Abstract

We theoretically analyze the performance of free-space optical (FSO) systems using rectangular quadrature-amplitude modulation (QAM) and an avalanche photodiode (APD) receiver over atmospheric turbulence channels. Both log-normal and gamma–gamma channel models are used in the analysis for the cases of weak/moderate and strong atmospheric turbulence. The system bit error rate, when Gray code mapping is employed, is theoretically derived taking into account various link conditions and system parameters, including the APD shot noise, thermal noise, channel attenuation and geometrical loss, atmospheric turbulence strengths, and link distances. The numerical results show that using APD with a proper selection of the average gain could greatly benefit the performance of the system; as a matter of fact, in the case of optimal gain, the system using an APD receiver could provide 7 dB gain in comparison with the one with a positive-instrinsic-negative receiver. We also quantitatively discuss the impact of link conditions and system parameters on the selection of optimal APD gain.

© 2013 Optical Society of America

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References

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  1. A. K. Majumdar, and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances. Springer, 2007.
  2. D. O’Brien and M. Katz, “Optical wireless communications within fourth-generation wireless systems,” J. Opt. Netw., vol.  4, no. 6, pp. 312–322, June 2005.
    [CrossRef]
  3. S. Karp, Optical Channels: Fibers, Clouds, Water, and the Atmosphere. Plenum, 1988.
  4. K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, no. 9, pp. 1455–1461, Sept. 2005.
    [CrossRef]
  5. 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, no. 5, pp. 1368–1377, May 2011.
    [CrossRef]
  6. Q. Lu, Q. Liu, and G. S. Mitchell, “Performance analysis for optical wireless communication systems using subcarrier PSK intensity modulation through turbulent atmospheric channel,” in IEEE Global Telecommunications Conf. (GLOBECOM), 2004, vol. 3, pp. 1872–1875.
  7. Q. Liu and Q. Lu, “Subcarrier PSK intensity modulation for optical wireless communications through turbulent atmospheric channel,” in IEEE Int. Conf. Communications (ICC), 2005, vol. 3, pp. 1761–1765.
  8. A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.
  9. K. P. Peppas and C. K. Datsikas, “Average symbol error probability of general-order rectangular quadrature amplitude modulation of optical wireless communication systems over atmospheric turbulence channels,” J. Opt. Commun. Netw., vol.  2, no. 2, pp. 102–110, Feb. 2010.
    [CrossRef]
  10. M. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw., vol.  4, no. 6, pp. 522–532, June 2012.
    [CrossRef]
  11. N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
    [CrossRef]
  12. N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
    [CrossRef]
  13. M. Cole and K. Kiasaleh, “Receiver architectures for the detection of spatially correlated optical field using avalanche photodiode detector arrays,” Opt. Eng., vol.  47, no. 2, 025008, 2008.
    [CrossRef]
  14. G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley-Interscience, 2002.
  15. J. W. Strohbehn, Laser Beam Propagations in the Atmosphere. Springer, 1978.
  16. M. 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, no. 8, pp. 1554–1562, 2001.
    [CrossRef]
  17. K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol.  50, no. 7, pp. 1074–1080, July 2002.
    [CrossRef]
  18. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
    [CrossRef]
  19. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 2000.

2012 (1)

2011 (1)

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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

2010 (1)

2009 (1)

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
[CrossRef]

2008 (2)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
[CrossRef]

M. Cole and K. Kiasaleh, “Receiver architectures for the detection of spatially correlated optical field using avalanche photodiode detector arrays,” Opt. Eng., vol.  47, no. 2, 025008, 2008.
[CrossRef]

2007 (1)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
[CrossRef]

2005 (2)

D. O’Brien and M. Katz, “Optical wireless communications within fourth-generation wireless systems,” J. Opt. Netw., vol.  4, no. 6, pp. 312–322, June 2005.
[CrossRef]

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, no. 9, pp. 1455–1461, Sept. 2005.
[CrossRef]

2002 (1)

K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol.  50, no. 7, pp. 1074–1080, July 2002.
[CrossRef]

2001 (1)

M. 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, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley-Interscience, 2002.

Al-Habash, M. A.

M. 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, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Andrews, L. C.

M. 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, no. 8, pp. 1554–1562, 2001.
[CrossRef]

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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
[CrossRef]

Brandt-Pearce, M.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
[CrossRef]

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
[CrossRef]

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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

Cheng, J.

Cheng, Z.

A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.

Cho, K.

K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol.  50, no. 7, pp. 1074–1080, July 2002.
[CrossRef]

Cole, M.

