Abstract

The outage capacity of slow-fading free-space optical channels is analyzed for a multiple-input/single-output configuration in the presence of atmospheric and misalignment fading. A spatial repetition code is considered at the transmitter and a closed-form expression for the outage capacity is developed. In addition, a simple asymptotic closed-form expression is derived at high signal-to-noise ratio. Two methods are considered for system design using the derived outage capacity results with different beam configurations. The outage capacity is optimized over a predetermined set using numerical techniques. Using the asymptotic form of the outage capacity, however, a closed form for the optimum beamwidth is derived. A comparison of simulation results in both cases gives very similar performance indicating the effectiveness of the asymptotic form to provide a near-optimum expression for the beamwidth.

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References

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  1. J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265‒298 (1997).
    [CrossRef]
  2. X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 293‒1300 (2002).
  3. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
    [CrossRef]
  4. M. Razavi and J. H. Shapiro, "Wireless optical communications via diversity reception and optical preamplification," IEEE Trans. Wireless Commun. 4, 975‒983 (2005).
    [CrossRef]
  5. S. M. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346‒1356 (2003).
    [CrossRef]
  6. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
    [CrossRef]
  7. fSONA Optical Wireless [Online]. Available: http://www.fsona.com
  8. A. A. Farid and S. Hranilovic, "Diversity gain and outage probability for MIMO free-space optical links with misalignment," IEEE Trans. Commun. (in press).
  9. A. A. Farid and S. Hranilovic, "Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment," Proc. IEEE GlobeCom, Dec. 2010, Miami, FL, USA, pp. 1015‒1019.
  10. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
    [CrossRef]
  11. S. Arnon, "Effects of atmospheric turbulence and building sway on optical wireless communication systems," Opt. Lett. 28, 129‒131 (2003).
    [CrossRef] [PubMed]
  12. A. A. Farid and S. Hranilovic, "Outage capacity optimization for free-space optical links with pointing errors," J. Lightwave Technol. 25, 1702‒1710 (2007).
    [CrossRef]
  13. MRV Communications Inc. [Online]. Available: http://www.mrv.com
  14. S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
    [CrossRef]
  15. C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379‒423 (1948).
  16. L. C. Andrews, Special Functions of Mathematics for Engineers, McGraw-Hill, New York, USA, 1992.
  17. D. Bushuev and S. Arnon, "Analysis of the performance of a wireless optical multi-input to multi-output communication system," J. Opt. Soc. Am. A 23, 1722‒1730 (2006).
    [CrossRef]
  18. I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).
  19. S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components," Bell Syst. Tech. J. 61, 1441‒1462 (1982).

2009 (1)

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

2007 (3)

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
[CrossRef]

A. A. Farid and S. Hranilovic, "Outage capacity optimization for free-space optical links with pointing errors," J. Lightwave Technol. 25, 1702‒1710 (2007).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

2006 (1)

2005 (2)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

M. Razavi and J. H. Shapiro, "Wireless optical communications via diversity reception and optical preamplification," IEEE Trans. Wireless Commun. 4, 975‒983 (2005).
[CrossRef]

2003 (3)

S. M. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346‒1356 (2003).
[CrossRef]

S. Arnon, "Effects of atmospheric turbulence and building sway on optical wireless communication systems," Opt. Lett. 28, 129‒131 (2003).
[CrossRef] [PubMed]

I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).

2002 (1)

X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 293‒1300 (2002).

1997 (1)

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265‒298 (1997).
[CrossRef]

1982 (1)

S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components," Bell Syst. Tech. J. 61, 1441‒1462 (1982).

1948 (1)

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379‒423 (1948).

Andrews, L. C.

L. C. Andrews, Special Functions of Mathematics for Engineers, McGraw-Hill, New York, USA, 1992.

Anguita, J. A.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
[CrossRef]

Arnon, S.

Barry, J. R.

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265‒298 (1997).
[CrossRef]

Brandt-Pearce, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Bushuev, D.

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Farid, A. A.

A. A. Farid and S. Hranilovic, "Outage capacity optimization for free-space optical links with pointing errors," J. Lightwave Technol. 25, 1702‒1710 (2007).
[CrossRef]

A. A. Farid and S. Hranilovic, "Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment," Proc. IEEE GlobeCom, Dec. 2010, Miami, FL, USA, pp. 1015‒1019.

A. A. Farid and S. Hranilovic, "Diversity gain and outage probability for MIMO free-space optical links with misalignment," IEEE Trans. Commun. (in press).

Haas, S. M.

S. M. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346‒1356 (2003).
[CrossRef]

Hranilovic, S.

