Abstract

An optical wireless communication system using subcarrier intensity modulation is analyzed for gamma–gamma turbulence channels with pointing errors. We study the error rate performance of such a system employing M-ary phase-shift keying, differential phase-shift keying, and noncoherent frequency-shift keying. Highly accurate error rate approximations are derived using a series expansion approach. Furthermore, outage probability expressions are obtained for such a system. Asymptotic error rate and outage probability analyses are also presented. Our asymptotic analysis reveals some unique transmission characteristics of such a system. Our findings suggest that pointing error compensation is necessary as pointing errors can severely degrade the error rate and outage probability performance of an uncompensated system.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  32. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Bellingham, WA: SPIE, 2001.
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    [CrossRef]
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2012

2011

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express, vol.  19, pp. 13480–13496, July 2011.
[CrossRef]

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol.  59, pp. 1368–1377, May 2011.
[CrossRef]

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in gamma-gamma fading channels,” IEEE Photon. Technol. Lett., vol.  23, pp. 269–271, Feb. 2011.
[CrossRef]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun., vol.  59, pp. 669–674, Mar. 2011.
[CrossRef]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett., vol.  15, pp. 875–877, Aug. 2011.
[CrossRef]

2010

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett., vol.  14, pp. 468–470, May 2010.
[CrossRef]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express, vol.  18, pp. 12824–12831, June 2010.
[CrossRef]

2009

2008

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol.  12, pp. 44–46, Jan. 2008.
[CrossRef]

2007

J. Li, J. Q. Liu, and D. P. Tayler, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  55, pp. 1598–1606, Aug. 2007.
[CrossRef]

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightwave Technol., vol.  25, pp. 1702–1710, July 2007.
[CrossRef]

2005

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol.  53, pp. 1402–1412, Aug. 2005.
[CrossRef]

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, pp. 1455–1461, Sept. 2005.
[CrossRef]

2003

2002

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  50, pp. 1293–1300, Aug. 2002.
[CrossRef]

2001

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, pp. 1554–1562, Aug. 2001.
[CrossRef]

1993

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol.  E76-B, pp. 1169–1177, Sept. 1993.

Agrawal, G. P.

G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed.New York: Wiley, 2002.

Al-Habash, A.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, pp. 1554–1562, Aug. 2001.
[CrossRef]

Andrews, L. C.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, pp. 1554–1562, Aug. 2001.
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Bellingham, WA: SPIE, 2001.

Arnon, S.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol.  59, pp. 1368–1377, May 2011.
[CrossRef]

D. Kedar and S. Arnon, “Optical wireless communication through fog in the presence of pointing errors,” Appl. Opt., vol.  42, pp. 4946–4954, Aug. 2003.
[CrossRef]

S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett., vol.  28, pp. 129–131, Jan. 2003.
[CrossRef]

S. Arnon, “Optimization of urban optical wireless communications systems,” IEEE Trans. Wireless Commun., vol.  2, pp. 626–629, July 2003.
[CrossRef]

Bloom, S.

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., vol.  53, pp. 1402–1412, Aug. 2005.
[CrossRef]

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., vol.  53, pp. 1402–1412, Aug. 2005.
[CrossRef]

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol.  59, pp. 1368–1377, May 2011.
[CrossRef]

Cheng, J.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol.  16, pp. 540–543, Apr. 2012.
[CrossRef]

Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw., vol.  4, pp. 522–532, June 2012.
[CrossRef]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express, vol.  18, pp. 12824–12831, June 2010.
[CrossRef]

Md. Z. Hassan, X. Song, and J. Cheng, “Error rate analysis of subcarrier intensity modulation using rectangular QAM in gamma-gamma turbulence,” in IEEE Global Communications Conf. (GLOBECOM 2012), Anaheim, CA, Dec. 2012.

X. Song and J. Cheng, “Subcarrier intensity modulated optical communications over K-distributed channels,” in 2012 Photonics Society Summer Topical Meetings, Seattle, WA, July 2012.

X. Song and J. Cheng, “Subcarrier intensity modulated optical communications in strong atmospheric turbulence,” in IEEE Int. Conf. on Communications in China (ICCC 2012), Beijing, China, Aug. 2012.

X. Song and J. Cheng, “Performance of subcarrier intensity modulated MIMO wireless optical communications,” in IEEE 26th Biennial Symp. on Communications, Kingston, Canada, May 2012.

