Abstract

Establishing a link between a ground station and a geosynchronous orbiting satellite can be aided greatly with the use of a beacon on the satellite. A tracker, or even an adaptive optics system, can use the beacon during communication or tracking activities to correct beam pointing for atmospheric turbulence and mount jitter effects. However, the pointing lead-ahead required to illuminate the moving object and an aperture mismatch between the tracking and the pointing apertures can limit the effectiveness of the correction, as the sensed tilt will not be the same as the tilt required for optimal transmission to the satellite. We have developed an analytical model that addresses the combined impact of these tracking issues in a ground-to-satellite optical link. We present these results for different tracker/pointer configurations. By setting the low-pass cutoff frequency of the tracking servo properly, the tracking errors can be minimized. The analysis considers geosynchronous Earth orbit satellites as well as low Earth orbit satellites.

© 2008 Optical Society of America

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References

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  1. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52-61 (1982).
    [CrossRef]
  2. A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).
  3. L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
    [CrossRef]
  4. G. Valley and S. Wandzura, “Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence,” J. Opt. Soc. Am. 69, 712-717 (1979).
    [CrossRef]
  5. K. Winick and D. Marquis, “Stellar scintillation technique for the measurement of tilt anisoplanatism,” J. Opt. Soc. Am. A 5, 1929-1936 (1988).
    [CrossRef]
  6. D. L. Fried, “Varieties of isoplanatism,” Proc. SPIE 75, 20-29 (1976).
  7. J. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65-68 (1975).
    [CrossRef]
  8. D. Greenwood, “Tracking turbulence-induced tilt errors with shared and adjacent apertures,” J. Opt. Soc. Am. 67, 282-290 (1977).
    [CrossRef]
  9. D. L. Fried, “Statistics of a geometric representation of a wave-front distortion,” J. Opt. Soc. Am. 55, 1427-1435 (1965).
    [CrossRef]
  10. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, 1971).
  11. G. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358-367 (1994).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1964), Eq. (11.4.24).
  13. S. Basu and D. G. Voelz, “Analysis of a ground to satellite optical link with a cooperative satellite beacon,” Proc. SPIE 6304, 63041M1-63041M7 (2006).

2006 (2)

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

S. Basu and D. G. Voelz, “Analysis of a ground to satellite optical link with a cooperative satellite beacon,” Proc. SPIE 6304, 63041M1-63041M7 (2006).

2005 (1)

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

1994 (1)

1988 (1)

1982 (1)

1979 (1)

1977 (1)

1976 (1)

D. L. Fried, “Varieties of isoplanatism,” Proc. SPIE 75, 20-29 (1976).

1975 (1)

1971 (1)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, 1971).

1965 (1)

1964 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1964), Eq. (11.4.24).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1964), Eq. (11.4.24).

Alonso, A.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Andrews, L.

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

Basu, S.

S. Basu and D. G. Voelz, “Analysis of a ground to satellite optical link with a cooperative satellite beacon,” Proc. SPIE 6304, 63041M1-63041M7 (2006).

Comeron, A.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Dios, F.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Fried, D. L.

Greenwood, D.

Marquis, D.

Parenti, R.

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

Phillips, R.

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

Reyes, M.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Rodriguez, A.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Rubio, J. A.

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Sasiela, R.

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

Shapiro, J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1964), Eq. (11.4.24).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, 1971).

Tyler, G.

Valley, G.

Voelz, D. G.

S. Basu and D. G. Voelz, “Analysis of a ground to satellite optical link with a cooperative satellite beacon,” Proc. SPIE 6304, 63041M1-63041M7 (2006).

Wandzura, S.

Winick, K.

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Opt. Eng. (Bellingham) (1)

L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. (Bellingham) 45, 076001 (2006).
[CrossRef]

Proc. SPIE (3)

S. Basu and D. G. Voelz, “Analysis of a ground to satellite optical link with a cooperative satellite beacon,” Proc. SPIE 6304, 63041M1-63041M7 (2006).

D. L. Fried, “Varieties of isoplanatism,” Proc. SPIE 75, 20-29 (1976).

A. Comeron, F. Dios, A. Rodriguez, J. A. Rubio, M. Reyes, and A. Alonso, “Modeling of power fluctuations induced by refractive turbulence in a multiple-beam ground-to-satellite optical uplink,” Proc. SPIE 5892, 58920O1-58920O10 (2005).

