Abstract

A novel measurement scheme for multiple high-frequency vibrations has been demonstrated by combining phase-sensitive optical time domain reflectometry (Ф-OTDR) and Mach-Zehnder interferometer (MZI) based on frequency division multiplexing. The light source is directly launched into the MZI structure, while it was modulated by an acoustic optical modulator (AOM) with a frequency shift of 200 MHz for the Ф-OTDR part. The vibration frequency is obtained by demodulating the interference signal obtained by the MZI structure, while the vibration position is located by Ф-OTDR system. The spatial resolution of 10m is obtained over 3 km sensing fiber. And the detectable vibration frequency reaches up to 40 kHz. Compared to the previous schemes, this system works without dead zone in the detectable frequency range. Furthermore, the frequency spectrum mapping method has been adopted to determine multiple high-frequency vibrations simultaneously. The experimental results prove the concept and match well with the theoretical analysis.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution

Tao Zhu, Qian He, Xianghui Xiao, and Xiaoyi Bao
Opt. Express 21(3) 2953-2963 (2013)

A distributed fiber vibration sensor utilizing dispersion induced walk-off effect in a unidirectional Mach-Zehnder interferometer

Qingming Chen, Chao Jin, Yuan Bao, Zhaohui Li, Jianping Li, Chao Lu, Liang Yang, and Guifang Li
Opt. Express 22(3) 2167-2173 (2014)

Enhanced phase-sensitive OTDR system with pulse width modulation Brillouin amplification

Haijun He, Bin Luo, Xihua Zou, Wei Pan, and Lianshan Yan
Opt. Express 26(18) 23714-23727 (2018)

References

  • View by:
  • |
  • |
  • |

  1. M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett. 21(10), 758–760 (1996).
    [Crossref] [PubMed]
  2. Z. Li, L. Yan, L. Shao, W. Pan, and B. Luo, “Coherent BOTDA sensor with intensity modulated local light and IQ demodulation,” Opt. Express 23(12), 16407–16415 (2015).
    [Crossref] [PubMed]
  3. L. Jiang, J. Yang, S. Wang, B. Li, and M. Wang, “Fiber Mach-Zehnder interferometer based on microcavities for high-temperature sensing with high sensitivity,” Opt. Lett. 36(19), 3753–3755 (2011).
    [Crossref] [PubMed]
  4. B. J. Vakoc, M. J. F. Digonnet, and G. S. Kino, “A novel fiber-optic sensor array based on the Sagnac interferometer,” J. Lightwave Technol. 17(11), 2316–2326 (1999).
    [Crossref]
  5. X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt. 50(22), 4333–4338 (2011).
    [Crossref] [PubMed]
  6. 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]
  7. 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]
  8. X. Fang, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14(10), 2250–2254 (1996).
    [Crossref]
  9. R. Bernini, A. Minardo, and L. Zeni, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett. 34(17), 2613–2615 (2009).
    [Crossref] [PubMed]
  10. Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16(14), 10240–10247 (2008).
    [Crossref] [PubMed]
  11. H. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” United States Patent 5,194 847 (March 16, 1993).
  12. Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
    [Crossref]
  13. X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (2012).
    [Crossref] [PubMed]
  14. T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
    [Crossref] [PubMed]
  15. Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
    [Crossref]
  16. Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).
  17. Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
    [Crossref]
  18. F. Marvasti, Nonuniform Sampling: Theory and Practice (Springer Science and Business Media, 2012).

2015 (2)

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Z. Li, L. Yan, L. Shao, W. Pan, and B. Luo, “Coherent BOTDA sensor with intensity modulated local light and IQ demodulation,” Opt. Express 23(12), 16407–16415 (2015).
[Crossref] [PubMed]

2014 (1)

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

2013 (2)

T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
[Crossref] [PubMed]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

2012 (1)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (2012).
[Crossref] [PubMed]

2011 (3)

2009 (1)

2008 (2)

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16(14), 10240–10247 (2008).
[Crossref] [PubMed]

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]

1999 (1)

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, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14(10), 2250–2254 (1996).
[Crossref]

M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett. 21(10), 758–760 (1996).
[Crossref] [PubMed]

Bao, X.

