Abstract

A high-quality hybrid time-division-multiplexing (TDM)/dense wavelength-division-multiplexing (DWDM)-based fiber optic sensor array with extremely low crosstalk is proposed and experimentally demonstrated in this paper. The array is based on a novel wavelength-cross-combination (WCC) method, aiming to combat the annoying problem of crosstalk between adjacent sensors. The core idea of this method is to reassign optical pulses of different wavelengths rather than of the same wavelength to the same photodetector to eliminate the possibility of coherent optical pulse overlapping and interference. Based on the WCC method and a so-called rectangular-pulse binary phase demodulation scheme, a sensor array with 4 TDM time windows and 16 DWDM wavelengths is set up and experimentally tested. Test results show that even utilizing an optical pulse modulator with less than 28 dB extinction ratio, the obtained crosstalk between adjacent sensors can be still suppressed down to less than −60 dB. This method provides an excellent choice for a practical sensor array application where the sensing performance is of the top consideration.

© 2017 Optical Society of America

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References

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2018 (1)

Z. Ren, J. Li, and R. Zhu, “Phase-shifting optical fiber sensing with rectangular-pulse binary phase modulation,” Opt. Lasers Eng. 100, 170–175 (2018).

2017 (1)

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

2016 (1)

2015 (1)

G. Fang, T. Xu, and F. Li, “Heterodyne interrogation system for TDM interferometric fiber optic sensors array,” Opt. Commun. 341(341), 74–78 (2015).

2014 (1)

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

2013 (1)

2012 (1)

2011 (2)

D. Y. Wang, Y. Wang, J. Gong, and A. Wang, “Fully distributed fiber-optic temperature sensing using acoustically-induced rocking grating,” Opt. Lett. 36(17), 3392–3394 (2011).
[PubMed]

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

2009 (3)

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

S. Liang, C. Zhang, W. Lin, L. Li, C. Li, X. Feng, and B. Lin, “Fiber-optic intrinsic distributed acoustic emission sensor for large structure health monitoring,” Opt. Lett. 34(12), 1858–1860 (2009).
[PubMed]

2008 (2)

T. Liu, J. Cui, D. Chen, L. Xiao, and D. Sun, “A new demodulation technique for optical fiber interferometric sensors with [3 × 3] directional couplers,” Chin. Opt. Lett. 6(1), 12–15 (2008).

O. Waagaard, E. Rønnekleiv, S. Forbord, and D. Thingbø, “Reduction of crosstalk in inline sensor arrays using inverse scattering,” Proc. SPIE 7004, 70044Z (2008).

2003 (1)

G. A. Cranch and P. J. Nash, “Large-Scale Remotely Interrogated Arrays of Fiber-Optic Interferometric Sensors for Underwater Acoustic Applications,” IEEE Sens. J. 3(1), 19–30 (2003).

2001 (1)

2000 (1)

K. T. V. Grattan and D. T. Sun, “Fiber optic sensor technology: an overview,” Sens. Actuators A Phys. 82(1), 40–61 (2000).

1999 (1)

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

1996 (1)

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

1995 (1)

1989 (1)

1984 (1)

1982 (2)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation schemes for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 x 3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616–618 (1982).

1980 (1)

1979 (1)

Asundi, A.

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

Austin, E.

Baere, I. D.

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Chao, H.

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

Chen, D.

Chen, M.

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

Cranch, G. A.

G. A. Cranch and P. J. Nash, “Large-Scale Remotely Interrogated Arrays of Fiber-Optic Interferometric Sensors for Underwater Acoustic Applications,” IEEE Sens. J. 3(1), 19–30 (2003).

G. A. Cranch and P. J. Nash, “Large-Scale multiplexing of interferometric fiber-optic sensors using TDM and DWDM,” J. Lightwave Technol. 19(5), 687–699 (2001).

Cui, J.

Cui, K.

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

Dai, Y.

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

Dandridge, A.

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 x 3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616–618 (1982).

