Abstract

We study the performance of diversity combining techniques applied to synchronous laser communication through the turbulent atmosphere. We assume that a single information-bearing signal is transmitted over two or more statistically independent fading channels, and that the multiple replicas are combined at the receiver to improve detection efficiency. We consider the effects of log-normal amplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise. We study the effect of various parameters, including the ratio of receiver aperture diameter to wavefront coherence diameter, the scintillation index, and the number of independent diversity branches combined at the receiver. We consider both maximal-ratio combining (MRC) and selective combining (SC) diversity schemes. We derive expressions for the outage Shannon capacity, thus placing upper bounds on the spectral efficiency achievable using these techniques.

© 2009 OSA

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Corrections

Aniceto Belmonte and Joseph M. Kahn, "Capacity of coherent free-space optical links 
using diversity-combining techniques: errata," Opt. Express 18, 17748-17748 (2010)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-17-17748

References

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  1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55(1), 57–77 (1967).
    [CrossRef]
  2. J. Proakis, and M. Salehi, Digital Communications, (Mc Graw-Hill, 2007).
  3. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16(18), 14151–14162 (2008).
    [CrossRef] [PubMed]
  4. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using atmospheric compensation techniques,” Opt. Express 17(4), 2763–2773 (2009).
    [CrossRef] [PubMed]
  5. S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003).
    [CrossRef]
  6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004).
    [CrossRef]
  7. 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(8), 1402–1412 (2005).
    [CrossRef]
  8. J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
    [CrossRef]
  9. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
    [CrossRef]
  10. N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008).
    [CrossRef]
  11. K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
    [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. Select. Areas. Commun. 26(3), 3–12 (2008).
    [CrossRef]
  13. C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
  14. R. M. Gagliardi, and S. Karp, Optical Communications (John Wiley & Sons, 1995).
  15. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
    [CrossRef]
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976).
    [CrossRef]
  17. J. D. Parsons, “Diversity techniques in communications receivers,” in Advanced Signal Processing, D. A. Creasey, ed. (Peregrinus, 1985), Chap. 6.
  18. N. Kong, T. Eng, and L. B. Milstein, “A selection combining scheme for rake receivers,” in Proc. ICUPC, Japan, 1995, pp. 426–429.

2009 (1)

2008 (4)

A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16(18), 14151–14162 (2008).
[CrossRef] [PubMed]

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008).
[CrossRef]

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
[CrossRef]

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. Select. Areas. Commun. 26(3), 3–12 (2008).
[CrossRef]

2006 (1)

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[CrossRef]

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(8), 1402–1412 (2005).
[CrossRef]

J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[CrossRef]

2004 (1)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004).
[CrossRef]

2003 (1)

S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003).
[CrossRef]

1976 (1)

1975 (1)

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
[CrossRef]

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55(1), 57–77 (1967).
[CrossRef]

1948 (1)

C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Anguita, J. A.

Belmonte, A.

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. Select. Areas. Commun. 26(3), 3–12 (2008).
[CrossRef]

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(8), 1402–1412 (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. 53(8), 1402–1412 (2005).
[CrossRef]

Chakraborty, K.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
[CrossRef]

Chan, V. W. S.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004).
[CrossRef]

Cowley, W.

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (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. Select. Areas. Commun. 26(3), 3–12 (2008).
[CrossRef]

Dey, S.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
[CrossRef]

Djordjevic, I. B.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[CrossRef]

J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[CrossRef]

Franceschetti, M.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
[CrossRef]

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55(1), 57–77 (1967).
[CrossRef]

Haas, S.

S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003).
[CrossRef]

Holland, I.

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008).
[CrossRef]

Kahn, J. M.

Khan, J.

Lee, E. J.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004).
[CrossRef]

Letzepis, N.

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008).
[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. 53(8), 1402–1412 (2005).
[CrossRef]

Neifeld, M.

Neifeld, M. A.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[CrossRef]

Noll, R. J.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Shapiro, J. H.

S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003).
[CrossRef]

Speck, J. P.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
[CrossRef]

Vasic, B.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[CrossRef]

Vasic, B. V.

Wang, T.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
[CrossRef]

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. Select. Areas. Commun. 26(3), 3–12 (2008).
[CrossRef]

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(8), 1402–1412 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

IEEE J. Sel. Areas Comm. (1)

S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[CrossRef]

IEEE Trans. Commun. (1)

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(8), 1402–1412 (2005).
[CrossRef]

IEEE Trans. Inf. Theory (1)

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008).
[CrossRef]

IEEE Trans. Wirel. Comm. (1)

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008).
[CrossRef]

J. Opt. Netw. (1)

J. Opt. Soc. Am. (1)

J. Select. Areas Commun. (1)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004).
[CrossRef]

J. Select. 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. Select. Areas. Commun. 26(3), 3–12 (2008).
[CrossRef]

Opt. Express (2)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55(1), 57–77 (1967).
[CrossRef]

Radio Sci. (1)

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975).
[CrossRef]

Other (4)

J. D. Parsons, “Diversity techniques in communications receivers,” in Advanced Signal Processing, D. A. Creasey, ed. (Peregrinus, 1985), Chap. 6.

