Abstract

A novel Brillouin optical time-domain analysis (BOTDA) system without frequency sweep operation is proposed using intensity-modulated direct detection optical orthogonal frequency division multiplexing (IM-DD-OOFDM) probe signal. The influence of peak to average power ratio (PAPR) of OFDM probe signal on the recovery of Brillouin gain spectrum (BGS) is analyzed in theory and experiment. The complex BGS is reconstructed by channel estimation algorithm and Brillouin frequency shift (BFS) is located by curve fitting of intensity spectrum. The IM-DD-OOFDM BOTDA is demonstrated experimentally with 25 m spatial resolution over 10 km standard single mode fibers within much less measurement time.

© 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. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
    [Crossref] [PubMed]
  3. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
    [Crossref] [PubMed]
  4. Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express 21(9), 10697–10705 (2013).
    [Crossref] [PubMed]
  5. D. Ba, B. Wang, D. Zhou, M. Yin, Y. Dong, H. Li, Z. Lu, and Z. Fan, “Distributed measurement of dynamic strain based on multi-slope assisted fast BOTDA,” Opt. Express 24(9), 9781–9793 (2016).
    [Crossref] [PubMed]
  6. D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
    [Crossref]
  7. A. Voskoboinik, J. Wang, B. Shamee, S. R. Nuccio, L. Zhang, M. Chitgarha, A. E. Willner, and M. Tur, “SBS Based Fiber Optical Sensing Using Frequency-Domain Simultaneous Tone Interrogation,” J. Lightwave Technol. 29(11), 1729–1735 (2011).
    [Crossref]
  8. A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
    [Crossref] [PubMed]
  9. C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
    [Crossref] [PubMed]
  10. X. Yi, W. Shieh, and Y. Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [Crossref]
  11. W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express 15(2), 350–356 (2007).
    [Crossref] [PubMed]
  12. J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), paper AT5C.3.
  13. X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
    [Crossref]
  14. B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
    [Crossref] [PubMed]
  15. L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
    [Crossref]
  16. C. Wei, H. Y. Chen, H. Chu, Y. Chen, C. Song, I. Lu, and J. Chen, “32-dB Loss Budget High-Capacity OFDM Long-Reach PON over 60-km Transmission without Optical Amplifier,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3G.1.
    [Crossref]
  17. C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
    [Crossref]
  18. 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]
  19. L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
    [Crossref] [PubMed]
  20. Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
    [Crossref]
  21. A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
    [Crossref]

2017 (1)

2016 (1)

2015 (2)

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

2013 (4)

2012 (3)

2011 (3)

2008 (1)

2007 (1)

1999 (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

1997 (1)

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[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]

Ba, D.

Bao, X.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Bao, Y.

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
[Crossref] [PubMed]

Bremner, T. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Brown, A. W.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Chen, W.

Chi, N.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Chitgarha, M.

DeMerchant, M. D.

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Deng, L.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Dong, Y.

Fan, Z.

Feng, X.

Feng, Y.

Feng, Z.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Fu, S.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Gui, T.

Guo, B.

Guo, N.

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]

Ji, Y.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Jiang, T.

Jin, C.

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]

Kressel, I.

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]

Lau, A. P. T.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Li, H.

Li, J.

Li, Z.

Liang, H.

Lin, J.

Lin, R.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Liu, D.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Liu, J.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Liu, S.

Lu, C.

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

Lu, Z.

Ma, Y.

Mafang, S. F.

Motil, A.

Nuccio, S. R.

Peled, Y.

Pendock, G.

Qiu, K.

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

Shamee, B.

Shieh, W.

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]

Shum, P.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Soto, M. A.

Tang, M.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Tao, L.

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

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]

Tellambura, C.

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[Crossref]

Thévenaz, L.

Tucker, R. S.

Tur, M.

Voskoboinik, A.

Wang, B.

Wang, J.

Wang, L.

Wang, R.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Willner, A. E.

Willner, A. W.

Wu, Q.

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

Xu, P.

