Abstract

A high-precision inter-satellite velocity measurement method based on two-one-way laser Doppler is presented in this paper. This method’s working principle and signal-to-noise ratio’s effect under different measurement times of signal on velocity precision are analyzed theoretically. This method is also tested by laboratory experiments and 1 mm/s velocity precision is achieved in 1 ms integrating time. The proposed method potentially contributes to inter-satellite velocity measurement, especially for the relative velocity measurement between two satellites in high dynamic motion and a long distance apart.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Su, “precise orbit determination of global navigation satellite system of second generation (GNSS-2),” Doctor Dissertation.Munich, University FAF Munich(2000).
  2. D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
    [Crossref]
  3. P. D. Angelo, A. Fernández, and T. Guardabrazo, “Enhancement of GNSS navigation function by the use of Inter-Satellite Links,” in Proceeding of IEEE Conference on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (IEEE, 2012),pp. 1–6.
  4. F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).
  5. G. Wubbena, “The GPS adjustment software package GEONAP concepts and models,” in Proceedings of the fifth international symposium on satellite positioning (1989), pp. 452–461.
  6. J. Kim and B. D. Tapley, “Simulation of dual one-way ranging measurement,” J. Spacecr. Rockets 40(3), 419–425 (2003).
    [Crossref]
  7. P. Mistra and P. Enge, Global Positioning system-signals, measurements and performance, (Ganga Jamuma Press, 2006), Chap. 6.
  8. G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).
  9. P. Gao and Z. You, “Characters of laser ranging on the inter-satellite relative position measurement,” in Proceedings of the 3rd international conference on recent advances in space technologies (IEEE, 2007), pp. 614–616.
  10. D. M. R. Wuchenich, “Inter-satellite Laser Interferometry,” phD thesis, Australian National University(2014).
  11. B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
    [Crossref]
  12. W.M.Folkner, P.L.Bender, and R.T.Stebbins. “LISA mission concept study laser interferometer space antenna for the detection and observation of gravitational waves,” Nasa Sti/recon Technical report N (1998).
  13. R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
    [Crossref]
  14. A. Yariv and P. Yeh.Optical electronics in modern communications (2006).
  15. D. Slepian, “Estimation of signal parameters in the presence of noise,” Transactions of IRE Professional Group on Information Theory, 3(3), 68–89(1954).
    [Crossref]

2017 (1)

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

2016 (1)

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

2012 (1)

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

2011 (1)

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

2010 (1)

F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).

2003 (1)

J. Kim and B. D. Tapley, “Simulation of dual one-way ranging measurement,” J. Spacecr. Rockets 40(3), 419–425 (2003).
[Crossref]

Angelo, P. D.

P. D. Angelo, A. Fernández, and T. Guardabrazo, “Enhancement of GNSS navigation function by the use of Inter-Satellite Links,” in Proceeding of IEEE Conference on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (IEEE, 2012),pp. 1–6.

Bachman, B.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Chen, J.

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

Danzmann, K.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Dickson, J.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Fernández, A.

P. D. Angelo, A. Fernández, and T. Guardabrazo, “Enhancement of GNSS navigation function by the use of Inter-Satellite Links,” in Proceeding of IEEE Conference on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (IEEE, 2012),pp. 1–6.

Folkner, W. M.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Gao, P.

P. Gao and Z. You, “Characters of laser ranging on the inter-satellite relative position measurement,” in Proceedings of the 3rd international conference on recent advances in space technologies (IEEE, 2007), pp. 614–616.

Guardabrazo, T.

P. D. Angelo, A. Fernández, and T. Guardabrazo, “Enhancement of GNSS navigation function by the use of Inter-Satellite Links,” in Proceeding of IEEE Conference on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (IEEE, 2012),pp. 1–6.

Heinzel, G.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Kim, J.

J. Kim and B. D. Tapley, “Simulation of dual one-way ranging measurement,” J. Spacecr. Rockets 40(3), 419–425 (2003).
[Crossref]

Klipstein, W.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Klipstein, W. M.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Krainak, M.

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

Li, G.

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

Lu, W.

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

Mckenzie, K.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Meng, F. Z.

F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).

Ou, G.

F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).

