Abstract

We characterize the performance of the optical signal propagation model of multi-level pulse amplitude modulation (M-PAM) based on avalanche photodiode (APD) detector in the free-space link for the first time. When the number of photons absorbed by the active surface of the APD is large enough, the bit error rate (BER) performance relationship of the systems based on the signal intensity and the photon characteristics are depicted. We use the Gamma-Gamma (G-G) channel model to analysis the communication systems with joint parameter constraints, and demonstrate the atmospheric turbulence intensity, link lengths, optical wavelength, symbol transmission rate, temperature of APD and pointing errors (PEs) impact on the system average bit error rate (ABER) performance. Moreover, the relationship between signal-to-noise-ratio (SNR) and ABER rate is given. The numerical results show that the 4-PAM free-space optical (FSO) communication is suitable for medium-to-weak turbulence, and the high gain of APD can mitigate the influence of ABER. The best detection condition of the 4-PAM optical signal is at least 20 dB SNR, when the ABER is under the 7% forward error correction (FEC) limit of 3.8 × 10−3. This work provides a reference for parameter designing and evaluating in M-PAM FSO communication systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (7)

H.-H. Lu, C.-Y. Li, C.-M. Ho, M.-T. Cheng, X.-Y. Lin, Z.-Y. Yang, and H.-W. Chen, “64 Gb/s PAM4 VCSEL-based FSO link,” Opt. Express 25, 5749–5757 (2017).
[Crossref] [PubMed]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “BER performance of an optically pre-amplified FSO system under turbulence and pointing errors with ASE noise,” J. Opt. Commun. Netw. 9, 498–510 (2017).
[Crossref]

A. Jaiswal, M. R. Bhatnagar, and V. K. Jain, “Performance of optical space shift keying over Gamma-Gamma fading with pointing error,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

W. Lianghua, P. Yang, Y. Kangjian, C. Shanqiu, W. Shuai, L. Wenjing, and B. Xu, “Synchronous model-based approach for wavefront sensorless adaptive optics system,” Opt. Express 25, 20584–20597 (2017).
[Crossref] [PubMed]

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

N. A. M. Nor, Z. F. Ghassemlooy, J. Bohata, P. Saxena, M. Komanec, S. Zvanovec, M. R. Bhatnagar, and M.-A. Khalighi, “Experimental investigation of all-optical relay-assisted 10 Gb/s FSO link over the atmospheric turbulence channel,” J. Lightwave Technol. 35, 45–53 (2017).
[Crossref]

A. Garg, M. R. Bhatnagar, O. Berder, and B. Vrigneau, “Imperfect-quantized-feedback-based beamforming for an FSO MISO system over Gamma-Gamma fading with pointing errors,” J. Opt. Commun. Netw. 9, 1005–1018 (2017).
[Crossref]

2016 (3)

2015 (1)

2013 (2)

C. Chen and H. Yang, “Temporal spectrum of beam wander for Gaussian Shell-model beams propagating in atmospheric turbulence with finite outer scale,” Opt. Lett. 38, 1887–1889 (2013).
[Crossref] [PubMed]

O. Kharraz and D. Forsyth, “Performance comparisons between PIN and APD photodetectors for use in optical communication systems,” Optik - International Journal for Light and Electron Optics 124(13), 1493–1498 (2013).
[Crossref]

2012 (2)

2011 (2)

K. Szczerba, P. Westbergh, J. S. Gustavsson, A. Haglund, J. Karout, M. Karlsson, P. A. Andrekson, E. Agrell, and A. Larsson, “30 Gbps 4-PAM transmission over 200m of MMF using an 850 nm VCSEL,” Opt. Express 19, 1–3 (2011).
[Crossref]

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

2009 (1)

2008 (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 Journal on Selected Areas in Communications 26(3), 3–12 (2008).
[Crossref]

2003 (1)

1991 (1)

J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Optics 30, 1982–1994 (1991).
[Crossref]

Agrell, E.

Andrekson, P. A.

Berder, O.

