Abstract

We analyze the dual random phase encoding technique in the temporal domain to evaluate its potential application for secure data transmission in fiber optic links. To take into account the optical fiber multiplexing capabilities, the noise content of the signal is restricted when multiple channels are transmitted over a single fiber optic link. We also discuss some mechanisms for producing encoded time-limited as well as bandwidth-limited signals and a comparison with another recently proposed technique is made. Numerical simulations have been carried out to analyze the system performance. The results indicate that this multiplexing encryption method could be a good alternative compared with other well-established methods.

© 2008 Optical Society of America

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References

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  1. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767-769 (1995).
  2. N. Towghi, B. Javidi, and Z. Luo, “Fully phase encrypted image processor,” J. Opt. Soc. Am. A 16, 1915-1927 (1999).
    [CrossRef]
  3. B. Javidi, N. Towghi, N. Maghzi, and C. Verrall, “Error-reduction techniques and error analysis for fully phase- and amplitude-based encryption,” Appl. Opt. 39, 4117-4130 (2000).
  4. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762-764 (1999).
  5. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random encoding in the fractional Fourier domain,” Opt. Lett. 25, 887-889 (2000).
    [CrossRef]
  6. G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853-2859 (2000).
  7. S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
    [CrossRef]
  8. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584-1586 (2004).
    [CrossRef]
  9. G. Situ and J. Zhang, “A lensless optical security system based on computer generated phase only,” Opt. Commun. 232, 115-122 (2004).
  10. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
  11. A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3-13 (1994).
  12. J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
    [CrossRef]
  13. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
    [CrossRef]
  14. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
    [CrossRef]
  15. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
    [CrossRef]
  16. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  17. N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
    [CrossRef]
  18. A. W. Lohmann and D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212-6219 (1992).
  19. A. E. Willner and Y. Xie, “Wavelength domain multiplexed (WDM) fiber optic communication networks,” in Fiber Optics Handbook, M. Bass and E. W. Van Stryland, eds. (McGraw-Hill, 2002), pp. 13.1-13.31.
  20. B. M. Hennelly, T. J. Naughton, J. McDonald, J. T. Sheridan, G. Unnikrishnan, D. P. Kelly, and B. Javidi, “Spread-space spread-spectrum technique for secure multiplexing,” Opt. Lett. 32, 1060-1062 (2007).
    [CrossRef]
  21. L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459-2463 (1996).
  22. C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).
  23. C. Cuadrado-Laborde, “Time-variant signal encryption by lensless dual random phase encoding applied to fiber optic links,” Opt. Lett. 32, 2867-2869 (2007).
    [CrossRef]

2007 (2)

2006 (3)

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

2004 (2)

G. Situ and J. Zhang, “A lensless optical security system based on computer generated phase only,” Opt. Commun. 232, 115-122 (2004).

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584-1586 (2004).
[CrossRef]

2000 (4)

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853-2859 (2000).

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random encoding in the fractional Fourier domain,” Opt. Lett. 25, 887-889 (2000).
[CrossRef]

B. Javidi, N. Towghi, N. Maghzi, and C. Verrall, “Error-reduction techniques and error analysis for fully phase- and amplitude-based encryption,” Appl. Opt. 39, 4117-4130 (2000).

1999 (3)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762-764 (1999).

N. Towghi, B. Javidi, and Z. Luo, “Fully phase encrypted image processor,” J. Opt. Soc. Am. A 16, 1915-1927 (1999).
[CrossRef]

1996 (1)

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459-2463 (1996).

1995 (2)

S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
[CrossRef]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767-769 (1995).

1994 (2)

A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3-13 (1994).

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).

1992 (1)

Atkins, S.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

Azaña, J.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

Barrera, J. F.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

Berger, N. K.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

Bolognini, N.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

Chen, L. R.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

Costanzo-Caso, P.

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).

Cuadrado-Laborde, C.

C. Cuadrado-Laborde, “Time-variant signal encryption by lensless dual random phase encoding applied to fiber optic links,” Opt. Lett. 32, 2867-2869 (2007).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).

Duchowicz, R.

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).

Fischer, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Granieri, S.

S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
[CrossRef]

Henao, R.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

Hennelly, B. M.

Javidi, B.

Joseph, J.

Kelly, D. P.

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).

Levit, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

Lohmann, A. W.

Luo, Z.

Maghzi, N.

Matoba, O.

McDonald, J.

Mendlovic, D.

Muriel, M. A.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

Naughton, T. J.

Neto, L. G.

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459-2463 (1996).

Papoulis, A.

Réfrégier, P.

Sheng, Y.

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459-2463 (1996).

Sheridan, J. T.

Sicre, E. E.

S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).

Singh, K.

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random encoding in the fractional Fourier domain,” Opt. Lett. 25, 887-889 (2000).
[CrossRef]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853-2859 (2000).

Situ, G.

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584-1586 (2004).
[CrossRef]

G. Situ and J. Zhang, “A lensless optical security system based on computer generated phase only,” Opt. Commun. 232, 115-122 (2004).

Smith, P. W. E.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

Tebaldi, M.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

Torroba, R.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

Towghi, N.

Trabocchi, O.

S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
[CrossRef]

Unnikrishnan, G.

Verrall, C.

Willner, A. E.

A. E. Willner and Y. Xie, “Wavelength domain multiplexed (WDM) fiber optic communication networks,” in Fiber Optics Handbook, M. Bass and E. W. Van Stryland, eds. (McGraw-Hill, 2002), pp. 13.1-13.31.

Xie, Y.

A. E. Willner and Y. Xie, “Wavelength domain multiplexed (WDM) fiber optic communication networks,” in Fiber Optics Handbook, M. Bass and E. W. Van Stryland, eds. (McGraw-Hill, 2002), pp. 13.1-13.31.

