Abstract

In order to ensure the communication link stability in mobile FSO system, a new omni-directional optical antenna is designed. Being aimed at discontinuous tracking, a novel beam control method based on the error correction Kalman prediction algorithm (EC-KPA) is proposed. The comparison of EC-KPA and the conventional Kalman prediction algorithm (KPA) is given. Numerical simulations about beam control method are carried out. The results show that the prediction accuracy of EC-KPA is improved about 77% than that of KPA in Gaussian noise situation, and that the increase is up to 12.92 times in strong noise situation. Therefore, the beam control method is feasible, and this optical antenna can meet the demands of fast mobile FSO.

© 2013 OSA

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References

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  1. T. Plank, E. Leitgeb, and M. Loeschnigg, “Recent developments on free space optical links and wavelength analysis,” In Proceedings of Space Optical Systems and Applications (Santa Monica, CA, 2011), pp. 14–20.
  2. L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
    [CrossRef]
  3. W. Hameed, S. S. Muhammad, and N. M. Sheikh, “Integration scenarios for Free Space Optics in next generation (4G) wireless networks,” In Processing of IEEE 7th International Symposium on Communication Systems Networks and Digital Signal (Newcastle upon Tyne, 2010), pp. 571 – 575.
  4. Ch. Chlestil, E. Leitgeb, and N. P. Schmitt, S. Sheikh Muhammad, K. Zettl, and W. Rehm, “Reliable optical wireless links within UAV swarms,” In Processing of IEEE Conference on Transparent Optical Networks International (Nottingham, 2006), 4, pp. 39 – 42.
  5. F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
    [CrossRef]
  6. K. Davaslioğlu, E. Cağiral, and M. Koca, “Free space optical ultra-wideband communications over atmospheric turbulence channels,” Opt. Express18(16), 16618–16627 (2010).
    [CrossRef] [PubMed]
  7. G. A. Cap, H. H. Refei, and J. J. Sluss, “Optical tracking and auto-alignment transceiver system,” IEEE Aerosp. Electron. Syst. Mag.25(9), 26–34 (2010).
    [CrossRef]
  8. J. Cunningham, D. Foulke, D. W. Yong, J. C. Juarez, J. E. Sluz, and J. L. Riggins, “Long Range Field Testing of Free Space Optical Communications Terminal on Mobile platforms,” In Processing of IEEE Conference on Military Communication (Boston, MA, 2009), pp. 1–7.
  9. M. Zhang and Y. Liang, “Compound tracking in ATP system for free space optical communication,” In Processing of IEEE International Conference on Mechatronic Science, Electric Engineering and Computer (Jilin, China, 2011). pp. 454 – 456.
  10. K. Yoshida and T. Tsujimura, “Tracking Control of the Mobile Terminal in an Active Free-Space Optical Communication system,” In Processing of SICE-ICASE International Joint Conference (Busan, Korea, 2006), pp. 369–374.
  11. D. Zhou and H. H. Refai, “Control algorithm development for mobile FSO Node Alignment,” In Processing of 28th IEEE/AIAA Digital Avionics system Conference (Orlando, FL, 2009), pp. 6.A.3–1–6.A.3–12.
  12. B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
    [CrossRef]
  13. A. Sevincer, M. Bilgi, M. Yuksel, and N. Pala, “Prototyping Multi-Transceiver Free-Space Optical Communication Structures,” In Processing of IEEE International Conference on Communications (Cape Town, South Africa, 2010). pp. 1–5.
  14. K. H. Heng, W. D. Zhong, and T. H. Cheng, “Multipoint free-space optics system for short-range communications between flight platforms,” Appl. Opt.49(2), 258–266 (2010).
    [CrossRef] [PubMed]
  15. Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).
  16. T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).
  17. H. Heffes, “The effect of erroneous models on the Kalman filter response,” IEEE Trans. Automat. Contr.11(3), 541–543 (1966).
    [CrossRef]
  18. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Engrg, Trans. ASME.35–45 (1960).
  19. N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
    [CrossRef]
  20. E. Dedrick and D. Lau, “A Kalman-filtering approach to high dynamic range imaging for measurement applications,” IEEE Trans. Image Process.21(2), 527–536 (2012).
    [CrossRef] [PubMed]
  21. M. Y. Fu, Z. H. Deng, and J. W. Zhang, “The principle of Kalman filtering theory” in Kalman filtering theory and its application in navigation system. (Science Press, China, 2003), Chap. 2.
  22. W. Xie and J. Tang, “Analysis on Characterization of Atmospheric Optical Wireless Communication System Based on Turbo Code,” J. Lasers in Chinese30(9), 835–838 (2003).