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation schemes for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

D. A. Jackson, A. Dandridge, and S. K. Sheem, “Measurement of small phase shifts using a single-mode optical-fiber interferometer,” Opt. Lett. 5(4), 139–141 (1980).
[PubMed]

Degrieck, J.

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Deng, G.

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

Fang, G.

G. Fang, T. Xu, and F. Li, “Heterodyne interrogation system for TDM interferometric fiber optic sensors array,” Opt. Commun. 341(341), 74–78 (2015).

Feng, X.

Forbord, S.

O. Waagaard, E. Rønnekleiv, S. Forbord, and D. Thingbø, “Reduction of crosstalk in inline sensor arrays using inverse scattering,” Proc. SPIE 7004, 70044Z (2008).

Gerges, A. S.

Giallorenzi, T. G.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation schemes for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Gong, J.

Grattan, K. T. V.

K. T. V. Grattan and D. T. Sun, “Fiber optic sensor technology: an overview,” Sens. Actuators A Phys. 82(1), 40–61 (2000).

Hocker, G. B.

Hu, Y.

Hu, Z.

Huang, S.

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

Hung, S.

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

Jackson, D. A.

Jiang, P.

Jones, J. D. C.

Kim, B. Y.

Kingsley, S. A.

Koo, K. P.

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 x 3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616–618 (1982).

Lai, S.

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Lam, Y. L.

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

Leng, J.

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

Li, C.

Li, F.

G. Fang, T. Xu, and F. Li, “Heterodyne interrogation system for TDM interferometric fiber optic sensors array,” Opt. Commun. 341(341), 74–78 (2015).

Li, J.

Z. Ren, J. Li, and R. Zhu, “Phase-shifting optical fiber sensing with rectangular-pulse binary phase modulation,” Opt. Lasers Eng. 100, 170–175 (2018).

Li, L.

Li, P.

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

Li, S.

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

Li, X.

Liang, S.

Liao, Y.

Y. Liao, E. Austin, P. J. Nash, S. A. Kingsley, and D. J. Richardson, “Highly Scalable Amplified Hybrid TDM/DWDM Array Architecture for Interferometric Fiber-Optic Sensor Systems,” J. Lightwave Technol. 31(6), 882–888 (2013).

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Lim, T. K.

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

Lin, B.

Lin, W.

S. Liang, C. Zhang, W. Lin, L. Li, C. Li, X. Feng, and B. Lin, “Fiber-optic intrinsic distributed acoustic emission sensor for large structure health monitoring,” Opt. Lett. 34(12), 1858–1860 (2009).
[PubMed]

S. Huang, W. Lin, M. Chen, S. Hung, and H. Chao, “Crosstalk analysis and system of time-division multiplexing of polarization-insensitive fiber optic Michlson interferometric sensors,” J. Lightwave Technol. 14(6), 1488–1500 (1996).

Lin, Y.

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

Liu, D.

Liu, T.

Liu, Y.

Y. Dai, P. Li, Y. Liu, A. Asundi, and J. Leng, “Integrated real-time monitoring system for strain/temperature distribution based on simultaneous wavelength and time division multiplexing technique,” Opt. Lasers Eng. 59(5), 19–24 (2014).

Y. Dai, Y. Liu, J. Leng, G. Deng, and A. Asundi, “A novel time-division multiplexing fiber Bragg grating sensor interrogator for structural health monitoring,” Opt. Lasers Eng. 47(10), 1028–1033 (2009).

Luyckx, G.

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Ma, L.

Ma, X.

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Nash, P. J.

Newson, T. P.

Pechstedt, R. D.

Ren, Z.

Z. Ren, J. Li, and R. Zhu, “Phase-shifting optical fiber sensing with rectangular-pulse binary phase modulation,” Opt. Lasers Eng. 100, 170–175 (2018).

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

Richardson, D. J.

Rønnekleiv, E.