Bhatnagar, M. R.

A. Garg, M. R. Bhatnagar, O. Berder, and B. Vrigneau, “Imperfect-quantized-feedback-based beamforming for an FSO MISO system over Gamma-Gamma fading with pointing errors,” J. Opt. Commun. Netw. 9, 1005–1018 (2017).
[Crossref]

A. Jaiswal, M. R. Bhatnagar, and V. K. Jain, “Performance of optical space shift keying over Gamma-Gamma fading with pointing error,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “BER performance of an optically pre-amplified FSO system under turbulence and pointing errors with ASE noise,” J. Opt. Commun. Netw. 9, 498–510 (2017).
[Crossref]

N. A. M. Nor, Z. F. Ghassemlooy, J. Bohata, P. Saxena, M. Komanec, S. Zvanovec, M. R. Bhatnagar, and M.-A. Khalighi, “Experimental investigation of all-optical relay-assisted 10 Gb/s FSO link over the atmospheric turbulence channel,” J. Lightwave Technol. 35, 45–53 (2017).
[Crossref]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under Gamma-Gamma turbulence with pointing errors and ASE noise,” in “2017 IEEE 85th Vehicular Technology Conference (VTC Spring),” (2017), pp. 1–5.

A. Mathur, P. Saxena, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under ASE noise with pointing errors,” in “2016 IEEE Annual India Conference (INDICON),” (2016), pp. 1–6.

P. Saxena, A. Mathur, M. R. Bhatnagar, and Z. Ghassemlooy, “BER of an optically pre-amplified FSO system under Malaga turbulence, pointing errors, and ASE noise,” in “2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC),” (2017), pp. 1–6.

Bloom, S.

Bohata, J.

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 Journal on Selected Areas in Communications 26(3), 3–12 (2008).
[Crossref]

Brink, S.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

Brychkov, A. P. P. Y. A.

A. P. P. Y. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 3Gordon and BreachNew York, 1990).

Castillovazquez, B.

Castillovazquez, C.

Chen, A.

Chen, C.

Chen, H. W.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Chen, H.-W.

Cheng, M. T.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Cheng, M.-T.

Chew, P.-H.

Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

Churnside, J. H.

J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Optics 30, 1982–1994 (1991).
[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 Journal on Selected Areas in Communications 26(3), 3–12 (2008).
[Crossref]

Du, T.

Forsyth, D.

O. Kharraz and D. Forsyth, “Performance comparisons between PIN and APD photodetectors for use in optical communication systems,” Optik - International Journal for Light and Electron Optics 124(13), 1493–1498 (2013).
[Crossref]

Fu, S.

Garciazambrana, A.

Garg, A.

Ghassemlooy, Z.

W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol. 27, 967–973 (2009).
[Crossref]

P. Saxena, A. Mathur, M. R. Bhatnagar, and Z. Ghassemlooy, “BER of an optically pre-amplified FSO system under Malaga turbulence, pointing errors, and ASE noise,” in “2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC),” (2017), pp. 1–6.

Ghassemlooy, Z. F.

Gustavsson, J. S.

Haglund, A.

Haglund, Å.

Ho, C. M.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Ho, C.-M.

Hu, R.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

Huang, X.-H.

Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

Huebner, B.

Idler, W.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

Jain, V. K.

A. Jaiswal, M. R. Bhatnagar, and V. K. Jain, “Performance of optical space shift keying over Gamma-Gamma fading with pointing error,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

Jaiswal, A.

A. Jaiswal, M. R. Bhatnagar, and V. K. Jain, “Performance of optical space shift keying over Gamma-Gamma fading with pointing error,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

Jheng, Y.-B.

Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

Kam, P. Y.

T. Song and P. Y. Kam, “Efficient direct detection of M-PAM sequences with implicit CSI acquisition for the FSO system,” in “2014 IEEE Globecom Workshops (GC Wkshps),” (2014), pp. 475–480.

Kangjian, Y.

Karlsson, M.

Karout, J.