Other (2)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, 1971).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1964), Eq. (11.4.24).

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Figures (24)

Fig. 1
Fig. 1

Illustration of lead-ahead and aperture mismatch in a tracking system, where θ is the lead-ahead angle.

Fig. 2
Fig. 2

Filled configuration: (a) on axis and (b) contiguous off axis.

Fig. 3
Fig. 3

Annular configuration.

Fig. 4
Fig. 4

Interior configuration.

Fig. 5
Fig. 5

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled on-axis configuration with no lead-ahead.

Fig. 6
Fig. 6

Normalized mean-square tilt error versus cutoff frequency for a LEO: filled on-axis configuration with no lead-ahead.

Fig. 7
Fig. 7

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled contiguous off-axis configuration with no lead-ahead.

Fig. 8
Fig. 8

Normalized mean-square tilt error versus cutoff frequency for a LEO: filled contiguous off-axis configuration with no lead-ahead.

Fig. 9
Fig. 9

Normalized mean-square tilt error versus cutoff frequency for a GEO: annular configuration with no lead-ahead.

Fig. 10
Fig. 10

Normalized mean-square tilt error versus cutoff frequency for a LEO: annular configuration with no lead-ahead.

Fig. 11
Fig. 11

Normalized mean-square tilt error versus cutoff frequency for a GEO: interior configuration with no lead-ahead.

Fig. 12
Fig. 12

Normalized mean-square tilt error versus cutoff frequency for a LEO: interior configuration with no lead-ahead.

Fig. 13
Fig. 13

Normalized mean-square tilt error versus cutoff frequency for a GEO: lead-ahead with no aperture mismatch.

Fig. 14
Fig. 14

Normalized mean-square tilt error versus cutoff frequency for a LEO: lead-ahead with no aperture mismatch.

Fig. 15
Fig. 15

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled on-axis configuration with lead-ahead.

Fig. 16
Fig. 16

Normalized mean-square tilt error versus cutoff frequency for a LEO: filled on-axis configuration with lead-ahead.

Fig. 17
Fig. 17

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled contiguous off-axis configuration with lead-ahead.

Fig. 18
Fig. 18

Normalized mean-square tilt error versus cutoff frequency for a LEO: filled contiguous off-axis configuration with lead-ahead.

Fig. 19
Fig. 19

Normalized mean-square tilt error versus cutoff frequency for a GEO: annular configuration with lead-ahead.

Fig. 20
Fig. 20

Normalized mean-square tilt error versus cutoff frequency for a LEO: annular configuration with lead-ahead.

Fig. 21
Fig. 21

Normalized mean-square tilt error versus cutoff frequency for a GEO: interior configuration with lead-ahead.

Fig. 22
Fig. 22

Normalized mean-square tilt error versus cutoff frequency for a LEO: interior configuration with lead-ahead.

Fig. 23
Fig. 23

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled on-axis configuration for a geometry similar to ARTEMIS/OGS.

Fig. 24
Fig. 24

Normalized mean-square tilt error versus cutoff frequency for a GEO: filled contiguous off-axis configuration for a geometry similar to ARTEMIS/OGS.

Tables (5)

Tables Icon

Table 1 Tracking Performance for a GEO Satellite in the Presence of Only Aperture Mismatch a

Tables Icon

Table 2 Tracking Performance for a LEO Satellite in the Presence of Only Aperture Mismatch a

Tables Icon

Table 3 Tracking Performance for a GEO Satellite in the Presence of Lead-Ahead and Aperture Mismatch a

Tables Icon

Table 4 Tracking Performance for a LEO Satellite in the Presence of Lead-Ahead and Aperture Mismatch a

Tables Icon

Table 5 Tracking Performance for a Geometry Similar to ARTEMIS/OGS a

Equations (73)

