Abstract

Random pointing errors in coherent ladar tend to cause a reduction in measured signal power due to misalignment among the transmitter, receiver, and (hard) target. A simple model for the size of this impact, in terms of the size of the pointing error, would be useful in the design and evaluation of coherent ladar systems. To be most applicable to monostatic systems, the model should also include correlation between transmitter and receiver pointing errors. We derive an analytic expression for the reduction in average signal power, which we call pointing efficiency, based on Gaussian beam coherent ladar with Gaussian pointing errors that includes arbitrary correlation between transmitter and receiver pointing errors.

© 2011 Optical Society of America

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References

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  1. D. G. Youmans and R. Robertson, “Modelocked-laser laser radar performance in the detection of TMD and NMD targets,” ADA329046 (Defense Technology Information Center, 1997).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

1994

1991

1989

C. Chen and C. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

1984

1983

D. M. Papurt, J. H. Shapiro, and S. T. Lau, “Measured turbulence and speckle effects in laser radar target returns,” Proc. SPIE 415, 166–178 (1983).

1978

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Chen, C.

C. Chen and C. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

Frehlich, R. G.

Fukumitsu, O.

Gardner, C.

C. Chen and C. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

Halldorsson, T.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Kavaya, Michael J.

Langerholc, J.

Lau, S. T.

D. M. Papurt, J. H. Shapiro, and S. T. Lau, “Measured turbulence and speckle effects in laser radar target returns,” Proc. SPIE 415, 166–178 (1983).

Papurt, D. M.

D. M. Papurt, J. H. Shapiro, and S. T. Lau, “Measured turbulence and speckle effects in laser radar target returns,” Proc. SPIE 415, 166–178 (1983).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Robertson, R.

D. G. Youmans and R. Robertson, “Modelocked-laser laser radar performance in the detection of TMD and NMD targets,” ADA329046 (Defense Technology Information Center, 1997).

Shapiro, J. H.

D. M. Papurt, J. H. Shapiro, and S. T. Lau, “Measured turbulence and speckle effects in laser radar target returns,” Proc. SPIE 415, 166–178 (1983).

Takenaka, T.

Tanaka, K.

Wang, J. Y.

Youmans, D. G.

D. G. Youmans and R. Robertson, “Modelocked-laser laser radar performance in the detection of TMD and NMD targets,” ADA329046 (Defense Technology Information Center, 1997).

Yura, H. T.

Appl. Opt.

IEEE Trans. Commun.

C. Chen and C. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

Proc. SPIE

D. M. Papurt, J. H. Shapiro, and S. T. Lau, “Measured turbulence and speckle effects in laser radar target returns,” Proc. SPIE 415, 166–178 (1983).

Other

D. G. Youmans and R. Robertson, “Modelocked-laser laser radar performance in the detection of TMD and NMD targets,” ADA329046 (Defense Technology Information Center, 1997).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Pointing efficiency versus normalized pointing jitter, uncorrelated pointing errors, no pointing bias.

Fig. 2
Fig. 2

Pointing efficiency versus normalized pointing jitter, totally correlated pointing errors, no pointing bias.

Fig. 3
Fig. 3

Pointing efficiency versus normalized pointing jitter, uncorrelated pointing errors, transmit pointing bias [ ( μ x t , μ y t ) / ω = ( 1 , 1 ) ]).

Fig. 4
Fig. 4

Pointing efficiency versus normalized pointing jitter, totally correlated pointing errors, transmit pointing bias [ ( μ x t , μ y t ) / ω = ( 1 , 1 ) ].

Fig. 5
Fig. 5

Pointing efficiency versus correlation, modest pointing jitter [ σ / ω = 1 / 2 ], no pointing bias.

Fig. 6
Fig. 6

Pointing efficiency versus correlation, modest pointing jitter ( σ / ω = 1 / 2 ), transmit pointing bias [ ( μ x t , μ y t ) / ω = ( 1 , 1 ) ].

Fig. 7
Fig. 7

Pointing efficiency versus round-trip time, modest pointing jitter ( σ / ω = 1 / 2 ), no pointing bias.

