Abstract

This paper proposes a novel method based on differential π-phase-shift long-pulse-width pair to narrow the Brillouin gain spectrum for improving the frequency accuracy in Brillouin optical time-domain analysis (BOTDA) system. This is the first approach to reduce the bandwidth of Brillouin gain spectrum for distributed sensing to the best of our knowledge. The temporal and spectral Brillouin responses for the proposal are analytically solved, and the key parameters such as pulse width and the length of π-phase-shift pulse section are investigated. Theoretical analysis and experimental results demonstrate that the proposal could achieve 17 MHz Brillouin gain spectrum bandwidth and a fixed spatial resolution of 2.5 m simultaneously, without signal-to-noise ratio penalty. This Brillouin gain spectrum is 3 times narrower than that obtained using standard single-pulse based BOTDA method with same spatial resolution, resulting in3times frequency accuracy improvement. Furthermore, such a significantly narrowed Brillouin gain spectrum provides more tolerance to the small temperature change when hot spots are introduced, giving rise to the sharper rising/falling edge of the Brillouin frequency shift profile along the sensing fiber. This way, more precise temperature/strain measurement can be obtained.

© 2017 Optical Society of America

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References

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  1. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
    [Crossref]
  2. A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
    [Crossref]
  3. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17(3), 1248–1255 (2009).
    [Crossref] [PubMed]
  4. M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
    [Crossref] [PubMed]
  5. A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
    [Crossref]
  6. R. W. Boyd, Nonlinear Optics (Academic Press, 2008).
  7. A. Loayssa and F. J. Lahoz, “Broad-band RF photonic phase shifter based on stimulated Brillouin scattering and single-sideband modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
    [Crossref]
  8. B. Vidal, M. A. Piqueras, and J. Martí, “Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering,” Opt. Lett. 32(1), 23–25 (2007).
    [Crossref] [PubMed]
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    [Crossref]
  10. T. Schneider, “Wavelength and line width measurement of optical sources with femtometre resolution,” Electron. Lett. 41(22), 1234–1235 (2005).
    [Crossref]
  11. A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
    [Crossref]
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  14. S. Preussler, A. Wiatrek, K. Jamshidi, and T. Schneider, “Ultrahigh-resolution spectroscopy based on the bandwidth reduction of stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(16), 1118–1120 (2011).
    [Crossref]
  15. S. Preussler, A. Zadok, A. Wiatrek, M. Tur, and T. Schneider, “Enhancement of spectral resolution and optical rejection ratio of Brillouin optical spectral analysis using polarization pulling,” Opt. Express 20(13), 14734–14745 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  19. T. Horiguchi, R. Muroi, A. Iwasaka, K. Wakao, and Y. Miyamoto, “Negative Brillouin Gain and its Application to Distributed Fiber Sensing,” in 33rd European Conference and Exhibition of Optical Communication (2007), paper P018.
    [Crossref]
  20. J. C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express 19(8), 7381–7397 (2011).
    [Crossref] [PubMed]
  21. M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
    [Crossref]
  22. S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).
  23. Z. Yang, X. Hong, W. Lin, and J. Wu, “Evaluating and overcoming the impact of second echo in Brillouin echoes distributed sensing,” Opt. Express 24(2), 1543–1558 (2016).
    [Crossref] [PubMed]

2016 (2)

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Z. Yang, X. Hong, W. Lin, and J. Wu, “Evaluating and overcoming the impact of second echo in Brillouin echoes distributed sensing,” Opt. Express 24(2), 1543–1558 (2016).
[Crossref] [PubMed]

2015 (1)

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

2014 (1)

M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
[Crossref]

2013 (1)

2012 (3)

2011 (3)

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2006 (1)

A. Loayssa and F. J. Lahoz, “Broad-band RF photonic phase shifter based on stimulated Brillouin scattering and single-sideband modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

2005 (2)

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

T. Schneider, “Wavelength and line width measurement of optical sources with femtometre resolution,” Electron. Lett. 41(22), 1234–1235 (2005).
[Crossref]

1995 (1)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Bao, X.

Bergman, A.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Beugnot, J.

Beugnot, J. C.

Chen, L.

Chin, S.

M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
[Crossref]

Domingo, J. M. S.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Facchini, M.

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

Fellay, A.

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

Foaleng, S. M.

He, Z.

Heras, C. D.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Hong, X.

Horiguchi, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Hotate, K.

Jamshidi, K.

Koyamada, Y.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Kurashima, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Lahoz, F. J.

A. Loayssa and F. J. Lahoz, “Broad-band RF photonic phase shifter based on stimulated Brillouin scattering and single-sideband modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

Le Floch, S.

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

Li, W.

Li, Y.