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α p ( θ 2 , t ) = 32 λ π 2 D 4 d r P ( r ) r ϕ ( r , θ 2 , t ) ,
P ( r ) = { 1 , r 0.5 D 0 , r 0.5 D ,
α t ( θ 1 , t ) = 32 λ π 2 d 4 d r P d ( r , r 1 ) ( r r 1 ) ϕ ( r , θ 1 , t ) ,
P d ( r , r 1 ) = { 1 , r r 1 0.5 d 0 , r r 1 0.5 d ,
e ( θ , t ) = α p ( θ 2 , t ) α t ( θ 1 , t ) * h ( t ) ,
σ e 2 ( θ , f c o ) = W p ( f ) d f + H ( f , f c o ) W t ( f ) d f 2 H 1 2 ( f , f c o ) W p t ( f , θ ) d f ,
C p t ( τ , θ ) = α p ( θ 2 , t 2 ) α t ( θ 1 , t 1 ) = ( 32 λ π 2 ) 2 1 D 4 d 4 d r d r P ( r ) P d ( r , r 1 ) r ( r r 1 ) ϕ ( r , θ 2 , t 2 ) ϕ ( r , θ 1 , t 1 ) ,
ϕ ( r , θ , t ) = k d z n [ r V ( z ) t + θ z , z ] ,
C p t ( τ , θ ) = ( 32 λ π 2 ) 2 1 D 4 d 4 k 2 d r d r d z 1 d z 2 P ( r ) P d ( r , r 1 ) r ( r r 1 ) n [ r V ( z 2 ) t 2 + θ 2 z 2 , z 2 ] n [ r V ( z 1 ) t 1 + θ 1 z 1 , z 1 ] .
C p t ( τ , θ ) = ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z 1 d z 2 P ( r ) P d ( r , r 1 ) r ( r r 1 ) { n [ r V ( z 2 ) t 2 + θ 2 z 2 , z 2 ] n [ r V ( z 1 ) t 1 + θ 1 z 1 , z 1 ] } 2 .
{ n [ r V ( z 2 ) t 2 + θ 2 z 2 , z 2 ] n [ r V ( z 1 ) t 1 + θ 1 z 1 , z 1 ] } 2 = C n 2 ( z 1 + z 2 2 ) [ r r V ( z 2 ) t 2 + V ( z 1 ) t 1 + θ 2 z 2 θ 1 z 1 2 + z 2 z 1 2 ] 1 3 .
C p t ( τ , θ ) = ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z 1 d z 2 P ( r ) P d ( r , r 1 ) r ( r r 1 ) C n 2 ( z 1 + z 2 2 ) { [ r r V ( z 2 ) t 2 + V ( z 1 ) t 1 + θ 2 z 2 θ 1 z 1 2 + z 2 z 1 2 ] 1 3 z 2 z 1 2 3 } ,
z + = 1 2 ( z 1 + z 2 )
z = z 2 z 1 ,
C p t ( τ , θ ) = ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z + d z P ( r ) P d ( r , r 1 ) r ( r r 1 ) C n 2 ( z + ) ( [ r r V ( t 2 t 1 ) + z + ( θ 2 θ 1 ) + z ( θ 1 + θ 2 2 ) 2 + ( z ) 2 ] 1 3 ( z ) 2 3 ) = ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z + d z P ( r ) P d ( r , r 1 ) r ( r r 1 ) C n 2 ( z + ) ( [ r r V τ + z + θ + z ( θ 1 + θ 2 2 ) 2 + ( z ) 2 ] 1 3 ( z ) 2 3 ) .
d x [ ( A 2 + x 2 ) 1 3 x 2 3 ] = 2.91 A 5 3 ,
C p t ( τ , θ ) = 2.91 ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z P ( r ) P d ( r , r 1 ) r ( r r 1 ) C n 2 ( z ) r r V τ + z θ 5 3 .
C p t ( τ , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 { 4 π β D [ θ h sec ( ψ ) V τ r 1 ] } ,
C p t ( τ , θ ) = 2.91 ( 32 λ π 2 ) 2 1 2 D 4 d 4 k 2 d r d r d z P ( r ) P d ( r , r 1 ) r ( r r 1 ) C n 2 ( z ) r r + z ( θ + ω τ ) 5 3 .
C p t ( τ , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 { 4 π β D [ ( θ + ω τ ) h sec ( ψ ) r 1 ] } .
C p ( τ ) = α p ( θ 2 , t 2 ) α p ( θ 2 , t 1 ) = 0.124 D 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 ( 4 π β V D τ ) ,
