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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 12550–12562
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Impact of pointing errors on the performance of generalized atmospheric optical channels

Antonio Jurado-Navas, José María Garrido-Balsells, José Francisco Paris, Miguel Castillo-Vázquez, and Antonio Puerta-Notario  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 12550-12562 (2012)
http://dx.doi.org/10.1364/OE.20.012550


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Abstract

Recently, a new and generalized statistical model, called ℳ or Málaga distribution, was proposed to model the irradiance fluctuations of an unbounded optical wavefront (plane and spherical waves) propagating through a turbulent medium under all irradiance fluctuation conditions in homogeneous, isotropic turbulence. Málaga distribution was demonstrated to have the advantage of unifying most of the proposed statistical models derived until now in the bibliography in a closed-form expression providing, in addition, an excellent agreement with published plane wave and spherical wave simulation data over a wide range of turbulence conditions (weak to strong). Now, such a model is completed by including the adverse effect of pointing error losses due to misalignment. In this respect, the well-known effects of aperture size, beam width and jitter variance are taken into account. Accordingly, after presenting the analytical expressions for the combined distribution of scintillation and pointing errors, we derive its centered moments of the overall probability distribution. Finally, we obtain the analytical expressions for the average bit error rate performance for the ℳ distribution affected by pointing errors. Numerical results show the impact of misalignment on link performance.

© 2012 OSA

1. Introduction

Over the years, many irradiance pdf models have been proposed with different degrees of success. Perhaps the most successful models are the lognormal [2

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

] and the gamma-gamma [3

3. 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. 40, 1554–1562 (2001). [CrossRef]

] ones. The scope of the lognormal model is restricted under weak irradiance fluctuations. Conversely, the gamma-gamma pdf was suggested by Andrews et al. as a reasonable alternative to Beckmann’s pdf [4

4. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987). [CrossRef]

] because of its much more tractable mathematical model. Nevertheless, both lognormal and gamma-gamma models are particular cases of the new and recently proposed ℳ (Málaga) distribution model [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

]. ℳ distribution was validated by comparing its pdf with published simulation data, showing that it unifies in an analytical expression most of the irradiance statistical models proposed in literature by the scientific community for more than four decades. So, for instance: Rice-Nakagami [1

1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).

], gamma [3

3. 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. 40, 1554–1562 (2001). [CrossRef]

], shadowed-Rician [6

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003). [CrossRef]

], K [3

3. 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. 40, 1554–1562 (2001). [CrossRef]

, 7

7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980). [CrossRef]

], homodyned-K [7

7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980). [CrossRef]

], exponential [3

3. 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. 40, 1554–1562 (2001). [CrossRef]

] or Gamma-Rician [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

] pdfs are considered as particular cases of this Málaga distribution.

2. System model

Finally, and as in [11

11. 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]

], it is considered that the pointing error is due to building sway, with sub-Hz bandwidth (correlation times of a few seconds), so that the scintillation process must have a smaller correlation time (10–100 ms) as compared with building sways. Hence, the two above impairments are stated to be independent.

3. Generalized atmospheric distribution model

Fig. 1 Proposed propagation geometry for a laser beam in a Málaga model to form the small-scale fluctuations [5].

Accordingly, the total observed field when atmospheric turbulence is the only adverse effect considered in h can be written as:
U=(UL+USC+USG)exp(χ+jS)
(2)
where UL=GΩexp(jϕA), USC=Gρ2b0exp(jϕB) and USG=(1ρ)US, being USC and USG statistically independent stationary random processes. Of course, UL and USG are also independent random processes. In Eq. (2), G is a real variable following a gamma distribution with E[G]=1. It represents the slow fluctuation of the LOS component. Following the notation of [6

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003). [CrossRef]

], the parameter Ω =E[|UL|2] represents the average power of the LOS term whereas the average power of the total scatter components is denoted by 2b0=E[|USC|2+|USG|2]. ϕA and ϕB are the deterministic phases of the LOS and the coupled-to-LOS scatter terms, respectively. On another note, 0≤ρ≤1 is the factor expressing the amount of scattering power coupled to the LOS component. Finally, US is a circular Gaussian complex random variable, and χ and S are real random variables representing the log-amplitude and phase perturbation of the field induced by the atmospheric turbulence, respectively. A plausible justification for the coupled-to-LOS scattering component, USC, is given in [16

