Abstract

A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in N  log  N time, where N is the number of input samples. A space–bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  58. L. Onural, M. F. Erden, and H. M. Ozaktas, “Extensions to common Laplace and Fourier transforms,” IEEE Signal Process. Lett. 4, 310–312 (1997).
    [CrossRef]
  59. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  62. 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]
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    [CrossRef]
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    [CrossRef]
  65. 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]
  66. A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
    [CrossRef]
  67. A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
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    [CrossRef] [PubMed]

2010

2009

D. Dragoman, “Classical versus complex fractional Fourier transformation,” J. Opt. Soc. Am. A 26, 274–277 (2009).
[CrossRef]

S. Baskal and Y. S. Kim, “ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics,” J. Opt. Soc. Am. A 26, 2049–2054 (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]

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

2008

2007

2006

2005

2004

A. A. Malyutin, “Complex-order fractional Fourier transforms in optical schemes with Gaussian apertures,” Quantum Electron. 34, 960–964 (2004).
[CrossRef]

2003

S. Baskal and Y. S. Kim, “Lens optics as an optical computer for group contractions,” Phys. Rev. E 67, 056601 (2003).
[CrossRef]

E. Georgieva and Y. S. Kim, “Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics,” Phys. Rev. E 68, 026606 (2003).
[CrossRef]

A. Torre, “Linear and radial canonical transforms of fractional order,” J. Comput. Appl. Math. 153, 477–486 (2003).
[CrossRef]

U. Sümbül and H. M. Ozaktas, “Fractional free space, fractional lenses, and fractional imaging systems,” J. Opt. Soc. Am. A 20, 2033–2040 (2003).
[CrossRef]

2002

C. Wang and B. Lü, “Implementation of complex-order Fourier transforms in complex ABCD optical systems,” Opt. Commun. 203, 61–66 (2002).
[CrossRef]

2001

D. J. Griffiths and C. A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137–154 (2001).
[CrossRef]

1998

1997

L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
[CrossRef]

L. Onural, M. F. Erden, and H. M. Ozaktas, “Extensions to common Laplace and Fourier transforms,” IEEE Signal Process. Lett. 4, 310–312 (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]

1996

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
[CrossRef]

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

L. M. Bernardo and O. D. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
[CrossRef] [PubMed]

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]

1995

C.-C. Shih, “Optical interpretation of a complex-order Fourier transform,” Opt. Lett. 20, 1178–1180 (1995).
[CrossRef] [PubMed]

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

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

1994

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]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (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]

S. Abe and J. T. Sheridan, “Optical operations on wavefunctions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

1993

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

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

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

1992

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[CrossRef]

1982

C. Jung and H. Kruger, “Representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators,” J. Phys. A 15, 3509–3523 (1982).
[CrossRef]

1979

1978

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

B. Simon, “Resonances and complex scaling: A rigorous overview,” Int. J. Quantum Chem. 14, 529–542 (1978).
[CrossRef]

J. N. Bardsley, “Complex scaling: An introduction,” Int. J. Quantum Chem. 14, 343–352 (1978).
[CrossRef]

1977

K. B. Wolf, “On self-reciprocal functions under a class of integral transforms,” J. Math. Phys. 18, 1046–1051 (1977).
[CrossRef]

1974

K. B. Wolf, “Canonical transformations I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

K. B. Wolf, “Canonical transformations II. Complex radial transforms,” J. Math. Phys. 15, 2102–2111 (1974).
[CrossRef]

1973

M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM J. Appl. Math. 25, 193–212 (1973).
[CrossRef]

1971

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[CrossRef]

1967

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part II,” Commun. Pure Appl. Math. 20, 1–101 (1967).
[CrossRef]

1961

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

Abe, S.

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]

S. Abe and J. T. Sheridan, “Optical operations on wavefunctions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

Alieva, T.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arikan, O.

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

Bardsley, J. N.

J. N. Bardsley, “Complex scaling: An introduction,” Int. J. Quantum Chem. 14, 343–352 (1978).
[CrossRef]

Bargmann, V.

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part II,” Commun. Pure Appl. Math. 20, 1–101 (1967).
[CrossRef]

V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform, part I,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

Barriuso, A. G.

L. L. Sanchez-Soto, J. F. Carinena, A. G. Barriuso, and J. J. Monzon, “Vector-like representation of one-dimensional scattering,” Eur. J. Phys. 26, 469–480 (2005).
[CrossRef]

Barshan, B.

Baskal, S.

Bastiaans, M. J.

Bernardo, L. M.

L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
[CrossRef]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
[CrossRef]

L. M. Bernardo and O. D. D. Soares, “Optical fractional Fourier transforms with complex orders,” Appl. Opt. 35, 3163–3166 (1996).
[CrossRef] [PubMed]

Boudreaux-Bartels, G. F.

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdağı, “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]

Carinena, J. F.