C t ( τ ) = α t ( θ 1 , t 2 ) α t ( θ 1 , t 1 ) = 0.124 d 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 ( 4 π β V d τ ) .
C p ( τ ) = 0.124 D 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 { 4 π β ω h sec ( ψ ) D τ } ,
C t ( τ ) = 0.124 d 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 { 4 π β ω h sec ( ψ ) d τ } .
W p t ( f , θ ) = d τ C p t ( τ , θ ) exp ( 2 π i f τ ) .
W p t ( f , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) d τ J 0 { 4 π β D [ θ h sec ( ψ ) V τ r 1 ] } exp ( 2 π i f τ ) .
t = 4 π β D [ V τ + r 1 θ h sec ( ψ ) ]
w = f D 2 V β
d t J n ( t ) exp ( i w t ) = { 2 ( i ) n T n ( w ) ( 1 w 2 ) 1 2 for w 2 1 0 for w 2 1 ,
T n [ cos ( ϑ ) ] = cos ( n ϑ ) ,
W p t ( f , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) exp ( 2 π i f r 1 V ) d h C n 2 ( h ) exp [ 2 π i f θ h V sec ( ψ ) ] × f D 2 V d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) 2 π β V D { 1 [ f D ( 2 β V ) ] 2 } 1 2 .
x = f D 2 β V ,
W p t ( f , θ ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 exp ( 2 π i f r 1 V ) d h C n 2 ( h ) exp [ 2 π i f θ h V sec ( ψ ) ] ( V D ) 11 3 0 1 d x x 11 3 ( 1 x 2 ) 1 2 J 2 ( π f D V x ) J 2 ( π f d V x ) .
W p t ( f , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) d τ J 0 { 4 π β D [ ( θ + ω τ ) h sec ( ψ ) r 1 ] } exp ( 2 π i f τ ) .
t = 4 π β D [ ( θ + ω τ ) h sec ( ψ ) r 1 ]
w = f D 2 ω β h sec ( ψ )
W p t ( f , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) exp ( 2 π i f θ ω ) d h C n 2 ( h ) exp [ 2 π i f r 1 ω h sec ( ψ ) ] f D 2 ω h sec ( ψ ) d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) 2 π β ω h sec ( ψ ) D { 1 [ f D 2 β ω h sec ( ψ ) ] 2 } 1 2 .
x = f D 2 ω β h sec ( ψ ) ,
W p t ( f , θ ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 exp ( 2 π i f θ ω ) d h C n 2 ( h ) exp [ 2 π i f r 1 ω h sec ( ψ ) ] 0 1 d x x 11 3 ( 1 x 2 ) 1 2 [ ω h sec ( ψ ) D ] 11 3 J 2 [ π f D ω h sec ( ψ ) x ] J 2 [ π f d ω h sec ( ψ ) x ] .
W p ( f ) = 0.501 D 1 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) ( V D ) 11 3 0 1 d x x 11 3 1 x 2 J 2 2 ( π f D V x ) .
W t ( f ) = 0.501 d 1 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) ( V d ) 11 3 0 1 d x x 11 3 1 x 2 J 2 2 ( π f d V x ) .
W p ( f ) = 0.501 D 1 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) [ ω h sec ( ψ ) D ] 11 3 × 0 1 d x x 11 3 1 x 2 J 2 2 [ π f D x ω h sec ( ψ ) ] .
W t ( f ) = 0.501 d 1 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) [ ω h sec ( ψ ) d ] 11 3 × 0 1 d x x 11 3 1 x 2 J 2 2 [ π f d x ω h sec ( ψ ) ] .
α t A ( θ 1 , t ) = 32 λ π 2 ( D 4 d 4 ) [ d r P ( r ) r ϕ ( r , θ 1 , t ) d r P d ( r , 0 ) r ϕ ( r , θ 1 , t ) ] ,
α t A ( θ 1 , t ) = 1 D 4 d 4 [ D 4 α D ( θ 1 , t ) d 4 α d ( θ 1 , t ) ] ,
