Abstract

Free-space optical (FSO) communication provides rapidly deployable, dynamic communication links that are capable of very high data rates compared with those of radio-frequency systems. As such, FSO communication is ideal for mobile platforms, for platforms that require the additional security afforded by the narrow divergence of a laser beam, and for systems that must be deployed in a relatively short time frame. In clear-weather conditions the data rate and utility of FSO communication links are primarily limited by fading caused by microscale atmospheric temperature variations that create parts-per-million refractive-index fluctuations known as atmospheric turbulence. Typical communication techniques to overcome turbulence-induced fading, such as interleavers with sophisticated codes, lose viability as the data rate is driven higher or the delay tolerance is driven lower. This paper, along with its companion [J. Opt. Commun. Netw. 4, 947 (2012)], present communication systems and techniques that exploit atmospheric reciprocity to overcome turbulence that are viable for high data rate and low delay tolerance systems. Part I proves that reciprocity is exhibited under rather general conditions and derives the optimal power-transfer phase compensation for far-field operation. Part II presents capacity-achieving architectures that exploit reciprocity to overcome the complexity and delay issues that limit state-of-the-art FSO communications.

© 2013 Optical Society of America

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References

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  1. J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol., vol.  19, pp. 410–414, Aug. 1971.
    [CrossRef]
  2. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt., vol.  10, no. 7, pp. 1652–1658, July 1971.
    [CrossRef]
  3. J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. Berlin: Springer-Verlag, 1978.
  4. J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am., vol.  61, no. 4, pp. 492–495, Apr. 1971.
    [CrossRef]
  5. J. H. Shapiro and A. L. Puryear, “Reciprocity-enhanced optical communication through atmospheric turbulence—Part I: Reciprocity proofs and far-field power transfer optimization,” J. Opt. Commun. Netw., vol.  4, no. 12, pp. 947–954, Dec. 2012.
    [CrossRef]
  6. J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
    [CrossRef]
  7. D. Tse and P. Viswanath, Fundamentals of Wireless Communication.New York: Cambridge University, 2005.
  8. A. Goldsmith, Wireless Communications.New York: Cambridge University, 2005.
  9. J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol.  7464, 746409, 2009.
    [CrossRef]
  10. J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
    [CrossRef]
  11. T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
    [CrossRef]
  12. R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observation of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol.  20, no. 19, pp. 21635–21644, Sept. 2012.
    [CrossRef]
  13. F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.
  14. G. S. Smith, “A direct derivation of a single-antenna reciprocity relation for the time domain,” IEEE Trans. Antennas Propag., vol.  52, no. 6, pp. 1568–1577, June 2004.
    [CrossRef]
  15. M. Guillaud, D. T. M. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” Proc. Eighth Int. Symp. on Signal Processing and Its Applications, Aug. 2005, vol. 1, pp. 403–406.
  16. R. L. Fante, “Electromagnetic beam propagation in turbulent media: An update,” Proc. IEEE, vol.  68, no. 11, pp. 1424–1443, 1980.
    [CrossRef]

2013 (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

2012 (2)

2010 (1)

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

2009 (3)

J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol.  7464, 746409, 2009.
[CrossRef]

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

2004 (1)

G. S. Smith, “A direct derivation of a single-antenna reciprocity relation for the time domain,” IEEE Trans. Antennas Propag., vol.  52, no. 6, pp. 1568–1577, June 2004.
[CrossRef]

1980 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: An update,” Proc. IEEE, vol.  68, no. 11, pp. 1424–1443, 1980.
[CrossRef]

1971 (3)

Crucioli, D. A.

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

Dolfi, D.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: An update,” Proc. IEEE, vol.  68, no. 11, pp. 1424–1443, 1980.
[CrossRef]

Goldsmith, A.

A. Goldsmith, Wireless Communications.New York: Cambridge University, 2005.

Greco, J. A.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observation of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol.  20, no. 19, pp. 21635–21644, Sept. 2012.
[CrossRef]

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol.  7464, 746409, 2009.
[CrossRef]

Guillaud, M.

M. Guillaud, D. T. M. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” Proc. Eighth Int. Symp. on Signal Processing and Its Applications, Aug. 2005, vol. 1, pp. 403–406.

Henion, S. R.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Knopp, R.

M. Guillaud, D. T. M. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” Proc. Eighth Int. Symp. on Signal Processing and Its Applications, Aug. 2005, vol. 1, pp. 403–406.