L. L. Sanchez-Soto, J. F. Carinena, A. G. Barriuso, and J. J. Monzon, “Vector-like representation of one-dimensional scattering,” Eur. J. Phys. 26, 469–480 (2005).
[CrossRef]

Cohen, L.

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

Davies, B.

B. Davies, Integral Transforms and Their Applications (Springer, 1978).

Dorsch, R. G.

Dragoman, D.

Erden, M. F.

L. Onural, M. F. Erden, and H. M. Ozaktas, “Extensions to common Laplace and Fourier transforms,” IEEE Signal Process. Lett. 4, 310–312 (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]

Fan, H.

Ferreira, C.

Georgieva, E.

E. Georgieva and Y. S. Kim, “Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics,” Phys. Rev. E 68, 026606 (2003).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths and C. A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137–154 (2001).
[CrossRef]

Healy, J.

Healy, J. J.

Hennelly, B. M.

Hesselink, L.

Hlawatsch, F.

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[CrossRef]

Hu, L.

Jung, C.

C. Jung and H. Kruger, “Representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators,” J. Phys. A 15, 3509–3523 (1982).
[CrossRef]

Kim, Y. S.

S. Baskal and Y. S. Kim, “ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics,” J. Opt. Soc. Am. A 26, 2049–2054 (2009).
[CrossRef]

E. Georgieva and Y. S. Kim, “Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics,” Phys. Rev. E 68, 026606 (2003).
[CrossRef]

S. Baskal and Y. S. Kim, “Lens optics as an optical computer for group contractions,” Phys. Rev. E 67, 056601 (2003).
[CrossRef]

Koç, A.

Kramer, P.

P. Kramer, M. Moshinsky, and T. H. Seligman, Complex Extensions of Canonical Transformations and Quantum Mechanics, Vol. 3 of Group Theory and Its Applications, E.M.Loebl, ed. (Academic, 1975), pp. 249–332.

Kruger, H.

C. Jung and H. Kruger, “Representation of quantum mechanical wavefunctions by complex valued extensions of classical canonical transformation generators,” J. Phys. A 15, 3509–3523 (1982).
[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]

A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
[CrossRef]

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

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

Lohmann, A. W.

Lü, B.

C. Wang and B. Lü, “Implementation of complex-order Fourier transforms in complex ABCD optical systems,” Opt. Commun. 203, 61–66 (2002).
[CrossRef]

Malyutin, A. A.

A. A. Malyutin, “Complex-order fractional Fourier transforms in optical schemes with Gaussian apertures,” Quantum Electron. 34, 960–964 (2004).
[CrossRef]

Martorell, J.

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Figures (6)

Fig. 1
Fig. 1

The effect of the CCM operation on the WD.

Fig. 2
Fig. 2

Example function F 4 .

Fig. 3
Fig. 3

Example function F 5 .

Fig. 4
Fig. 4

Transform ( T 1 ) of F 1 , F 2 , F 3 , F 4 , and F 5 . The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close.

Fig. 5
Fig. 5

Transform ( T 3 ) of F 1 , F 2 , F 3 , F 4 , and F 5 . The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close.

Fig. 6
Fig. 6

CFRT with order 0.8 i 0.2 of F 1 , F 2 , F 3 , F 4 , and F 5 . The results obtained with the presented algorithm and the reference result have been plotted with dotted and solid lines, respectively. However, the two types of lines are almost indistinguishable since the results are very close.

Tables (2)

Tables Icon

Table 1 Summary of the Conditions to Have Bounded R R CLCTs

Tables Icon

Table 2 Percentage Errors for Different Functions F and Transforms T

Equations (66)

Equations on this page are rendered with MathJax. Learn more.