X. Song and J. Cheng, “Alamouti-type STBC for subcarrier intensity modulated wireless optical communications,” in IEEE Global Communications Conf. (GLOBECOM 2012), Anaheim, CA, Dec. 2012.

Farid, A. A.

Gappmair, W.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett., vol.  15, pp. 875–877, Aug. 2011.
[CrossRef]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett., vol.  14, pp. 468–470, May 2010.
[CrossRef]

García-Zambrana, A.

Ghassemlooy, Z.

W. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol.  27, pp. 967–973, Apr. 2009.
[CrossRef]

I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. Uysal, “Performance analysis of free space optical links over turbulence and misalignment induced fading channels,” in 8th Int. Symp. on Communication Systems, Networks & Digital Signal Processing (CSNDSP), Poznan, Poland, July 2012.

I. E. Lee, Z. Ghassemlooy, and W. P. Ng, “Effects of aperture averaging and beam width on Gaussian free space optical links in the presence of atmospheric turbulence and pointing error,” in 14th Int. Conf. on Transparent Optical Networks (ICTON), Coventry, England, July 2012.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed.San Diego, CA: Academic, 2000.

Hassan, Md. Z.

Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw., vol.  4, pp. 522–532, June 2012.
[CrossRef]

Md. Z. Hassan, X. Song, and J. Cheng, “Error rate analysis of subcarrier intensity modulation using rectangular QAM in gamma-gamma turbulence,” in IEEE Global Communications Conf. (GLOBECOM 2012), Anaheim, CA, Dec. 2012.

Hinedi, S. M.

M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection. Englewood Cliffs, NJ: Prentice-Hall, 1995.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Bellingham, WA: SPIE, 2001.

Hranilovic, S.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett., vol.  15, pp. 875–877, Aug. 2011.
[CrossRef]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett., vol.  14, pp. 468–470, May 2010.
[CrossRef]

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightwave Technol., vol.  25, pp. 1702–1710, July 2007.
[CrossRef]

Huang, W.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol.  E76-B, pp. 1169–1177, Sept. 1993.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  50, pp. 1293–1300, Aug. 2002.
[CrossRef]

Karagiannidis, G. K.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol.  59, pp. 1368–1377, May 2011.
[CrossRef]

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol., vol.  27, pp. 4440–4445, Oct. 2009.
[CrossRef]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol.  12, pp. 44–46, Jan. 2008.
[CrossRef]

Kedar, D.

Kiasaleh, K.

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, pp. 1455–1461, Sept. 2005.
[CrossRef]

Korevaar, E.

Lee, E.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in gamma-gamma fading channels,” IEEE Photon. Technol. Lett., vol.  23, pp. 269–271, Feb. 2011.
[CrossRef]

Lee, I. E.

I. E. Lee, Z. Ghassemlooy, and W. P. Ng, “Effects of aperture averaging and beam width on Gaussian free space optical links in the presence of atmospheric turbulence and pointing error,” in 14th Int. Conf. on Transparent Optical Networks (ICTON), Coventry, England, July 2012.

I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. Uysal, “Performance analysis of free space optical links over turbulence and misalignment induced fading channels,” in 8th Int. Symp. on Communication Systems, Networks & Digital Signal Processing (CSNDSP), Poznan, Poland, July 2012.

Leitgeb, E.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett., vol.  15, pp. 875–877, Aug. 2011.
[CrossRef]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett., vol.  14, pp. 468–470, May 2010.
[CrossRef]

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., vol.  53, pp. 1402–1412, Aug. 2005.
[CrossRef]

Li, J.

J. Li, J. Q. Liu, and D. P. Tayler, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  55, pp. 1598–1606, Aug. 2007.
[CrossRef]

Lindsey, W. C.

M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection. Englewood Cliffs, NJ: Prentice-Hall, 1995.

Lioumpas, A. S.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol.  59, pp. 1368–1377, May 2011.
[CrossRef]

Liu, J. Q.

J. Li, J. Q. Liu, and D. P. Tayler, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  55, pp. 1598–1606, Aug. 2007.
[CrossRef]

Nakagawa, M.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol.  E76-B, pp. 1169–1177, Sept. 1993.

Ng, W. P.

I. E. Lee, Z. Ghassemlooy, and W. P. Ng, “Effects of aperture averaging and beam width on Gaussian free space optical links in the presence of atmospheric turbulence and pointing error,” in 14th Int. Conf. on Transparent Optical Networks (ICTON), Coventry, England, July 2012.