M. Cole and K. Kiasaleh, “Receiver architectures for the detection of spatially correlated optical field using avalanche photodiode detector arrays,” Opt. Eng., vol.  47, no. 2, 025008, 2008.
[CrossRef]

Cvijetic, N.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
[CrossRef]

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
[CrossRef]

Datsikas, C. K.

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 2000.

Guo, S.

A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.

Hassan, M. Z.

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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

Karp, S.

S. Karp, Optical Channels: Fibers, Clouds, Water, and the Atmosphere. Plenum, 1988.

Katz, M.

Kiasaleh, K.

M. Cole and K. Kiasaleh, “Receiver architectures for the detection of spatially correlated optical field using avalanche photodiode detector arrays,” Opt. Eng., vol.  47, no. 2, 025008, 2008.
[CrossRef]

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, no. 9, pp. 1455–1461, Sept. 2005.
[CrossRef]

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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

Liu, Q.

Q. Liu and Q. Lu, “Subcarrier PSK intensity modulation for optical wireless communications through turbulent atmospheric channel,” in IEEE Int. Conf. Communications (ICC), 2005, vol. 3, pp. 1761–1765.

Q. Lu, Q. Liu, and G. S. Mitchell, “Performance analysis for optical wireless communication systems using subcarrier PSK intensity modulation through turbulent atmospheric channel,” in IEEE Global Telecommunications Conf. (GLOBECOM), 2004, vol. 3, pp. 1872–1875.

Lu, Q.

Q. Lu, Q. Liu, and G. S. Mitchell, “Performance analysis for optical wireless communication systems using subcarrier PSK intensity modulation through turbulent atmospheric channel,” in IEEE Global Telecommunications Conf. (GLOBECOM), 2004, vol. 3, pp. 1872–1875.

Q. Liu and Q. Lu, “Subcarrier PSK intensity modulation for optical wireless communications through turbulent atmospheric channel,” in IEEE Int. Conf. Communications (ICC), 2005, vol. 3, pp. 1761–1765.

Majumdar, A. K.

A. K. Majumdar, and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances. Springer, 2007.

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
[CrossRef]

Mitchell, G. S.

Q. Lu, Q. Liu, and G. S. Mitchell, “Performance analysis for optical wireless communication systems using subcarrier PSK intensity modulation through turbulent atmospheric channel,” in IEEE Global Telecommunications Conf. (GLOBECOM), 2004, vol. 3, pp. 1872–1875.

O’Brien, D.

Peppas, K. P.

Pham, A. T.

A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.

Phillips, R. L.

M. 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, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Ricklin, J. C.

A. K. Majumdar, and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances. Springer, 2007.

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 2000.

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
[CrossRef]

Song, X.

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagations in the Atmosphere. Springer, 1978.

Thang, T. C.

A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.

Wilson, S. G.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
[CrossRef]

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
[CrossRef]

Yoon, D.

K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol.  50, no. 7, pp. 1074–1080, July 2002.
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun., vol.  26, no. 3, pp. 3–12, Apr. 2008.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Technol. Lett., vol.  19, no. 2, pp. 103–105, Jan. 2007.
[CrossRef]

IEEE Trans. Commun. (4)

K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol.  50, no. 7, pp. 1074–1080, July 2002.
[CrossRef]

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, 2009.
[CrossRef]

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun., vol.  53, no. 9, pp. 1455–1461, Sept. 2005.
[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, no. 5, pp. 1368–1377, May 2011.
[CrossRef]

J. Opt. Commun. Netw. (2)

J. Opt. Netw. (1)

Opt. Eng. (2)

M. 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, no. 8, pp. 1554–1562, 2001.
[CrossRef]

M. Cole and K. Kiasaleh, “Receiver architectures for the detection of spatially correlated optical field using avalanche photodiode detector arrays,” Opt. Eng., vol.  47, no. 2, 025008, 2008.
[CrossRef]

Other (8)

G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley-Interscience, 2002.

J. W. Strohbehn, Laser Beam Propagations in the Atmosphere. Springer, 1978.

Q. Lu, Q. Liu, and G. S. Mitchell, “Performance analysis for optical wireless communication systems using subcarrier PSK intensity modulation through turbulent atmospheric channel,” in IEEE Global Telecommunications Conf. (GLOBECOM), 2004, vol. 3, pp. 1872–1875.

Q. Liu and Q. Lu, “Subcarrier PSK intensity modulation for optical wireless communications through turbulent atmospheric channel,” in IEEE Int. Conf. Communications (ICC), 2005, vol. 3, pp. 1761–1765.