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
[Crossref] [PubMed]

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (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 Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16(14), 10240–10247 (2008).
[Crossref] [PubMed]

Bernini, R.

Cai, H.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Chen, L.

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (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 Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Chtcherbakov, A. A.

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]

Diao, D.

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

Digonnet, M. J. F.

Fang, X.

X. Fang, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14(10), 2250–2254 (1996).
[Crossref]

Fang, Z.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Guo, H.

He, Q.

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
[Crossref] [PubMed]

Hong, X.

Jiang, L.

Kino, G. S.

Li, B.

Li, Z.

Liu, D.

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]

Luo, B.

Minardo, A.

Niklès, M.

Pan, W.

Pan, Z.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Qin, Z.

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

Qu, R.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Robert, P. A.

Shao, L.

Spammer, S. J.

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.

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.

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]

Thévenaz, L.

Vakoc, B. J.

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]

Wang, M.

Wang, S.

Wang, Z.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Wu, J.

Xiao, X.

T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
[Crossref] [PubMed]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

Xu, K.

Yan, L.

Yang, J.

Ye, Q.

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Zeni, L.

Zhang, B.

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

Zhang, Z.

Zhou, J.

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Zhu, T.

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
[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 Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Zuo, C.

Appl. Opt. (1)

IEEE Photonics Technol. Lett. (3)

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

Q. He, T. Zhu, X. Xiao, B. Zhang, D. Diao, and X. Bao, “All fiber distributed vibration sensing using modulated time-difference pulse,” IEEE Photonics Technol. Lett. 25(20), 1955–1957 (2013).
[Crossref]

Q. He, T. Zhu, J. Zhou, D. Diao, and X. Bao, “Frequency response enhancement by periodical nonuniform sampling in distributed sensing,” IEEE Photonics Technol. Lett. 27(20), 2158–2161 (2015).
[Crossref]

J. Lightwave Technol. (3)

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]

X. Fang, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14(10), 2250–2254 (1996).
[Crossref]

B. J. Vakoc, M. J. F. Digonnet, and G. S. Kino, “A novel fiber-optic sensor array based on the Sagnac interferometer,” J. Lightwave Technol. 17(11), 2316–2326 (1999).
[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. (3)

Proc. SPIE (1)

Z. Pan, Z. Wang, Q. Ye, H. Cai, R. Qu, and Z. Fang, “High sampling rate multi-pulse phase-sensitive OTDR employing frequency division multiplexing,” Proc. SPIE 9157, 91576X (2014).

Sensors (Basel) (1)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (2012).
[Crossref] [PubMed]

Other (2)

H. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” United States Patent 5,194 847 (March 16, 1993).