I. E. Lee, Z. Ghassemlooy, W. P. Ng, and M. Uysal, “Performance analysis of free space optical links over turbulence and misalignment induced fading channels,” in 8th Int. Symp. on Communication Systems, Networks & Digital Signal Processing (CSNDSP), Poznan, Poland, July 2012.

Niu, M.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol.  16, pp. 540–543, Apr. 2012.
[CrossRef]

Park, J.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in gamma-gamma fading channels,” IEEE Photon. Technol. Lett., vol.  23, pp. 269–271, Feb. 2011.
[CrossRef]

Phillips, R. L.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, pp. 1554–1562, Aug. 2001.
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Bellingham, WA: SPIE, 2001.

Popoola, W.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed.San Diego, CA: Academic, 2000.

Sakanaka, T.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol.  E76-B, pp. 1169–1177, Sept. 1993.

Sandalidis, H. G.

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun., vol.  59, pp. 669–674, Mar. 2011.
[CrossRef]

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol., vol.  27, pp. 4440–4445, Oct. 2009.
[CrossRef]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol.  12, pp. 44–46, Jan. 2008.
[CrossRef]

Schuster, J.

Simon, M. K.

M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection. Englewood Cliffs, NJ: Prentice-Hall, 1995.

M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002.

Song, X.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol.  16, pp. 540–543, Apr. 2012.
[CrossRef]

Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw., vol.  4, pp. 522–532, June 2012.
[CrossRef]

X. Song and J. Cheng, “Performance of subcarrier intensity modulated MIMO wireless optical communications,” in IEEE 26th Biennial Symp. on Communications, Kingston, Canada, May 2012.

X. Song and J. Cheng, “Alamouti-type STBC for subcarrier intensity modulated wireless optical communications,” in IEEE Global Communications Conf. (GLOBECOM 2012), Anaheim, CA, Dec. 2012.

Md. Z. Hassan, X. Song, and J. Cheng, “Error rate analysis of subcarrier intensity modulation using rectangular QAM in gamma-gamma turbulence,” in IEEE Global Communications Conf. (GLOBECOM 2012), Anaheim, CA, Dec. 2012.

X. Song and J. Cheng, “Subcarrier intensity modulated optical communications over K-distributed channels,” in 2012 Photonics Society Summer Topical Meetings, Seattle, WA, July 2012.

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Figures (6)

Fig. 1.
Fig. 1.

Block diagram of a subcarrier intensity modulated OWC system.

Fig. 2.
Fig. 2.

SERs of subcarrier QPSK and 8PSK modulated OWC systems over the gamma–gamma turbulence channels with pointing errors for ωz/r=10 and σs/r=1.

Fig. 3.
Fig. 3.

BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the gamma–gamma turbulence channels with pointing errors for ωz/r=10, σs/r=1.

Fig. 4.
Fig. 4.

BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the gamma–gamma turbulence channels with pointing errors for α=4.03, β=3.45, ωz/r=10, and σs/r=3 (α>φ2).

Fig. 5.
Fig. 5.

BER comparison of subcarrier BPSK modulated OWC systems over the gamma–gamma turbulence channels with or without pointing errors for ωz/r=10, σs/r=1.

Fig. 6.
Fig. 6.

Outage probability of subcarrier intensity modulated OWC systems over the gamma–gamma turbulence channels with or without pointing errors.

Equations (56)

