Abstract

In a recent publication [J. Opt. Soc. Am. A 25, 1594–1608 (2008)] we developed expressions for the tilt errors that arise from the effects of lead-ahead and aperture mismatch when transmitting a laser beam from the ground to a satellite. We extend these results to examine the fade statistics of the irradiance at the satellite due to these tilt errors and turbulence induced scintillation. The system concept is that the light from a beacon on the satellite is received by the ground station and a derived signal is used to drive a tracking/pointing system for the uplink beam. However, the beam must be pointed ahead along the satellite track to intercept the satellite (lead-ahead), and physical constraints may require that the beam transmit aperture is different in size or location than the aperture receiving the beacon signal (aperture mismatch). These two issues cause the light entering the receiving aperture (tracker) and the beam exiting the transmit aperture (pointer) to traverse somewhat different turbulence volumes, which limits the ability of the tracking/pointing system to place the maximum flux on the satellite.

© 2009 Optical Society of America

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References

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  1. S. Basu and D. Voelz, “Tracking in a ground-to-satellite optical link: effects due to lead-ahead and aperture mismatch including temporal tracking response,” J. Opt. Soc. Am. A 25, 1594-1608 (2008).
    [CrossRef]
  2. I. Last, M. Tamir, U. Halavee, and E. Azoulay, “Level crossing and fading probabilities in an atmospheric communications link for a non-Markov process,” Appl. Opt. 25, 1149-1154(1986).
    [CrossRef] [PubMed]
  3. P. A. Pincus, R. A. Elliott, and J. R. Kerr, “Conditional fading statistics of scintillation,” J. Opt. Soc. Am. 68, 756-760(1978).
    [CrossRef]
  4. H. T. Yura and W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication system,” Appl. Opt. 22, 3353-3358 (1983).
    [CrossRef] [PubMed]
  5. L. C. Andrews, R. L. Phillips, and P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742-7751 (1995).
    [CrossRef] [PubMed]
  6. D. L. Fried, “Statistics of laser beam fade induced by pointing jitter,” Appl. Opt. 12, 422-423 (1973).
    [CrossRef] [PubMed]
  7. L. Andrews, R. Phillips, R. Sasiela, and R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
    [CrossRef]
  8. F. Dios, J. A. Rubio, A. Rodriguez, and A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866-3873 (2004).
    [CrossRef] [PubMed]
  9. A. Rodriguez-Gomez, F. Dios, J. A. Rubio, and A. Comeron, “Temporal statistics of the beam-wander contribution to scintillation in ground-to-satellite optical links: an analytical approach,” Appl. Opt. 44, 4574-4581 (2005).
    [CrossRef] [PubMed]
  10. F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780-3789 (2007).
    [CrossRef] [PubMed]
  11. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  12. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967).

2008 (1)

2007 (1)

2006 (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. 45, 076001 (2006).
[CrossRef]

2005 (1)

2004 (1)

1995 (1)

1986 (1)

1983 (1)

1978 (1)

1973 (1)

Andrews, L.

F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780-3789 (2007).
[CrossRef] [PubMed]

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

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Andrews, L. C.

Azoulay, E.

Basu, S.

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967).

Comeron, A.

Dios, F.

Elliott, R. A.

Fried, D. L.

Halavee, U.

Kerr, J. R.

Last, I.

McKinley, W. G.

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. 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. 45, 076001 (2006).
[CrossRef]

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Phillips, R. L.

Pincus, P. A.

Rodriguez, A.

Rodriguez-Gomez, A.

Rubio, J. A.

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. 45, 076001 (2006).
[CrossRef]

Tamir, M.

Vetelino, F. S.

Voelz, D.

Young, C.

Yu, P. T.

Yura, H. T.

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (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. 45, 076001 (2006).
[CrossRef]

Other (2)

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967).

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Figures (20)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Filled configurations: (a) on-axis, (b) contiguous off-axis.

Fig. 3
Fig. 3

Probability of fade versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 4
Fig. 4

Probability of fade versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 5
Fig. 5

Probability of fade versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 6
Fig. 6

Probability of fade versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 7
Fig. 7

Probability of fade versus threshold irradiance level I T for GEO: on-axis configuration with wavelength of 2 μm .

