Abstract

Monitoring the optical phase change in a fiber enables a wide range of applications where fast phase variations are induced by acoustic signals or by vibrations in general. However, the quality of the estimated fiber response strongly depends on the method used to modulate the light sent to the fiber and capture the variations of the optical field. In this paper, we show that distributed optical fiber sensing systems can advantageously exploit techniques from the telecommunication domain, as those used in coherent optical transmission, to enhance their performance in detecting mechanical events, while jointly offering a simpler setup than widespread pulse-cloning or spectral-sweep based schemes with acousto-optic modulators. We periodically capture an overall fiber Jones matrix estimate thanks to a novel probing technique using two mutually orthogonal complementary (Golay) pairs of binary sequences applied simultaneously in phase and quadrature on two orthogonal polarization states. A perfect channel response estimation of the sensor array is achieved, subject to conditions detailed in the paper, thus enhancing the sensitivity and bandwidth of coherent ϕ-OTDR systems. High sensitivity, linear response, and bandwidth coverage up to 18 kHz are demonstrated with a sensor array composed of 10 fiber Bragg gratings (FBGs).

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

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References

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  1. A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
    [Crossref] [PubMed]
  2. L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on Rayleigh scattering,” The Open Optics Journal 7(1), 104–127 (2013).
    [Crossref]
  3. Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
    [Crossref] [PubMed]
  4. 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 Photonics Journal 8(3), 1–12 (2016).
  5. X. Fan, G. Yang, S. Wang, Q. Liu, and Z. He, “Distributed Fiber-Optic Vibration Sensing Based on Phase Extraction From Optical Reflectometry,” J. Lightwave Technol. 35(16), 3281–3288 (2017).
    [Crossref]
  6. D. Chen, Q. Liu, X. Fan, and Z. He, “Distributed fiber-optic acoustic sensor with enhanced response bandwidth and high signal-to-noise ratio,” J. Lightwave Technol. 35(10), 2037–2043 (2017).
    [Crossref]
  7. H. F. Martins, K. Shi, B. C. Thomsen, S. M. Lopez, M. G. 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]
  8. Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).
  9. K. Kikuchi, “Fundamentals of coherent optical fiber communications,” J. Lightwave Technol. 34(1), 157–179 (2016).
    [Crossref]
  10. F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
    [Crossref]
  11. F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).
  12. M. Golay, “Complementary series,” IRE Transactions on Information Theory 7(2), 82–87 (1961).
    [Crossref]
  13. M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
    [Crossref]
  14. X. Huang, “Complementary Properties of Hadamard Matrices,” in International Conference on Communications, Circuits and Systems, 588–592 (2006).
    [Crossref]
  15. R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
    [Crossref]

2017 (2)

2016 (4)

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

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 Photonics Journal 8(3), 1–12 (2016).

K. Kikuchi, “Fundamentals of coherent optical fiber communications,” J. Lightwave Technol. 34(1), 157–179 (2016).
[Crossref]

H. F. Martins, K. Shi, B. C. Thomsen, S. M. Lopez, M. G. 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]

2015 (2)

Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
[Crossref] [PubMed]

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

2013 (1)

L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on Rayleigh scattering,” The Open Optics Journal 7(1), 104–127 (2013).
[Crossref]

2000 (1)

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
[Crossref]

1989 (1)

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

1961 (1)

M. Golay, “Complementary series,” IRE Transactions on Information Theory 7(2), 82–87 (1961).
[Crossref]

Chen, D.

Fan, X.

Feng, H.

Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
[Crossref] [PubMed]

Foster, S.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Giffard, R.P.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Golay, M.

M. Golay, “Complementary series,” IRE Transactions on Information Theory 7(2), 82–87 (1961).
[Crossref]

He, Z.

Herraez, M. G.

Huang, X.

X. Huang, “Complementary Properties of Hadamard Matrices,” in International Conference on Communications, Circuits and Systems, 588–592 (2006).
[Crossref]

Johnson, G. A.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
[Crossref]

Kikuchi, K.

Li, X.

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

Liu, D.

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

Liu, Q.

Liu, T.

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

Lopez, S. M.

Martins, H. F.

Masoudi, A.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

Moberly, D.S.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Nazarathy, M.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Newson, T. P.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

Newton, S.A.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Palmieri, L.

L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on Rayleigh scattering,” The Open Optics Journal 7(1), 104–127 (2013).
[Crossref]

Posey, R.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
[Crossref]

Savory, S. J.

Schenato, L.

