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
Atmospheric optical communication (AOC) is receiving considerable attention recently for use in high data rate wireless links. However, even in clear sky conditions, wireless optical links experience fluctuations in both the intensity and the phase of an optical wave propagating through this medium  due to time varying inhomogeneities in the refractive index of the atmosphere. The reliability of an optical system operating in such an environment can be deduced from a mathematical model for the probability density function (pdf) of the randomly fading irradiance signal (scintillation). For that reason, one of the goals in studying optical wave propagation through turbulence is the identification of a tractable pdf of the irradiance under all intensity fluctuation regimes.
Over the years, many irradiance pdf models have been proposed with different degrees of success. Perhaps the most successful models are the lognormal  and the gamma-gamma  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  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 . 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 , gamma , shadowed-Rician , K [3, 7], homodyned-K , exponential  or Gamma-Rician  pdfs are considered as particular cases of this Málaga distribution.
In addition to the scintillation effect, misalignment between transmitter and receiver due to building sway causes vibrations of the transmitted beam and, thus, pointing errors that limit the performance of atmospheric free-space optical links. The impact of pointing error (jitter) was widely investigated for intersatellite space-based free-space optical links [8, 9] although without providing closed-form expressions. In a further step, Kiasaleh  deduced approximated expressions for the pdf of optical signal intensity in an optical communication channel impaired by motion-induced beam jitter and turbulence, assuming that the beam scintillation is governed by a lognormal pdf and a K distribution. Recently, a new procedure to derive the statistical distribution of pointing errors was proposed in . Results derived in this last work were employed by Sandalidis et al.  to study the average bit error rate (ABER) expression for an intensity-modulation/direct detection (IM/DD) atmospheric optical system with on-off keying (OOK), assuming pointing errors, but restricted to strong turbulence.
Now, through this paper we widely generalize the results obtained in  for all turbulence regimes, thus completing our works reported in [5, 13] with ℳ distributed turbulence, by including the impact of misalignment fading on the performance of AOC systems. In this sense, we present a statistical AOC channel model considering the joint effects of ℳ distributed turbulence and pointing errors. As detailed in , ℳ distribution is valid under any turbulence regime, unifying in a closed-form expression the link performance of most of the irradiance statistical models proposed in literature. In addition, we derive the centered moments of the overall probability distribution and, finally, we provide closed-form expressions for the ABER performance of an IM/DD system over the combined effect of Málaga turbulence and misalignment fading, showing the impairment in performance of pointing errors in atmospheric optical links.
2. System model
In this paper, we consider IM/DD channels using on-off keying (OOK) modulation, which is widely employed in practical systems. Laser beams propagate along a horizontal path through a turbulence channel governed by an ℳ distribution in the presence of pointing errors. An additive white Gaussian noise (AWGN) is assumed that includes any shot noise caused by ambient light that may be much stronger than the desired signal as well as front-end thermal noise in the electronics following the photodetector. The receiver integrates the photocurrent signal which is related to the incident optical power by the detector responsivity, R, for each bit period. As explained above, the received optical power, PR(t), suffers from a fluctuation in signal intensity due to atmospheric turbulence and misalignment, as well as additive noise, and can be modeled, from , as:11]. This channel state is considered to be a product of two random components with a third and deterministic factor. The random components that take part in h are geometric spread and pointing errors, denoted by hp and atmospheric turbulence, represented by ha, following the same notation employed in . The deterministic component that must be included in h arises due to atmospheric path loss, hl. This atmospheric attenuation, hl, is described by the exponential Beers-Lambert Law and can be expressed in terms of the visibility . However, although weather-induced attenuation can also degrade the performance of AOC systems in the way shown in , due to its deterministic nature hl acts as a scaling factor as shown in  and for this reason, it is not considered in this manuscript. Thus, through this paper, h(t) = ha(t) ·hp(t), and each of these two factors are explained in detail through Sections 3 and 4, respectively.
