Abstract

The numerical approximation of the linear canonical transforms (LCTs) is important in modeling coherent wave field propagation through first-order optical systems and in many digital signal processing applications. The continuous LCTs are unitary, but discretization can destroy this property. We present a sufficient condition on the sampling rates chosen in the discretization to ensure unitarity. We discuss the various subsets of the unitary matrices examined in this paper that have been proposed elsewhere. We offer a proof of the existence of all of the unitary matrices we discuss. We examine the consequences of these results, particularly in relation to the use of discrete transforms in iterative phase retrieval applications.

© 2013 Optical Society of America

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  1. M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution-Theory and Applications in the Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, 1997).
  2. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  3. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  4. B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their application,” Inf. Sci. 49, 592–603 (2006).
    [CrossRef]
  5. 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]
  6. A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
    [CrossRef]
  7. Y. Zhang, B.-Z. Dong, B.-Y. Gu, and G.-Z. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
    [CrossRef]
  8. F.-C. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
    [CrossRef]
  9. B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
    [CrossRef]
  10. D.-Y. Wang, J. Zhao, F.-C. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47, D12–D20 (2008).
    [CrossRef]
  11. P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).
  12. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).
  13. 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]
  14. G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008).
    [CrossRef]
  15. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  17. R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).
  18. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).
  19. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  20. J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).
  21. J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).
    [CrossRef]
  22. J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
    [CrossRef]
  23. 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]
  24. D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).
  25. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
    [CrossRef]
  26. 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]
  27. D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
    [CrossRef]
  28. J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).
    [CrossRef]
  29. J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.
  30. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177(1970).
    [CrossRef]
  31. F. S. Oktem and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space—bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895(2010).
    [CrossRef]
  32. J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).
    [CrossRef]
  33. H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).
  34. M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, 1997), Chap. 2.
  35. While this statement is true it requires clarification. We note that no proof has been provided that unitary DLCTs form a group. In fact, the DLCTs do not (cannot) form a faithful representation of the group of 2×2 matrices M. What has been proven here is that the manifold of N×N matrices, i.e., defined by Eq. (4b), contains points where they are unitary.
  36. http://mathworld.wolfram.com/RelativelyPrime.html .
  37. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), p. 16.
  38. R. Bracewell, The Fourier Transform & Its Applications (McGraw-Hill, 1999), p. 261.
  39. K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

2011 (4)

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).
[CrossRef]

2010 (3)

2009 (2)

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

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

2008 (4)

2006 (2)

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

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

2005 (3)

2004 (1)

1998 (1)

1982 (1)

1981 (1)

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

1979 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1970 (1)

Aitken, A. C.

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

Almoro, P. F.

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

Bastiaans, M. J.

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

Bracewell, R.

R. Bracewell, The Fourier Transform & Its Applications (McGraw-Hill, 1999), p. 261.

Collins, S. A.

Corballis, A.

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
[CrossRef]

Deng, B.

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

Ding, J.-J.

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

Dong, B.-Z.

Fannin, D. R.

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

Gopinathan, U.

Gori, F.

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

Gu, B.-Y.

Hanson, S. G.

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

Healy, J. J.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).
[CrossRef]

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).
[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]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).
[CrossRef]

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]

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

Heath, M. T.

M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, 1997), Chap. 2.

Hennelly, B. M.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

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]

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]

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
[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]

Kelly, D.

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
[CrossRef]

Kelly, D. P.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

Kujawinska, M.

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

Kutay, M. A.

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

McElhinney, C.

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Oktem, F. S.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

Osten, W.

Ozaktas, H. M.

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Patorski, K.

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

Pedrini, G.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Ryle, J. P.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

Sheridan, J. T.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790(2011).
[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]

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–29 (2010).
[CrossRef]

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]

G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008).
[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]

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
[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]

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

Situ, G.

Stern, A.

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

Tao, R.

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

Tranter, W. H.

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

Turnbull, H. W.

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

Wang, D.-Y.

Wang, Y.

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

Wolf, K. B.

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), p. 16.

Yamaguchi, I.

Yang, G.-Z.

Yaroslavsky, L. P.

Zalevsky, Z.

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

Zhang, F.-C.

Zhang, Y.

Zhao, J.

Zhao, L.

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

Ziemer, R. E.

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

Appl. Opt. (3)

Inf. Sci. (1)

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

J. Eur. Opt. Soc. Rapid (2)

P. F. Almoro and S. G. Hanson, “Object wave reconstruction by speckle illumination and phase retrieval,” J. Eur. Opt. Soc. Rapid 4, 09002 (2011).

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5 D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid 6, 11034(2011).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Phys. A Math. Theory (1)

J. J. Healy and K. B. Wolf, “Discrete canonical transforms that are Hadamard matrices,” J. Phys. A Math. Theory 44, 265302 (2011).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (1)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SPIE (2)

B. M. Hennelly, D. Kelly, A. Corballis, and J. T. Sheridan, “Phase retrieval using theoretically unitary discrete fractional Fourier transform,” Proc. SPIE 5908, 59080D (2005).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and C. McElhinney, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008).

Signal Process. (2)

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

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

Other (14)

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

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

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 2009).

R. E. Ziemer, W. H. Tranter, and D. R. Fannin, Signals and Systems: Continuous and Discrete (Prentice-Hall, 1983).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

H. W. Turnbull and A. C. Aitken, An Introduction to the Theory of Canonical Matrices (Dover, 1961).

M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, 1997), Chap. 2.

While this statement is true it requires clarification. We note that no proof has been provided that unitary DLCTs form a group. In fact, the DLCTs do not (cannot) form a faithful representation of the group of 2×2 matrices M. What has been proven here is that the manifold of N×N matrices, i.e., defined by Eq. (4b), contains points where they are unitary.

http://mathworld.wolfram.com/RelativelyPrime.html .

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), p. 16.

R. Bracewell, The Fourier Transform & Its Applications (McGraw-Hill, 1999), p. 261.

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993).

J. T. Sheridan, L. Zhao, and J. J. Healy, “The condition number and first order optical systems,” in Imaging and Applied Optics Conference, OSA Technical Digest (Optical Society of America, 2012), paper ITu3C.4.

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

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