F. Marvasti, Nonuniform Sampling: Theory and Practice (Springer Science and Business Media, 2012).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1 The evolution of sampling theorem. (a) The time domain of the band-limited signal and (b) the frequency spectrum with band width of f0 ; (c) The time domain signal of the sampled function and (d) the frequency spectrum with repetition of fs ; (e) and (f) the time domain signal and the frequency spectrum of the obtained signal, respectively.
Fig. 2
Fig. 2 The evolution of under-sampling. (a) The frequency spectrum of the band-limited signal Xa(f) with a narrow bandwidth; (b) the frequency spectrum with range from kfs-fs/2 to kfs; (c) the frequency spectrum with range from kfs to kfs + fs/2;. (d) the evolution of the frequency spectrum with range from kfs-f1 to kfs + f2 when the signal is sampled by an unit impulse function; (e) the frequency spectrum with range from kfs-f1 to kfs + f2 ; (e) the frequency spectrum with range from kfs + fs/2-f1 to kfs + fs/2 + f2 .
Fig. 3
Fig. 3 Simulated results in over-sampling and under-sampling. (a) The real and (b) the displayed frequency spectrum with a single peak of 320 MHz and 20 MHz, respectively; (c) the real frequency spectrum of band-limited signal with center frequency of 285 MHz; (d) the displayed frequency spectrum in under-sampling, which corresponds to (c); (e) the real frequency spectrum and (f) the aliasing one.
Fig. 4
Fig. 4 Experimental setup of phase-sensitive OTDR merged MZI system. NLL: narrow linewidth laser; AOM: acoustic-optic modulator; AFG: arbitrary function generator; PC: polarization controller; FUT: fiber under test; BPD: balanced photo detector; LNA: low noise amplifier; BPF: bandpass filter; LPF: lowpass filter; OSC: oscilloscope.
Fig. 5
Fig. 5 (a) The Ф-OTDR traces demodulate form the raw data; (b) The detail of the vibration position; (c) Superimposed differential signals of 100 traces at 2.04 km; (d) the spatial resolution of vibration detection with 100 ns modulated pulses.
Fig. 6
Fig. 6 (a) The time domain signal and (b) frequency spectrum of the vibration with the peak of 20 kHz; (c) The time domain signal and (d) frequency spectrum of the vibration with the peak of 30 kHz; (e) The time domain signal and (f) frequency spectrum of the vibration with the peak of 40 kHz.
Fig. 7
Fig. 7 (a) The locating signal of two vibrations; (b) the frequency spectrum of the vibrations with the peak of 23 kHz and 40 kHz.
Fig. 8
Fig. 8 (a) Superimposed signals of 100 consecutive traces with amplitude change at the position of 2.04 km and 2.76 km (the insets show the details of the vibration information); (b) Superimposed differential signals of 100 traces at 2.04 km and 2.76 km;
Fig. 9
Fig. 9 (a) The time domain signal measurement by MZI structure; (b) the frequency spectrum of the vibrations with the peak of 23 kHz and 40 kHz; (c) and (d) the simulation results with sample rate of 1 MS/s and 25 kS/s; (e) and (f) the frequency spectrum of test point A (2 kHz) and B (10 kHz) measured by Ф-OTDR traces, respectively.
Fig. 10
Fig. 10 (a) The time domain signal of double vibrations measurement by MZI structure; (b) the frequency spectrum of the vibrations with the peak of 29 kHz and 39 kHz; (c) and (d) the simulation results with sample rate of 1 MS/s and 25 kS/s; (e) and (f) the frequency spectrums of test point A (4 kHz) and B (11 kHz) measured by Ф-OTDR traces, respectively.
Fig. 11
Fig. 11 (a) The real frequency spectrum of two vibrations measured by MZI structure; (b) and (c) the test results simulated with sample rate of 1 MS/s and 25 kS/s, respectively; (d) and (e) the fake frequency spectrums detected by Φ-OTDR structure at the point A and point B, respectively.

Equations (9)

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

X a ( t ) X a ( f )
P δ ( t ) = n = δ ( t n T ) P δ ( f ) = 1 T k = δ ( f k f s )
S ( t ) = n = X a ( t ) δ ( t n t ) S ( f ) = 1 2 π T k = X a ( j f j k f s ) k z
f d i s p l a y = | f r e a l k f s | ( k z , | f r e a l k f s | < f s / 2 )
A f d i s p l a y = A f r e a l _ d i s p l a y
I ( t ) E R ( t ) 2 + E L O ( t ) 2 + 2 E R ( t ) E L O ( t ) cos θ R L cos ( 2 π Δ f t + φ ( t ) )
I M ( t ) E M ( t ) 2 + E L O ( t ) 2 + 2 E M ( t ) E L O ( t ) cos θ M L cos φ ( t )
I n o i s e ( t ) E R ( t ) 2 + E M ( t ) 2 + 2 E R ( t ) E M ( t ) cos θ R M cos ( 2 π Δ f t + φ n o i s e ( t ) )
S ( t ) 2 E R ( t ) [ E L O ( t ) cos θ R L cos ( 2 π Δ f t + φ ( t ) ) + E M ( t ) cos θ R M cos ( 2 π Δ f t + φ n o i s e ( t ) ) ]

Metrics