Yaron, L.

Yi, X.

Yilmaz, O. F.

Yin, M.

Yu, C.

Zhang, H.

Zhang, L.

Zhou, D.

Electron. Lett. (1)

C. Tellambura, “Upper bound on peak factor of N-multiple carriers,” Electron. Lett. 33(19), 1608–1609 (1997).
[Crossref]

IEEE Netw. (1)

L. Tao, Y. Ji, J. Liu, A. P. T. Lau, N. Chi, and C. Lu, “Advanced modulation formats for short reach optical communication systems,” IEEE Netw. 27(6), 6–13 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (2)

X. Yi, Z. Li, Y. Bao, and K. Qiu, “Characterization of Passive Optical Components by DSP-Based Optical Channel Estimation,” IEEE Photonics Technol. Lett. 24(6), 443–445 (2012).
[Crossref]

Z. Feng, M. Tang, S. Fu, L. Deng, Q. Wu, R. Lin, R. Wang, P. Shum, and D. Liu, “Performance-enhanced direct detection optical OFDM transmission with CAZAC equalization,” IEEE Photonics Technol. Lett. 27(14), 1507–1510 (2015).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (11)

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
[Crossref] [PubMed]

A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
[Crossref] [PubMed]

Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012).
[Crossref] [PubMed]

B. Guo, T. Gui, Z. Li, Y. Bao, X. Yi, J. Li, X. Feng, and S. Liu, “Characterization of passive optical components with ultra-fast speed and high-resolution based on DD-OFDM,” Opt. Express 20(20), 22079–22086 (2012).
[Crossref] [PubMed]

Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express 21(9), 10697–10705 (2013).
[Crossref] [PubMed]

L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
[Crossref] [PubMed]

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]

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

D. Ba, B. Wang, D. Zhou, M. Yin, Y. Dong, H. Li, Z. Lu, and Z. Fan, “Distributed measurement of dynamic strain based on multi-slope assisted fast BOTDA,” Opt. Express 24(9), 9781–9793 (2016).
[Crossref] [PubMed]

D. Zhou, Y. Dong, B. Wang, T. Jiang, D. Ba, P. Xu, H. Zhang, Z. Lu, and H. Li, “Slope-assisted BOTDA based on vector SBS and frequency-agile technique for wide-strain-range dynamic measurements,” Opt. Express 25(3), 1889–1902 (2017).
[Crossref]

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express 15(2), 350–356 (2007).
[Crossref] [PubMed]

Proc. SPIE (1)

A. W. Brown, M. D. DeMerchant, X. Bao, and T. W. Bremner, “Precision of a Brillouin-scattering-based distributed strain sensor,” Proc. SPIE 3670, 359–365 (1999).
[Crossref]

Other (2)

C. Wei, H. Y. Chen, H. Chu, Y. Chen, C. Song, I. Lu, and J. Chen, “32-dB Loss Budget High-Capacity OFDM Long-Reach PON over 60-km Transmission without Optical Amplifier,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3G.1.
[Crossref]

J. Fang, P. Xu, and W. Shieh, “Single-shot measurement of stimulated Brillouin spectrum by using OFDM probe and coherent detection,” in Photonics and Fiber Technology 2016 (ACOFT, BGPP, NP), OSA Technical Digest (online) (Optical Society of America, 2016), paper AT5C.3.

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Figures (9)