Ozawa, T.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Robison, D.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Shaddock, D.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Shaddock, D. A.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Sheard, B. S.

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

Slepian, D.

D. Slepian, “Estimation of signal parameters in the presence of noise,” Transactions of IRE Professional Group on Information Theory, 3(3), 68–89(1954).
[Crossref]

Spero, R.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Sun, X.

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

Sutton, A.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Tang, C. P.

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

Tapley, B. D.

J. Kim and B. D. Tapley, “Simulation of dual one-way ranging measurement,” J. Spacecr. Rockets 40(3), 419–425 (2003).
[Crossref]

Vine, G.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Ware, B.

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

Wu, X. Y.

F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).

Wubbena, G.

G. Wubbena, “The GPS adjustment software package GEONAP concepts and models,” in Proceedings of the fifth international symposium on satellite positioning (1989), pp. 452–461.

Yang, D.

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

Yang, J.

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh.Optical electronics in modern communications (2006).

Yeh, P.

A. Yariv and P. Yeh.Optical electronics in modern communications (2006).

Ying, G.

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

You, Z.

P. Gao and Z. You, “Characters of laser ranging on the inter-satellite relative position measurement,” in Proceedings of the 3rd international conference on recent advances in space technologies (IEEE, 2007), pp. 614–616.

Zhou, Y.

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

Class. Quantum Gravity (1)

R. Spero, B. Bachman, G. Vine, J. Dickson, W. Klipstein, T. Ozawa, K. Mckenzie, D. Shaddock, D. Robison, A. Sutton, and B. Ware, “Progress in interferometry for LISA at JPL,” Class. Quantum Gravity 28(9), 94007 (2011).
[Crossref]

GPS Solut. (1)

D. Yang, J. Yang, G. Li, Y. Zhou, and C. P. Tang, “Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements,” GPS Solut. 21(3), 1395–1404 (2017).
[Crossref]

J. Geod. (1)

B. S. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. M. Klipstein, and W. M. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J. Geod. 86(12), 1083–1095 (2012).
[Crossref]

J. Spacecr. Rockets (1)

J. Kim and B. D. Tapley, “Simulation of dual one-way ranging measurement,” J. Spacecr. Rockets 40(3), 419–425 (2003).
[Crossref]

J. Spacecr. Technol. (1)

F. Z. Meng, X. Y. Wu, and G. Ou, “Autonomous Orbit Determination of Navigation Constellation Based on Inter-Satellite Ranging and Ranging Rate,” J. Spacecr. Technol. 29(4), 89–94 (2010).

Proc. SPIE (1)

G. Ying, W. Lu, X. Sun, J. Chen, and M. Krainak, “Innovative free space optical communication and navigation system with high data rate communication, precision ranging, ranging rate measurements and accurate spacecraft pointing,” Proc. SPIE 9739, 97390K (2016).

Other (9)

P. Gao and Z. You, “Characters of laser ranging on the inter-satellite relative position measurement,” in Proceedings of the 3rd international conference on recent advances in space technologies (IEEE, 2007), pp. 614–616.

D. M. R. Wuchenich, “Inter-satellite Laser Interferometry,” phD thesis, Australian National University(2014).

P. Mistra and P. Enge, Global Positioning system-signals, measurements and performance, (Ganga Jamuma Press, 2006), Chap. 6.

G. Wubbena, “The GPS adjustment software package GEONAP concepts and models,” in Proceedings of the fifth international symposium on satellite positioning (1989), pp. 452–461.

P. D. Angelo, A. Fernández, and T. Guardabrazo, “Enhancement of GNSS navigation function by the use of Inter-Satellite Links,” in Proceeding of IEEE Conference on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (IEEE, 2012),pp. 1–6.

W.M.Folkner, P.L.Bender, and R.T.Stebbins. “LISA mission concept study laser interferometer space antenna for the detection and observation of gravitational waves,” Nasa Sti/recon Technical report N (1998).

A. Yariv and P. Yeh.Optical electronics in modern communications (2006).

D. Slepian, “Estimation of signal parameters in the presence of noise,” Transactions of IRE Professional Group on Information Theory, 3(3), 68–89(1954).
[Crossref]

M. Su, “precise orbit determination of global navigation satellite system of second generation (GNSS-2),” Doctor Dissertation.Munich, University FAF Munich(2000).