Khalighi, M.-A.

Kharraz, O.

O. Kharraz and D. Forsyth, “Performance comparisons between PIN and APD photodetectors for use in optical communication systems,” Optik - International Journal for Light and Electron Optics 124(13), 1493–1498 (2013).
[Crossref]

Kocot, C.

Komanec, M.

Korevaar, E.

Larsson, A.

Leven, A.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

Li, C.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

Li, C. Y.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Li, C.-Y.

H.-H. Lu, C.-Y. Li, C.-M. Ho, M.-T. Cheng, X.-Y. Lin, Z.-Y. Yang, and H.-W. Chen, “64 Gb/s PAM4 VCSEL-based FSO link,” Opt. Express 25, 5749–5757 (2017).
[Crossref] [PubMed]

Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

Li, H.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

Li, X.

Lianghua, W.

Lin, X.-Y.

Lou, Y.

Lu, C. K.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Lu, H. H.

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

Lu, H.-H.

H.-H. Lu, C.-Y. Li, C.-M. Ho, M.-T. Cheng, X.-Y. Lin, Z.-Y. Yang, and H.-W. Chen, “64 Gb/s PAM4 VCSEL-based FSO link,” Opt. Express 25, 5749–5757 (2017).
[Crossref] [PubMed]

Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

Luo, M.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

Marichev, O. I.

A. P. P. Y. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 3Gordon and BreachNew York, 1990).

Mathur, A.

P. Saxena, A. Mathur, and M. R. Bhatnagar, “BER performance of an optically pre-amplified FSO system under turbulence and pointing errors with ASE noise,” J. Opt. Commun. Netw. 9, 498–510 (2017).
[Crossref]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under Gamma-Gamma turbulence with pointing errors and ASE noise,” in “2017 IEEE 85th Vehicular Technology Conference (VTC Spring),” (2017), pp. 1–5.

A. Mathur, P. Saxena, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under ASE noise with pointing errors,” in “2016 IEEE Annual India Conference (INDICON),” (2016), pp. 1–6.

P. Saxena, A. Mathur, M. R. Bhatnagar, and Z. Ghassemlooy, “BER of an optically pre-amplified FSO system under Malaga turbulence, pointing errors, and ASE noise,” in “2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC),” (2017), pp. 1–6.

Motaghiannezam, R.

Nor, N. A. M.

Pham, T.

Phillips, R. L.

R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).

Popoola, W. O.

Saxena, P.

N. A. M. Nor, Z. F. Ghassemlooy, J. Bohata, P. Saxena, M. Komanec, S. Zvanovec, M. R. Bhatnagar, and M.-A. Khalighi, “Experimental investigation of all-optical relay-assisted 10 Gb/s FSO link over the atmospheric turbulence channel,” J. Lightwave Technol. 35, 45–53 (2017).
[Crossref]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “BER performance of an optically pre-amplified FSO system under turbulence and pointing errors with ASE noise,” J. Opt. Commun. Netw. 9, 498–510 (2017).
[Crossref]

P. Saxena, A. Mathur, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under Gamma-Gamma turbulence with pointing errors and ASE noise,” in “2017 IEEE 85th Vehicular Technology Conference (VTC Spring),” (2017), pp. 1–5.

A. Mathur, P. Saxena, and M. R. Bhatnagar, “Performance of optically pre-amplified FSO system under ASE noise with pointing errors,” in “2016 IEEE Annual India Conference (INDICON),” (2016), pp. 1–6.

P. Saxena, A. Mathur, M. R. Bhatnagar, and Z. Ghassemlooy, “BER of an optically pre-amplified FSO system under Malaga turbulence, pointing errors, and ASE noise,” in “2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC),” (2017), pp. 1–6.

Schmalen, L.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

Schuster, J.

Shanqiu, C.

Shuai, W.

Song, T.

T. Song and P. Y. Kam, “Efficient direct detection of M-PAM sequences with implicit CSI acquisition for the FSO system,” in “2014 IEEE Globecom Workshops (GC Wkshps),” (2014), pp. 475–480.