Equations (56)

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P sig J xmt ( x , y ) J bplo ( x , y ) ρ Trg ( x , y ) d x d y ,
P sig ( ξ , ζ ) = J xmt ( x , y ; ξ ) J bplo ( x , y ; ζ ) ρ Trg ( x , y ) d x d y ,
P sig = E [ J xmt ( x , y ; ξ ) J bplo ( x , y ; ζ ) ρ Trg ( x , y ) d x d y ] ,
η = E [ J xmt ( x , y ; ξ ) J bplo ( x , y ; ζ ) ρ Trg ( x , y ) d x d y ] J xmt ( x , y ; 0 ) J bplo ( x , y ; 0 ) ρ Trg ( x , y ) d x d y ,
η = η x η y ,
η x = exp [ 2 [ ( ω x t 2 + 4 σ x t 2 ) μ x b 2 + ( ω x b 2 + 4 σ x b 2 ) μ x t 2 8 ρ x σ x t σ x b μ x t μ x b + ω x T 2 ( μ x t μ x b ) 2 ] ω x t 2 ω x b 2 + 4 ω x t 2 σ x b 2 + 4 ω x b 2 σ x t 2 + 16 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 σ x t 2 + 4 σ x b 2 8 ρ x σ x t σ x b ) ] ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) 1 / 2 ( ω x t 2 ω x b 2 + 4 ω x t 2 σ x b 2 + 4 ω x b 2 σ x t 2 + 16 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 σ x t 2 + 4 σ x b 2 8 ρ x σ x t σ x b ) ) 1 / 2 ,
η = exp [ 2 [ ( ω 2 + 4 σ 2 ) ( μ x t 2 + μ x b 2 + μ y t 2 + μ y b 2 ) 8 ρ σ 2 ( μ x t μ x b + μ y t μ y b ) + ω T 2 ( ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ) ] ω 4 + 8 ω 2 σ 2 + 16 σ 4 ( 1 ρ 2 ) + 2 ω T 2 ( ω 2 + 4 σ 2 ( 1 ρ ) ) ] ω 2 ( 2 ω T 2 + ω 2 ) ω 4 + 8 ω 2 σ 2 + 16 σ 4 ( 1 ρ 2 ) + 2 ω T 2 ( ω 2 + 4 σ 2 ( 1 ρ ) ) .
η = exp [ 2 [ ( ω 2 + 4 σ 2 ) ( μ x t 2 + μ x b 2 + μ y t 2 + μ y b 2 ) 8 ρ σ 2 ( μ x t μ x b + μ y t μ y b ) ] ( ω 2 + 4 σ 2 ) 2 16 ρ 2 σ 4 ] ω 4 ( ω 2 + 4 σ 2 ) 2 16 ρ 2 σ 4 .
η = exp [ 2 ( r t 2 + r b 2 ) ( ω 2 + 4 σ 2 ) ] ω 4 ( ω 2 + 4 σ 2 ) 2 ,
η = exp [ 2 [ ω 2 ( μ x t 2 + μ x b 2 + μ y t 2 + μ y b 2 ) + 4 σ 2 ( ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ) ] ω 2 ( ω 2 + 8 σ 2 ) ] ω 2 ( ω 2 + 8 σ 2 ) .
η = exp [ 2 [ ω 2 ( μ x t 2 + μ x b 2 + μ y t 2 + μ y b 2 ) + 4 σ 2 ( ( μ x t + μ x b ) 2 + ( μ y t + μ y b ) 2 ) ] ω 2 ( ω 2 + 8 σ 2 ) ] ω 2 ( ω 2 + 8 σ 2 ) .
η = exp [ ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ω 2 + 4 σ 2 ( 1 ρ ) ] ω 2 ω 2 + 4 σ 2 ( 1 ρ ) .
η = exp [ ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ω 2 + 4 σ 2 ] ω 2 ω 2 + 4 σ 2 .
η = exp [ ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ω 2 ] .
η = exp [ ( μ x t μ x b ) 2 + ( μ y t μ y b ) 2 ω 2 + 8 σ 2 ] ω 2 ω 2 + 8 σ 2 .
ρ ( t ) = exp [ t 2 t c 2 ] ,
η = E [ J xmt ( x , y ; ξ ) J bplo ( x , y ; ζ ) ρ Trg ( x , y ) d x d y ] J xmt ( x , y ; 0 ) J bplo ( x , y ; 0 ) ρ Trg ( x , y ) d x d y ,
J xmt ( x , y ; x t , y t ) = e 2 ( x x t ) 2 ω x t 2 e 2 ( y y t ) 2 ω y t 2 .
ρ Trg ( x , y ) = e 2 x 2 ω x T 2 e 2 y 2 ω y T 2 .