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Figures (6)

Fig. 1
Fig. 1 Block diagram for the inter-satellite velocity measurement using two-one-way laser Doppler.
Fig. 2
Fig. 2 The effect of SNRf and measure time T on velocity accuracy.
Fig. 3
Fig. 3 The experimental set-up of the two-one-way laser Doppler velocity measurement system.
Fig. 4
Fig. 4 Oscillograph of the output signals from the two detectors.
Fig. 5
Fig. 5 The frequency spectrum of the two output signals.
Fig. 6
Fig. 6 Velocity measurement error under different measurement time.

Tables (1)

Tables Icon

Table 1 Velocity measurements results.

Equations (51)

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v= ( f d1 f d2 )c f 1 + f 2
E 1 = A 1 ( t )exp[ j( 2π f 1 t+ φ 1 ( t ) ) ]
E 2 = A 2 ( t )exp[ j( 2π f 2 t+ φ 2 ( t ) ) ]
E L1 = A 1 ( t )exp[ j( 2π f 1 t+ φ 1 ( t ) ) ]
E s1 = A s2 ( t )exp[ j( 2π f 2 t k 2 s 1 ( t )+ φ 2 ( t τ 2 1 ) ) ]
i 11 ( t )= R 1 2 A s2 2 ( t )+ R 1 2 A 1 ( t ) A s 2 ( t )cos( 2π( ( f 2 f 1 )t k 2 s 1 ( t ) )+ φ 2 ( t τ 2 1 ) φ 1 ( t ) )+ R 1 2 A 1 2 ( t )+ i SN11
i 12 ( t )= R 1 2 A s2 2 ( t ) R 1 2 A 1 ( t ) A s 2 ( t )cos( 2π( ( f 2 f 1 )t k 2 s 1 ( t ) )+ φ 2 ( t τ 2 1 ) φ 1 ( t ) )+ R 1 2 A 1 2 ( t )+ i SN12
i 1 ( t )= R 1 A 1 ( t ) A s 2 ( t )cos( 2π( ( f 2 f 1 )t k 2 s 1 ( t ) )+ φ 2 ( t τ 2 1 ) φ 1 ( t ) )+ i SN11 + i SN12
i 2 ( t' )= R 1 A 2 ( t' ) A s 1 ( t' )cos( 2π( ( f 1 f 2 )t'+ k 1 s 2 ( t' ) )+ φ 1 ( t' τ 1 2 ) φ 2 ( t' ) )+ i SN21 + i SN22
s 1 ( t )= s 1 ( t 0 )+ s 1 ' ( t 0 )( t t 0 )+ s 1 ( n ) ( t 0 ) ( t t 0 ) n
s T ( t )= s 1 ( t 0 )+v( t t 0 )
f d1 ( t )=( f 2 f 1 + v c f 2 )+ f n_laser21 ( t )+ f n_BD1 ( t )
f d2 ( t+Δ t clk )=( f 2 f 1 v c f 1 )+ f n_laser12 ( t+Δ t clk )+ f n_BD2 ( t+Δ t clk )
v= c f 1 + f 2 [ f d2 ( t+Δtclk ) f d1 ( t )+ f n_laser21 ( t )+ f n_BD1 ( t )+ f n_laser12 ( t+Δ t clk )+ f n_BD2 ( t+Δ t clk ) ]
f 2 f 1 = c 2c+v [ f d2 ( t+Δtclk )+ f d1 ( t )+ f n_laser21 ( t )+ f n_BD1 ( t )+ f n_laser12 ( t+Δ t clk )+ f n_BD2 ( t+Δ t clk ) ]