16. R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970). [CrossRef]

]. There, it is said that if the turbulent medium is so thin that multiple scattering can be ignored, the multipath delays of the scattered radiation collected by a diffraction-limited receiver will usually be small relative to the signal bandwidth. Then the scattered field will combine coherently with the unscattered field and there will be no-“interfering” signal component of the field, in a similar way as USC combines with UL in our proposed model. Of course, when the turbulent medium becomes so thick, then the unscattered component of the field can be neglected.

Now, and from Eq. (2), the observed irradiance of the proposed propagation model (ha, as we have adopted the notation given in [11

11. 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]

]) can be expressed as:
ha=|UL+USC+USG|2exp(2χ)=YX,{Y|UL+USC+USG|2(small-scalefluctuations)Xexp(2χ)(large-scalefluctuations);
(3)
whereas the lowpass-equivalent complex envelope can be rewritten as:
R(t)=(UL+USC+USG)=G[Ωexp(jϕA)+ρ2b0exp(jϕB)]+(1ρ)US,
(4)
so that we have the identical shadowed Rice single model employed in [6

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003). [CrossRef]

], composed by the sum of a Rayleigh random phasor (the independent scatter component, US) and a Nakagami distribution, G. Then, we can apply the same procedure exposed in [6

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003). [CrossRef]

] consisting in calculating the expectation of the Rayleigh component with respect to the Nakagami distribution and then deriving the pdf of the instantaneous power. Hence, the pdf of Y is given by:
fY(y)=1γ[γβγβ+Ω]βexp[yγ]1F1(β;1;1γΩ(γβ+Ω)y)
(5)
where β ≜ (E[G])2/Var[G] is the amount of fading parameter with Var[·] being the variance operator, where | · | denotes the absolute value. For simplicity, we have denoted γ=E[|USG|2]=2b0(1ρ) whereas Ω=Ω+ρ2b0+22b0Ωρcos(ϕAϕB)represents the average power from the coherent contributions, as detailed in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

]. Finally, 1F1 (a; c; x) is the Kummer confluent hypergeometric function of the first kind.

Thus to obtain the statistical characterization of the intensity, ha, the mixture of fX (x) and fY (y) must be accomplished. After the corresponding and non-obvious algebraic manipulation, extensively detailed in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

], the pdf of the irradiance is represented by:
fha(ha)=Ak=1βakhaα+k21Kαk(2αβhaγβ+Ω)
(7)
where
{A2αα2γ1+α2Γ(α)(γβγβ+Ω)β+α2;ak(β1k1)(γβ+Ω)1k2(k1)!(Ωγ)k1(αβ)k2.
(8)
with β being a natural number. In Eq. (7), Kν (·) is the modified Bessel function of the second kind and order ν.

A generalized expression of Eq. (7) was also given in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

] with β being a real number:
fha(ha)=A(G)k=1ak(G)haα+k21Kαk(2αhaγ),
(9)
where
{A(G)2αα2γ1+α2Γ(α)(γβγβ+Ω)β;ak(G)(β)k1(αγ)k2[(k1)!]2γk1(Ω+γβ)k1.
(10)

In Eq. (10), (β)k represents the Pochhammer symbol.

4. Misalignment fading model

By considering independent identical Gaussian distributions for the elevation and the horizontal displacement (sway), the radial displacement, r, at the receiver follows a Rayleigh distribution.
fr(r)=rσs2exp(r22σs2),r>0;
(14)
where σs2 is the jitter variance at the receiver. Then, from Eqs. (12) and (14), the pdf of hp is given by:
fhp(hp)=g2A0g2hpg21,0hpA0;
(15)
where g = wzeq/(2σs) is the ratio between the equivalent beam radius at the receiver, wzeq, and the pointing error displacement standard deviation at the receiver, σs. See [11

11. 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]

] for more details.