Fig. 8
Fig. 8

Probability of fade versus threshold irradiance level I T for LEO: on-axis configuration with wavelength of 2 μm .

Fig. 9
Fig. 9

Expected number of fades per second versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 10
Fig. 10

Expected number of fades per second versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 11
Fig. 11

Expected number of fades per second versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 12
Fig. 12

Expected number of fades per second versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 13
Fig. 13

Expected number of fades per second versus threshold irradiance level I T for GEO: on-axis configuration with wavelength of 2 μm .

Fig. 14
Fig. 14

Expected number of fades per second versus threshold irradiance level I T for LEO: on-axis configuration with wavelength of 2 μm .

Fig. 15
Fig. 15

Mean fade duration (in seconds) versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 16
Fig. 16

Mean fade duration (in seconds) versus threshold parameter for GEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 17
Fig. 17

Mean fade duration (in seconds) versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 2 μm .

Fig. 18
Fig. 18

Mean fade duration (in seconds) versus threshold parameter for LEO: on-axis and off-axis configurations with wavelength of 3 μm .

Fig. 19
Fig. 19

Mean fade duration versus threshold irradiance level I T for GEO: on-axis configuration with wavelength of 2 μm .

Fig. 20
Fig. 20

Mean fade duration versus threshold irradiance level I T for LEO: on-axis configuration with wavelength of 2 μm .

Tables (2)

Tables Icon

Table 1 Pointing Parameters for a GEO Satellite a

Tables Icon

Table 2 Pointing Parameters for a LEO Satellite a

Equations (68)