A 1 ( t )= A 1 +Δ A 1 ( t )
A s2 ( t )= A s2 +Δ A s2 ( t )
P( t )=( A s1 +Δ A s1 ( t ) )( A s2 +Δ A s2 ( t ) )
ΔP ( t ) 2 ¯ = [ P( t ) P( t ) ¯ ] 2 ¯ =2 P 1 Δ A s2 ( t ) 2 ¯ +2 P s2 Δ A 1 ( t ) 2 ¯ + Δ A s2 ( t ) 2 Δ A 1 ( t ) 2 ¯
i NL 2 ¯ = R 1 2 ΔP ( t ) 2 ¯ = R 1 2 ( 2 P 1 Δ A s2 ( t ) 2 ¯ +2 P s2 Δ A 1 ( t ) 2 ¯ + Δ A s2 ( t ) 2 Δ A 1 ( t ) 2 ¯ )
Δ φ 1 ( t,τ )= φ 1 ( t+τ ) φ 1 ( t )
Δ φ 2 ( t τ 2 1 ,τ )= φ 2 ( t τ 2 1 +τ ) φ 2 ( t τ 2 1 )
c 1 ( τ )= i 1 ( t ) i 1 ( t+τ ) ¯ = ( R 1 A 1 A S2 ) 2 4 × { exp{ i[ 2π f d1 τ+Δ φ 2 ( t τ 2 1 ,τ )Δ φ 1 ( t,τ ) ] }+exp{ i[ 2π f d1 τ+Δ φ 2 ( t τ 2 1 ,τ )Δ φ 1 ( t,τ ) ] } } ¯
exp{ ±i[ Δ φ 2 ( t τ 2 1 ,τ )Δ φ 1 ( t,τ ) ] } ¯ =exp{ [ Δ φ 2 ( t τ 2 1 ,τ )Δ φ 1 ( t,τ ) ] 2 ¯ 2 }
[ Δ φ 1 ( t,τ ) ] 2 ¯ = | τ | t coh1
[ Δ φ 2 ( t τ 2 1 ,τ ) ] 2 ¯ = | τ | t coh2
[ Δ φ 1 ( t,τ )Δ φ 2 ( t τ 2 1 ,τ ) ] 2 ¯ =0
c 1 ( τ )= ( R 1 A 1 A S2 ) 2 4 exp[ ( | τ | t coh1 + | τ | t coh2 ) ][ exp( i2π f d1 τ )+exp( i2π f d1 τ ) ]
S c 1 ( f )= ( R 1 A 1 A s2 ) 2 2π ( Δ υ 1 +Δ υ 2 ) [ ( ( Δ υ 1 +Δ υ 2 ) )/2 ] 2 + ( f f d1 ) 2
f d1 = f 2 +Δ f 2 ( t τ 2 1 ) f 1 Δ f 1 ( t )+ v c f 2
f d2 = f 2 +Δ f 2 ( t ) f 1 Δ f 1 ( t τ 1 2 ) v c f 1
σ v = Δ f 2 ( t τ 2 1 )Δ f 2 ( t )+Δ f 1 ( t τ 1 2 )Δ f 1 ( t ) f 1 + f 2 c
i SN 2 ¯ =2e B d1 [ i 11 ( t ) ¯ + i 12 ( t ) ¯ ]=2 R 1 e B d1 P 1 ( 1+ P s2 P 1 )
i SN 2 ¯ =2 R 1 e B d1 P 1
i T1 2 ¯ = 4 k B T em B d1 R L
SN R 1 = i 1 2 ¯ i SN 2
i 1 2 ¯ = 1 2 R 1 2 P 1 P s2
i 1 ( t )= i S ( t )+ i SN ( t )
i S ( t )= R 1 P 1 P s2 cos( 2π f d1 t )
f d1 = f 2 f 1 v c f 2
0<t T 1
I 1 ( f )= I S ( f )+ I SN ( f )
δ f d1 = σ f 2 0 B 1 ( i S ( f ) f ) 2 df
0 B 1 ( i S ( f ) f ) 2 df = 0 T 1 ( 2πt ) 2 [ i S ( t ) ] 2 dt
δ f d1 = 3 ( 2π ) 2 T 1 3 ( SNR 1 f )
SNR 1 f = | I s ( f d1 ) | 2 σ f 2 =( T 1 × B 1 )×SN R 1
δ f d2 = 3 ( 2π ) 2 T 2 3 ( SNR 2 f )
SNR 2 f =( T 2 × B 2 )×SN R 2
σ v = ( δ f d1 +δ f d2 )c f 1 + f 2
i noise 2 =NE P 2 × R 2 × B d
P SNL =12.5mW

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