L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on Rayleigh scattering,” The Open Optics Journal 7(1), 104–127 (2013).
[Crossref]

Shi, K.

Shi, Y.

Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
[Crossref] [PubMed]

Sischka, F.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Sun, F.A.Q.

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

Thomsen, B. C.

Tian, M.

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

Trutna, W.R.

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

Vohra, S. T.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
[Crossref]

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 Photonics Journal 8(3), 1–12 (2016).

Wang, S.

X. Fan, G. Yang, S. Wang, Q. Liu, and Z. He, “Distributed Fiber-Optic Vibration Sensing Based on Phase Extraction From Optical Reflectometry,” J. Lightwave Technol. 35(16), 3281–3288 (2017).
[Crossref]

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 Photonics Journal 8(3), 1–12 (2016).

Wu, X.

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

Xia, L.

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

Xu, Y.

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

Yan, Q.

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

Yan, Z.

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

Yang, G.

X. Fan, G. Yang, S. Wang, Q. Liu, and Z. He, “Distributed Fiber-Optic Vibration Sensing Based on Phase Extraction From Optical Reflectometry,” J. Lightwave Technol. 35(16), 3281–3288 (2017).
[Crossref]

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 Photonics Journal 8(3), 1–12 (2016).

Yang, Q.

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

Zeng, Z.

Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
[Crossref] [PubMed]

Zhang, W.

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

Zhang, X.

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

Zhang, Y.

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

Zhu, F.

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

Electronics Letters (1)

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electronics Letters 36(20), 1688–1689 (2000).
[Crossref]

IEEE Photonics Journal (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 Photonics Journal 8(3), 1–12 (2016).

IRE Transactions on Information Theory (1)

M. Golay, “Complementary series,” IRE Transactions on Information Theory 7(2), 82–87 (1961).
[Crossref]

J. Lightw. Technol. (1)

M. Nazarathy, S.A. Newton, R.P. Giffard, D.S. Moberly, F. Sischka, W.R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightw. Technol. 7(1), 24–38 (1989).
[Crossref]

J. Lightwave Technol (1)

F. Zhu, Y. Zhang, L. Xia, X. Wu, and X. Zhang, “Improved ϕ-OTDR sensing system for high-precision dynamic strain measurement based on ultra-weak fiber Bragg grating array,” J. Lightwave Technol.  33(23), 4775–4780 (2015).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (1)

Rev. Sci. Instrum. (1)

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

Sensors (1)

Y. Shi, H. Feng, and Z. Zeng, “A long distance phase-sensitive optical time domain reflectometer with simple structure and high locating accuracy,” Sensors 15(9), 21957–21970 (2015).
[Crossref] [PubMed]

The Open Optics Journal (1)

L. Palmieri and L. Schenato, “Distributed optical fiber sensing based on Rayleigh scattering,” The Open Optics Journal 7(1), 104–127 (2013).
[Crossref]

Other (3)

Q. Yan, M. Tian, X. Li, Q. Yang, and Y. Xu, “Coherent ϕ-OTDR based on polarization-diversity integrated coherent receiver and heterodyne detection,” in IEEE 25th Optical Fiber Sensors Conference (OFS), 1–4 (2017).

F.A.Q. Sun, W. Zhang, T. Liu, Z. Yan, and D. Liu, “Wideband fully-distributed vibration sensing by using UWFBG based coherent OTDR,” in IEEE/OSA Optical Fiber Communications Conference and Exhibition (OFC), 1–3 (2017).

X. Huang, “Complementary Properties of Hadamard Matrices,” in International Conference on Communications, Circuits and Systems, 588–592 (2006).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 (a) PDM-BPSK sequences. (b) Auto- and cross- correlations with PDM-BPSK.
Fig. 2
Fig. 2 (a) PDM-QPSK sequences. (b) Auto- and cross- correlations with PDM-QPSK.
Fig. 3
Fig. 3 Experimental Setup (PEA: piezoelectric actuator).
Fig. 4
Fig. 4 (a) Measured intensities at the receiver side. (b) Estimated phases in static mode.
Fig. 5
Fig. 5 (a) Standard deviation of estimated phases as a function of code length. (b) Standard deviation of estimated phases as a function of signal power at the receiver input.
Fig. 6
Fig. 6 Standard deviation of estimated phases as a function of reach.
Fig. 7
Fig. 7 (a) Distributed sensing capability showing low crosstalk between sensors. (b) Crosstalk level at other sensors when only a single one is excited.
Fig. 8
Fig. 8 (a) Dynamic range: peak-to-peak phase magnitude versus peak-to-peak voltage for a 1 kHz sine wave. (b) Sensitivity in rad / Hz for sine waves between [100: 18000] Hz.
Fig. 9
Fig. 9 Power spectral response of the system measured over the audio bandwidth.