5. Combined channel statistical model

In this section, we derive a complete statistical model of a stochastic atmospheric optical channel taking into account both turbulence-induced scintillation and misalignment-induced fading. Therefore, the unconditional pdf, fh(h), for the channel state, h, is obtained by calculating the mixture of the two distributions presented above in Eqs. (7) or (9) and (15):
fh(h)=fh|ha(h|ha)fha(ha)dha,
(16)
where fh|ha (h|ha) is the conditional probability given a turbulence state, ha, and it is expressed as:
fh|ha(h|ha)=1hafhp(hha)=g2A0g2ha(hha)g21,0hA0ha;
(17)
and where fha (ha) was written in Eqs. (7) and (9) depending on whether parameter β is a natural number or a real one, respectively. For the first case, we substitute Eq. (7) into the integral expression given by Eq. (16) to finally obtain fh(h):
fh(h)=g2AA0g2hg21k=1βakh/A0haα+k21g2Kαk(2αβhaγβ+Ω)dha
(18)
where the parameters A and ak were already defined in Eq. (8). From [19, Eq. (07.34.03.0605.01)], the modified Bessel function of the second kind, Kv(·) can be expressed as a special case of the Meijer G function, given by the following relationship:
G0,22,0(x|a,b)=2x(a+b)/2Kab(2x),
(19)
where a definition of the Meijer G function can be found at [19, Eq. (07.34.02.0001.01)]. Taking a = −b = (αk)/2 in Eq. (19) and substituting such an equation into Eq. (18), then:
fh(h)=g2A2A0g2hg21k=1βakh|A0haα+k21g2G0,22,0(αβhaγβ+Ω|αk2,αk2)dha.
(20)

Finally, to solve the last integrate, we can employ Eq. (07.34.21.0085.01) in [19] and, thus, the unconditional pdf, fh(h), for the channel state, h, can be written in a closed-form expression:
fh(h)=g2A2h1k=1βak(αβγβ+Ω)α+k2G1,33,0(αβγβ+ΩhA0|g2+1g2,α,k),
(21)
just after having simplified the G-Meijer function using the relationship
xrGp,qm,n(x|apcq)=Gp,qm,n(x|ap+rcq+r)
(22)
gathered in [19, Eq. (07.34.17.0011.01)].

For the generic case of Eq. (9) with β being a real number, the obtention of fh(h) is not so evident because it is firstly mandatory to verify if the infinite summation of Eq. (9) and the integrate operator that appears when building fh(h) can be interchanged. To prove that, by normalization, it is possible to express Eq. (9) as a discrete mixture [20

20. C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005) [CrossRef]

] in the form of f(ha)=k=1ωkfk(ha) where ωk0k,k=1ωk=1 and with fk(·) being a Gamma-Gamma pdf. Since the cumulative distribution function (CDF) can be written as F(ha)=k=1ωkFk(ha), where Fk(ha) is the CDF of each fk(·), then we can build a new CDF in the form: ϕn(ha)=k=1nωkFk(ha). By applying the ratio criterion [21

21. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).

] it is easy to test that Fk(ha) converges absolutely so ϕn(ha) is said to converge weakly to F(ha). This condition and the fact that Eq. (17) is bounded and continuous allow us to apply the Helly-Bray theorem [22

22. P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)

]. By proceeding in an analogous manner as in [13

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

], we can conclude that all the conditions needed to apply the Helly-Bray theroem [22

22. P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)

] are satisfied. Thus, operating in a similar way to obtain Eq. (21), then fh(h) is written as an infinite series of Meijer-G functions as
fh(h)=g2A(G)2h1k=1ak(G)(αγ)α+k2G1,33,0(αγhA0|g2+1g2,α,k).
(23)

5.1. Moments of the combined ℳ probability distribution

The kth centered moments of ha, denoted by mk(ha), following an ℳ distribution was derived in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

]. Based on assumptions of statistical independence for the underlying random processes, X and Y, then:
mk(ha)E[hak]=E[Xk]E[Yk]=mk(X)mk(Y),
(24)
whereas the operator E[·] stands for ensemble average. From [23