M R = [ a b c d ] ,
M C = [ a b c d ] ,
( C M R f ) ( u ) = K R ( u , u ) f ( u ) d u ,
K R ( u , u ) = e i π / 4 β   exp [ i π ( α u 2 2 β u u + γ u 2 ) ] ,
M R = [ a b c d ] = [ γ / β 1 / β β + α γ / β α / β ] = [ α / β 1 / β β α γ / β γ / β ] 1 .
( C M C f ) ( u ) = K C ( u , u ) f ( u ) d u ,
K C ( u , u ) = e i π / 4 β ¯   exp [ i π ( α ¯ u 2 2 β ¯ u u + γ ¯ u 2 ) ] ,
M C = [ a b c d ] = [ a r + i a c b r + i b c c r + i c c d r + i d c ] = [ γ ¯ / β ¯ 1 / β ¯ β ¯ + α ¯ γ ¯ / β ¯ α ¯ / β ¯ ] ,
K C ( u , u ) = e i π / 4 β r + i β c e i π ( α r u 2 2 β r u u + γ r u 2 ) e π ( α c u 2 2 β c u u + γ c u 2 ) .
α r = d r b r + d c b c b r 2 + b c 2 ,     α c = d c b r d r b c b r 2 + b c 2 ,
β r = b r b r 2 + b c 2 ,     β c = b c b r 2 + b c 2 ,
γ r = a r b r + a c b c b r 2 + b c 2 ,     γ c = a c b r a r b c b r 2 + b c 2 ,
a r = β c γ c + β r γ r β r 2 + β c 2 ,     a c = β r γ c β c γ r β r 2 + β c 2 ,
b r = β r β r 2 + β c 2 ,     b c = β c β r 2 + β c 2 ,
d r = β c α c + β r α r β r 2 + β c 2 ,     d c = α c β r α r β c β r 2 + β c 2 .
W f ( u , μ ) = f ( u + u / 2 ) f ( u u / 2 ) e 2 π i μ u d u .
W f M ( u , μ ) = W f ( d u b μ , c u + a μ ) .
W h ( u , μ μ ) W f ( u , μ ) d μ .
C Q i q f ( u ) = Q i q f ( u ) = e π q u 2 f ( u ) ,
Q i q = [ 1 0 i q 1 ] = [ 1 0 i q 1 ] 1 ,
W h ( u , μ ) = 2 q e 2 π q u 2 e ( 2 π / q ) μ 2 ,     q < 0.
g 1 = 16 / π | q | ,
g 2 = 16 | q | / π
G t f ( u ) = 1 / t e π ( u u ) 2 / t f ( u ) d u .
R i r = [ 1 i r 0 1 ] ,
C R i r f ( u ) = R i r f ( u ) = f ( u ) e i π / 4 1 / r   exp ( π u 2 / r ) .
F lc i b = [ cosh ( b π / 2 ) i   sinh ( b π / 2 ) i   sinh ( b π / 2 ) cosh ( b π / 2 ) ] ,
C F lc i b f ( u ) = F lc i b f ( u ) = e b π / 4 F i b f ( u ) .
( C M C f ) ( u ) = 1 a e j c y 2 / 2 a f ( y / a ) .
a r 0 ,
d = 1 / a ,
a c = 0 ,
a r c c 0 ,
M C = [ a r 0 c 1 / a r ] = [ 1 0 c / a r 1 ] [ a r 0 0 1 / a r ] = [ 1 0 c r / a r 1 ] [ 1 0 i c c / a r 1 ] [ a r 0 0 1 / a r ] .
M C = [ 1 0 q 3 r 1 ] [ 1 0 i q 3 c 1 ] [ 0 1 1 0 ] [ 1 0 q 2 r 1 ] [ 1 0 i q 2 c 1 ] [ 0 1 1 0 ] [ 1 0 q 1 r 1 ] [ 1 0 i q 1 c 1 ] .
M C = [ 1 0 ( q 3 r + i q 3 c ) 1 ] [ 0 1 1 0 ] [ 1 0 ( q 2 r + i q 2 c ) 1 ] [ 0 1 1 0 ] [ 1 0 ( q 1 r + i q 1 c ) 1 ] .
M C = [ 1 0 ( q 3 r + i q 3 c ) 1 ] [ 1 ( q 2 r + i q 2 c ) 0 1 ] [ 1 0 ( q 1 r + i q 1 c ) 1 ] ,
q 1 r = b r b r a r a c b c b r 2 + b c 2 ,
q 1 c = b c a r b c b r a c b r 2 + b c 2 ,
q 2 r = b r ,
q 2 c = b c ,
q 3 r = b r b r d r d c b c b r 2 + b c 2 ,
q 3 c = b c d r b c b r d c b r 2 + b c 2 .
C M = Q q 3 r Q i q 3 c F lc 1 Q q 2 r Q i q 2 c F lc Q q 1 r Q i q 1 c .
q 1 r = b r b r 2 + b c 2 ,
q 1 c = b c b r 2 + b c 2 ,
q 2 r = b r ,
q 2 c = b c ,
q 3 r = b r b r d r d c b c b r 2 + b c 2 ,
q 3 c = b c d r b c b r d c b r 2 + b c 2 .
b c b r 2 + b c 2 0 ,
b c 0 ,
b c d r b c b r d c b r 2 + b c 2 0.
q 1 r = 1 / b r ,
q 1 c = 0 ,
q 2 r = b r ,
q 2 c = 0 ,
q 3 r = ( 1 d r ) / b r 2 ,
q 3 c = d c / b r ,
M C = [ 1 0 ( 1 d r ) / b r 1 ] [ 1 0 i d c / b r 1 ] [ 0 1 1 0 ] [ 1 0 b r 1 ] [ 0 1 1 0 ] [ 1 0 1 / b r 1 ] .
b c 0 ,
b c a r b r a c b c ,
b c d r b r d c b c ,
β c 0 ,
α c β c ,
γ c β c ,

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