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Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 ,
Θ = 1 + L F = Θ 0 Θ 0 2 + Λ 0 2 , Λ = 2 L k W 2 = Λ 0 Θ 0 2 + Λ 0 2 .
W e = W [ 1 + 4.35 μ Λ 5 6 k 7 6 H 5 6 sec 11 6 ( ψ ) ] 0.5 ,
μ = 0 H d h C n 2 ( h ) ( 1 h H ) 5 3 ,
r ( t ) = L { [ α p x ( t ) α t x ( t ) * h ( t ) ] 2 + [ α p y ( t ) α t y ( t ) * h ( t ) ] 2 } 0.5 ,
X ( t ) = L [ α p x ( t ) α t x ( t ) * h ( t ) ] ,
Y ( t ) = L [ α p y ( t ) α t y ( t ) * h ( t ) ] ,
r ( t ) = ( X ( t ) ) 2 + ( Y ( t ) ) 2 .
Φ p x ( f ) = Φ p y ( f ) = 0.5 Φ p ( f ) ,
Φ t x ( f ) = Φ t y ( f ) = 0.5 Φ t ( f ) ,
Φ p t x ( f ) = Φ p t y ( f ) = 0.5 Φ p t ( f ) .
σ x 2 = L 2 [ Φ p x ( f ) d f + H ( f ) Φ t x ( f ) d f 2 H 1 2 ( f ) Φ p t x ( f ) d f ] = 0.5 L 2 [ Φ p ( f ) d f + H ( f ) Φ t ( f ) d f 2 H 1 2 ( f ) Φ p t ( f ) d f ] ,
H ( f ) = [ 1 + ( f / f co ) 2 ] 1 .
b x 2 = 4 π 2 f 2 Φ x ( f ) = 2 π 2 L 2 [ f 2 Φ p ( f ) d f + f 2 H ( f ) Φ t ( f ) d f 2 f 2 H 1 2 ( f ) Φ p t ( f ) d f ] ,
b x 2 = b y 2 .
I ( t ) = K T exp [ r 2 ( t ) 2 Ω T 2 ] exp [ 2 χ ( t ) ] ,
K T = W 0 2 W e 2 ,
Ω T = 0.5 W e .
ln [ I ( t ) K T ] = 2 χ ( t ) r 2 ( t ) 2 Ω T 2 .
Z ( t ) = C ( t ) Q ( t ) .
p ( χ | q ) = 1 σ χ ( q ) 2 π exp { [ χ m χ ( q ) ] 2 2 σ χ 2 ( q ) } ,
p ( c | q ) = 1 2 σ χ ( q ) 2 π exp { [ 0.5 c m χ ( q ) ] 2 2 σ χ 2 ( q ) } .
p ( q ) = Ω T 2 σ x 2 exp ( - Ω T 2 q σ x 2 ) , q > 0.
p ( z | q ) = 1 2 σ χ ( q ) 2 π exp { [ 0.5 ( z + q ) m χ ( q ) ] 2 2 σ χ 2 ( q ) } .
m χ ( q ) = σ χ 2 ( q ) .
p ( z | q ) = 1 2 σ χ ( q ) 2 π exp { [ 0.5 ( z + q ) + σ χ 2 ( q ) ] 2 2 σ χ 2 ( q ) } .
p ( I | q ) = 1 2 I σ χ ( q ) 2 π exp ( { 0.5 [ ln ( I / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) , I > 0.
p ( I ) = 0 p ( I | q ) p ( q ) d q = 0 Ω T 2 σ x 2 1 2 I σ χ ( q ) 2 π exp ( Ω T 2 q σ x 2 ) exp ( { 0.5 [ ln ( I / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) d q , I > 0.
P f ( I T ) = 0 I T p ( I ) d I ,
P f ( I T ) = 0 d q Ω T 2 σ x 2 exp ( Ω T 2 q σ x 2 ) 0 I T d I 1 2 I σ χ ( q ) 2 π exp ( { 0.5 [ ln ( I / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) .
P f ( I T | q ) = 0 I T d I 1 2 I σ χ ( q ) 2 π exp ( { 0.5 [ ln ( I / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) = 0.5 { 1 + erf [ ln ( I T / K T ) + q + 2 σ χ 2 ( q ) 2 2 σ χ ( q ) ] } .
P f ( I T ) = 0 d q Ω T 2 2 σ x 2 exp ( Ω T 2 q σ x 2 ) { 1 + erf [ ln ( I T / K T ) + q + 2 σ χ 2 ( q ) 2 2 σ χ ( q ) ] } .
σ χ 2 ( q ) = 2.175 k 7 6 H 5 6 sec 11 6 ( ψ ) μ 1 + 2.975 Ω T 2 ( 2 W 0 r 0 ) 5 3 q W 2 , 0 q W 2 2 Ω T 2 ,
μ 1 = Re 0 H C n 2 ( h ) { ξ 5 6 [ Λ ξ + i ( 1 Θ ¯ ξ ) ] 5 6 Λ 5 6 ξ 5 3 } d h ,