2012

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

E. Dedrick and D. Lau, “A Kalman-filtering approach to high dynamic range imaging for measurement applications,” IEEE Trans. Image Process.21(2), 527–536 (2012).
[CrossRef] [PubMed]

2010

2009

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

2008

L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
[CrossRef]

Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).

2003

W. Xie and J. Tang, “Analysis on Characterization of Atmospheric Optical Wireless Communication System Based on Turbo Code,” J. Lasers in Chinese30(9), 835–838 (2003).

1966

H. Heffes, “The effect of erroneous models on the Kalman filter response,” IEEE Trans. Automat. Contr.11(3), 541–543 (1966).
[CrossRef]

1960

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Engrg, Trans. ASME.35–45 (1960).

Awan, M. S.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Bilgi, M.

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

Cagiral, E.

Cap, G. A.

G. A. Cap, H. H. Refei, and J. J. Sluss, “Optical tracking and auto-alignment transceiver system,” IEEE Aerosp. Electron. Syst. Mag.25(9), 26–34 (2010).
[CrossRef]

Cheng, T. H.

Davaslioglu, K.

Dedrick, E.

E. Dedrick and D. Lau, “A Kalman-filtering approach to high dynamic range imaging for measurement applications,” IEEE Trans. Image Process.21(2), 527–536 (2012).
[CrossRef] [PubMed]

Heffes, H.

H. Heffes, “The effect of erroneous models on the Kalman filter response,” IEEE Trans. Automat. Contr.11(3), 541–543 (1966).
[CrossRef]

Hella, M.

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

Heng, K. H.

Jia, J. J.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Engrg, Trans. ASME.35–45 (1960).

Kandus, G.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Koca, M.

Kvicera, V.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Lau, D.

E. Dedrick and D. Lau, “A Kalman-filtering approach to high dynamic range imaging for measurement applications,” IEEE Trans. Image Process.21(2), 527–536 (2012).
[CrossRef] [PubMed]

Lee, G.

L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
[CrossRef]

Leitgeb, E.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Li, W. X.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Li, Y.

Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).

Liu, N.

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

Muhammad, S.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Nadeem, F.

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

Nakhkoob, B.

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

Refei, H. H.

G. A. Cap, H. H. Refei, and J. J. Sluss, “Optical tracking and auto-alignment transceiver system,” IEEE Aerosp. Electron. Syst. Mag.25(9), 26–34 (2010).
[CrossRef]

Shang, T.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Shen, D.

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

Sluss, J. J.

G. A. Cap, H. H. Refei, and J. J. Sluss, “Optical tracking and auto-alignment transceiver system,” IEEE Aerosp. Electron. Syst. Mag.25(9), 26–34 (2010).
[CrossRef]

Stadler, B.

L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
[CrossRef]

Stotts, L. B.

L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
[CrossRef]

Tang, J.

W. Xie and J. Tang, “Analysis on Characterization of Atmospheric Optical Wireless Communication System Based on Turbo Code,” J. Lasers in Chinese30(9), 835–838 (2003).

Wang, X.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Wu, X.

Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).

Xie, W.

W. Xie and J. Tang, “Analysis on Characterization of Atmospheric Optical Wireless Communication System Based on Turbo Code,” J. Lasers in Chinese30(9), 835–838 (2003).

Yang, Y.T.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Yuksel, M.

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

Zhang, J.

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

Zhang, L. R.

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

Zhong, W. D.

Zhu, Y.

Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).

Appl. Opt.

Electron. Lett.

N. Liu, L. R. Zhang, J. Zhang, and D. Shen, “Direction finding of MIMO radar through ESPRIT and Kalman filter,” Electron. Lett.45(17), 908–910 (2009).
[CrossRef]

IEEE Aerosp. Electron. Syst. Mag.

G. A. Cap, H. H. Refei, and J. J. Sluss, “Optical tracking and auto-alignment transceiver system,” IEEE Aerosp. Electron. Syst. Mag.25(9), 26–34 (2010).
[CrossRef]

IEEE J. Sel. Areas Comm.