Lin, W.

Loayssa, A.

A. Loayssa and F. J. Lahoz, “Broad-band RF photonic phase shifter based on stimulated Brillouin scattering and single-sideband modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

Mafang, S. F.

Martí, J.

Motil, A.

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Niklès, M.

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

Pelayo, J.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Pellejer, E.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Perez Garcia, J.

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

Piqueras, M. A.

Preussler, S.

Robert, P.

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

Sauser, F.

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

Scandale, W.

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

Schneider, T.

Shimizu, K.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Soto, M. A.

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
[Crossref]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

Tateda, M.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Thévenaz, L.

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
[Crossref]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

J. C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express 19(8), 7381–7397 (2011).
[Crossref] [PubMed]

S. M. Foaleng, M. Tur, J. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[Crossref]

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

Tur, M.

Vidal, B.

Villuendas, F.

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

Wiatrek, A.

Wu, J.

Yang, Z.

Z. Yang, X. Hong, W. Lin, and J. Wu, “Evaluating and overcoming the impact of second echo in Brillouin echoes distributed sensing,” Opt. Express 24(2), 1543–1558 (2016).
[Crossref] [PubMed]

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

Zadok, A.

Zou, W.

Electron. Lett. (1)

T. Schneider, “Wavelength and line width measurement of optical sources with femtometre resolution,” Electron. Lett. 41(22), 1234–1235 (2005).
[Crossref]

IEEE Photonics Technol. Lett. (3)

J. M. S. Domingo, J. Pelayo, F. Villuendas, C. D. Heras, and E. Pellejer, “Very high resolution optical spectrometry by stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 17(4), 855–857 (2005).
[Crossref]

S. Preussler, A. Wiatrek, K. Jamshidi, and T. Schneider, “Ultrahigh-resolution spectroscopy based on the bandwidth reduction of stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 23(16), 1118–1120 (2011).
[Crossref]

A. Loayssa and F. J. Lahoz, “Broad-band RF photonic phase shifter based on stimulated Brillouin scattering and single-sideband modulation,” IEEE Photonics Technol. Lett. 18(1), 208–210 (2006).
[Crossref]

J. Lightwave Technol. (2)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

S. M. Foaleng, M. Tur, J. Beugnot, and L. Thévenaz, “High spatial and spectral resolution long-range sensing using Brillouin echoes,” J. Lightwave Technol. 28(20), 2993–3003 (2010).
[Crossref]

Opt. Express (7)

Opt. Laser Technol. (1)

A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Opt. Lett. (3)

Proc. SPIE (2)

M. A. Soto, S. Chin, and L. Thévenaz, “Double-pulse Brillouin distributed optical fiber sensors: analytical model and experimental validation,” Proc. SPIE 8421, 842124 (2014).
[Crossref]

S. Le Floch, Z. Yang, F. Sauser, M. A. Soto, and L. Thévenaz, “Differential chirped-pulse pair for sub-meter spatial resolution Brillouin distributed fiber sensing,” Proc. SPIE 9634, 96341D (2015).

Other (4)

T. Horiguchi, R. Muroi, A. Iwasaka, K. Wakao, and Y. Miyamoto, “Negative Brillouin Gain and its Application to Distributed Fiber Sensing,” in 33rd European Conference and Exhibition of Optical Communication (2007), paper P018.
[Crossref]

A. Fellay, L. Thévenaz, J. Perez Garcia, M. Facchini, W. Scandale, and P. Robert, “Brillouin-based temperature sensing in optical fibers down to 1 K,” in Proceedings of 15th International Conference on Optical Fiber Sensors (2002), 301–304.
[Crossref]

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in Proceedings of 12th International Conference on Optical Fiber Sensors (1997), pp. 324–327.
[Crossref]

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

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Figures (10)