23. M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

, Eq. (2.23)], the moment of a Nakagami-m pdf is given by:
mk(X)=Γ(α+k)Γ(α)αk,
(25)

In addition, the moment of a shadowed-Rician distribution with the restriction of β being a natural number was derived in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

]. Then:
mk(Y)=1γ(γβγβ+Ω)βr=0β1(β1r)1r!(Ωγ(γβ+Ω))rΓ(k+r+1)(βγβ+Ω)k+r+1.
(26)

Hence, when performing the product of Eq. (25) by Eq. (26), we certainly obtain:
mk(ha)=Γ(α+k)Γ(α)αkγ(γβγβ+Ω)βr=0β1(β1r)1r!(Ωγ(γβ+Ω))rΓ(k+r+1)(βγβ+Ω)k+r+1,
(27)
as indicated in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

].

And for the generic case of Eq. (9) with the parameter β being a real number, the moment of a shadowed-Rician distribution was given in [23

23. M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

, Eq. (2.69)]:
mk(Y)=(γβγβ+Ω)βγkΓ(k+1)2F1(k+1,β;1;Ωγβ+Ω),
(28)
and, thus, Eq. (27) is now generalized as:
mk(G)(ha)=Γ(α+k)Γ(α)αk(γβγβ+Ω)βγkΓ(k+1)2F1(k+1,β;1;Ωγβ+Ω).
(29)

On the other hand, as discussed in [11

11. 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]

] and indicated in Section 2, it is considered that pointing errors and turbulence-induced scintillation are independent processes. As in the case of ha, we can obtain the kth centered moments of the channel state, h, as
mk(h)E[hk]=E[hak]E[hpk]=mk(ha)mk(hp).
(30)

To calculate mk (hp) we must solve:
mk(hp)=E[hpk]=hpkfhp(hp)dhp.
(31)

Inserting Eq. (15) into Eq. (31), then
mk(hp)=g2A0g20A0hpg2+k1dhp=g2g2+kA0k.
(32)

Hence, for the case of β being a natural number, we can obtain the kth centered moments of h by performing the product of Eqs. (27) and (32):
mk(h)=Γ(α+k)Γ(α)αkγ(γβγβ+Ω)βg2g2+kA0k××r=0β1(β1r)1r!(Ωγ(γβ+Ω))rΓ(k+r+1)(βγβ+Ω)k+r+1.
(33)

Finally, for the generic case of β being a real number, the kth centered moments of h is obtained by the product of Eqs. (29) and (32):
mk(h)=g2A0kg2+kΓ(α+k)Γ(α)αk(γβγβ+Ω)βγkΓ(k+1)2F1(k+1,β;1;Ωγβ+Ω).
(34)

6. Average BER

Finally, using [19, Eq. (07.34.21.0013.01)], the ABER is derived in a closed-form expression as:
Pb(e)=2αg2ABα232ππk=1β2kBk2akG7,42,6(8R2P2A02σN2B2|1g22,2g22,1α2,2α2,1k2,2k2,10,12,g22,1g22)
(39)
with B=γβ+Ωαβ.

For the generic case of the irradiance pdf represented by Eq. (9) with β being a real number, we verify again that the infinite summation and the integrate operator can be interchanged. Operating in a similar way as to obtain Eq. (39), the ABER is written as:
Pb(e)=2αg2A(G)32ππ(γα)α2k=12k(γα)k2ak(G)××G7,42,6(8R2P2A02σN2γ2α2|1g22,2g22,1α2,2α2,1k2,2k2,10,12,g22,1g22).
(40)

7. Numerical results and discussions

Some numerical results are shown in Figs. 23, where the IM/DD AOC system is evaluated for its error performance capabilities against average SNR, and in terms of the Rytov variance, σR2, as in [12

12. 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. 12, 44–46 (2008). [CrossRef]

, 13

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

]. Effectively, as indicated in [24

24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). [CrossRef]