B. Nakhkoob, M. Bilgi, M. Yuksel, and M. Hella, “Multi-transceiver optical wireless spherical structures for MANETs,” IEEE J. Sel. Areas Comm.27(9), 1612–1622 (2009).
[CrossRef]

F. Nadeem, V. Kvicera, M. S. Awan, E. Leitgeb, S. Muhammad, and G. Kandus, “Weather effects on hybrid FSO/RF communication link,” IEEE J. Sel. Areas Comm.27(9), 1687–1697 (2009).
[CrossRef]

IEEE Trans. Automat. Contr.

H. Heffes, “The effect of erroneous models on the Kalman filter response,” IEEE Trans. Automat. Contr.11(3), 541–543 (1966).
[CrossRef]

IEEE Trans. Image Process.

E. Dedrick and D. Lau, “A Kalman-filtering approach to high dynamic range imaging for measurement applications,” IEEE Trans. Image Process.21(2), 527–536 (2012).
[CrossRef] [PubMed]

J. Basic Engrg, Trans. ASME.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Engrg, Trans. ASME.35–45 (1960).

J. Lasers in Chinese

W. Xie and J. Tang, “Analysis on Characterization of Atmospheric Optical Wireless Communication System Based on Turbo Code,” J. Lasers in Chinese30(9), 835–838 (2003).

J. Microwaves in Chinese

Y. Zhu, Y. Li, and X. Wu, “Design for Optical Phased-Array Antenna,” J. Microwaves in Chinese24(1), 47–51 (2008).

Opt. Eng.

T. Shang, Y.T. Yang, W. X. Li, J. J. Jia, and X. Wang, “Beam control method based on omnidirectional regular icosahedron-shaped optical antenna for mobile free-space optical communication,” Opt. Eng.51(10), 105007 (2012).

Opt. Express

Proc. SPIE

L. B. Stotts, B. Stadler, and G. Lee, “Free space optical communications: coming of age,” Proc. SPIE6951, 69510W (2008).
[CrossRef]

Other

W. Hameed, S. S. Muhammad, and N. M. Sheikh, “Integration scenarios for Free Space Optics in next generation (4G) wireless networks,” In Processing of IEEE 7th International Symposium on Communication Systems Networks and Digital Signal (Newcastle upon Tyne, 2010), pp. 571 – 575.

Ch. Chlestil, E. Leitgeb, and N. P. Schmitt, S. Sheikh Muhammad, K. Zettl, and W. Rehm, “Reliable optical wireless links within UAV swarms,” In Processing of IEEE Conference on Transparent Optical Networks International (Nottingham, 2006), 4, pp. 39 – 42.

T. Plank, E. Leitgeb, and M. Loeschnigg, “Recent developments on free space optical links and wavelength analysis,” In Proceedings of Space Optical Systems and Applications (Santa Monica, CA, 2011), pp. 14–20.

M. Y. Fu, Z. H. Deng, and J. W. Zhang, “The principle of Kalman filtering theory” in Kalman filtering theory and its application in navigation system. (Science Press, China, 2003), Chap. 2.

A. Sevincer, M. Bilgi, M. Yuksel, and N. Pala, “Prototyping Multi-Transceiver Free-Space Optical Communication Structures,” In Processing of IEEE International Conference on Communications (Cape Town, South Africa, 2010). pp. 1–5.

J. Cunningham, D. Foulke, D. W. Yong, J. C. Juarez, J. E. Sluz, and J. L. Riggins, “Long Range Field Testing of Free Space Optical Communications Terminal on Mobile platforms,” In Processing of IEEE Conference on Military Communication (Boston, MA, 2009), pp. 1–7.

M. Zhang and Y. Liang, “Compound tracking in ATP system for free space optical communication,” In Processing of IEEE International Conference on Mechatronic Science, Electric Engineering and Computer (Jilin, China, 2011). pp. 454 – 456.

K. Yoshida and T. Tsujimura, “Tracking Control of the Mobile Terminal in an Active Free-Space Optical Communication system,” In Processing of SICE-ICASE International Joint Conference (Busan, Korea, 2006), pp. 369–374.

D. Zhou and H. H. Refai, “Control algorithm development for mobile FSO Node Alignment,” In Processing of 28th IEEE/AIAA Digital Avionics system Conference (Orlando, FL, 2009), pp. 6.A.3–1–6.A.3–12.

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Figures (11)

Fig. 1
Fig. 1

Diagram of the communication antenna.

Fig. 2
Fig. 2

Distribution of the array elements.