Zhang, J.

G. Situ and J. Zhang, “A lensless optical security system based on computer generated phase only,” Opt. Commun. 232, 115-122 (2004).

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584-1586 (2004).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (2)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223-2224 (1999).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro optic phase modulation,” Electron. Lett. 36, 1644-1646 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).

J. Opt. Soc. Am. A (2)

Opt. Commun. (5)

S. Granieri, O. Trabocchi, and E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275-278 (1995).
[CrossRef]

G. Situ and J. Zhang, “A lensless optical security system based on computer generated phase only,” Opt. Commun. 232, 115-122 (2004).

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532-536 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data using polarization architectures,” Opt. Commun. 260, 109-112 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiple image encryption using an aperture-modulated optical system,” Opt. Commun. 261, 29-33 (2006).
[CrossRef]

Opt. Eng. (2)

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853-2859 (2000).

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459-2463 (1996).

Opt. Lett. (6)

Other (3)

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Ultrafast optical temporal processing using phase-space signal representations,” in Optics Research Trends, P. V. Gallico, ed. (Nova Science, 2007).

A. E. Willner and Y. Xie, “Wavelength domain multiplexed (WDM) fiber optic communication networks,” in Fiber Optics Handbook, M. Bass and E. W. Van Stryland, eds. (McGraw-Hill, 2002), pp. 13.1-13.31.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (7)

Fig. 1
Fig. 1

(a) Scheme of the single channel temporal DRPE optical device. To encode f ( t ) into a white stationary sequence ψ ( t ) , the signal is phase modulated by p ( t ) = exp [ i 2 π n ( t ) ] and then it is convolved with q ( t ) . To decode, we just partially reverse the encoding process. (b) Scheme of the convolution optical device. The convolution operation is performed by translating to the time domain the well-known 4 f system. To this end, two Fourier transforms are necessary; each of them is composed of two dispersive mediums ( Φ 20 ) and a PM acting as a time lens. Between both Fourier transforms the signal is phase modulated by the key signal b ( ν ) .

Fig. 2
Fig. 2

(a) Scheme of the multiplexing DRPE optical device. The encoding process of each channel can be followed from the Fig. 1 caption. At the end, the several encoded signals ψ j ( t ) are mixed and transmitted through a single fiber optic link. (b) Scheme of the multiple channel decoding device of the temporal DRPE technique. At the output of the jth channel, the decoded signal g j ( t ) = f j ( t ) p j ( t ) is obtained.

Fig. 3
Fig. 3

Single channel behavior of the temporal DRPE technique. (a) Input optical signal, (b) transmitted encoded signal, (c) decoded signal by the correct key, and (d) eavesdropper measurement, i.e., an incorrect key has been used. As a result of the spreading in the time domain, the vertical axis in (b) and (d) has been enhanced in order to better visualize these signals.

Fig. 4
Fig. 4

Multiple channel behavior of the temporal DRPE technique in which five signals are simultaneously transmitted: (a), (c), (e), (g), and (i) input optical signals and (b), (d), (f), (h), and (j) decoded signals.

Fig. 5
Fig. 5

SNR versus channel numbers in the multiple channel DRPE; the decaying exponential dashed curve is a guide for the eyes.

Fig. 6
Fig. 6

Single channel decoding behavior as a function of the chopper widths: (a)  τ ψ = 20 × T 1 = 4 ns , (b)  τ ψ = 10 × T 1 = 2 ns , and (c)  τ ψ = 5 × T 1 = 1 ns . The input signal used was the same signal represented in Fig. 5(a). As a result of the different spreading involved, scales of (b) and (c) have been enhanced by 2 × and 4 × , respectively.

Fig. 7
Fig. 7

Single channel behavior of the temporal DRPE technique with a quasi-white noise key. (a) Input optical signal, (b) transmitted encoded signal, (c) decoded signal by the correct key, and (d) eavesdropper measurement, i.e., an incorrect key has been used. As a result of the spreading in the time domain, the vertical axes in (b) and (d) have been enhanced by 4 × and 10 × , respectively, in order to better visualize these signals.

Equations (11)

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

ψ ( t ) = { f ( t ) p ( t ) } * q ( t ) ,
Ψ ( t ) = j = 1 n { f j ( t ) p j ( t ) } * q j ( t ) .
g D ( t ) = f m ( t ) p m ( t ) + j = 1 n , j m { f j ( t ) p j ( t ) } * q j ( t ) .
SNR = 10 log ( | f ( t ) | 2 d t [ | f ( t ) | 2 | g ( t ) | 2 ] d t ) .
g ( t ) 0     | t | > Δ t g / 2 ,
n TLS ( t ) = n ( t ) rect ( t / τ n ) , b TLS ( t ) = b ( t ) rect ( t / τ b ) ,
ψ TLS ( t ) = ψ ( t ) rect ( t / τ ψ ) .
{ f ( t ) , F ( ν ) } 0 ,     { | t | , | ν | } > { Δ t f / 2 , Δ ν f / 2 } ,
{ p ( t ) , P ( ν ) } 0 ,     { | t | , | ν | } > { Δ t p / 2 , Δ ν p / 2 } , { q ( t ) , Q ( ν ) } 0 ,     { | t | , | ν | } > { Δ t q / 2 , Δ ν q / 2 } .
Δ t ψ = Δ t f + c Δ ν q , Δ ν ψ = Δ ν f + Δ ν p
Δ t ψ = Δ t f + 2 π ( Φ 20 ( 1 ) + Φ 20 ( 2 ) ) ( Δ ν f + Δ ν p ) + 2 π Φ 20 ( 2 ) Δ ν q , Δ ν ψ = Δ ν f + Δ ν p + Δ ν q .

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