], atmospheric turbulence is well characterized by such a Rytov variance when using the optical wave models of an infinite plane wave or a spherical wave (point source). The Rytov variance is defined, for a plane wave, by [24

24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). [CrossRef]

, 11

11. 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]

]:
σR2=1.23Cn2k7/6L11/6.
(41)
where Cn2(m2/3) is the index of refraction structure parameter of the atmosphere, k = 2π/λ is the optical wave number, λ (m) is the wavelength, and L (m) is the propagation path length between transmitter and receiver. To obtain typical values of σR2, we have employed some experimental data of Cn2 provided by University of Waseda, in Japan, on October 15th, 2009, obtained from atmospheric parameters. Thus, near midday, Cn2 parameter oscillated from 2.8 to 11 · 10−14 m−2/3, and the temperature oscillated from 22 °C to 26.4 °C, and the received power was recorded from −22.52 dBm to −20.48 dBm. At night (near 1 a.m.), Cn2 parameter registered its minimum value (∼ 7 ·10−15 m−2/3), for a temperature of 15.1 °C and a received power of −22.91 dBm. At sunrise (6.45 a.m.), Cn2=1.21014m2/3, for a temperature of 15.9 °C and with a received power of −23.47 dBm. The wavelength employed in all cases was 785 nm and the receiver aperture had a diameter of 100 mm. The propagation path length is L = 1 km at a height of 25 m. The optical power transmitted is 11.5 dBm with a responsivity of 0.8 A/W. With such a set of parameters, if we consider Cn2 to 7.2 ·10−15 m−2/3 (night), then σR20.32; secondly, for the case of sunrise when Cn2=1.2m2/3, then σR20.52; finally, for a Cn2=2.81014m2/3 (near midday), then σR21.2. This will be the values of Rytov variance employed through this section. As a final remark, all cases presented were obtained by employing the same normalized average optical power (Ω + 2b0 = 1).

Fig. 2 Average optical BER against average SNR for different values of ρ and wz/a. In all curves, α = 10, β = 5 and the transmitted power is normalized, i.e., Ω + 2b0 = 1. The case of ρ = 1 corresponds to the Gamma-Gamma distribution.
Fig. 3 Average optical BER against average SNR for different values of α, β, ρ assuming σR2=0.36 and wz/a = 20. In all curves, the transmitted power is normalized, i.e., Ω + 2b0 = 1.

Figure 2 shows the ABER for various values of the normalized beamwidth: wz/a=10, 20, and 25. Parameters of ℳ turbulence model were fixed to: α = 10 and β = 5 in all cases. Moreover, for each wz/a value, three different behaviors were obtained: ρ= 1, which corresponds to a Gamma-Gamma distribution; ρ = 0.75, and ρ = 0.25. For such values, the associated Rytov variance is σR2=0.32,0.52 and 1.2, respectively, coinciding with the values selected above. As a reference, the behavior of the system in the absence of misalignment fading is also provided (dotted curves) for the three intensities of turbulence mentioned before [13

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

]. Hence, it is straightforward to check the impairment in performance when pointing errors are considered: for instance, the penalties in SNR with respect to the ideal case where no pointing errors are considered are 17.03, 23.02 and 24.95 optical dB, for wz/a = 10, 20 and 25, respectively, for a bit error rate of 10−6.

In Fig. 3, different ABERs are displayed for wz/a = 20 and a same intensity of turbulence ( σR2=0.36, Cn2=0.831014m2/3, measured on October 15th at 23:10), a typical value in terrestrial horizontal links. It turns out that the behavior of each curve is different depending on ρ. Thus, when ρ = 1 the overall scattering power travels through USC, so it is entirely coupled to the LOS component, minimizing its adverse effect, as displayed in Fig. 3. When ρ is getting lower then the scattering power is distributed both into USC and into USG. The presence of a higher power from the USG component has a more harmful effect in the performance of an AOC system because it is not correlated to the LOS contribution, as corroborated in Fig. 3. Obviously, obtained results are consistent with the ones achieved in [13

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

], being the results shown in this paper a 23.03 optical dB displaced version with reference to the ones displayed in Fig. 3 of [13

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

]. For this reason, same conclusions derived there are applicable when pointing errors are considered.