E [ e 2 ( x x t ) 2 ω x t 2 e 2 ( y y t ) 2 ω y t 2 e 2 ( x x b ) 2 ω x b 2 e 2 ( y y b ) 2 ω y b 2 e 2 x 2 ω x T 2 e 2 y 2 ω y T 2 d x d y ] ,
e 2 x 2 ω x t 2 e 2 y 2 ω y t 2 e 2 x 2 ω x b 2 e 2 y 2 ω y b 2 e 2 x 2 ω x T 2 e 2 y 2 ω y T 2 d x d y
η = E [ e 2 ( x x t ) 2 ω x t 2 e 2 ( y y t ) 2 ω y t 2 e 2 ( x x b ) 2 ω x b 2 e 2 ( y y b ) 2 ω y b 2 e 2 x 2 ω x T 2 e 2 y 2 ω y T 2 d x d y ] e 2 x 2 ω x t 2 e 2 x 2 ω y t 2 e 2 x 2 ω x b 2 e 2 x 2 ω y b 2 e 2 x 2 ω x T 2 e 2 y 2 ω y T 2 d x d y .
e 2 ( x x t ) 2 ω x t 2 e 2 ( x x b ) 2 ω x b 2 e 2 x 2 ω x T 2 d x .
( x x t ) 2 ω x t 2 + ( x x b ) 2 ω x b 2 + x 2 ω x T 2 = ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) x 2 2 ω x T 2 ( ω x t 2 x b + ω x b 2 x t ) x + ω x T 2 ( ω x t 2 x b 2 + ω x b 2 x t 2 ) ω x T 2 ω x t 2 ω x b 2 .
α ( x 2 2 β α x ) + γ = α ( x β α ) 2 β 2 α + γ = α ( x β α ) 2 + α γ β 2 α .
α γ β 2 α = ω x T 2 ω x t 2 ω x b 2 ( ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ) ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 .
e 2 ( x x t ) 2 ω x t 2 e 2 ( x x b ) 2 ω x b 2 e 2 x 2 ω x T 2 d x = e 2 ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 e 2 α ( x β α ) 2 ω x T 2 ω x t 2 ω x b 2 d x = 2 π ω x T ω x t ω x b 2 ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) 1 / 2 e 2 ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 .
η x = E [ e 2 ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ] .
η = E [ e 2 ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 e 2 ω y T 2 ( y t y b ) 2 + ω y t 2 y b 2 + ω y b 2 y t 2 ω y T 2 ω y t 2 + ω y T 2 ω y b 2 + ω y t 2 ω y b 2 ] .
f ( x t , x b , y t , y b ) = e 2 ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 e 2 ω y T 2 ( y t y b ) 2 + ω y t 2 y b 2 + ω y b 2 y t 2 ω y T 2 ω y t 2 + ω y T 2 ω y b 2 + ω y t 2 ω y b 2 ,
p ( x t , y t , x b , y b ) = 1 2 π | Σ x | 1 2 π | Σ y | exp ( 1 2 ( x μ x ) T Σ x 1 ( x μ x ) ) exp ( 1 2 ( y μ y ) T Σ y 1 ( y μ y ) ) ,
Σ x = ( σ x t 2 ρ x σ x t σ x b ρ x σ x t σ x b σ x b 2 )
η = f ( x t , x b , y t , y b ) 1 2 π | Σ x | 1 2 π | Σ y | e 1 2 ( x μ x ) T Σ x 1 ( x μ x ) e 1 2 ( y μ y ) T Σ y 1 ( y μ y ) d x t d x b d y t d y b .
μ n = e 2 n ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 e 2 n ω y T 2 ( y t y b ) 2 + ω y t 2 y b 2 + ω y b 2 y t 2 ω y T 2 ω y t 2 + ω y T 2 ω y b 2 + ω y t 2 ω y b 2 1 2 π | Σ x | 1 2 π | Σ y | e 1 2 ( x μ x ) T Σ x 1 ( x μ x ) e 1 2 ( y μ y ) T Σ y 1 ( y μ y ) d x t d x b d y t d y b .
μ n , x = 1 2 π | Σ x | e 2 n ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 e 1 2 ( x μ x ) T Σ x 1 ( x μ x ) d x t d x b .
