Abstract

In order to solve fading problem and realize sub-meter spatial resolution in DAS, this paper proposes a novel configuration of time-gated digital optical frequency domain reflectometry (TGD-OFDR) based on optical intensity modulator (IM). IM has a large modulation bandwidth and the positive and negative harmonics can be fully used to suppress fading while the spatial resolution remains unchanged. In experiments, with fading suppressed, the spatial resolution of DAS is 0.8 m and the strain resolution is about 245.6 √Hz along the total 9.8-km sensing fiber. The response bandwidth of vibration is 5 kHz, only/limited by the fiber length.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

2016 (6)

2015 (2)

2014 (3)

F. Peng, N. Duan, Y. Rao, and J. Li, “Real-time position and speed monitoring of trains using phase-sensitive OTDR,” IEEE Photon. Technol. Lett. 26 (20), 2055–2057 (2014).
[Crossref]

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

F. Peng, H. Wu, X. Jia, Y. Rao, Z. Wang, and Z. Peng, “Ultra-long high-sensitivity phi-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22 (11), 13804–13810 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

Z. Qin, L. Chen, and X. Bao, “Continuous wavelet transform for non-stationary vibration detection with phase-OTDR,” Opt. Express 20(18), 20459–20465 (2012).
[Crossref] [PubMed]

Z. Qin, L. Chen, and X. Bao, “Wavelet denoising method for improving detection performance of distributed vibration sensor,” IEEE Photon. Technol. Lett. 24 (7), 542–544 (2012).
[Crossref]

2011 (1)

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

2010 (1)

2006 (1)

2005 (1)

1989 (1)

L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7(2), 279–292 (1989).
[Crossref]

Andres, M. V.

Bao, X.

M. Ren, P. Lu, L. Chen, and X. Bao, “Theoretical and experimental analysis of phi-otdr based on polarization diversity detection,” IEEE Photon. Technol. Lett. 28(6), 697–700 (2016).
[Crossref]

Z. Qin, L. Chen, and X. Bao, “Continuous wavelet transform for non-stationary vibration detection with phase-OTDR,” Opt. Express 20(18), 20459–20465 (2012).
[Crossref] [PubMed]

Z. Qin, L. Chen, and X. Bao, “Wavelet denoising method for improving detection performance of distributed vibration sensor,” IEEE Photon. Technol. Lett. 24 (7), 542–544 (2012).
[Crossref]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-otdr,” J. Lightwave Technol. 28(22), 3243–3249 (2010).

Berlang, W.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Bostick, T.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Cai, H.

Chen, D.

Chen, L.

M. Ren, P. Lu, L. Chen, and X. Bao, “Theoretical and experimental analysis of phi-otdr based on polarization diversity detection,” IEEE Photon. Technol. Lett. 28(6), 697–700 (2016).
[Crossref]

Z. Qin, L. Chen, and X. Bao, “Continuous wavelet transform for non-stationary vibration detection with phase-OTDR,” Opt. Express 20(18), 20459–20465 (2012).
[Crossref] [PubMed]

Z. Qin, L. Chen, and X. Bao, “Wavelet denoising method for improving detection performance of distributed vibration sensor,” IEEE Photon. Technol. Lett. 24 (7), 542–544 (2012).
[Crossref]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-otdr,” J. Lightwave Technol. 28(22), 3243–3249 (2010).

Chen, X.

Cox, B.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Detomo, R.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Dong, Y.

Duan, N.

F. Peng, N. Duan, Y. Rao, and J. Li, “Real-time position and speed monitoring of trains using phase-sensitive OTDR,” IEEE Photon. Technol. Lett. 26 (20), 2055–2057 (2014).
[Crossref]

Eyal, A.

H. Gabai and A. Eyal, “On the sensitivity of distributed acoustic sensing,” Opt. Lett. 41(24), 5648–5651 (2016).
[Crossref] [PubMed]

A. Eyal, H. Gabai, and I. Shpatz, “Distributed acoustic sensing: how to make the best out of the Rayleigh-backscattered energy?” in 25th Optical Fiber Sensors Conference (OFS), Proc. SPIE10323, 103230I (2017).

Fan, M.

Fan, X.

Fang, Z.

J. Zhou, Z. Pan, Q. Ye, H. Cai, R. Qu, and Z. Fang, “Characteristics and explanations of interference fading of a ϕ-OTDR with a multi-frequency source,” J. Lightwave Technol. 31(17), 2947–2954 (2013).
[Crossref]

Z. Pan, Q. Ye, H. Cai, R. Qu, and Z. Fang, “Phase-sensitive otdr system based on digital coherent detection,” in Asia Communications and Photonics Conference and Exhibition 2011, (Optical Society of America, 2011) paper 83110S.