A. T. Pham, T. C. Thang, S. Guo, and Z. Cheng, “Performance bounds for turbo-coded SC-PSK/FSO communications over strong turbulence channels,” in Int. Conf. Advanced Technologies for Communications (ATC), 2011, pp. 161–164.

S. Karp, Optical Channels: Fibers, Clouds, Water, and the Atmosphere. Plenum, 1988.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 2000.

A. K. Majumdar, and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances. Springer, 2007.

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Figures (10)

Fig. 1.
Fig. 1.

System diagram of subcarrier intensity-modulated rectangular QAM/FSO system over an atmospheric turbulence channel.

Fig. 2.
Fig. 2.

Signal-space diagram for the 8×4 QAM scheme.

Fig. 3.
Fig. 3.

BER versus average APD gain for weak turbulence and different QAM schemes, transmitted power per bit Pb=3dBm, and L=1000m.

Fig. 4.
Fig. 4.

BER versus average APD gain for strong turbulence and different QAM schemes, transmitted power per bit Pb=0dBm, and L=1000m.

Fig. 5.
Fig. 5.

BER versus average APD gain for 8×4 QAM scheme with different turbulence strengths, transmitted power per bit Pb=3 dBm, and Cn2=1015m2/3.

Fig. 6.
Fig. 6.

BER versus average APD gain for 8×4 QAM scheme with different noise temperatures, transmitted power per bit Pb=3 dBm, and Cn2=1015m2/3.

Fig. 7.
Fig. 7.

BER versus average transmitted power per bit for 8×4 QAM scheme with different channel distances, Cn2=1015m2/3, and g¯=10.

Fig. 8.
Fig. 8.

BER versus average transmitted power per bit for 8×4 QAM scheme with different turbulence strengths, L=1000m, and g¯=10.

Fig. 9.
Fig. 9.

BER versus channel distance for 8×4 QAM scheme and different values of average transmitted power per bit, Cn2=1015m2/3, and g¯=10.

Fig. 10.
Fig. 10.

BER versus transmitted power for 8×4 QAM scheme with different bit rates, Cn2=1015m2/3, and g¯=10.

