Abstract

A novel distributed fiber-optic localization algorithm with high sensitivity and precision based on merged Michelson-Sagnac interferometer is proposed and demonstrated. By performing simple addition and subtraction processing on the two phase differences of the two interferometers, two superimposed signals with a fixed delay can be obtained. The time delay can be calculated by a cross-correlation algorithm. Combined with the polynomial fitting interpolation, the disturbance location resolution is significantly improved. The total sensing distance can reach 120 km, and the localization errors are within ±35 m and ±160 m in the contact and non-contact disturbance experiment, respectively.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2016 (2)

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

2015 (1)

2014 (3)

2013 (1)

2012 (1)

K. K. K. Annamdas and V. G. M. Annamdas, “Review on developments in fiber optical sensors and applications,” Proc. SPIE 1(1), 1–16 (2012).
[Crossref]

2011 (1)

2008 (1)

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

2007 (1)

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

2003 (1)

F. Viola and W. F. Walker, “A comparison of the performance of time-delay estimators in medical ultrasound,” IEEE Trans. Sonics Ultrason. 50(4), 392–401 (2003).
[Crossref]

2001 (1)

1998 (2)

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach–Zehnder and Modified Sagnac-Distributed Fiber-Optic Impact Sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref]

1997 (1)

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-Michelson Interferometer for Distributed Disturbance Detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

1996 (2)

X. Fang, “Fiber-optic distributed sensing by a two-loop Sagnac interferometer,” Opt. Lett. 21(6), 444–446 (1996).
[Crossref]

E. Udd, “Distributed fiber optic strain sensor based on Sagnac and Michelson interferometers,” Proc. SPIE 2719, 210–212 (1996).
[Crossref]

1963 (1)

A. W. Morley and W. D. Bryce, “Natural Vibration with Damping Force Proportional to a Power of the Velocity,” J. R. Aeronaut. Soc. 67(630), 381–385 (1963).
[Crossref]

Annamdas, K. K. K.

K. K. K. Annamdas and V. G. M. Annamdas, “Review on developments in fiber optical sensors and applications,” Proc. SPIE 1(1), 1–16 (2012).
[Crossref]

Annamdas, V. G. M.

K. K. K. Annamdas and V. G. M. Annamdas, “Review on developments in fiber optical sensors and applications,” Proc. SPIE 1(1), 1–16 (2012).
[Crossref]

Bao, X.

Bao, Y.

Baoquan, J.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Bo, J.

Bottacchi, S.

S. Bottacchi, Noise Principles in Optical Fiber Communication (John Wiley & Sons, Ltd, 2008), Chap. 2.

Brady, K. R. C.

Bryce, W. D.

A. W. Morley and W. D. Bryce, “Natural Vibration with Damping Force Proportional to a Power of the Velocity,” J. R. Aeronaut. Soc. 67(630), 381–385 (1963).
[Crossref]

Chen, Q.

Chtcherbakov, A. A.

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach–Zehnder and Modified Sagnac-Distributed Fiber-Optic Impact Sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-Michelson Interferometer for Distributed Disturbance Detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Ciurapinski, W.

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Dakin, J. P.

Dan, H.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Ding, Z.

Dong, W.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Dong, Z.

H. Wu, F. Yang, H. Xu, and Z. Dong, “A new demodulation method to improve the sensitivity and dynamic range of fiber optic interferometric system,” in Proceedings of Optical Communications and Networks (IET, 2010), 70–72.

Fang, X.

Guo, H.

He, C.

He, Q.

Hong, X.

Hu, H.

Jia, B.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Jian, Z.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Jiang, J.

Jin, C.

Ka, N. P.

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Kondrat, M.

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Lacquet, B. M.

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

Li, G.

Li, J.

Li, X.

Li, Z.

Liu, D.

H. Wang, Q. Sun, X. Li, J. Wo, P. P. Shum, and D. Liu, “Improved location algorithm for multiple intrusions in distributed Sagnac fiber sensing system,” Opt. Express 22(7), 7587–7597 (2014).
[Crossref]

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

Liu, F.

Liu, H.

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

Liu, K.

Liu, T.

Lu, C.

Ma, C.

Mohanan, B.

B. Mohanan, P. J. Shaija, and S. Varghese, “Studies on merged Sagnac-Michelson interferometer for detecting phase sensitive events on fiber optic cables,” in Proceedings of IEEE Conference on Automation, Computing, Communication, Control and Compressed Sensing (iMac4s) (IEEE, 2013), pp. 84–89.

Morley, A. W.

A. W. Morley and W. D. Bryce, “Natural Vibration with Damping Force Proportional to a Power of the Velocity,” J. R. Aeronaut. Soc. 67(630), 381–385 (1963).
[Crossref]

Pang, B.

Peng, H.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Qian, X.

Qing, B.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Russell, S. J.