Fu, C.

Gabai, H.

H. Gabai and A. Eyal, “On the sensitivity of distributed acoustic sensing,” Opt. Lett. 41(24), 5648–5651 (2016).
[Crossref] [PubMed]

A. Eyal, H. Gabai, and I. Shpatz, “Distributed acoustic sensing: how to make the best out of the Rayleigh-backscattered energy?” in 25th Optical Fiber Sensors Conference (OFS), Proc. SPIE10323, 103230I (2017).

Gonzalez-Herraez, M.

Grandi, S.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

He, Z.

Hedekvist, P. O.

Hornman, K.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Jia, X.

Juarez, J. C.

Kashikar, S.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Kazovsky, L. G.

L. G. Kazovsky, “Phase- and polarization-diversity coherent optical techniques,” J. Lightwave Technol. 7(2), 279–292 (1989).
[Crossref]

Kiyashchenko, D.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Krebber, K.

Kuvshinov, B.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Kyoo Nam, C.

Li, J.

F. Peng, N. Duan, Y. Rao, and J. Li, “Real-time position and speed monitoring of trains using phase-sensitive OTDR,” IEEE Photon. Technol. Lett. 26 (20), 2055–2057 (2014).
[Crossref]

Liehr, S.

Liu, E.

Liu, Q.

Lopez, J.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Lu, B.

Lu, P.

M. Ren, P. Lu, L. Chen, and X. Bao, “Theoretical and experimental analysis of phi-otdr based on polarization diversity detection,” IEEE Photon. Technol. Lett. 28(6), 697–700 (2016).
[Crossref]

Lu, Y.

Lu, Z.

Maier, E. W.

Martin-Lopez, S.

Martins, H. F.

Masoudi, A.

Mateeva, A.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Mestayer, J.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Miller, D.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Muanenda, Y. S.

Munzenberger, S.

Newson, T. P.

Pan, Z.

Parker, T.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Peng, F.

Peng, Z.

Perez-Millan, P.

Potters, H.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Qian, X.

Qin, Z.

Z. Qin, L. Chen, and X. Bao, “Wavelet denoising method for improving detection performance of distributed vibration sensor,” IEEE Photon. Technol. Lett. 24 (7), 542–544 (2012).
[Crossref]

Z. Qin, L. Chen, and X. Bao, “Continuous wavelet transform for non-stationary vibration detection with phase-OTDR,” Opt. Express 20(18), 20459–20465 (2012).
[Crossref] [PubMed]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Qu, R.

Rao, J.

Rao, Y.

Ren, M.

M. Ren, P. Lu, L. Chen, and X. Bao, “Theoretical and experimental analysis of phi-otdr based on polarization diversity detection,” IEEE Photon. Technol. Lett. 28(6), 697–700 (2016).
[Crossref]

Savory, S. J.

Shi, K.

Shpatz, I.

A. Eyal, H. Gabai, and I. Shpatz, “Distributed acoustic sensing: how to make the best out of the Rayleigh-backscattered energy?” in 25th Optical Fiber Sensors Conference (OFS), Proc. SPIE10323, 103230I (2017).

Sun, W.

Taylor, H. F.

Thomsen, B. C.

Todorov, M.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Wang, B.

G. Yang, X. Fan, S. Wang, B. Wang, Q. Liu, and Z. He, “Long-range distributed vibration sensing based on phase extraction from phase-sensitive otdr,” IEEE Photon. J. 8(3), 1–12 (2016).

Wang, S.

Wang, Z.

Wiberg, A.

Wills, P.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Wu, H.

Xue, N.

Yang, G.

G. Yang, X. Fan, S. Wang, B. Wang, Q. Liu, and Z. He, “Long-range distributed vibration sensing based on phase extraction from phase-sensitive otdr,” IEEE Photon. J. 8(3), 1–12 (2016).

Yang, Z.

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

Ye, Q.

Zhang, H.

Zhang, L.

Zheng, H.

Zhou, J.

Zhu, T.

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-otdr,” J. Lightwave Technol. 28(22), 3243–3249 (2010).