Fig. 1
Fig. 1 Simplified BOTDA. Pump pulse (with power of Pp) interacts with continuous probe (with power of Ps) through stimulated Brillouin effect and gives rise to a local power transfer ΔPs.
Fig. 2
Fig. 2 Experimental setup: ECL: external cavity laser, PC: polarization controller, EOM: electro-optic modulator, MS: microwave source, AWG: arbitrary waveform generator, ISO: optical isolator, SOA: semiconductor optical amplifier, EDFA: Erbium doped fiber amplifier, BPF: optical band-pass filter, PS: polarization controller, FUT: fiber under test, PD: photo detector, TDO: time-domain oscilloscope, DSP: digital signal processing.
Fig. 3
Fig. 3 Time-domain OFDM probe signal: TS: Training symbol, SS: Sensing symbol.
Fig. 4
Fig. 4 (a) Normalized baseband OFDM signal with PAPR of 15 dB. (b) Normalized baseband OFDM signal with PAPR of 9 dB.
Fig. 5
Fig. 5 (a) Brillouin intensity gain spectrum for PAPR of 15 dB. (b) Brillouin intensity gain spectrum for PAPR of 9 dB. (c) Brillouin phase spectrum for PAPR of 15 dB. (d) Brillouin phase spectrum for PAPR of 9 dB.
Fig. 6
Fig. 6 (a) Reconstructed spectrogram along the fiber. (b) Measured and fit BFS as a function of temperature.
Fig. 7
Fig. 7 Spatial resolution at the end of 2km fiber.
Fig. 8
Fig. 8 (a) 3D plot of Brillouin gain spectrum along 10 km fiber. (b) Measured and fit BFS as a function of temperature at the fiber end.
Fig. 9
Fig. 9 Spatial resolution at the end of 10km fiber.

Equations (16)

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

s ( t ) = e j 2 π f 0 t + α e j 2 π f 0 t s B ( t )
s B ( t ) = k = 0 N 1 c k e j 2 π f k t
r ( t ) = s ( t τ ) h ( τ ) d τ = e j 2 π f 0 t h ( τ ) e j 2 π f 0 τ d τ + α e j 2 π f 0 t k = 0 N 1 c k e j 2 π f k t h ( τ ) e j 2 π ( f 0 + f k ) τ d τ
H ( f ) = h ( τ ) e j 2 π f t d τ
r ( t ) = H ( f 0 ) e j 2 π f 0 t + α e j 2 π f 0 t k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t
I ( t ) | r ( t ) | 2 | H ( f 0 ) | 2 + 2 α H ( f 0 ) Re { k = 0 N 1 [ [ | H ( f 0 + f k ) | e j ϕ ( f k ) c k ] r k e j 2 π f k t ] } + α 2 ( k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t ) ( k = 0 N 1 H ( f 0 + f k ) c k e j 2 π f k t )
H ( f 0 + f k ) = r k c k = H ( f 0 + f k ) H S B S
H S B S = exp ( G S B S + j φ S B S ) ( 1 + G S B S ) exp ( j φ S B S ) = ( 1 + g B Δ v B 2 Δ v B 2 + 4 Δ v 2 ) exp ( j 2 g B Δ v Δ v B Δ v B 2 + 4 Δ v 2 )
P A P R ( d B ) = 10 log 10 max ( | s B ( t ) | 2 ) E { | s B ( t ) | 2 }
P A P R ( d B ) = 10 log 10 max ( | s B ( t ) | 2 ) N
P ( t ) = | s B ( t ) | 2 = s B ( t ) s B ( t ) = i = 0 N 1 k = 0 N 1 c i c k exp [ j 2 π f ( i k ) t ] = N + 2 Re { i = 0 N 2 k = i + 1 N 1 c i c k exp [ j 2 π f ( i k ) t ] } = N + 2 Re { m = 1 N 1 exp ( j 2 π f m t ) i = 0 N 1 m c i c ( i + m ) }
P ( t ) N + 2 m = 1 N 1 | ρ ( m ) |
Δ P s ( z ) = P s ( z ) [ exp ( g B ( z ) A e f f P p ( z ) Δ z ) 1 ]
Δ P s ( z ) = g B ( z ) A e f f P p ( z ) P s ( z ) Δ z
Δ P ( z ) = g B ( z ) ¯ A e f f P p ( z ) P ( z ) Δ z
σ ν ( z ) = σ ( z ) 3 δ Δ ν B 8 2 ( 1 η ) 3 / 2 = 1 S N R ( z ) 3 δ Δ ν B 8 2 ( 1 η ) 3 / 2

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