, D. A. B.

Nazarathy, M.

Newsam, G.

Oktem, F. S.

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).

Onural, L.

Ozaktas, H.

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]

Ozaktas, H. M.

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

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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]

Pierri, R.

Piestun, R.

Ran, Q.-W.

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]

Rodrigo, J. A.

Ronchi, L.

Sari, I.

Shamir, J.

Sharma, K. K.

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]

Sheridan, J. T.

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

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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.

Tan, L.-Y.

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]

Tao, R.

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).
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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]

Toraldo di Francia, G.

Torres, R.

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).
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Torres, Y.

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).
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Wang, Y.

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).
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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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Wang, Z.

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.

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R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
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K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9: Construction and properties of canonical transforms.

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

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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).
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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).
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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).
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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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IEEE Signal Process. Lett. (5)

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).
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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).
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IEEE Trans. Commun. (1)

T. Erseghe, N. Laurenti, and V. Cellini, “A multicarrier architecture based upon the affine Fourier transform,” IEEE Trans. Commun. 53, 853–862 (2005).
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IEEE Trans. Signal Process. (7)

H. Ozaktas, O. Arikan, M. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
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T. Erseghe, P. Kraniauskas, and G. Carioraro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
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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).
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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).
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J. Opt. Soc. Am. (10)

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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).
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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).
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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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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).
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K. K. Sharma and S. D. Joshi, “Signal separation using linear canonical and fractional Fourier transforms,” Opt. Commun. 265, 454–460 (2006).
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K. K. Sharma and S. D. Joshi, “Signal reconstruction from the undersampled signal samples,” Opt. Commun. 268, 245–252 (2006).
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B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
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J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).
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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).
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T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).
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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).
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M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett. 31, 2414–2416 (2006).
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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).
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[CrossRef]

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

Signal Process. (7)

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

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

Other (15)

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).

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

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).

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Figures (5)

Fig. 1
Fig. 1

Rectangular space–frequency support with area equal to the space–bandwidth product Δ u Δ μ .

Fig. 2
Fig. 2

Space–frequency support when finite extents are specified in two LCT domains. The area of the parallelogram is equal to Δ u T 1 Δ u T 2 | β 1 , 2 | . This figure will be revisited in Section 5.

Fig. 3
Fig. 3

a th -order fractional Fourier domain.

Fig. 4
Fig. 4

Space-frequency support of f ( u ) (left) and f T ( u ) (right) for space- and LCT-limited signals. The area of both parallelograms is equal to Δ u Δ u T | β | .

Fig. 5
Fig. 5

Parallelogram shaped space–frequency support with area equal to the bicanonical width product Δ u T 1 Δ u T 2 | β 1 , 2 | , which is smaller than the space–bandwidth product Δ u Δ μ .

Equations (28)

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N Δ u Δ μ ,
N Δ u Δ u T | β | .
N Δ u T 1 Δ u T 2 | β 1 , 2 | ,
f T ( u ) ( C T f ) ( u ) C T ( u , u ) f ( u ) d u ,
C T ( u , u ) 1 B e i π 4 e i π ( D B u 2 2 1 B u u + A B u 2 ) ,
T = [ A B C D ] = [ γ β 1 β β + α γ β α β ] .
f a ( u ) ( F a f ) ( u ) K a ( u , u ) f ( u ) d u ,
K a ( u , u ) A ϕ e i π ( cot ϕ u 2 2 csc ϕ u u + cot ϕ u 2 ) ,
A ϕ = 1 i cot ϕ , ϕ = a π 2
F a = [ cos ϕ sin ϕ sin ϕ cos ϕ ] ,
Q q = [ 1 0 q 1 ] .
R r = [ 1 r 0 1 ] .
M M = [ M 0 0 1 M ] .
T = [ A B C D ] = [ 1 0 q 1 ] [ M 0 0 1 M ] [ cos ϕ sin ϕ sin ϕ cos ϕ ] .
f T ( u ) = exp [ i π q u 2 ] 1 M f a ( u M ) .
a = { 2 π arctan ( B A ) , if A 0 2 π arctan ( B A ) + 2 , if A < 0 } ,
M = A 2 + B 2 ,
q = { C A B A A 2 + B 2 , if A 0 D B , if A = 0 } .
W f ( u , μ ) = f ( u + u 2 ) f * ( u u 2 ) e i 2 π μ u d u .
W f T ( u , μ ) = W f ( D u B μ , C u + A μ ) .
W f a ( u , μ ) = W f ( u cos ϕ μ sin ϕ , u sin ϕ + μ cos ϕ ) .
{ R D N ϕ [ W f ( u , μ ) ] } ( u a ) = | f a ( u a ) | 2 ,
1 M { R D N ϕ [ W f ( u , μ ) ] } ( u T M ) = | f T ( u T ) | 2 ,
Area = Δ u T 1 M 1 Δ u T 2 M 2 | csc ( ϕ 2 ϕ 1 ) |
= Δ u T 1 Δ u T 2 M 1 M 2 | sin ϕ 2 cos ϕ 1 cos ϕ 2 sin ϕ 1 |
= Δ u T 1 Δ u T 2 | A 1 B 2 B 1 A 2 |
= Δ u T 1 Δ u T 2 | β 1 β 2 | | γ 1 γ 2 |
= Δ u T 1 Δ u T 2 | β 1 , 2 | ,

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