Finally, and as in , 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
In this section, we detail the analytical development for the factor ha introduced in Section 2. The starting point is the new statistical model for the irradiance fluctuations of an unbounded optical wavefront propagating through a turbulent medium under all irradiance fluctuation conditions in homogeneous, isotropic turbulence that was presented in . For this reason, a brief description of such a model is achieved through this section in order to improve the overall understanding of this paper. The proposed model, labeled Málaga or simply ℳ, is based on a physical model of the scattering processes that is seen as an extension of the previous work developed by Churnside and Clifford . The main advantage of the Málaga distribution associated to the proposed physical model is that it does reduce to a simple closed-form analytical formula valid from every turbulence regime.
Thus, assume an electromagnetic wave propagating through a turbulent atmosphere with a random refractive index. As the wave passes through this medium, part of the energy is scattered and the form of the irradiance probability distribution is determined by the type of scattering involved. The basis of the ℳ distribution is the conception of an alternative physical model in the generation of small-scale fluctuations. The propagation scheme is illustrated in Fig. 1. As detailed in  the observed field at the receiver is supposed to consist of three terms: the first one is the line-of-sight (LOS) contribution, UL, the second one is the component which is quasi-forward scattered by the eddies on the propagation axis, , and coupled to the LOS contribution; whereas the third term, , is due to energy which is scattered to the receiver by off-axis eddies, this latter contribution being statistically independent from the previous two other terms. The inclusion of the coupled to the LOS scattering component, , is the main novelty of the model and it can be justified by the high directivity and the narrow beamwidths of laser beams in atmospheric optical communications.
Accordingly, the total observed field when atmospheric turbulence is the only adverse effect considered in h can be written as: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 , 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 . 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, U′S 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, , is given in . 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 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.6], composed by the sum of a Rayleigh random phasor (the independent scatter component, U′S) and a Nakagami distribution, . Then, we can apply the same procedure exposed in  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: 5]. Finally, 1F1 (a; c; x) is the Kummer confluent hypergeometric function of the first kind.
Otherwise, the large-scale fluctuations, X ≜ exp (2χ), is widely accepted to be a lognormal amplitude  but, as in [3, 6], this distribution is approximated by a gamma one, this latter with a more favorable analytical structure. For further details about the fact that the gamma distribution closely approximates the lognormal distribution, we suggest to consult references [17, 18]. Then:3].
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 , the pdf of the irradiance is represented by:Eq. (7), Kν (·) is the modified Bessel function of the second kind and order ν.
In Eq. (10), (β)k represents the Pochhammer symbol.
4. Misalignment fading model
As explained in Section 2, the channel fading coefficient, h(t) is considered through this paper to be a product of two random components: the scintillation, ha, detailed in Section 3, and geometric spread and pointing errors, denoted by hp. This last factor is now analytically detailed through this section. Thus, in atmospheric optical links, pointing accuracy is an important issue in determining link performance. For this reason, Farid and Hranilovic  derived its statistical nature, fhp (hp), assuming a Gaussian spatial intensity profile of beam waist, wz, at distance z from the transmitter:
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.Eqs. (12) and (14), the pdf of hp is given by: 11] 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):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): 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: 19, Eq. (07.34.02.0001.01)]. Taking a = −b = (α−k)/2 in Eq. (19) and substituting such an equation into Eq. (18), then:
Finally, to solve the last integrate, we can employ Eq. (07.34.21.0085.01) in  and, thus, the unconditional pdf, fh(h), for the channel state, h, can be written in a closed-form expression: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  in the form of where and with fk(·) being a Gamma-Gamma pdf. Since the cumulative distribution function (CDF) can be written as , where Fk(ha) is the CDF of each fk(·), then we can build a new CDF in the form: . By applying the ratio criterion  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 . By proceeding in an analogous manner as in , we can conclude that all the conditions needed to apply the Helly-Bray theroem  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
5.1. Moments of the combined ℳ probability distribution
In this subsection, the kth moment of the channel state, h, following a combined ℳ probability distribution, fh(h), as given in Eq. (23) is derived. As we indicated in Section 2, the channel state is considered to be a product of hp and ha, where hp represents the random effect of pointing errors whereas ha denotes the turbulence-induced scintillation suffered by the observed irradiance. Furthermore, ha can be decomposed into two other components, as indicated in Eq. (3): ha = YX. The first component, denoted by Y and following a shadowed-Rician distribution , is associated to small-scale fluctuations of the irradiance, i.e., the small-scale contributions to scintillation associated with turbulent cells smaller than either the first Fresnel zone or the transverse spatial coherence radius, whichever is smallest; whereas the second component of ha, denoted by X and following a gamma distribution, is related to to large-scale fluctuations of the irradiance that are generated by turbulent cells larger than that of either the Fresnel zone or the so-called “scattering disk”, whichever is largest.