Shaija, P. J.

B. Mohanan, P. J. Shaija, and S. Varghese, “Studies on merged Sagnac-Michelson interferometer for detecting phase sensitive events on fiber optic cables,” in Proceedings of IEEE Conference on Automation, Computing, Communication, Control and Compressed Sensing (iMac4s) (IEEE, 2013), pp. 84–89.

Shi, X.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Shum, P. P.

Song, Q.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Spammer, S. J.

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach–Zehnder and Modified Sagnac-Distributed Fiber-Optic Impact Sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-Michelson Interferometer for Distributed Disturbance Detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Sun, Q.

H. Wang, Q. Sun, X. Li, J. Wo, P. P. Shum, and D. Liu, “Improved location algorithm for multiple intrusions in distributed Sagnac fiber sensing system,” Opt. Express 22(7), 7587–7597 (2014).
[Crossref]

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

Swart, P. L.

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach–Zehnder and Modified Sagnac-Distributed Fiber-Optic Impact Sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref]

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-Michelson Interferometer for Distributed Disturbance Detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Szustakowski, M.

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Tian, M.

Udd, E.

E. Udd, “Distributed fiber optic strain sensor based on Sagnac and Michelson interferometers,” Proc. SPIE 2719, 210–212 (1996).
[Crossref]

Varghese, S.

B. Mohanan, P. J. Shaija, and S. Varghese, “Studies on merged Sagnac-Michelson interferometer for detecting phase sensitive events on fiber optic cables,” in Proceedings of IEEE Conference on Automation, Computing, Communication, Control and Compressed Sensing (iMac4s) (IEEE, 2013), pp. 84–89.

Viola, F.

F. Viola and W. F. Walker, “A comparison of the performance of time-delay estimators in medical ultrasound,” IEEE Trans. Sonics Ultrason. 50(4), 392–401 (2003).
[Crossref]

Walker, W. F.

F. Viola and W. F. Walker, “A comparison of the performance of time-delay estimators in medical ultrasound,” IEEE Trans. Sonics Ultrason. 50(4), 392–401 (2003).
[Crossref]

Wang, H.

Wang, J.

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

Wo, J.

Wu, H.

H. Wu, F. Yang, H. Xu, and Z. Dong, “A new demodulation method to improve the sensitivity and dynamic range of fiber optic interferometric system,” in Proceedings of Optical Communications and Networks (IET, 2010), 70–72.

Wu, J.

Xianyang, Q.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Xiao, Q.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Xiao, X.

Xin, C.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Xin, L.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Xu, H.

H. Wu, F. Yang, H. Xu, and Z. Dong, “A new demodulation method to improve the sensitivity and dynamic range of fiber optic interferometric system,” in Proceedings of Optical Communications and Networks (IET, 2010), 70–72.

Xu, K.

Yang, F.

H. Wu, F. Yang, H. Xu, and Z. Dong, “A new demodulation method to improve the sensitivity and dynamic range of fiber optic interferometric system,” in Proceedings of Optical Communications and Networks (IET, 2010), 70–72.

Yang, L.

Yczkowski, M.

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Yonggang, G.

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Yu, W.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Yuan, W.

Yuncai, W.

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Zhang, X.

Zhou, P.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Zhou, S.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

Zhu, T.

Zuo, C.

Appl. Opt. (2)

IEEE Trans. Sonics Ultrason. (1)

F. Viola and W. F. Walker, “A comparison of the performance of time-delay estimators in medical ultrasound,” IEEE Trans. Sonics Ultrason. 50(4), 392–401 (2003).
[Crossref]

J. Lightwave Technol. (4)

J. R. Aeronaut. Soc. (1)

A. W. Morley and W. D. Bryce, “Natural Vibration with Damping Force Proportional to a Power of the Velocity,” J. R. Aeronaut. Soc. 67(630), 381–385 (1963).
[Crossref]

Noise Control (1)

X. Shi, Q. Xianyang, Z. Jian, H. Dan, C. Xin, and G. Yonggang, “A Comparative Study of Ground and Underground Vibrations Induced by Bench Blasting,” Noise Control 2016(1), 1–9 (2016).
[Crossref]

Opt. Commun. (1)

Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Opto-Electron. Rev. (1)

M. Kondrat, M. Szustakowski, N. P. Ka, W. Ciurapiński, and M. Yczkowski, “A Sagnac-Michelson fibre optic interferometer: Signal processing for disturbance localization,” Opto-Electron. Rev. 15(3), 127–132 (2007).
[Crossref]

Proc. SPIE (3)

E. Udd, “Distributed fiber optic strain sensor based on Sagnac and Michelson interferometers,” Proc. SPIE 2719, 210–212 (1996).
[Crossref]

A. A. Chtcherbakov, P. L. Swart, S. J. Spammer, and B. M. Lacquet, “Modified Sagnac/Mach-Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64 (1998).
[Crossref]