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Vacondio, F.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

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Y.-C. Wang, P.-H. Chew, Y.-B. Jheng, C.-Y. Li, H.-H. Lu, X.-H. Huang, and W.-S. Tsai, “A 84 Gb/s VSB-PAM8 VCSEL-Based Fiber-FSO convergence,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2018), p. M1F.1.

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[Crossref]

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C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

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[Crossref]

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N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE Journal on Selected Areas in Communications 26(3), 3–12 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23, 1547–1549 (2011).
[Crossref]

IEEE Photonics J. (3)

H. H. Lu, C. Y. Li, H. W. Chen, C. M. Ho, M. T. Cheng, Z. Y. Yang, and C. K. Lu, “A 56 Gb/s PAM4 VCSEL-Based LiFi transmission with two-stage injection-locked technique,” IEEE Photonics J. 9, 1–8 (2017).

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-Based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8, 1–7 (2016).

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Figures (9)

Fig. 1
Fig. 1 Likelihood function of 2-PAM and schematic of optimal decision domain
Fig. 2
Fig. 2 ABER as a function of G for M level PAM, where λ = 1550 nm, L = 1500 m, C n 2 = 8 × 10 15 m 2 / 3, T = 300 K, Symbol rate = 1 GB/s.
Fig. 3
Fig. 3 ABER of the M-PAM as a function of G for the atmospheric structure constant C n 2, (a) and (b) represent 2-PAM and 4-PAM, respectively. Where λ = 1550 nm, L = 3000 m, T = 300 K, Symbol rate = 1 GB/s.
Fig. 4
Fig. 4 BER of the 4-PAM as a function of G for the link lengths L, where λ = 1550 nm, C n 2 = 8 × 10 15 m 2 / 3, T = 300 K, Symbol rate = 1 GB/s.
Fig. 5
Fig. 5 ABER of the 4-PAM as a function of G for the wavelength λ, where L = 1500 m, C n 2 = 8 × 10 15 m 2 / 3, T = 300 K, Symbol rate = 1 GB/s.
Fig. 6
Fig. 6 BER of the 4-PAM as a function of G for symbol rate, where λ = 1550 nm, L = 1500 m, C n 2 = 8 × 10 15 m 2 / 3, T = 300 K.
Fig. 7
Fig. 7 BER of the 4-PAM as a function of G for the temperature of APD K, where λ = 1550nm, L = 1500m, C n 2 = 8 × 10 15, Symbol rate = 1GB/s.
Fig. 8
Fig. 8 ABER-PEs of the M-PAM as a function of G for the atmospheric structure constant C n 2, (a) and (b) represent ε = 0.93 and ε = 1.2, respectively. Where A0 = 1, λ = 1550 nm, L = 3000 m, T = 300 K, Symbol rate = 1 GB/s.
Fig. 9
Fig. 9 ABER of the 4-PAM as a function of SNR for the atmospheric structure constant C n 2, where λ = 1550 nm, L = 1500 m, T = 300 K, Symbol rate = 1 GB/s.

Equations (22)