A. A. Farid and S. Hranilovic, "Outage capacity optimization for free-space optical links with pointing errors," J. Lightwave Technol. 25, 1702‒1710 (2007).
[CrossRef]

A. A. Farid and S. Hranilovic, "Diversity gain and outage probability for MIMO free-space optical links with misalignment," IEEE Trans. Commun. (in press).

A. A. Farid and S. Hranilovic, "Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment," Proc. IEEE GlobeCom, Dec. 2010, Miami, FL, USA, pp. 1015‒1019.

Kahn, J.

X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 293‒1300 (2002).

Kahn, J. M.

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265‒298 (1997).
[CrossRef]

Karagiannidis, G. K.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

Kavehrad, M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

Kim, I.

I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).

Korevaar, I.

I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).

Leveque, J. H.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

McArthur, B.

I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).

Navidpour, S. M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

Neifeld, M. A.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
[CrossRef]

Razavi, M.

M. Razavi and J. H. Shapiro, "Wireless optical communications via diversity reception and optical preamplification," IEEE Trans. Wireless Commun. 4, 975‒983 (2005).
[CrossRef]

Sandalidis, H. G.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

Schwartz, S. C.

S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components," Bell Syst. Tech. J. 61, 1441‒1462 (1982).

Shannon, C. E.

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379‒423 (1948).

Shapiro, J. H.

M. Razavi and J. H. Shapiro, "Wireless optical communications via diversity reception and optical preamplification," IEEE Trans. Wireless Commun. 4, 975‒983 (2005).
[CrossRef]

S. M. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346‒1356 (2003).
[CrossRef]

Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

Uysal, M.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

Vasic, B. V.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
[CrossRef]

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Yeh, Y. S.

S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components," Bell Syst. Tech. J. 61, 1441‒1462 (1982).

Zhu, X.

X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 293‒1300 (2002).

Bell Syst. Tech. J. (2)

C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379‒423 (1948).

S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components," Bell Syst. Tech. J. 61, 1441‒1462 (1982).

IEEE J. Sel. Areas Commun. (1)

S. M. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346‒1356 (2003).
[CrossRef]

IEEE Trans. Commun. (3)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 293‒1300 (2002).

A. A. Farid and S. Hranilovic, "Diversity gain and outage probability for MIMO free-space optical links with misalignment," IEEE Trans. Commun. (in press).

IEEE Trans. Wireless Commun. (3)

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical wireless links with spatial diversity over strong atmospheric turbulence channels," IEEE Trans. Wireless Commun. 8, 951‒957 (2009).
[CrossRef]

M. Razavi and J. H. Shapiro, "Wireless optical communications via diversity reception and optical preamplification," IEEE Trans. Wireless Commun. 4, 975‒983 (2005).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

J. Appl. Opt. (1)

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link," J. Appl. Opt. 46, 6561‒6571 (2007).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Lett. (1)

Proc. IEEE (1)

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265‒298 (1997).
[CrossRef]

Proc. SPIE (1)

I. Korevaar, I. Kim, and B. McArthur, "Atmospheric propagation characteristics of highest importance to commercial free space optics," Proc. SPIE 4976, 1‒12 (2003).

Other (4)

L. C. Andrews, Special Functions of Mathematics for Engineers, McGraw-Hill, New York, USA, 1992.

MRV Communications Inc. [Online]. Available: http://www.mrv.com

A. A. Farid and S. Hranilovic, "Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment," Proc. IEEE GlobeCom, Dec. 2010, Miami, FL, USA, pp. 1015‒1019.

fSONA Optical Wireless [Online]. Available: http://www.fsona.com

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Figures (7)

Fig. 1
Fig. 1

Transmitter #1 footprint at the receiver plane. The beamwidth is w and the aperture radius is a. P 1 is the center of the footprint in the absence of misalignment. The random misalignment displacements in the x- and y-axis are X and Y , respectively. Q 1 is the center of the footprint in the presence of misalignment. A detector of radius a is centered at the origin. The displacement between Q 1 and the detector center is R 1 .

Fig. 2
Fig. 2

(Color online) Probability of outage versus SNR for a 4 × 1 MISO FSO system. Numerical simulation (, Eq. (22)), closed-form expression (*, Eq. (24)) and the asymptotic outage probability at high SNR (, Eq. (28)).

Fig. 3
Fig. 3

(Color online) Probability of outage versus SNR for 4 × 1 and 2 × 1 MISO and 1 × 1 SISO FSO systems for R 0 = 0 . 5 , w o = 2 cm , σ X 2 = 0 . 2 , and misalignment variance σ s 2 = 0 . 1 .

Fig. 4
Fig. 4

(Color online) Probability of outage versus SNR for a 4 × 1 MISO FSO system for tilted and parallel beam configurations with σ s 2 = 0 . 02 and 0 . 1 .