Lutomirski, R. F.

Magliocco, R. J.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Michael, S.

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Minet, J.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

Moores, J. D.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Murphy, R. J.

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

Parenti, R. R.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observation of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol.  20, no. 19, pp. 21635–21644, Sept. 2012.
[CrossRef]

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

Polnau, E.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

Puryear, A. L.

Roth, J. M.

Shapiro, J. H.

Slock, D. T. M.

M. Guillaud, D. T. M. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” Proc. Eighth Int. Symp. on Signal Processing and Its Applications, Aug. 2005, vol. 1, pp. 403–406.

Smith, G. S.

G. S. Smith, “A direct derivation of a single-antenna reciprocity relation for the time domain,” IEEE Trans. Antennas Propag., vol.  52, no. 6, pp. 1568–1577, June 2004.
[CrossRef]

Taylor, J. A.

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

Tse, D.

D. Tse and P. Viswanath, Fundamentals of Wireless Communication.New York: Cambridge University, 2005.

Viswanath, P.

D. Tse and P. Viswanath, Fundamentals of Wireless Communication.New York: Cambridge University, 2005.

Volpicelli, A. M.

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Vorontsov, M. A.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

Walther, F. G.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observation of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol.  20, no. 19, pp. 21635–21644, Sept. 2012.
[CrossRef]

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

Wilcox, W. E.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

Williams, T. H.

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

Yura, H. T.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

G. S. Smith, “A direct derivation of a single-antenna reciprocity relation for the time domain,” IEEE Trans. Antennas Propag., vol.  52, no. 6, pp. 1568–1577, June 2004.
[CrossRef]

IEEE Trans. Commun. Technol. (1)

J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol., vol.  19, pp. 410–414, Aug. 1971.
[CrossRef]

J. Opt. (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt., vol.  15, no. 2, 022401, Feb. 2013.
[CrossRef]

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. (1)

Opt. Express (1)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: An update,” Proc. IEEE, vol.  68, no. 11, pp. 1424–1443, 1980.
[CrossRef]

Proc. SPIE (4)

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol.  7814, 78140Y, 2010.

J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol.  7464, 746409, 2009.
[CrossRef]

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox, A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol.  7464, 746404, 2009.
[CrossRef]

T. H. Williams, R. J. Murphy, F. G. Walther, A. M. Volpicelli, W. E. Wilcox, and D. A. Crucioli, “A free-space optical terminal for fading channels,” Proc. SPIE, vol.  7464, 74640W, 2009.
[CrossRef]

Other (4)

D. Tse and P. Viswanath, Fundamentals of Wireless Communication.New York: Cambridge University, 2005.

A. Goldsmith, Wireless Communications.New York: Cambridge University, 2005.

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. Berlin: Springer-Verlag, 1978.

M. Guillaud, D. T. M. Slock, and R. Knopp, “A practical method for wireless channel reciprocity exploitation through relative calibration,” Proc. Eighth Int. Symp. on Signal Processing and Its Applications, Aug. 2005, vol. 1, pp. 403–406.

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Figures (9)

Fig. 1.
Fig. 1.

Pictorial diagram of single-mode fiber-coupled reciprocal system.

Fig. 2.
Fig. 2.

Pictorial diagram of single-mode fiber-coupled system that is partially reciprocal.

Fig. 3.
Fig. 3.

Pictorial diagram of the optimal reciprocity architecture for peak-power-limited systems. The encoder-decoder pairs are selected based on the optimal estimate of the turbulence state given the observation of the z=L to z=0 channel. We show the z=L receiver as having knowledge of the turbulence state estimated by the z=0 transmitter—this may be accomplished by appending each codeword with information regarding the encoder selection. If each realizable turbulence state can be well approximated as an AWGN channel, each encoder-decoder pair can simply implement an AWGN channel code.

Fig. 4.
Fig. 4.

Pictorial diagram of the optimal reciprocity architecture for peak-power-limited systems. The encoder-decoder pairs are selected based on the optimal estimate of the turbulence state given the observation of the z=L to z=0 channel. We show the z=L receiver as having knowledge of the turbulence state estimated by the z=0 transmitter—this may be accomplished by appending each codeword with information regarding the encoder selection. If each realizable turbulence state can be well approximated as an AWGN channel, each encoder-decoder pair can simply implement an AWGN channel code.