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

q(t)=sI(t)cos(2πfct)sQ(t)sin(2πfct),
s(t)=Ps{1+m[sI(t)cos(2πfct)sQ(t)sin(2πfct)]},
r(t)=avX(t)s(t).
r(t)=avX(t)Ps{1+m[sI(t)cos(2πfct)sQ(t)sin(2πfct)]}.
re(t)=g¯RavX(t)Ps{m[sI(t)cos(2πfct)sQ(t)sin(2πfct)]}+n(t).
n(t)=iSh(t)+iTh(t),
σTh2=(4kBTRL)FnΔf,
σSh2=2qg¯2FARavxPsmΔf,
σn2=2qg¯2FARavxPsmΔf+(4kBTRL)FnΔf.
γ=(g¯RavxPs)22qg¯2FARavxPsmΔf+(4kBTRL)FnΔf.
av=Aπ(ΘL2)2e(βvL),
fX(x)=1xσS2πexp[lnx+σS22]22σS2,
σS2=exp[0.49σ22(1+0.18d2+0.56σ212/5)7/6+0.51σ22(1+0.69σ212/5)5/6(1+0.90d2+0.62d2σ212/5)]1,
σ22=0.492Cn2k7/6L11/6,
fX(x)=2(αβ)(α+β)/2Γ(α)Γ(β)x(α+β)/21Kαβ(2αβx),
α={exp[0.49σ22(1+0.18d2+0.56σ212/5)7/6]1}1,
β={exp[0.51σ22(1+0.69σ212/5)5/6(1+0.9d2+0.62d2σ212/5)]1}1.
BER=0BERinstfX(x)dx.
BERinst=1log2(8×4)(k=1log28P8(k)+l=1log24P4(l)),
P8(1)=18[erfc(dI2σn2)+erfc(3dI2σn2)+erfc(5dI2σn2)+erfc(7dI2σn2)].
P8(2)=18[2·erfc(dI2σn2)+2·erfc(3dI2σn2)+erfc(5dI2σn2)+erfc(7dI2σn2)erfc(9dI2σn2)erfc(11dI2σn2)],
P8(3)=18[4·erfc(dI2σn2)+3·erfc(3dI2σn2)3erfc(5dI2σn2)+2erfc(7dI2σn2)2erfc(9dI2σn2)+erfc(11dI2σn2)erfc(13dI2σn2)].
P4(1)=14[erfc(dQ2σn2)+erfc(3dQ2σn2)],
P4(2)=14[2·erfc(dQ2σn2)+erfc(3dQ2σn2)erfc(5dQ2σn2)].
BERinst=1log2(MIMQ)(k=1log2MIPMI(k)+l=1log2MQPMQ(l)),
dI2σn2=3γ2((MI21)+ζ2(MQ21)),
dQ2σn2=3ζ2γ2((MI21)+ζ2(MQ21)).
PMI(k)=1MIi=0(12k)MI1{(1)i·2k1MI(2k1i·2k1MI+12)×erfc((2i+1)dI2σn2)},
PMQ(l)=1MQj=0(12l)MQ1{(1)j·2l1MQ(2l1j·2l1MQ+12)×erfc((2i+1)dQ2σn2)}.
PMI(k)=1MIi=0(12k)MI1{(1)i·2k1MI(2k1i·2k1MI+12)×erfc((2i+1)3γ2((MI21)+ζ2(MQ21)))},
PMQ(l)=1MQj=0(12l)MQ1{(1)j·2l1MQ(2l1j·2l1MQ+12)×erfc((2j+1)3ζ2γ2((MI21)+ζ2(MQ21)))}.
PI=k=1log2MI0PMI(k)fX(x)dx,
PQ=l=1log2MQ0PMQ(l)fX(x)dx,
BER=1log2(MIMQ)(PI+PQ).
PMI(k)=i=0(12k)MI1Cierfc(Aiγ),
PMQ(l)=j=0(12l)MQ1Cjerfc(Ajγ),
Cj1MQ(1)j·2l1MQ(2l1j·2l1MQ+12)
Aj=(2j+1)32((MI21)+ζ2(MQ21)).
γ=ax2bx+c.
PI=k=1log2MIi=0(12k)MI1Ci1π×exp(y2)erfc(Aiaexp(22σSyσS2)bexp(2σSyσS2/2)+c)dy.
PIk=1log2MIi=0(12k)MI1t=N;t0NCi1π×wterfc(Aiaexp(22σSytσS2)bexp(2σSytσS2/2)+c),
PQl=1log2MQj=0(12l)MQ1t=N;t0NCj1π×wterfc(Ajaexp(22σSytσS2)bexp(2σSytσS2/2)+c),
fX(x)=n=0(an(α,β)xn+β1+an(β,α)xn+α1),
an(α,β)=π(αβ)n+βsin[π(αβ)]Γ(α)Γ(β)Γ(nα+β+1)n!.
erfc(Aiγ)16eAi2γ+12e43Ai2γ,
PMI(k)i=0(12k)MI1Ci(16eAi2γ+12e43Ai2γ).
fγ(γ)=n=0m=0(D(α,β)g(n+β1m2,n+β1+m2)+D(β,α)g(n+α1m2,n+α1+m2))×(b2a+b24aacg(12,12)+c4acg(12,12)),
D(α,β)=an(α,β)1(2a)n+β1(n+β1m)bm(4ac)n+β1m2,
g(z1,z2)=[γb24ac+1]z1γz2.
PI16k=1log2MIi=0(12k)MI1Cin=0m=0(b2aD(α,β)h(Ai2,n+β1m2,n+β+1+m2)+b2aD(β,α)h(Ai2,n+α1m2,n+α+1+m2)+b24aacD(α,β)h(Ai2,n+β2m2,n+β+2+m2)+b24aacD(β,α)h(Ai2,n+α2m2,n+α+2+m2)+c4acD(α,β)h(Ai2,n+β2m2,n+β+m2)+c4acD(β,α)h(Ai2,n+α2m2,n+α+m2))+12k=1log2MIi=0(12k)MI1Cin=0m=0(b2aD(α,β)h(4Ai23,n+β1m2,n+β+1+m2)+b2aD(β,α)h(4Ai23,n+α1m2,n+α+1+m2)+b24aacD(α,β)h(4Ai23,n+β2m2,n+β+2+m2)+b24aacD(β,α)h(4Ai23,n+α2m2,n+α+2+m2)+c4acD(α,β)h(4Ai23,n+β2m2,n+β+m2)+c4acD(β,α)h(4Ai23,n+α2m2,n+α+m2)),
h(p,v,q)=0epγg(v,q1)dγ.
h(p,v,q)=π2pqΓ(v)sin[π(qv)]×[(pz)vLvvq(pz)sin(πv)Γ(1q)(pz)qLqqv(pz)sin(πq)Γ(1v)],