N. Kong, T. Eng, and L. B. Milstein, “A selection combining scheme for rake receivers,” in Proc. ICUPC, Japan, 1995, pp. 426–429.

R. M. Gagliardi, and S. Karp, Optical Communications (John Wiley & Sons, 1995).

J. Proakis, and M. Salehi, Digital Communications, (Mc Graw-Hill, 2007).

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Figures (3)

Fig. 1
Fig. 1

ε-outage spectral efficiency vs. normalized receiver aperture diameter D/r 0 for coherent detection and AWGN. In (a), MRC combining is employed. In (b), a SC combiner is considered. In all cases, the outage probability is fixed at ε=0.001, and the channel capacity per unit bandwidth is shown for different values of the number L of combiner branches. The case L=1 corresponds to no receive diversity (blue lines). The area πD2 describes the combined, multi-aperture system equivalent aperture. When no receive diversity is considered, D equals the receiver aperture diameter. The turbulence-free SNR per symbol γ0 is proportional to the square of the aperture diameter D. For the smallest aperture considered, we assume γ0 equal to 10 photons per symbol. Solid lines neglect amplitude fluctuations by assuming σβ 2=0. In this case, turbulence is characterized by the phase coherence length r 0. The dotted lines consider the ε-outage spectral efficiency when the scintillation index is not neglected but fixed at σβ 2 = 1. The AWGN Shannon limit is indicated by black lines.

Fig. 3
Fig. 3

Probability of outage versus channel capacity per unit bandwidth for coherent detection and additive white Gaussian noise (AWGN). We consider both MRC (in a) and SC (in b) combining of the received signal and the trade-off between the outage probability and the maximum achievable rate is analyzed for different values of the number of branches L in the combiner. The case L=1 corresponds to no receive diversity (blue lines). In all cases, we assume the number of photons per symbol γ0 equals to 100. Amplitude fluctuations are neglected by assuming σβ 2=0. Turbulence is characterized by a moderate phase coherence length r 0 such that D/r 0=2. The area πD2 describes the combined, multi-aperture system equivalent aperture. The AWGN Shannon limit corresponding to γ0 = 100 photons-per-symbol is indicated by vertical black lines.

Fig. 2
Fig. 2

ε-outage spectral efficiency vs. turbulence-free photons per symbol γ0 for coherent detection and additive white Gaussian noise (AWGN). In (a), MRC combining is employed. In (b), a SC combiner is considered. In all cases, the outage probability is fixed at ε=0.001, and the channel capacity per unit bandwidth is shown for different values of the number L of combiner branches. The case L=1 corresponds to no receive diversity (blue lines). Amplitude fluctuations are neglected by assuming σβ 2=0. Turbulence is characterized by a moderate phase coherence length r 0 such as D/r 0=2. The area πD2 describes the combined, multi-aperture system equivalent aperture. The AWGN Shannon limit is indicated by black lines. The dotted lines consider the ε-outage spectral efficiency when the scintillation index is not neglected but fixed at σβ 2 = 1.

Equations (13)

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pγ(γ)=1+rγ¯exp(r)exp[(1+r)γγ¯]I0[2(1+r)rγγ¯],γ0,
σχ2=loge(1+σβ2)σϕ2=1.0299(Dr0)53
γMRC=l=1Lγl   .
γ¯l=γ¯for all     l     {1,2,...,L}   ,
pγ(γMRC)=(1+rγ¯)L+12(1Lr)L12exp(Lr)exp[(1+r)γMRCγ¯]IL1[2L(1+r)rγMRCγ¯].
FMRC(γt)=0γtdγMRCpγ(γMRC)=1(12Lr)L12QL(2Lr,2(1+r)γ¯γt),
FSC(γt)=[1Q(2r, 2 (1+r)γ¯ γt)] L   ,
CεB=log2(1+γR)=log2[1+F1(ε)]   .
Fc(ε)=Γ(L,εγ¯)Γ(L)   .
CεB=log2{1+γ¯ [ ε Γ(1+L) ] 1L}   .
Fc(γR)=exp(L γγ¯)   .
CεB=log2(1+γ¯L ε)   ,
ΔPdB=L (ε 1LL!) 1L   ,

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