Fig. 5.
Fig. 5.

Downlink optical diagram showing the data-transmission beam (indicated by the arrows projecting from the left side of this figure) and upward-propagating tracking beams (indicated by the arrows projecting from the right side of this figure). Separate high-bandwidth trackers were used to stabilize the pointing direction of each of the laser sources and correct angle-of-arrival errors at the detectors.

Fig. 6.
Fig. 6.

In the FOCAL experiment the performance of a 2.7Gb/s link between a transmitter mounted in a Twin Otter aircraft and a ground-based receiver was evaluated. For most of the tests, the aircraft was flown in a semicircular pattern centered on the location of the ground terminal. Tests were performed at ranges between 25 and 80 km.

Fig. 7.
Fig. 7.

Joint probability distribution of the z=0 (aircraft) and z=L (ground) observations for FOCAL experiment exhibiting a correlation coefficient of 0.982. In the figure, the red solid line shows the optimal estimator for an outage probability ε0L of 0.1 while the green dashed line shows the optimal estimator for an outage probability ε0L of 0.01. The intensity scale for the joint probability distribution is logarithmic.

Fig. 8.
Fig. 8.

Joint probability distribution of z=0 (aircraft) and z=L (ground) observations for FOCAL experiment exhibiting a correlation coefficient of 0.982. The dot is the optimal single encoder-decoder operating point for an outage probability of 0.0024. For the example, the system is on 53% of the time, the system is appropriately off 35% of the time, and the system is off but could be successfully communicating (missed opportunity) 11% of the time. The intensity scale for the joint probability distribution is logarithmic.

Fig. 9.
Fig. 9.

Margin for optimal estimator and single encoder-decoder estimator for FOCAL experiment exhibiting a correlation coefficient of 0.982.

Equations (37)