Θ ¯ = 1 Θ ,
ξ = 1 h / H ,
r 0 = [ 0.423 k 2 sec ( ψ ) 0 H C n 2 ( h ) d h ] 3 5 .
P f min ( I T ) = 0 W 2 2 Ω T 2 P f ( I T | q ) p ( q ) d q + W 2 2 Ω T 2 P f ( I T | q = W 2 / 2 Ω T 2 ) p ( q ) d q ,
P f min ( I T ) = 0 W 2 2 Ω T 2 d q Ω T 2 2 σ x 2 exp ( Ω T 2 q σ x 2 ) { 1 + erf [ ln ( I T / K T ) + q + 2 σ χ 2 ( q ) 2 2 σ χ ( q ) ] } + { 1 + erf [ ln ( I T / K T ) + ( W 2 / 2 Ω T 2 ) + 2 σ χ 2 ( W 2 / 2 Ω T 2 ) 2 2 σ χ ( W 2 / 2 Ω T 2 ) ] } W 2 2 Ω T 2 d q Ω T 2 2 σ x 2 exp ( Ω T 2 q σ x 2 ) .
P f max ( I T ) = 0 W 2 2 Ω T 2 P f ( I T | q ) p ( q ) d q + W 2 2 Ω T 2 P f ( I T | q ) p ( q ) d q ,
P f max ( I T ) = 0 W 2 2 Ω T 2 d q Ω T 2 2 σ x 2 exp ( Ω T 2 q σ x 2 ) { 1 + erf [ ln ( I T / K T ) + q + 2 σ χ 2 ( q ) 2 2 σ χ ( q ) ] } + W 2 2 Ω T 2 d q Ω T 2 σ x 2 exp ( Ω T 2 q σ x 2 ) .
I ˙ ( t ) = I ( t ) [ 2 χ ˙ ( t ) X ( t ) X ˙ ( t ) + Y ( t ) Y ˙ ( t ) Ω T 2 ] ,
p ( x , x ˙ ) = 1 2 π σ x b x exp [ ( x 2 2 σ x 2 + x ˙ 2 2 b x 2 ) ] .
Z 1 ( t ) = X ( t ) X ˙ ( t ) + Y ( t ) Y ˙ ( t ) ,
p ( z 1 ) = 1 2 σ x b x exp ( | z 1 | σ x b x ) , < z 1 < .
b χ 2 ( q ) = 0.90675 k 7 6 H 5 6 sec 11 6 ( ψ ) ( k V 2 L ) ( μ 2 + 2 3 μ 3 Ω T 2 Λ 1 6 q W 2 ) , 0 q W 2 2 Ω T 2 ,
μ 2 = Re 0 H C n 2 ( h ) { Λ 1 6 ξ 1 3 ξ 1 6 [ Λ ξ + i ( 1 Θ ¯ ξ ) ] 1 6 } d h ,
μ 3 = 0 H C n 2 ( h ) ξ 1 3 d h ,
b χ 2 ( q ) = 0.90675 k 7 6 H 5 6 sec 23 6 ( ψ ) ( k ω 2 L ) ( μ 4 + 2 3 μ 5 Ω T 2 Λ 1 6 q W 2 ) , 0 q W 2 2 Ω T 2 ,
μ 4 = Re 0 H h 2 C n 2 ( h ) { Λ 1 6 ξ 1 3 ξ 1 6 [ Λ ξ + i ( 1 Θ ¯ ξ ) ] 1 6 } d h ,
μ 5 = 0 H h 2 C n 2 ( h ) ξ 1 3 d h .
Z 2 ( t ) = 2 χ ˙ ( t ) X ( t ) X ˙ ( t ) + Y ( t ) Y ˙ ( t ) Ω T 2 .
p ( z 2 | q ) = Ω T 2 2 σ x b x exp ( | Ω T 2 g | σ x b x ) { 1 2 2 π b χ 2 ( q ) exp [ ( z 2 + g ) 2 8 b χ 2 ( q ) ] } d g ,
p ( z 2 | q ) = Ω T 2 4 σ x b x exp ( z 2 2 8 b χ 2 ( q ) ) ( exp { 2 b χ 2 ( q ) [ Ω T 2 σ x b x + z 2 4 b χ 2 ( q ) ] 2 } × erfc { 2 b χ ( q ) [ Ω T 2 σ x b x + z 2 4 b χ 2 ( q ) ] } + exp { 2 b χ 2 ( q ) [ z 2 4 b χ 2 ( q ) Ω T 2 σ x b x ] 2 } × erfc { 2 b χ ( q ) [ z 2 4 b χ 2 ( q ) Ω T 2 σ x b x ] } ) , < z 2 < .
I ˙ ( t ) = I ( t ) Z 2 ( t ) ,
p ( I ˙ | I , q ) = Ω T 2 4 σ x b x I exp ( I ˙ 2 8 I 2 b χ 2 ( q ) ) ( exp { 2 b χ 2 ( q ) [ Ω T 2 σ x b x + I ˙ 4 I b χ 2 ( q ) ] 2 } × erfc { 2 b χ ( q ) [ Ω T 2 σ x b x + I ˙ 4 I b χ 2 ( q ) ] } + exp { 2 b χ 2 ( q ) [ I ˙ 4 I b χ 2 ( q ) Ω T 2 σ x b x ] 2 } × erfc { 2 b χ ( q ) [ I ˙ 4 I b χ 2 ( q ) Ω T 2 σ x b x ] } ) , < I ˙ < .
p ( I ˙ , I | q ) = p ( I ˙ | I , q ) p ( I | q ) .