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Pt(t)=P[1+ξm(t)],
ir(t)=PRI(t)[1+ξm(t)]+n(t),
γ=(PRξ)2σn2I2=γ¯I2,
fIa(Ia)=2(αβ)α+β2Γ(α)Γ(β)Iaα+β21Kαβ(2αβIa),Ia>0,
fIp(Ip)=φ2A0φ2Ipφ21,0IpA0,
ωzeq=ωz[πerf(v)2vexp(v2)]12.
fI(I)=2φ2(αβ)α+β2A0φ2Γ(α)Γ(β)Iφ21I/A0Iaα+β2φ21Kαβ(2αβIa)dIa.
Kν(x)=π2sin(πν)p=0[(x/2)2pνΓ(pν+1)p!(x/2)2p+νΓ(p+ν+1)p!],νZ,
fI(I)=Λ(α,β,φ)Iφ21A0φ2p=0[(αβ)p+βI/A0Iap+βφ21dIaΓ(pα+β+1)p!(αβ)p+αI/A0Iap+αφ21dIaΓ(p+αβ+1)p!],
Λ(α,β,φ)πφ2Γ(α)Γ(β)sin[π(αβ)].
x=ωzeq2lnIA0Ia2,
fI(I)=Λ(α,β,φ)p=0[ap(α,β,φ,A0)gp(β)Ip+β1ap(β,α,φ,A0)gp(α)Ip+α1],
ap(α,β,φ,A0)(αβA0)p+βΓ(pα+β+1)p!
gp(x)0exp[(p+xφ2)t]dt.
MY(s)=E[exp(sY)]=Λ(α,β,φ)2×p=0[ap(α,β,φ,A0)gp(β)Γ(p+β2)sp+β2ap(β,α,φ,A0)gp(α)Γ(p+α2)sp+α2].
Pe=0Pe(I)fI(I)dI,
P2(I)=Q(2γ¯I2),
P2=1π00π/2exp(γ¯I2sin2θ)fI(I)dθdI=Λ(α,β,φ)2πp=0[Γ(p+β+12)ap(α,β,φ,A0)gp(β)p+βγ¯p+β2Γ(p+α+12)ap(β,α,φ,A0)gp(α)p+αγ¯p+α2].
0π/2(sinθ)p+xdθ=12B(12,p+x+12),
P2Λ(α,β,φ)2πp=0K[Γ(p+β+12)ap(α,β,φ,A0)(p+β)(φ2pβ)γ¯p+β2Γ(p+α+12)ap(β,α,φ,A0)(p+α)(φ2pα)γ¯p+α2],
P2=Λ(α,β,φ)(αβA0)βΓ(β+12)2πΓ(β+1α)(φ2β)βγ¯β2,
PM=1π0ηπMY(κγ¯sin2θ)dθ=Λ(α,β,φ)2πp=0[ap(α,β,φ,A0)gp(β)Γ(p+β2)(κγ¯)p+β20ηπ(sinθ)p+βdθap(β,α,φ,A0)gp(α)Γ(p+α2)(κγ¯)p+α20ηπ(sinθ)p+αdθ],
hp(x,η)0ηπ(sinθ)p+xdθ
hp(x,η)=πΓ(1+p+x2)2Γ(1+p+x2)cos(ηπ)F[12,1px2;32;cos2(ηπ)],
PM=Λ(α,β,φ)2πp=0[ap(α,β,φ,A0)gp(β)hp(β,η)Γ(p+β2)(κγ¯)p+β2ap(β,α,φ,A0)gp(α)hp(α,η)Γ(p+α2)(κγ¯)p+α2].
PMΛ(α,β,φ)2πp=0K[ap(α,β,φ,A0)hp(β,η)Γ(p+β2)φ2pβ(κγ¯)p+β2ap(β,α,φ,A0)hp(α,η)Γ(p+α2)φ2pα(κγ¯)p+α2].
PM=Λ(α,β,φ)h0(β,η)Γ(β2)(αβA0)β2πΓ(β+1α)(φ2β)(κγ¯)β2.
Pe,k(I)=12exp(γ¯kI2),
Pe,k=120exp(γ¯kI2)fI(I)dI.
Pe,k=Λ(α,β,φ)4p=0[Γ(p+β2)ap(α,β,φ,A0)gp(β)(γ¯k)p+β2Γ(p+α2)ap(β,α,φ,A0)gp(α)(γ¯k)p+α2].
Pe,kΛ(α,β,φ)4p=0K[Γ(p+β2)ap(α,β,φ,A0)φ2pβ(γ¯k)p+β2Γ(p+α2)ap(β,α,φ,A0)φ2pα(γ¯k)p+α2].
Pe,k=Λ(α,β,φ)(αβA0)βΓ(β2)4Γ(β+1α)(φ2β)(γ¯k)β2.
SNRDPSK-BPSK=10βlog[πβΓ(β2)2Γ(β+12)]2,