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f ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 K α β ( 2 α β I ) , I > 0 ,
I = I / E ( I ) .
s i ( t ) = a i g T ( t ) cos ( 2 π ν t ) , { i = 1 , 2 , M } ,
r = 0 T s ( s i ( t ) + n w ( t ) ) f ( t ) d t = a i 2 E g 0 T s g T 2 ( t ) cos 2 ( 2 π ν t ) d t + 0 T s n w ( t ) cos ( ν t ) d t = a i E g 2 + n ,
P g T ( t ) = { E g T s 0 T s 0 o t h e r s .
K s ¯ = η h v 0 T s s i 2 ( t ) d t ¯ = η h v × E g a i 2 2 ¯ = η h v × 6 ( log 2 M ) M 2 1 E b = η h v × 6 ( log 2 M ) M 2 1 T s I ,
E b A v T s I / E ( E b ) = T s I ,
E [ r | s i ] = m 0 m 1 a i G K s ,
D [ r | s i ] = σ 0 2 + σ 1 2 = [ G 2 F ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s q 2 R L ] × 2 B T s ,
p ( r | s i ) = 1 4 π [ G 2 F ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s q 2 R L ] B T s × exp ( ( r s i ) 2 4 π [ G 2 F ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s q 2 R L ] B T s ) .
P ( e r r o r | s 1 ) = 0 p ( r | s 1 ) d r = 1 2 e r f c ( a i G K s q 4 [ G 2 F q 2 ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s R L ] B T s ) ,
P ( e r r o r | s 1 ) = Q ( a i G K s q 2 [ G 2 F q 2 ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s R L ] B T s ) .
P S E R = P ( s 1 ) P ( e r r o r | s 1 ) + P ( s 2 ) P ( e r r o r | s 2 ) = P ( e r r o r | s 1 ) = P ( e r r o r | s 2 ) = Q ( a i G K s q 2 [ G 2 F q 2 ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s R L ] B T s ) .
P S E R = P ( s 1 ) P ( e r r o r | s 1 ) + P ( s 2 ) P ( e r r o r | s 2 ) + + P ( s M ) P ( e r r o r | s M ) = P ( e r r o r | s 1 ) = P ( e r r o r | s 2 ) = = P ( e r r o r | s M ) = 2 ( M 1 ) M Q ( a i G K s q 2 [ G 2 F q 2 ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s R L ] B T s ) .
P b = 2 ( M 1 ) M log 2 M Q ( a i G K s q 2 [ G 2 F q 2 ( a i K s + 2 K b ) + 2 I s T s q + 2 κ T T T s R L ] B T s ) .
P B E R ¯ = 0 f ( I ) P b ( I ) d I = 2 ( M 1 ) M log 2 M 0 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 K α β ( 2 α β I ) × Q ( 6 η G q ( log 2 M ) T s I 2 h v ( M 2 1 ) [ G 2 F q 2 ( 6 η ( log 2 M ) T s I + 2 h v ( M 2 1 ) K b + 2 h v ( M 2 1 ) ( I s T s q + κ T T T s R L ) ] B T s ) d I .
K n = 2 F K b + 2 ( I s T s R L q + κ T T T s ) G 2 q 2 .
P B E R ¯ = 0 f ( K s ¯ ) P b ( K s ¯ ) d K s ¯ = 2 ( M 1 ) M log 2 M 0 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) K s ¯ ( α + β ) / 2 1 K α β ( 2 α β K s ¯ ) × Q ( 6 η G q ( log 2 M ) K s ¯ 6 ( M 2 1 ) log 2 ( M ) F K s ¯ + ( M 2 1 ) 2 K n ) d K s ¯ ,
f P E s ( I ) = ε 2 A 0 ε 2 I ε 2 1 .
P B E R P E s ¯ = 0 [ f ( I ) P b ( I ) f P E s ( I ) d I = 2 ( M 1 ) M log 2 M ε 2 A 0 ε 2 0 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 + ε 2 2 K α β ( 2 α β I ) × Q ( 6 η G q ( log 2 M ) T s I 2 h v ( M 2 1 ) [ G 2 F q 2 ( 6 η ( log 2 M ) T s I + 2 h v ( M 2 1 ) K b + 2 h v ( M 2 1 ) ( I s T s q + κ T T T s R L ) ] B T s ) d I .
ξ = 2 ( M 1 ) M log 2 M , ρ = η 2 × 6 log 2 M M 2 1 , σ 2 = σ 0 2 + σ 1 2 , ϵ ε 2 A 0 ε 2 .
P B E R ¯ = ξ 0 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 K α β ( 2 α β I ) × Q ( ρ I 2 σ 2 ) d I = ξ 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) G 5 , 2 2 , 4 [ ( 2 α β ) 2 u S N R | 0 , 1 2 1 α 2 , 2 α 2 1 β 2 2 β 2 , 1 ] .

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