p ( I ˙ , I T | q ) = Ω T 2 8 2 π σ x b x σ χ ( q ) I T 2 exp ( I ˙ 2 8 I T 2 b χ 2 ( q ) ) exp ( { 0.5 [ ln ( I T / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) × ( exp { 2 b χ 2 ( q ) [ Ω T 2 σ x b x + I ˙ 4 I T b χ 2 ( q ) ] 2 } erfc { 2 b χ ( q ) [ Ω T 2 σ x b x + I ˙ 4 I T b χ 2 ( q ) ] } + exp { 2 b χ 2 ( q ) [ I ˙ 4 I T b χ 2 ( q ) Ω T 2 σ x b x ] 2 } erfc { 2 b χ ( q ) [ I ˙ 4 I T b χ 2 ( q ) Ω T 2 σ x b x ] } ) .
n ( I T ) = 0.5 | I ˙ | p ( I ˙ , I T ) d I ˙ = 0.5 d I ˙ | I ˙ | 0 d q p ( I ˙ , I T | q ) p ( q ) .
n ( I T ) = 0 d q p ( q ) d I ˙ [ 0.5 | I ˙ | p ( I ˙ , I T | q ) ] .
n ( I T | q ) = 0.5 | I ˙ | p ( I ˙ , I T | q ) d q = 1 4 2 π σ χ 2 ( q ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] exp ( { 0.5 [ ln ( I T / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) × { 2 2 π b χ ( q ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] + σ x b x Ω T 2 erfc [ 2 Ω T 2 b χ ( q ) σ x b x ] } .
n ( I T ) = 0.25 2 π 0 ( Ω T 2 σ x 2 σ χ ( q ) exp ( Ω T 2 q σ x 2 ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] exp ( { 0.5 [ ln ( I T / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) × { 2 2 π b χ ( q ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] + σ x b x Ω T 2 erfc [ 2 Ω T 2 b χ ( q ) σ x b x ] } ) d q .
n ( I T ) min = 0 W 2 2 Ω T 2 n ( I T | q ) p ( q ) d q + W 2 2 Ω T 2 n ( I T | q ) p ( q ) d q ,
n ( I T ) min = 0.25 2 π 0 W 2 2 Ω T 2 ( Ω T 2 σ x 2 σ χ ( q ) exp ( Ω T 2 q σ x 2 ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] × exp ( { 0.5 [ ln ( I T / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) × { 2 2 π b χ ( q ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] + σ x b x Ω T 2 erfc [ 2 Ω T 2 b χ ( q ) σ x b x ] } ) d q .
n ( I T ) max = 0 W 2 2 Ω T 2 n ( I T | q ) p ( q ) d q + W 2 2 Ω T 2 n ( I T | q = W 2 2 Ω T 2 ) p ( q ) d q ,
n ( I T ) max = 0.25 2 π 0 W 2 2 Ω T 2 ( Ω T 2 σ x 2 σ χ ( q ) exp ( Ω T 2 q σ x 2 ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] × exp ( { 0.5 [ ln ( I T / K T ) + q ] + σ χ 2 ( q ) } 2 2 σ χ 2 ( q ) ) × { 2 2 π b χ ( q ) exp [ 2 Ω T 4 b χ 2 ( q ) σ x 2 b x 2 ] + σ x b x Ω T 2 erfc [ 2 Ω T 2 b χ ( q ) σ x b x ] } ) d q + 0.25 2 π Ω T 2 σ x 2 σ χ ( W 2 / 2 Ω T 2 ) exp [ 2 Ω T 4 b χ 2 ( W 2 / 2 Ω T 2 ) σ x 2 b x 2 ] × exp ( { 0.5 [ ln ( I T / K T ) + ( W 2 / 2 Ω T 2 ) ] + σ χ 2 ( W 2 / 2 Ω T 2 ) } 2 2 σ χ 2 ( W 2 / 2 Ω T 2 ) ) × { 2 2 π b χ ( W 2 / 2 Ω T 2 ) exp [ 2 Ω T 4 b χ 2 ( W 2 / 2 Ω T 2 ) σ x 2 b x 2 ] + σ x b x Ω T 2 erfc [ 2 Ω T 2 b χ ( W 2 / 2 Ω T 2 ) σ x b x ] } × W 2 2 Ω T 2 exp ( Ω T 2 q σ x 2 ) d q .
t ( I T ) = P f ( I T ) n ( I T ) .
F T = 10 log ( K T / I T ) .

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