Appl. Opt. (1)

Geophys. Prospect. (1)

A. Mateeva, J. Lopez, H. Potters, J. Mestayer, B. Cox, D. Kiyashchenko, P. Wills, S. Grandi, K. Hornman, B. Kuvshinov, W. Berlang, Z. Yang, and R. Detomo, “Distributed acoustic sensing for reservoir monitoring with vertical seismic profiling,” Geophys. Prospect. 62(4), 679–692 (2014).
[Crossref]

IEEE Photon. J. (1)

G. Yang, X. Fan, S. Wang, B. Wang, Q. Liu, and Z. He, “Long-range distributed vibration sensing based on phase extraction from phase-sensitive otdr,” IEEE Photon. J. 8(3), 1–12 (2016).

IEEE Photon. Technol. Lett. (4)

F. Peng, N. Duan, Y. Rao, and J. Li, “Real-time position and speed monitoring of trains using phase-sensitive OTDR,” IEEE Photon. Technol. Lett. 26 (20), 2055–2057 (2014).
[Crossref]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Z. Qin, L. Chen, and X. Bao, “Wavelet denoising method for improving detection performance of distributed vibration sensor,” IEEE Photon. Technol. Lett. 24 (7), 542–544 (2012).
[Crossref]

M. Ren, P. Lu, L. Chen, and X. Bao, “Theoretical and experimental analysis of phi-otdr based on polarization diversity detection,” IEEE Photon. Technol. Lett. 28(6), 697–700 (2016).
[Crossref]

J. Lightwave Technol. (6)

Opt. Express (9)

S. Liehr, Y. S. Muanenda, S. Munzenberger, and K. Krebber, “Relative change measurement of physical quantities using dual-wavelength coherent otdr,” Opt. Express 25 (2), 720–729 (2017).
[Crossref] [PubMed]

H. F. Martins, K. Shi, B. C. Thomsen, S. Martin-Lopez, M. Gonzalez-Herraez, and S. J. Savory, “Real time dynamic strain monitoring of optical links using the backreflection of live psk data,” Opt. Express 24(19), 22303–22318 (2016).
[Crossref] [PubMed]

S. Wang, X. Fan, Q. Liu, and Z. He, “Distributed fiber-optic vibration sensing based on phase extraction from time-gated digital ofdr,” Opt. Express 23(26), 33301–33309 (2015).
[Crossref]

Q. Liu, X. Fan, and Z. He, “Time-gated digital optical frequency domain reflectometry with 1.6-m spatial resolution over entire 110-km range,” Opt. Express 23(20), 25988–25995 (2015).
[Crossref] [PubMed]

D. Chen, Q. Liu, and Z. He, “Phase-detection distributed fiber-optic vibration sensor without fading-noise based on time-gated digital OFDR,” Opt. Express 25(7), 8315–8325 (2017).
[Crossref] [PubMed]

F. Peng, H. Wu, X. Jia, Y. Rao, Z. Wang, and Z. Peng, “Ultra-long high-sensitivity phi-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22 (11), 13804–13810 (2014).
[Crossref] [PubMed]

A. Masoudi and T. P. Newson, “Analysis of distributed optical fibre acoustic sensors through numerical modelling,” Opt. Express 25(25), 32021–32040 (2017).
[Crossref] [PubMed]

Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, and Y. Rao, “Coherent phi-otdr based on i/q demodulation and homodyne detection,” Opt. Express 24 (2), 853–858 (2016).
[Crossref] [PubMed]

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Other (4)

A. Eyal, H. Gabai, and I. Shpatz, “Distributed acoustic sensing: how to make the best out of the Rayleigh-backscattered energy?” in 25th Optical Fiber Sensors Conference (OFS), Proc. SPIE10323, 103230I (2017).

D. Chen, Q. Liu, and Z. He, “Distributed fiber-optic acoustic sensor with sub-nano strain resolution based on time-gated digital OFDR,” in Asia Communications and Photonics Conference 2017, (Optical Society of America, 2017), paper S4A.2.

D. Miller, T. Parker, S. Kashikar, M. Todorov, and T. Bostick, “Vertical seismic profiling using a fibre-optic cable as a distributed acoustic sensor,” in 74th EAGE Conference and Exhibition incorporating EUROPEC 2012.

Z. Pan, Q. Ye, H. Cai, R. Qu, and Z. Fang, “Phase-sensitive otdr system based on digital coherent detection,” in Asia Communications and Photonics Conference and Exhibition 2011, (Optical Society of America, 2011) paper 83110S.