α p A ( θ 2 , t ) = α d ( θ 2 , t ) = 32 λ π 2 d 4 d r P d ( r , 0 ) r ϕ ( r , θ 2 , t ) .
C t A ( τ ) = α t A ( θ 1 , t 2 ) α t A ( θ 1 , t 1 ) = b 2 2 C D ( τ ) + b 1 2 C d ( τ ) 2 b 1 b 2 C d D ( τ ) ,
b 1 = [ ( D d ) 4 1 ] 1 ,
b 2 = [ 1 ( d D ) 4 ] 1 .
C d D ( τ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 ( 4 π β D V τ ) .
C p t A ( τ , θ ) = α p A ( θ 2 , t 2 ) α t A ( θ 1 , t 1 ) = b 2 C d D θ ( τ , θ ) b 1 C d d ( τ , θ ) ,
C d D θ ( τ , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 { 4 π β D [ V τ θ h sec ( ψ ) ] }
C d d ( τ , θ ) = 0.124 d 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 { 4 π β d [ V τ θ h sec ( ψ ) ] } .
W t A ( f ) = b 2 2 W D ( f ) + b 1 2 W d ( f ) 2 b 1 b 2 W d D ( f ) ,
W d D ( f ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) ( V D ) 11 3 0 1 d x x 11 3 ( 1 x 2 ) 1 2 J 2 ( π f D V x ) J 2 ( π f d V x ) .
W p t A ( f , θ ) = b 2 W d D θ ( f , θ ) b 1 W d d ( f , θ ) ,
W d D θ ( f , θ ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) exp [ 2 π i f θ h V sec ( ψ ) ] × ( V D ) 11 3 0 1 d x x 11 3 ( 1 x 2 ) 1 2 J 2 ( π f D V x ) J 2 ( π f d V x )
W d d ( f , θ ) = 0.501 d 1 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) exp [ 2 π i f θ h V sec ( ψ ) ] × ( V d ) 11 3 0 1 d x x 11 3 ( 1 x 2 ) 1 2 J 2 2 ( π f d V x ) .
C d D ( τ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 { 4 π β D [ ω τ h sec ( ψ ) ] } .
C d D θ ( τ , θ ) = 0.124 d 2 D 5 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 ( 2 π β ) J 2 ( 2 π β d D ) J 0 [ 4 π β D ( θ + ω τ ) h sec ( ψ ) ]
C d d ( τ , θ ) = 0.124 d 1 3 sec ( ψ ) d h C n 2 ( h ) 0 d β β 14 3 J 2 2 ( 2 π β ) J 0 [ 4 π β d ( θ + ω τ ) h sec ( ψ ) ] .
W d D ( f ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 d h C n 2 ( h ) 0 1 d x x 11 3 ( 1 x 2 ) 1 2 [ ω h sec ( ψ ) D ] 11 3 J 2 [ π f D ω h sec ( ψ ) x ] J 2 [ π f d ω h sec ( ψ ) x ] .
W d D θ ( f , θ ) = 0.501 d 2 D 5 3 sec ( ψ ) f 14 3 exp ( 2 π i f θ ω ) d h C n 2 ( h ) 0 1 d x x 11 3 ( 1 x 2 ) 1 2 [ ω h sec ( ψ ) D ] 11 3 J 2 [ π f D ω h sec ( ψ ) x ] J 2 [ π f d ω h sec ( ψ ) x ]
W d d ( f , θ ) = 0.501 d 1 3 sec ( ψ ) f 14 3 exp ( 2 π i f θ ω ) d h C n 2 ( h ) 0 1 d x x 11 3 ( 1 x 2 ) 1 2 [ ω h sec ( ψ ) d ] 11 3 J 2 2 [ π f d ω h sec ( ψ ) x ] .
C t I ( τ ) = C p A ( τ ) ,
C p I ( τ ) = C t A ( τ ) ,
C p t I ( τ , θ ) = C p t A ( τ , θ ) .
W t I ( f ) = W p A ( f ) ,
W p I ( f ) = W t A ( f ) ,
W p t I ( f , θ ) = W p t A ( f , θ ) .
σ n o r m 2 = 10 9 σ e 2 D 1 3 .
H ( f , f c o ) = [ 1 + ( f f c o ) 2 ] 1 .
R = min [ σ e 2 ( f c o ) ] σ e 2 ( 0 ) ,

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