Fig. 5
Fig. 5

(Color online) Probability of outage versus SNR for a 4 × 1 MISO FSO system with R 0 = 0 . 5 and w o W o = { 1 , 1 . 2 , , 3 } cm , where P out ( R 0 ) is given in Eq. (34) and σ s 2 = 0 . 1 .

Fig. 6
Fig. 6

(Color online) Probability of outage versus SNR for a 4 × 1 MISO FSO system. Both P out ( R 0 ) W o and P out Asy ( R 0 ) are presented. In all cases σ s 2 = 0 . 1 .

Fig. 7
Fig. 7

(Color online) The optimum beam waist w o W o = { 1 , 1 . 2 , , 3 } and the asymptotic beam waist w o Asy , Eq. (33), versus SNR for a 4 × 1 MISO FSO system.

Tables (1)

Tables Icon

Table I System Parameters

Equations (85)

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Y = h T x + Z ,
Y = H X + Z ,
H = m = 1 M H m .
H m = H m a H m p ,
SNR ( h ) = P h σ .
SNR 0 = P σ .
H m a = e X m ,
σ X 2 = 1 . 23 C n 2 k 7 / 6 L 11 / 6 ,
w L = w o ( Θ o + Λ o ) ( 1 + 1 . 63 σ R 12 / 5 Λ 1 ) ,
H m p A 0 e 2 R m 2 / w 2 ,
ζ = π a 2 w L , A 0 = erf ( ζ ) 2 , w 2 = w L 2 π erf ( ζ ) 2 v exp ( ζ 2 ) ,
w = M ( w o ) ,
H m = H m a H m p = A 0 e X m 2 R m 2 / w 2 ,
H = m = 1 M H m = m = 1 M A 0 e X m 2 R m 2 / w 2 .
Q m = X Y + P m .
P m 2 = d 2 = constant .
R m 2 = Q m 2 = X Y 2 R 0 2 + 2 X Y T P m + P m 2 d 2 ,
H = A 0 e 2 R 0 2 / w 2 e 2 d 2 / w 2 m = 1 M e X m U m ,
U m = 4 w 2 X Y T P m .
e G = m = 1 M e X m U m ,
H = A 0 e 2 d 2 / w 2 e G T ,
T = 2 R 0 2 w 2
f T ( t ) = γ 2 e γ 2 t ,
γ = w / ( 2 σ s ) .
H = A 0 e 2 d 2 / w 2 e V ,
f V ( v ) = 0 f V | T ( v | t ) f T ( t ) d t , = 0 1 2 π σ G e ( v ( μ G t ) ) 2 2 σ G 2 γ 2 e γ 2 t d t , = B 1 e γ 2 v erfc v + B 2 2 σ G ,
B 1 = γ 2 2 e γ 4 σ G 2 / 2 γ 2 μ G and B 2 = γ 2 σ G 2 μ G .
C ( SNR ( h ) ) = x f Y | H , X ( y | h , x ) p X ( x ) log 2 f Y | H , X ( y | h , x ) f Y | H ( y | h ) d y ,
p X ( x ) = 0 . 5 δ ( x ) + 0 . 5 δ ( x 2 P ) ,
f Y | H , X ( y | h , x ) = N ( h x , σ 2 ) ,
f Y | H ( y | h ) = x f Y | H , X ( y | h , x ) p X ( x ) ,
P out ( R 0 ) = Prob ( C ( SNR ( H ) ) < R 0 ) .
P out ( R 0 ) = Prob ( SNR ( H ) < SNR R 0 ) , = Prob H < SNR R 0 SNR 0 , = Prob A 0 e 2 d 2 / w 2 e V < SNR R 0 SNR 0 , = Prob ( V < log e 2 d 2 / w 2 A 0 SNR R 0 SNR 0 η ) .
P out ( R 0 ) = η B 1 e γ 2 v erfc v + B 2 2 σ G d v .
e b u erfc ( a u ) d u = 1 b e b u erfc ( a u ) e b 2 4 a 2 erf b 2 a a u
P out ( R 0 ) = B 1 e γ 2 B 2 γ 2 e γ 2 ( η + B 2 ) erfc η + B 2 2 σ G + e γ 4 σ G 2 / 2 erfc γ 2 σ G 2 η + B 2 2 σ G .