Fig. 3
Fig. 3

Schematic of the communication process between two communication antennas.

Fig. 4
Fig. 4

(a) Diagram of the communicate angle of a transmission unit during the antenna rotation, (b) diagram of mobile communication.

Fig. 5
Fig. 5

Calculation results of ω with different ν(t) and L(t) , the dotted lines express antenna rotation rate, and the solid lines express ν(t)/L(t) .

Fig. 6
Fig. 6

Performance of KPA (left) and the EC-KPA (right) with weak noise in tracking antenna, the asterisked line denotes actual trajectory, and the hollow circle line is predicted trajectory.

Fig. 7
Fig. 7

Performance of KPA (left) and the EC-KPA (right) with strong noise in tracking antenna, the asterisked line denotes actual trajectory, and the hollow circle line is predicted trajectory.

Fig. 8
Fig. 8

Diagram of prediction error of KPA with weak noise.

Fig. 9
Fig. 9

Diagram of prediction error of the EC-KPA with weak noise.

Fig. 10
Fig. 10

Influence of the EC-KPA’s prediction error on communication performance, while L(t) =800m.

Fig. 11
Fig. 11

Relationships among bandwidth utilization, communication distance and beam divergence angle.

Tables (1)

Tables Icon

Table 1 Parameters of Antenna in FSO System

Equations (13)

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

φ_com=2arctan( R L(t) )+θ 2R L(t) +θ.
φ_nocom={ 2π N ( 2R L(t) +θ),               (2R/L(t)+θ)N<2π 0,                                       (2R/L(t)+θ)N2π .
{ T (t) com_once =θ+2R/L(t)/ω'(t) T (t) nocom_once ={ 2π N ( 2R L(t) +θ)}/ω'(t) ω'(t)=ω+ν(t)cos(β)/L(t) .
ων(t)cos(β)/L(t).
{ η=( T com / T all )= N(θ+2R/L(t))/ω'(t) 2π/ω'(t) =N(θ+2R/L(t))/2π 2 r 1 N2πR .
{ X ^ k,k1 = Φ k,k1 X ^ k1 + W k1 Z ^ k,k1 = H k X ^ k,k1 + V k X ^ k = X ^ k,k1 + K k ( Z k H k X ^ k,k1 ) K k = P k,k1 H k T [ H k P k,k1 H k T + R k ] 1 P k,k1 =E[ ( X k X ^ k,k1 ) ( X k X ^ k,k1 ) T ]= Φ k,k1 P k1 Φ T k,k1 + Q k1 P k =E[ X k ~ X T k ~ ]=[I K k H k ] P k,k1 .
Φ k,k1 =[ 1 Δt 0 1 0 0 0 0 0 0 0 0 1 Δt 0 1 ], H k =[ 1 0 0 0 0 0 1 0 ].
ΔX ^ k,k1 = Φ k,k1 ΔX ^ k1 + D k .
ΔX ^ k,k1 = Φ k,k1 ΔX ^ k1 + D k =[ 1 Δt 0 1 0 0 0 0 0 0 0 0 1 Δt 0 1 ] [ Δ x k1 Δx v k1 Δ y k1 Δy v k1 ] T + D k =[ Δ x k1 +Δx v k1 Δt Δx v k1 Δ y k1 +Δy v k1 Δt Δy v k1 ]+ D k .
ΔX ^ k,k1_new = ΔX ^ k,k1 ΔX ^ k1 = Φ k,k1 ΔX ^ k1 + D k ΔX ^ k1 =[ Δ x k1 +Δx v k1 Δt Δx v k1 Δ y k1 +Δy v k1 Δt Δy v k1 ]+ D k [ Δ x k1 Δx v k1 Δ y k1 Δy v k1 ] =[ Δx v k1 Δt 0 Δy v k1 Δt 0 ]+ D k .
D c k ± a(t) ¯ 2 Δ 2 t=± a(t) ¯ 2 {[ 2π N ( 2R L ( t k ) 1 +θ)]/ω' ( t k ) 1 } 2 .
μ= D c k +Δx v k1 Δt θL( t k ) = 3 a(t) ¯ Δ 2 t 2θL( t k ) = 3 a(t) ¯ 2θL( t k ) {[ 2π N ( 2R L ( t k ) 1 +θ)]/ω' ( t k ) 1 } 2 .
S P T 20lg(θL(t)/(2 r 2 ))αL(t) A T A R M.

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