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Figures (4)

Fig. 1
Fig. 1 The experiment configuration. Red solid lines are polarization-maintaining (PM) fiber and black solid lines are normal single mode fiber. AWG: arbitrary waveform generator; Amp.: ratio-frequency amplifier; BPD: balanced photodetector; C: connector; DC: direct-current source; EDFA: erbium-doped fiber amplifier; IM: intensity modulator; LFM: linear frequency modulated; Osc.: oscilloscope; PZT: cylinder piezoelectric transducer; PDR: polarization diversity receiver; SG: signal generator.
Fig. 2
Fig. 2 (a) Normalized intensity traces with fading (red) and with fading suppressed (blue); (b) Normalized intensity traces zooming at an undisturbed area; (c) Phase SD traces with fading noise (red) and with fading noise suppressed (blue); (d) Phase SD traces zooming at an undisturbed area; (e)(f) Fresnel peak before fading suppressed (red) and after fading suppressed (blue); SD: standard deviation.
Fig. 3
Fig. 3 (a) Phase SD trace zooms on the vibration area; (b) The phase map with time and distance information zooms on the vibration area.
Fig. 4
Fig. 4 (a) Waveform of the first 200-Hz vibration; (b) waveform of the second 4.6-kHz vibration; (c) Their single-side power spectra, in which the spectral resolution is 50 Hz. PSD: power spectral density.

Equations (16)

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s ( t ) = π β 2 + π α 2 cos ( 2 π f 0 t + π κ t 2 ) , t [ 0 , τ p ] ,
E P ( t ) = exp j [ ω c t + s ( t ) ] + exp j [ ω c t s ( t ) ] , t [ 0 , τ p ] ,
E L ( t ) = exp j [ ω c t ] ,
E X ( t ) = i a i cos ( θ i ) E p ( t τ i ) ,
i X ( t ) = i X I ( t ) + j i X Q ( t ) = { E X ( t ) E L * ( t ) + j J { E X ( t ) E L * ( t ) } = i a i cos ( θ i ) { exp j [ ω c τ i + s ( t τ i ) ] + exp j [ ω c τ i s ( t τ i ) ] } ,
i X ( t ) = τ a ( τ ) cos [ θ ( τ ) ] { exp j [ ω c τ + s ( t τ ) ] + exp j [ ω c τ s ( t τ ) ] } d τ = { a ( t ) cos [ θ ( t ) ] exp j [ ω c t ] } { exp j [ s ( t ) ] + exp j [ s ( t ) ] } = h X ( t ) cos [ s ( t ) ] ,
cos [ s ( t ) ] = cos ( π β 2 ) J 0 ( π α 2 ) 2 sin ( π β 2 ) J 1 ( π α 2 ) cos [ φ ( t ) ] 2 cos ( π β 2 ) J 2 ( π α 2 ) cos [ 2 φ ( t ) ] + 2 sin ( π β 2 ) J 3 ( π α 2 ) cos [ 3 φ ( t ) ] + ,
cos [ s ( t ) ] J 1 ( π α 2 ) exp j [ φ ( t ) ] J 1 ( π α 2 ) exp j [ φ ( t ) ] .
r X + ( t ) = i X ( t ) exp j [ φ ( t ) ] = h X ( t ) R + ( t ) ,
r X ( t ) = i X ( t ) exp j [ φ ( t ) ] = h X ( t ) R ( t ) ,
R + ( t ) = cos [ s ( t ) ] exp j [ φ ( t ) ] J 1 ( π α 2 ) exp j [ φ ( t ) ] exp j [ φ ( t ) ] = J 1 ( π α 2 ) ( τ p | t | ) sin [ π κ t ( τ p | t | ) ] π κ t ( τ p | t | ) exp { j 2 π ( f 0 + κ τ p 2 ) t } = W ( t ) exp { j 2 π f 0 t } , t [ τ p , τ p ] ,
R ( t ) = cos [ s ( t ) ] exp j [ φ ( t ) ] J 1 ( π α 2 ) exp j [ φ ( t ) ] exp j [ φ ( t ) ] = [ R + ( t ) ] * .
Δ Z = v g 2 κ τ p ,
x m ( k ) = r m ( k ) + n m ( k ) = | x m ( k ) | exp j [ ϕ m ( k ) + Δ ϕ m ( k ) ] , m = 1 , , 4 , k = 1 , , K ,
x a ( k ) = m = 1 4 x m ( k ) x m * ( 1 ) | x m ( 1 ) | = | x a ( k ) | exp j [ Δ ϕ a ( k ) ] , k = 1 , , K ,
x ( k , t ) = x a ( k , t ) rect ( t τ 0 ) ,

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