The kth centered moments of ha, denoted by mk(ha), following an ℳ distribution was derived in . Based on assumptions of statistical independence for the underlying random processes, X and Y, then:23, Eq. (2.23)], the moment of a Nakagami-m pdf is given by:
In addition, the moment of a shadowed-Rician distribution with the restriction of β being a natural number was derived in . Then:5].Eq. (27) is now generalized as:
On the other hand, as discussed in  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
To calculate mk (hp) we must solve:
6. Average BER
The study of the ABER of the ℳ probability distribution in the absence of pointing errors was derived in . Now, in this section, we complete such a previous work by detailing the procedure to derive the analytical expressions for the ABER of the ℳ probability distribution in the presence of misalignment fading. In this sense, an IM/DD link using OOK signalling technique is considered again. For such links, each bit symbol is transmitted by pulsing the light source either on or off during each bit time. Since in most practical systems, the receiver signal-to-noise ratio (SNR) is limited by shot noise caused by ambient light that may be much stronger than the desired signal and/or by thermal noise in the electronics following the photodetector, then noise can usually be modeled to high accuracy as AWGN that is statistically independent of the desired signal . Because of AWGN, errors may be made at the receiver in determining the actual symbols transmitted. The overall probability of error, Pb(e) is expressed as the following weighted sum:24]. On another note, p(e|0) and p(e|1) denote the conditional bit error probabilities when the transmitted bit is “0” and “1”, respectively. We assume that either symbol (binary “0” or binary “1”) are equally likely to be transmitted, i.e., p0 = p1 = 1/2. Considering that p(e|0) = p(e|1), then it is straightforward to show that the conditional BER is expressed, from [13, 24], as 12, 13].
Thus, for the particular case of β being a natural number, the pdf of the channel state, fh(h), is taken from Eq. (21). Now we can express the Gaussian Q function appeared in Eq. (36) in terms of the complementary error function, erfc(x). Next, we can write both the erfc(·) and the modified Bessel integrands as Meijer G functions by employing [19, Eqs. (07.34.03.0619.01), (07.34.03.0605.01) ], respectively, as in [12, 13]. Such relationships are written here for added convenience: in fact, we have already written in Eq. (19) the relationship between a modified Bessel function and a Meijer G function. In the same way, the erfc(·) and the Meijer G function are related by:
Finally, using [19, Eq. (07.34.21.0013.01)], the ABER is derived in a closed-form expression as:
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:
7. Numerical results and discussions
Some numerical results are shown in Figs. 2–3, where the IM/DD AOC system is evaluated for its error performance capabilities against average SNR, and in terms of the Rytov variance, , as in [12, 13]. Effectively, as indicated in , 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, 11]:
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 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 . 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 ( , , 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 , 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 and into . The presence of a higher power from the 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 , being the results shown in this paper a 23.03 optical dB displaced version with reference to the ones displayed in Fig. 3 of . 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 .
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 , 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 , 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.
Finally, as can be deduced, the consideration of both factors, turbulence and misalignment, affecting the AOC systems performance, is an essential AOC link design criterion. Through this paper, we have derived a generalized analytical expression to represent the ABER of an AOC system including the presence of pointing errors, so that a designer can obtain in a straightforward manner the ABER related to a particular location affected for both turbulence-induced scintillation and pointing errors due to misalignment between the transmitter and receiver.
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 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]