O. Waagaard, E. Rønnekleiv, S. Forbord, and D. Thingbø, “Reduction of crosstalk in inline sensor arrays using inverse scattering,” Proc. SPIE 7004, 70044Z (2008).

Shaw, H. J.

Sheem, S. K.

Sun, D.

Sun, D. T.

K. T. V. Grattan and D. T. Sun, “Fiber optic sensor technology: an overview,” Sens. Actuators A Phys. 82(1), 40–61 (2000).

Sun, Q.

Thingbø, D.

O. Waagaard, E. Rønnekleiv, S. Forbord, and D. Thingbø, “Reduction of crosstalk in inline sensor arrays using inverse scattering,” Proc. SPIE 7004, 70044Z (2008).

Tveten, A. B.

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 x 3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616–618 (1982).

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation schemes for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Van Paepegem, W.

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Voet, E.

I. D. Baere, G. Luyckx, E. Voet, W. Van Paepegem, and J. Degrieck, “On the feasibility of optical fibre sensors for strain monitoring in thermoplastic composites under fatigue loading conditions,” Opt. Lasers Eng. 47(3–4), 403–411 (2009).

Waagaard, O.

O. Waagaard, E. Rønnekleiv, S. Forbord, and D. Thingbø, “Reduction of crosstalk in inline sensor arrays using inverse scattering,” Proc. SPIE 7004, 70044Z (2008).

Wang, A.

Wang, D. Y.

Wang, L.

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Wang, Y.

Wo, J.

Xiao, L.

Xu, T.

G. Fang, T. Xu, and F. Li, “Heterodyne interrogation system for TDM interferometric fiber optic sensors array,” Opt. Commun. 341(341), 74–78 (2015).

Yip, Y. M.

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

Zhang, C.

Zhang, M.

X. Li, Q. Sun, J. Wo, M. Zhang, and D. Liu, “Hybrid TDM/WDM-Based Fiber-Optic Sensor Network for Perimeter Intrusion Detection,” J. Lightwave Technol. 30(8), 1113–1120 (2012).

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Zhou, H.

M. Zhang, X. Ma, L. Wang, S. Lai, H. Zhou, and Y. Liao, “Photonic Sensors Review Progress of Optical Fiber Sensors and Its Application in Harsh Environment,” Photon. Sens. 1(1), 84–89 (2011).

Zhou, Y.

T. K. Lim, Y. Zhou, Y. Lin, Y. M. Yip, and Y. L. Lam, “Fiber optic acoustic hydrophone with double Mach–Zehnder interferometers for optical path length compensation,” Opt. Commun. 159(4–6), 301–308 (1999).

Zhu, R.

Z. Ren, J. Li, and R. Zhu, “Phase-shifting optical fiber sensing with rectangular-pulse binary phase modulation,” Opt. Lasers Eng. 100, 170–175 (2018).

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 x 3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616–618 (1982).

Chin. Opt. Lett. (1)

IEEE Sens. J. (2)

G. A. Cranch and P. J. Nash, “Large-Scale Remotely Interrogated Arrays of Fiber-Optic Interferometric Sensors for Underwater Acoustic Applications,” IEEE Sens. J. 3(1), 19–30 (2003).

K. Cui, S. Li, Z. Ren, and R. Zhu, “A Highly compact and efficient interrogation controller based on FPGA for fiber-optic sensor array using interferometric TDM,” IEEE Sens. J. 17(11), 3490–3496 (2017).