P out ( R 0 ) = 1 2 e γ 4 σ G 2 / 2 e γ 2 ( η + B 2 ) 2 erfc ( η + B 2 ) 2 σ G + e γ 4 σ G 2 / 2 erfc γ 2 σ G 2 η + B 2 2 σ G .
P out Asy ( R 0 ) 1 2 e γ 4 σ G 2 / 2 e γ 2 ( η + B 2 ) 2 e ( η + B 2 ) 2 2 σ G 2 + e γ 4 σ G 2 / 2 e γ 2 σ G 2 η + B 2 2 σ G 2 ,
P out Asy ( R 0 ) e γ 2 η + γ 4 σ G 2 / 2 γ 2 μ G ,
P out Asy ( R 0 ) e γ 4 σ G 2 / 2 γ 2 μ G e 2 d 2 / w 2 A 0 SNR R 0 SNR 0 γ 2 .
P out ( R 0 ) = min w P out ( R 0 ) .
w = arg min w P out ( R 0 ) .
P out ( R 0 ) w w = 0 .
log P out ( R 0 ) w w = 0 ,
log P out Asy γ 4 σ G 2 2 γ 2 μ G + γ 2 log e 4 d 2 / w 2 SNR R 0 A 0 SNR 0 .
log P out Asy K 1 γ 4 + K 2 γ 2 + γ 2 log γ 2 + d 2 2 σ s 2 ,
K 1 = 1 2 M e σ X 2 1 ,
K 2 = K 1 log M + log SNR R 0 SNR 0 + log 2 σ s 2 a 2 .
P out Asy ( R 0 ) γ γ = 0 .
2 K 1 γ 2 + ( K 2 + 1 ) + log γ 2 = 0 .
w Asy = 2 σ s γ = 4 σ s 2 2 K 1 W 2 K 1 e ( K 2 + 1 ) ,
w o Asy = M 1 ( w Asy ) .
P out ( R 0 ) | W o = min w o W o P out ( R 0 ) .
P out Asy ( R 0 ) = P out Asy ( R 0 ) | w = w Asy .
e G = m = 1 M e G m = m = 1 M e X m U m ,
μ G = log α / 1 + β 2 / α 2 ,
σ G 2 = log 1 + β 2 / α 2 ,
α = m = 1 M e μ m + ν m m 2 ,
β 2 = m = 1 M m = 1 M e μ m + μ m + ν m m + ν m m 2 ( e ν m m 1 ) .
μ m = E { G m } = E { X m U m } = μ X m = μ X ,
ν m m = cov ( G m , G m )
= cov ( X m U m , X m U m )
= cov ( X m , X m ) + cov ( U m , U m )
= σ X 2 δ m m + 16 w 4 σ s 2 P m T P m ,
δ i j = 1 if  i = j , 0 otherwise.
P m = d cos θ m d sin θ m , where θ m = 2 π m M , m = 1 , , M .
P = [ P 1 P 2 P M ] .
P T P = d 2 1 cos Δ θ cos ( M 1 ) Δ θ cos Δ θ 1 1 cos ( M 1 ) Δ θ 1 ,
P m T P m = d 2 cos ( m m ) Δ θ .
ν m m = σ X 2 δ m m + 16 w 4 σ s 2 d 2 cos ( m m ) Δ θ .
α = m = 1 M e μ m + ν m m 2 , = m = 1 M e μ X + 1 2 ( σ X 2 + 16 w 4 σ s 2 d 2 ) , = M e 8 w 4 σ s 2 d 2 .
β 2 = m = 1 M m = 1 M e ( μ m + ν m m 2 ) + ( μ m + ν m m 2 ) ( e ν m m 1 ) ,
= m = 1 M m = 1 M e 16 w 4 σ s 2 d 2 ( e ν m m 1 ) .
β 2 = e 16 w 4 σ s 2 d 2 M e σ X 2 + 16 w 4 σ s 2 d 2 1 + i = 1 M 1 2 ( M i ) e 16 w 4 σ s 2 d 2 cos i Δ θ 1 .
e x 1 + x for x 1 ,
β 2 e 16 w 4 σ s 2 d 2 [ M e σ X 2 1 + 16 w 4 σ s 2 d 2 1 + 16 w 4 σ s 2 d 2 i = 1 M 1 2 ( M i ) cos i Δ θ M ] .
β 2 α 2 1 M e σ X 2 1 1 + 16 w 4 σ s 2 d 2 .
σ G 2 log 1 + 1 M e σ X 2 1 1 + 16 w 4 σ s 2 d 2 .
log ( 1 + x ) x for x 1 ,
σ G 2 1 M e σ X 2 1 .
μ G = log α σ G 2 2 .
μ G log M σ G 2 2 .
log P out Asy K 1 γ 4 + K 2 γ 2 + γ 2 log γ 2 + d 2 2 σ s 2 ,
K 1 = 1 2 M e σ X 2 1 ,
K 2 = K 1 log M + log SNR R 0 SNR 0 + log 2 σ s 2 a 2 .