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

sLR(t)=α0Lν0L(t)s0T(tL/c)+wLR(t).
α0L=E[|A0ALq0T(ρ;t)h(ρ,ρ;t)qLR(ρ;t)dρdρ|2],q0T(ρ;t)=p0T(ρ)T0T(ρ;tL/c),qLR(ρ;t)=pLR(ρ)TLR(ρ;t),
ν0L(t)=A0ALq0T(ρ;t)h(ρ,ρ;t)qLR(ρ;t)dρdρα0L.
E[|wLR(t)|2]=λ2NLλΔλL,
μLR(t)=ηL|α0Lν0L(t)s0T(tL/c)+wLR(t)|2ω0,
s0R(t)=αL0νL0(t)sLT(tL/c)+w0R(t),
αL0=E[|A0ALq0R(ρ;t)h(ρ,ρ;t)qLT(ρ;t)dρdρ|2],q0R(ρ;t)=p0R(ρ)T0R(ρ;t),qLT(ρ;t)=TLT(ρ;tL/c)pLT(ρ),
νL0(t)=A0ALq0R(ρ;t)h(ρ,ρ;t)qLT(ρ;t)dρdραL0.
E[|w0R(t)|2]=λ2N0λΔλ0,
μ0R(t)=η0|αL0νL0(t)sLT(tL/c)+w0R(t)|2ω0,
ν(t)νL0(t)=ν0L(t),ααL0=α0L.
p0(ρ)p0T(ρ)=p0R(ρ)=F0ξ(ρf)g0(ρ,ρf)dρf,pL(ρ)pLT(ρ)=pLR(ρ)=FLξ(ρf)gL(ρ,ρf)dρf,
p0T(ρ)=F0ξ0T(ρf)g0T(ρ,ρf)dρf,p0R(ρ)=F0ξ0R(ρf)g0R(ρ,ρf)dρf,pLT(ρ)=FLξLT(ρf)gLT(ρ,ρf)dρf,pLR(ρ)=FLξLR(ρf)gLR(ρ,ρf)dρf.
C0Lergodic=E[C(|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2))].
ε0L=Pr(C(|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2))<R0Lε)
R0LLD(ε0L)=E[maxd<t0Cε0L(|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2))],
Cε0L(|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2))=maxPr(outage)<ε0LC(|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2)).
m0L(ε0L)=argβ{E[maxd<t0Cε0L(β|ν0L|2;ϒ0T(|ν0L|2),ϒLR(|ν0L|2))]=E[C(|ν0L|2;|νL0|2,|ν0L|2)]},
R0LLD(ε0L)=maxy0T(·):Pr(y0T(γ)>|ν0L|2|ϒ0T(|ν0L|2)=γ)<ε0LγC(y0T(γ))Pr(y0T(γ)<|ν0L|2|ϒ0T(|ν0L|2)=γ)fϒ0T(|ν0L|2)(γ)dγ,
R0LLD(ε0L)=maxy0T(·):F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ))<ε0LγC(y0T(γ))(1F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ)))fϒ0T(|ν0L|2)(γ)dγ,
R0LLD(ε0L)=maxy0T(·):F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ))<ε0Lγ(1ε0L)C(y0T(γ))fϒ0T(|ν0L|2)(γ)dγ.
y0Ts(ϒ0T(|ν0L|2))={y0Ts,0,ϒ0T(|ν0L|2)γ00,ϒ0T(|ν0L|2)<γ0,
R0LLD,s(ε0L)=maxy0Ts,0,γ0:F|ν0L|2|ϒ0T(|ν0L|2)(y0Ts(γ))<ε0Lγ(1ε0L)C(y0Ts(γ))fϒ0T(|ν0L|2)(γ)dγ,
R0LLD,s(ε0L)=maxy0Ts,0,γ0:F|ν0L|2|ϒ0T(|ν0L|2)(y0Ts(γ))<ε0Lγ(1ε0L)C(y0Ts,0)(1Fϒ0T(|ν0L|2)(γ0)).
F|ν0L|2|ϒ0T(|ν0L|2)(y0T(ϒ0T(|ν0L|2)))=F|ν0L|2||ν0L|2(|ν0L|2)=0
R0LLD(ε0L)=maxy0T(·):F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ))<ε0Lγ(1ε0L)C(y0T(γ))fϒ0T(|ν0L|2)(γ)dγ=C(γ)f|ν0L|2(γ)dγ=C0Lergodic.
R0LLD,s(ε0L)=maxγ0C(γ0)(1F|ν0L|2(γ0)),
F|ν0L|21(y)=infxR{F|ν0L|21(x)y}.
R0LLD(ε0L)=maxy0T(·):F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ))<ε0Lγ(1ε0L)C(y0T(γ))fϒ0T(|ν0L|2)(γ)dγ=(1ε0L)C(F|ν0L|21(ε0L))=(1ε0L)R0Lε.
R0LLD,s(ε0L)=R0LLD(ε0L),y0Ts(ϒ0T(|ν0L|2))=y0T(ϒ0T(|ν0L|2)).
R0LLD(ε0L)=maxy0T(·):F|ν0L|2|ϒ0T(|ν0L|2)(y0T(γ))<ε0Lγ(1ε0L)W0Llog(1+ηLα0LP0Tω0W0Ly0T(γ))fϒ0T(|ν0L|2)(γ)dγ.
f|ν0L|2(γ|ϒ0T(|ν0L|2)=y)=fϒ0T(|ν0L|2)||ν0L|2(y||ν0L|2=γ)f|ν0L|2(γ)fϒ0T(|ν0L|2)||ν0L|2(y||ν0L|2=γ)f|ν0L|2(γ)dγ,fϒ0T(|ν0L|2)(y)=fϒ0T(|ν0L|2)||ν0L|2(y||ν0L|2=γ)f|ν0L|2(γ)dγ.
μ0R(t)=η0αL0ω0|νL0(t)|2PLT,
Pr(ϒ0T(|ν0L|2)=k||ν0L|2=γ)=(η0αL0PLTγτ0/ω0)kk!eη0αL0PLTγτ0/ω0.
sLR1(t)=α0Lν0L1(t)s0T(tL/c)+wLR1(t),sLR2(t)=α0Lν0L2(t)s0T(tL/c)+wLR2(t),sLR3(t)=α0Lν0L3(t)s0T(tL/c)+wLR3(t),sLR4(t)=α0Lν0L4(t)s0T(tL/c)+wLR4(t).
s0R1(t)=αL0νL01(t)sLT1(tL/c)+w0R1(t),s0R2(t)=αL0νL02(t)sLT2(tL/c)+w0R2(t),s0R3(t)=αL0νL03(t)sLT3(tL/c)+w0R3(t),s0R4(t)=αL0νL04(t)sLT4(tL/c)+w0R4(t).
|ν0L|2=14(|ν0L1|2+|ν0L2|2+|ν0L3|2+|ν0L4|2),|νL0|2=14(|νL01|2+|νL02|2+|νL03|2+|νL04|2),