μ n , x = 1 2 π | Σ x | e 1 2 x T ( n A ) x e 1 2 ( x μ x ) T Σ x 1 ( x μ x ) d x t d x b ,
A = 4 ω x T 2 α ( 1 1 1 1 ) + 4 α ( ω x b 2 0 0 ω x t 2 ) ,
x T ( n A ) x + ( x μ x ) T Σ x 1 ( x μ x ) = ( x C μ x ) T ( n A + Σ x 1 ) ( x C μ x ) + μ x T B μ x ,
x T ( n A ) x = 4 n ω x T 2 ( x t x b ) 2 + ω x t 2 x b 2 + ω x b 2 x t 2 ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2
μ n , x = 1 2 π | Σ x | e 1 2 ( x C μ x ) T ( n A + Σ x 1 ) ( x C μ x ) e 1 2 μ x T B μ x d x t d x b = e 1 2 μ x T B μ x | n A + Σ x 1 | | Σ x | 1 2 π ( | n A + Σ x 1 | ) 1 e 1 2 ( x C μ x ) T ( n A + Σ x 1 ) ( x C μ x ) d x t d x b .
μ n , x = e 1 2 μ x T B μ x | n A + Σ x 1 | | Σ x | .
| Σ x | = ( 1 ρ x 2 ) σ x t 2 σ x b 2 .
Σ x 1 = 1 1 ρ x 2 ( 1 σ x t 2 ρ σ x t σ x b ρ σ x t σ x b 1 σ x b 2 ) .
n A + Σ x 1 = 1 α ( 1 ρ x 2 ) ( 4 n ( 1 ρ x 2 ) σ x t 2 ( ω x T 2 + ω x b 2 ) + α σ x t 2 4 n ω x T 2 σ x t σ x b ( 1 ρ x 2 ) α ρ x σ x t σ x b 4 n ω x T 2 σ x t σ x b ( 1 ρ x 2 ) α ρ x σ x t σ x b 4 n ( 1 ρ x 2 ) σ x b 2 ( ω x T 2 + ω x t 2 ) + α σ x b 2 ) .
| n A + Σ x 1 | = 1 α 2 ( 1 ρ x 2 ) 2 σ x t 2 σ x b 2 [ 16 n 2 ( 1 ρ x 2 ) 2 σ x t 2 σ x b 2 ( ω x T 2 + ω x t 2 ) ( ω x T 2 + ω x b 2 ) + 4 n α ( 1 ρ x 2 ) 2 σ x t 2 ( ω x T 2 + ω x b 2 ) + 4 n α ( 1 ρ x 2 ) 2 σ x b 2 ( ω x T 2 + ω x t 2 ) + α 2 16 n 2 ω x T 4 ( 1 ρ x 2 ) 2 σ x t 2 σ x b 2 α 2 ρ x 2 8 n α ( 1 ρ x 2 ) ρ x ω x T 2 σ x t σ x b ] = α ( 1 ρ x 2 ) ( α + 4 n σ x t 2 ( ω x T 2 + ω x b 2 ) + 4 n σ x b 2 ( ω x T 2 + ω x t 2 ) + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) 8 n ρ x ω x T 2 σ x t σ x b ) α 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) 2 = ω x t 2 ω x b 2 + 4 n ω x t 2 σ x b 2 + 4 n ω x b 2 σ x t 2 + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 n σ x t 2 + 4 n σ x b 2 8 n ρ x σ x t σ x b ) α σ x t 2 σ x b 2 ( 1 ρ x 2 ) ,
κ = ω x t 2 ω x b 2 + 4 n ω x t 2 σ x b 2 + 4 n ω x b 2 σ x t 2 + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 n σ x t 2 + 4 n σ x b 2 8 n ρ x σ x t σ x b ) ,
1 | n A + Σ x 1 | | Σ x | = α κ = ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) 1 / 2 ( ω x t 2 ω x b 2 + 4 n ω x t 2 σ x b 2 + 4 n ω x b 2 σ x t 2 + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 n σ x t 2 + 4 n σ x b 2 8 n ρ x σ x t σ x b ) ) 1 / 2 .
B Σ x ( n A + Σ x 1 ) = ( Σ x 1 Σ x 1 n ( A + Σ x 1 ) 1 Σ x 1 ) Σ x ( n A + Σ x 1 ) = ( I Σ x 1 ( n A + Σ x 1 ) 1 ) ( n A + Σ x 1 ) = n A + Σ x 1 Σ x 1 = n A .
B = n A ( n A + Σ x 1 ) 1 Σ x 1 .