fI(I)=2φ2(αβ)α+β2A0φ2Γ(α)Γ(β)Iφ21[I/A0I/A0+BIaα+β2φ21Kαβ(2αβIa)dIa+I/A0+BIaα+β2φ21Kαβ(2αβIa)dIa]=2φ2(αβ)α+β2A0φ2Γ(α)Γ(β)Iφ21[I/A0I/A0+BIaα+β2φ21Kαβ(2αβIa)dIa+RB]Λ(α,β,φ)A0φ2Iφ21p=0{(αβ)p+βΓ(p(αβ)+1)p!(p+βφ2)[(IA0+B)p+βφ2(IA0)p+βφ2](αβ)p+αΓ(p+(αβ)+1)p!(p+αφ2)[(IA0+B)p+αφ2(IA0)p+αφ2],
RB=I/A0+BIaα+β2φ21Kαβ(2αβIa)dIa.
RB<22φ2(α+β)+1(αβ)φ2(α+β)/2Kαβ(I/A0+B)(I/A0+B)αβ((αβ)+(αβ)2+(I/A0+B)2)αβexp((αβ)2+(I/A0+B)2)Γ(α+β2φ2,2αβ(I/A0+B)),
(x+y)μ=i=0(μi)xμiyi,μC,iZ,
(μi)=μ(μ1)(μi+1)i!=(μ)ii!
(x+y)n=i=0n(ni)xniyi,iZ.
(IA0+B)μ=i=0(μi)Bμi(IA0)i,iZ,
f(I)Λ(α,β,φ)A0φ2p=0{βp[i=0(p+βφ2i)1(A0B)iIi+φ211(A0B)p+βφ2Ip+β1]αp[i=0(p+αφ2i)1(A0B)iIi+φ211(A0B)p+αφ2Ip+α1]},
βp=(αβ)p+βBp+βφ2Γ(p(αβ)+1)p!(p+βφ2)
αp=(αβ)p+αBp+αφ2Γ(p+(αβ)+1)p!(p+αφ2).
P2=0Q(2γ¯I2)f(I)dIΛ(α,β,φ)2πA0φ2i=0{Γ(i+φ2+12)(A0B)i(i+φ2)p=0[(p+βφ2i)βp(p+αφ2i)αp]γ¯i+φ22[βiΓ(i+β+12)(A0B)i+βφ2(i+β)γ¯i+β2αiΓ(i+α+12)(A0B)i+αφ2(i+α)γ¯i+α2]}.
P2Λ(α,β,φ)2πA0φ2i=0N{Γ(i+φ2+12)(A0B)i(i+φ2)p=0J[(p+βφ2i)βp(p+αφ2i)αp]γ¯i+φ22[βiΓ(i+β+12)(A0B)i+βφ2(i+β)γ¯i+β2αiΓ(i+α+12)(A0B)i+αφ2(i+α)γ¯i+α2]}.
limy0fY(y)=φ2(αβ)α+β2A0φ2Γ(α)Γ(β)yφ2/21×0Iaα+β2φ21Kαβ(2αβIa)dIa,
Gc=[aΓ(t+3/2)2π(t+1)]1t+1,
a=φ2(αβ)α+β2A0φ2Γ(α)Γ(β)0Iaα+β2φ21Kαβ(2αβIa)dIa
P2=aΓ(t+3/2)2π(t+1)γ¯(t+1).
fγ(γ)=Λ(α,β,φ)2p=0[ap(α,β,φ,A0)gp(β)γ¯p+β2γp+β21ap(β,α,φ,A0)gp(α)γ¯p+α2γp+α21].
Po(γth)0γthfγ(γ)dγ=Λ(α,β,φ)×p=0[ap(α,β,φ,A0)gp(β)p+β(γthγ¯)p+β2ap(β,α,φ,A0)gp(α)p+α(γthγ¯)p+α2],
Po(γth)Λ(α,β,φ)p=0K[ap(α,β,φ,A0)(p+β)(φ2pβ)(γthγ¯)p+β2ap(β,α,φ,A0)(p+α)(φ2pα)(γthγ¯)p+α2],
Po(γth)=Λ(α,β,φ)(αβ)ββA0β(φ2β)Γ(βα+1)(γthγ¯)β2.
Po(γth)Λ(α,β,φ)A0φ2i=0{1(A0B)i(i+φ2)p=0[(p+βφ2i)βp(p+αφ2i)αp](γthγ¯)i+φ22[βi(A0B)i+βφ2(i+β)(γthγ¯)i+β2αi(A0B)i+αφ2(i+α)(γthγ¯)i+α2]},
Po(γth)Λ(α,β,φ)A0φ2i=0N{1(A0B)i(i+φ2)p=0J[(p+βφ2i)βp(p+αφ2i)αp](γthγ¯)i+φ22[βi(A0B)i+βφ2(i+β)(γthγ¯)i+β2αi(A0B)i+αφ2(i+α)(γthγ¯)i+α2]}.
Po(γth)=2aφ2(γthγ¯)φ22,