K. K. K. Annamdas and V. G. M. Annamdas, “Review on developments in fiber optical sensors and applications,” Proc. SPIE 1(1), 1–16 (2012).
[Crossref]

Sensors (1)

L. Xin, J. Baoquan, B. Qing, W. Yu, W. Dong, and W. Yuncai, “Distributed Fiber-Optic Sensors for Vibration Detection,” Sensors 16(8), 1164–1195 (2016).
[Crossref]

Other (4)

B. Mohanan, P. J. Shaija, and S. Varghese, “Studies on merged Sagnac-Michelson interferometer for detecting phase sensitive events on fiber optic cables,” in Proceedings of IEEE Conference on Automation, Computing, Communication, Control and Compressed Sensing (iMac4s) (IEEE, 2013), pp. 84–89.

H. Wu, F. Yang, H. Xu, and Z. Dong, “A new demodulation method to improve the sensitivity and dynamic range of fiber optic interferometric system,” in Proceedings of Optical Communications and Networks (IET, 2010), 70–72.

Q. Song, H. Peng, S. Zhou, P. Zhou, Q. Xiao, and B. Jia, “A Novel Weak-Scattering Michelson Interferometer Based on PBS for Long-Distance Disturbance Localization,” J. Lightwave Technol. (to be published) (10.1109/JLT.2019.2953134)

S. Bottacchi, Noise Principles in Optical Fiber Communication (John Wiley & Sons, Ltd, 2008), Chap. 2.

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Figures (14)

Fig. 1.
Fig. 1. A merged MI-SI sensor for disturbance localization with 120 km loop.
Fig. 2.
Fig. 2. Block diagram of the signal processing algorithm for localization.
Fig. 3.
Fig. 3. Simulation of (a) 10 Hz and (b) 200 Hz vibration signals with noise of the same standard. (c) and (d) are local amplification of (a) and (b), respectively.
Fig. 4.
Fig. 4. The influence of noise and frequency of vibration on localization precision.
Fig. 5.
Fig. 5. Cross correlation peak value under different time delay. The pentagrams in the figure show the peak value obtained by fitting curves.
Fig. 6.
Fig. 6. Comparison of time-delay estimate errors between fitting and no-fitting.
Fig. 7.
Fig. 7. Original interference signals of (a) MI and (b) SI.
Fig. 8.
Fig. 8. Signals processing of disturbance localization. (a) Demodulation of phase difference. (b) High pass filtering results of signals of (a). (c) Two signals obtained with time delay. (d) Cross-correlation of the two signals in (c). The pentagram in the figure show the peak value obtained by fitting curve.
Fig. 9.
Fig. 9. The signals of ϕ1(t) and ϕ2(t) when the disturbance act on (a) SOF1 and (b) SOF2.
Fig. 10.
Fig. 10. Localization errors after a series of disturbance.
Fig. 11.
Fig. 11. Experimental frequency response of system using PZT modulator.
Fig. 12.
Fig. 12. The non-contact disturbance act on (a) SOF1 and (b) SOF2
Fig. 13.
Fig. 13. Power spectrum of contact disturbance and non-contact disturbance.
Fig. 14.
Fig. 14. The influence of low cut-off frequency of HPF on location precision and SNR.

Equations (12)

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I P D 1 = 2 E 01 2 [ 1 + cos ( Δ φ MI ( t ) + φ 01 ]
I P D 3 = 2 E 03 2 [ 1 + cos ( Δ φ MI ( t ) + φ 03 ]
I P D 2 = 2 E 02 2 [ 1 + cos ( Δ φ SI ( t ) + φ 02 ]
I P D 4 = 2 E 04 2 [ 1 + cos ( Δ φ SI ( t ) + φ 04 ]
Δ φ MI ( t ) = [ φ ( t 2 τ x ) + φ ( t ) ] 0 = φ ( t 2 n L x / c ) + φ ( t )
Δ φ SI ( t ) = φ ( t 2 τ x τ 0 ) φ ( t ) = φ ( t 2 n L x / c n L 0 / c ) φ ( t )
ϕ 1 ( t ) = Δ φ MI ( t ) + Δ φ SI ( t ) = φ ( t 2 τ x ) + φ ( t 2 τ x τ 0 )
ϕ 2 ( t ) = Δ φ MI ( t τ 0 ) Δ φ SI ( t ) = φ ( t ) + φ ( t τ 0 )
ϕ 1 ( t ) = φ ( t 2 τ x ) φ ( t 2 τ x τ 0 )
ϕ 2 ( t ) = φ ( t ) φ ( t τ 0 )
L x = c τ x / 2 n
H ( j ) = j = ( N 1 ) N 1 k = 0 N 1 ϕ 1 ( k ) ϕ 2 ( j + k )

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