On the other hand, all BER curves tend to a same diversity order, where the diversity order is defined in [25

25. L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003). [CrossRef]

].

Finally, we must recall that Eq. (7) is a particularization of Eq. (9) valid when β is restricted to a natural number. Nevertheless, due to the high degree of freedom inherent to the ℳ distribution, demonstrated in the set of results shown in [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

], this particularization allows us to reproduce every turbulent scenario, avoiding the necessity of employing the infinite summation included in Eqs. (9) or (40).

8. Concluding remarks

In summary, in this paper, two analytical closed-form representations for the ABER performance of an AOC system operating over a generalized turbulence in the presence of pointing errors are derived. Due to the high degree of freedom inherent to the ℳ distribution [5

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

], the particularization given in Eq. (39) let us reproduce every turbulent scenario, avoiding the infinite summation included in Eq. (40).

As discussed in Section 7, the effect of pointing errors on the final performance of an AOC system can be seen as a penalty in SNR with respect to the ideal case where no pointing errors are considered, but with the special feature that such a penalty in SNR is seen as a constant value for every different turbulence regime when the ratio wz/a is fixed, and for a given value of bit error rate.

Acknowledgment

This work was supported by the Spanish Ministerio de Ciencia e Innovación, Project TEC2008-06598. The authors thank Prof. Mitsuji Matsumoto from University of Waseda, in Japan, for his kind support and for providing some experimental measurements of the Cn2 parameter needed to calculate realistic values of Rytov variance.

References and links

1.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).

2.

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

3.

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. 40, 1554–1562 (2001). [CrossRef]

4.

J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987). [CrossRef]

5.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]

6.

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003). [CrossRef]

7.

E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980). [CrossRef]

8.

S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997). [CrossRef]

9.

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626–629 (2003). [CrossRef]

10.

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994). [CrossRef]

11.

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]

12.

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. 12, 44–46 (2008). [CrossRef]

13.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011). [CrossRef] [PubMed]

14.

M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004). [CrossRef]

15.

ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010.

16.

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970). [CrossRef]

17.

T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).

18.

J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970). [CrossRef]

19.

Wolfram, http://functions.wolfram.com/

20.

C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005) [CrossRef]

21.

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).

22.

P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)

23.

M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

24.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). [CrossRef]

25.

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(290.5930) Scattering : Scintillation
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: March 9, 2012
Revised Manuscript: May 10, 2012
Manuscript Accepted: May 11, 2012
Published: May 18, 2012

Citation
Antonio Jurado-Navas, José María Garrido-Balsells, José Francisco Paris, Miguel Castillo-Vázquez, and Antonio Puerta-Notario, "Impact of pointing errors on the performance of generalized atmospheric optical channels," Opt. Express 20, 12550-12562 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12550


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References

  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).
  2. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun.50, 1293–1300 (2002). [CrossRef]
  3. 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.40, 1554–1562 (2001). [CrossRef]
  4. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A4, 1923–1930 (1987). [CrossRef]
  5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]
  6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun.2, 519–528 (2003). [CrossRef]
  7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A13, 31–48 (1980). [CrossRef]
  8. S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE85, 1646–1661 (1997). [CrossRef]
  9. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun.2, 626–629 (2003). [CrossRef]
  10. K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng.33, 3748–3757 (1994). [CrossRef]
  11. 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]
  12. 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.12, 44–46 (2008). [CrossRef]
  13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett.36, 4095–4097 (2011). [CrossRef] [PubMed]
  14. M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng.43, 319–329 (2004). [CrossRef]
  15. ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010.
  16. R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE58, 1651–1665 (1970). [CrossRef]
  17. T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).
  18. J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE58, 1964–1965 (1970). [CrossRef]
  19. Wolfram, http://functions.wolfram.com/
  20. C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005) [CrossRef]
  21. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).
  22. P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)
  23. M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).
  24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). [CrossRef]
  25. L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory49, 1073–1096 (2003). [CrossRef]

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