( n A + Σ x 1 ) 1 = 1 | n A + Σ x 1 | 1 α ( 1 ρ x 2 ) ( 4 n ( 1 ρ x 2 ) σ x b 2 ( ω x T 2 + ω x t 2 ) + α σ x b 2 4 n ω x T 2 σ x t σ x b ( 1 ρ x 2 ) + α ρ x σ x t σ x b 4 n ω x T 2 σ x t σ x b ( 1 ρ x 2 ) + α ρ x σ x t σ x b 4 n ( 1 ρ x 2 ) σ x t 2 ( ω x T 2 + ω x b 2 ) + α σ x t 2 ) = 1 κ ( 4 n σ x t 2 σ x b 2 ( 1 ρ x 2 ) ( ω x T 2 + ω x t 2 ) + α σ x t 2 4 n ω x T 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + α ρ x σ x t σ x b 4 n ω x T 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + α ρ x σ x t σ x b 4 n σ x t 2 σ x b 2 ( 1 ρ x 2 ) ( ω x T 2 + ω x b 2 ) + α σ x b 2 ) = 4 n σ x t 2 σ x b 2 ( 1 ρ x 2 ) κ ( ω x T 2 + ω x t 2 ω x T 2 ω x T 2 ω x T 2 + ω x b 2 ) + α κ ( σ x t 2 ρ x σ x t σ x b ρ x σ x t σ x b σ x b 2 ) .
( n A + Σ x 1 ) 1 = 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) κ ( n A ) 1 + α κ Σ x .
B = 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) κ Σ x 1 + n α κ A = 16 n 2 κ ( σ x b 2 ρ x σ x t σ x b ρ x σ x t σ x b σ x t 2 ) + 1 κ ( 4 n ( ω x T 2 + ω x b 2 ) 4 n ω x T 2 4 n ω x T 2 4 n ( ω x T 2 + ω x t 2 ) ) .
μ x T B μ x = 16 n 2 κ ( σ x b 2 μ x t 2 2 ρ x σ x t σ x b μ x t μ x b + σ x t 2 μ x b 2 ) + 1 κ ( 4 n ( ω x T 2 + ω x b 2 ) μ x t 2 8 n ω x T 2 μ x t μ x b + 4 n ( ω x T 2 + ω x t 2 ) μ x b 2 ) = 4 n κ [ ( ω x t 2 + 4 n σ x t 2 ) μ x b 2 + ( ω x b 2 + 4 n σ x b 2 ) μ x t 2 8 n ρ x σ x t σ x b μ x t μ x b + ω x T 2 ( μ x t μ x b ) 2 ] ,
exp ( 1 2 μ x T B μ x ) = exp ( 2 n κ [ ( ω x t 2 + 4 n σ x t 2 ) μ x b 2 + ( ω x b 2 + 4 n σ x b 2 ) μ x t 2 8 n ρ x σ x t σ x b μ x t μ x b + ω x T 2 ( μ x t μ x b ) 2 ] ) .
μ n , x = exp [ 2 n [ ( ω x t 2 + 4 n σ x t 2 ) μ x b 2 + ( ω x b 2 + 4 n σ x b 2 ) μ x t 2 8 n ρ x σ x t σ x b μ x t μ x b + ω x T 2 ( μ x t μ x b ) 2 ] ω x t 2 ω x b 2 + 4 n ω x t 2 σ x b 2 + 4 n ω x b 2 σ x t 2 + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 n σ x t 2 + 4 n σ x b 2 8 n ρ x σ x t σ x b ) ] ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) 1 / 2 ( ω x t 2 ω x b 2 + 4 n ω x t 2 σ x b 2 + 4 n ω x b 2 σ x t 2 + 16 n 2 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 n σ x t 2 + 4 n σ x b 2 8 n ρ x σ x t σ x b ) ) 1 / 2 .
η x = exp [ 2 [ ( ω x t 2 + 4 σ x t 2 ) μ x b 2 + ( ω x b 2 + 4 σ x b 2 ) μ x t 2 8 ρ x σ x t σ x b μ x t μ x b + ω x T 2 ( μ x t μ x b ) 2 ] ω x t 2 ω x b 2 + 4 ω x t 2 σ x b 2 + 4 ω x b 2 σ x t 2 + 16 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 σ x t 2 + 4 σ x b 2 8 ρ x σ x t σ x b ) ] ( ω x T 2 ω x t 2 + ω x T 2 ω x b 2 + ω x t 2 ω x b 2 ) 1 / 2 ( ω x t 2 ω x b 2 + 4 ω x t 2 σ x b 2 + 4 ω x b 2 σ x t 2 + 16 σ x t 2 σ x b 2 ( 1 ρ x 2 ) + ω x T 2 ( ω x t 2 + ω x b 2 + 4 σ x t 2 + 4 σ x b 2 8 ρ x σ x t σ x b ) ) 1 / 2 .

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