IEEE Trans. Microw. Theory Tech. (1)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation schemes for fiber optic sensors using phase generated carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

J. Lightwave Technol. (5)

Opt. Commun. (2)

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Figures (15)

Fig. 1
Fig. 1 N-time-windows TDM array.
Fig. 2
Fig. 2 An example system of N-time-windows TDM and M-wavelengths DWDM sensor array.
Fig. 3
Fig. 3 WCC-method-based combination pattern.
Fig. 4
Fig. 4 Implementation schematic of binary rectangular pulse phase modulation method.
Fig. 5
Fig. 5 Time and phase relationships between the modulated phase terms φ p ( t ),   φ p (tτ), and φ p (t) φ p (tτ). (a) φ p (t); (b) φ p (tτ); (c); φ p (t) φ p ( tτ ) (d) Sampling timing for the three phases.
Fig. 6
Fig. 6 Pulse train and sampling timing for the TDM array.
Fig. 7
Fig. 7 The structure of the sensor array prototype.
Fig. 8
Fig. 8 The structure of the signal receiver and recombination module.
Fig. 9
Fig. 9 The FPGA control terminal.
Fig. 10
Fig. 10 Photograph of the system. (a) Front side; (b) Back side.
Fig. 11
Fig. 11 (a) Light intensity versus time; (b) Three phase-shifting steps I 1 , I 2 , I 3 ; (c) Lissajous curves of I s and I c ; (d) Experimental result of the demodulated phase shifts.
Fig. 12
Fig. 12 (a) Vibration signal of the sensor 2; (b) Frequency spectrum of the Sensor 2; (c) Vibration signal of the sensor 1; (d) Frequency spectrum of the Sensor 2.
Fig. 13
Fig. 13 (a) Time-domain curve of the system noise; (b) the power spectrum of the system noise.
Fig. 14
Fig. 14 The sensors deployment of the vibration experiment.
Fig. 15
Fig. 15 Vibration phase signals of the 64 sensors.

Equations (10)

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I m =[ k=1 N ( E kr + E ks ) ]× [ k=1 N ( E kr + E ks ) ] ,
I m = I ms + I mc + I msc = I ms + I mc1 + I mc2 + I mc3 + I mc4 + I msc =2 A 2 [cos( 4πnl λ + φ ms φ mr )+η k=1,km N cos( 4πnL λ + φ kr φ mr ) +η k=1,km N cos( 2πn(kL+2l) λ + φ ks φ mr ) +η k=1,km N cos( 2πn(kL2l) λ + φ kr φ ms ) , +η k=1,km N cos( 2πnl λ + φ ks φ ms ) +2 η 2 k 1 , k 2 =1, k 1 , k 2 m N cos( 4πnl λ + φ k 1 s φ k 2 r ) ]
( S 1j S 2(j+1) S 3(j+2) ... S N(j+N1) )Mu x j (j=1, 2, ..., MN+1), ( S 1j S 2(j+1) ... S (M+1j)M S (M+2j)1 ... S N(j+NM1) )Mu x j (j=MN+2, MN+2, ..., M).
I m =2 A 2 [cos( 4πnl λ + φ ms φ mr )+ η 2 k=1,km N cos( 4πnl λ + φ ks φ kr ) ].
I d = A d + B d cos( φ 0 + φ e + φ d +Δ φ p ),
φ p (t)=π/2 , 0t<τ, φ p (t)=0, τt<T.
I d1 = A d + B d cos(θπ/2), τt<2τ, I d2 = A d + B d cos(θ), 2τt<T, I d3 = A d + B d cos(θ+π/2), 0t<τ.
θ=arctan(( I d1 I d3 )/(2 I d2 ( I d1 + I d3 ))),
I m =2 A 2 [cos( 4πnl λ + φ ms φ mr +Δ φ mp )+ η 2 k=1,km N cos( 4πnl λ + φ ks φ kr +Δ φ kp ) ],
I m1 =2 A 2 [cos( 4πnl λ + φ ms φ mr π 2 )+ η 2 k=1,km N cos( 4πnl λ + φ ks φ kr ) ], I m2 =2 A 2 [cos( 4πnl λ + φ ms φ mr )+ η 2 k=1,km N cos( 4πnl λ + φ ks φ kr ) ], I m3 =2 A 2 [cos( 4πnl λ + φ ms φ mr + π 2 )+ η 2 k=1,km N cos( 4πnl λ + φ ks φ kr ) ].

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