Fig. 1
Fig. 1 The schematic model of the proposal.
Fig. 2
Fig. 2 (a) The Brillouin temporal waveforms corresponding the reference and sensing pulses calculated by Eqs. (7) and (8) at Brillouin resonance frequency located at position zN; (b) The differential Brillouin temporal response in the proposal. gpeak denotes the peak Brillouin gain at Brillouin resonance frequency.
Fig. 3
Fig. 3 (a) The differential Brillouin temporal response in the proposal. g peak denotes the peak Brillouin gain at Brillouin resonance frequency; (b) the ratio between the contribution of the first section and second one versus τ.
Fig. 4
Fig. 4 (a) The simulated BGS corresponding to the reference pulse, sensing pulse and the subtraction response; (b) the simulated BGS of the proposal and the single-pulse method in cases of different pulse widths.
Fig. 5
Fig. 5 Experimental setup of the proposal. PM: phase modulator; EOM: electro-optic modulator; PG: pattern generator; PC: polarization controller; PS: polarization switch; EDFA: erbium-doped fiber amplifier; PD: photodetector; DWDM: dense wavelength division multiplexing.
Fig. 6
Fig. 6 The measured BGS induced by reference pulse, sensing pulse and the proposal respectively at the start of sensing fiber.
Fig. 7
Fig. 7 (a) The measured and quadratic fitted BGS at the end of sensing fiber; (b) the FWHM of BGS versus position for both methods.
Fig. 8
Fig. 8 (a) The frequency error versus position calculated from 20 consecutive measurements; (b) the BFS profiles around the 2.5-m long hotspot.
Fig. 9
Fig. 9 The BGS and the corresponding quadratic fitted curves inside and outside the hotspot obtained by using (a) standard single pulse BOTDA method; (b) our proposal.
Fig. 10
Fig. 10 The BFS profiles around the two 2.5-m long hotspots.

Equations (13)

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A p (z,t) z + 1 V g A p (z,t) t =i 1 2 g 2 A s (z,t)Q(z,t),
A s (z,t) z 1 V g A s (z,t) t =i 1 2 g 2 A p (z,t) Q * (z,t),
Q(z,t) t + Γ A Q(z,t)=i g 1 A p (z,t) A s * (z,t),
A p ( z,t )= A p 0 { u(t z V g )u(t t 1 z V g )+β[u(t t 1 z V g )u(t t 1 t 2 z V g )] },
a s ( z = 0 , z N , t ) = g ( z N ) I P 0 A S 0 2 Γ A * Δ z { [ 1 exp [ Γ A * ( t 2 z N V g ) ] ] u ( t 2 z N V g ) + [ ( β 1 ) [ 1 exp [ Γ A * ( t 2 z N V g ) ] ] + ( β 2 β ) [ 1 exp [ Γ A * ( t t 1 2 z N V g ) ] ] ] u ( t t 1 2 z N V g ) β [ [ 1 exp [ Γ A * ( t 2 z N V g ) ] ] + ( β 1 ) [ 1 exp [ Γ A * ( t t 1 2 z N V g ) ] ] ] u ( t t 1 t 2 2 z N V g ) } .
a s ( z = 0 , t ) = Re [ N = 0 L a s ( z = 0 , z N , t ) ] .
a s r e f e r e n c e ( z = 0 , z N , t ) = g ( z N ) I P 0 A S 0 2 Γ A * Δ z × [ 1 exp [ Γ A * ( t 2 z N V g ) ] ] [ u ( t 2 z N V g ) u ( t t 1 t 2 2 z N V g ) ] ,
a s s e n s i n g ( z = 0 , z N , t ) = g ( z N ) I P 0 A S 0 2 Γ A * Δ z × { [ 1 exp [ Γ A * ( t 2 z N V g ) ] ] [ u ( t 2 z N V g ) 2 u ( t t 1 2 z N V g ) + u ( t t 1 t 2 2 z N V g ) ] + 2 [ 1 exp [ Γ A * ( t t 1 2 z N V g ) ] ] [ u ( t t 1 2 z N V g ) u ( t t 1 t 2 2 z N V g ) ] } ,
a s p r o p o s a l ( z = 0 , z N , t ) = a s r e f e r e n c e ( z = 0 , z N , t ) a s s e n s i n g ( z = 0 , z N , t ) = g ( z N ) I P 0 A S 0 Γ A * Δ z × [ exp [ Γ A * ( t t 1 2 z N V g ) ] exp [ Γ A * ( t 2 z N V g ) ] ] × [ u ( t t 1 2 z N V g ) u ( t t 1 t 2 2 z N V g ) ] .
a s proposal ( z=0, z N ,t ) =g( z N ) I P 0 A S 0 Γ A * Δz×exp[ Γ A * (t t 1 2 z N V g )][ u(t t 1 2 z N V g )u(t t 1 t 2 2 z N V g ) ],
a s 1st ( z = 0 , t ) a s 2 n d ( z = 0 , t ) = R e [ t 1 + 2 z N V g t 1 + τ + 2 z N V g a s p r o p o s a l ( z = 0 , z N , t ) d t ] R e [ t 1 + τ + 2 z N V g t 1 +t 2 + 2 z N V g a s p r o p o s a l ( z = 0 , z N , t ) d t ] | Ω = Ω B = exp ( Γ A * τ ) 1 exp ( Γ A * t 2 ) exp ( Γ A * τ ) .
σ ν (z)= 1 SNR(z) 3 4 δ×Δ ν B ,
R = σ ν p ( z ) σ ν s ( z ) = Δ ν B p Δ ν B s = 3 .

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