Equations (16)

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

E t ( n ) = [ A t x ( n ) exp ( i ϕ t x ( n ) ) A t y ( n ) exp ( i ϕ t y ( n ) ) ] exp ( i 2 π ν 0 n T S + ϕ 0 ( n ) ) , n = [ 1 N ]
H = [ h x x h x y h y x h y y ]
G a 1 ( n ) G a 1 ( n ) + G b 1 ( n ) G b 1 ( n ) = δ ( n )
E t x , t y ( n + k N ) = { G x I , y I ( n ) + i G x Q , y Q ( n ) if n mod N N G 0 elsewhere
E r x ( n ) = h x x ( n ) E t x ( n ) + h x y ( n ) E t y ( n ) E r y ( n ) = h y x ( n ) E t x ( n ) + h y y ( n ) E t y ( n )
h x x ( n ) = E r x ( n ) ( G x I ( n ) + i G x Q ( n ) ) = ( h x x ( G x I + i G x Q ) + h x y ( G y I + i G y Q ) ) ( G x I + i G x Q ) = h x x ( G x I + i G x Q ) ( G x I + i G x Q ) + h x y ( G y I + i G y Q ) ( G x I + i G x Q ) = h x x ( G x I G x I + G x Q G x Q + i ( G x Q G x I G x I G x Q ) ) + h x y ( G y I G x I + G y Q G x Q + i ( G y Q G x I G y I G x Q ) ) = h x x ( n ) ( g 0 x ( n ) + i g 1 x ( n ) ) + h x y ( n ) ( g 2 x ( n ) + i g 3 x ( n ) )
g 0 x ( n ) = G x I ( n ) G x I ( n ) + G x Q ( n ) G x Q ( n ) g 1 x ( n ) = G x Q ( n ) G x I ( n ) G x I ( n ) G x Q ( n ) g 2 x ( n ) = G y I ( n ) G x I ( n ) + G y Q ( n ) G x Q ( n ) g 3 x ( n ) = G y Q ( n ) G x I ( n ) G y I ( n ) G x Q ( n )
g 0 x ( n ) = δ ( n ) , g 1 x ( n ) = g 2 x ( n ) = g 3 x ( n ) = 0
h x y ( n ) = E r x ( n ) ( G y I ( n ) + i G y Q ( n ) ) = h x x ( G x I G y I + G x Q G y Q + i ( G x Q G y I G x I G y Q ) ) + h x y ( G y I G y I + G y Q G y Q + i ( G y Q G y I G y I G y Q ) ) = h x x ( n ) ( g 2 y ( n ) + i g 3 y ( n ) ) + h x y ( n ) ( g 0 y ( n ) + i g 1 y ( n ) )
g 0 y ( n ) = δ ( n ) , g 1 y ( n ) = g 2 y ( n ) = g 3 y ( n ) = 0
G a 1 ( n ) G a 2 ( n ) + G b 1 ( n ) G b 2 ( n ) = 0 G a 1 ( n ) G b 1 ( n ) + G a 2 ( n ) G b 2 ( n ) = 0
G a 1 N G = [ G a 1 N G / 2 , G b 1 N G / 2 ] G b 1 N G = [ G a 1 N G / 2 , G b 1 N G / 2 ] G a 2 N G = [ G a 2 N G / 2 , G b 2 N G / 2 ] G b 2 N G = [ G a 2 N G / 2 , G b 2 N G / 2 ]
E t x ( n + k N ) = { G a 1 ( n ) 0 n < N G 0 N G n < N G + N s e p G b 1 ( n N G N s e p ) N G + N s e p n < 2 N G + N s e p 0 2 N G + N s e p n < N
h x x ( n ) = h x x ( n ) + i ( h x x ( n ) g 1 ( n ) + h x y ( n ) g 3 ( n ) ) h x y ( n ) = h x y ( n ) i ( h x y ( n ) g 1 ( n ) + h x x ( n ) g 3 ( n ) )
ϕ = 0.5 ( h x x h y y h x y h y x )
I I , X / Y P S , X / Y P L O , X / Y cos ( ϕ X , Y + ϕ L O ) + η I , X / Y I Q , X / Y P S , X / Y P L O , X / Y sin ( ϕ X , Y + ϕ L O ) + η Q , X / Y

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