**Upper Burst Error Bound for Atmospheric Correlated Optical Communications Using an Alternative Matrix Decomposition**

Antonio Jurado-Navas, José María Garrido-Balsells, Miguel Castillo-Vázquez and Antonio Puerta-Notario *Communications Engineering Department, University of Málaga Campus de Teatinos Málaga, Spain*

#### **1. Introduction**

Atmospheric optical communication has been receiving considerable attention recently for use in high data rate wireless links (Arnon, 2003; Haas et al., 2002; Juarez et al., 2006; Zhu & Kahn, 2002). Considering their narrow beamwidths and lack of licensing requirements as compared to microwave systems, atmospheric optical systems are appropriate candidates for secure, high data rate, cost-effective, wide bandwidth communications. Furthermore, the atmospheric optical communications are less susceptible to the radio interference than radio-wireless communications. Moreover, free space optical (FSO) communication systems represent a promising alternative to solve the "last mile" problem, above all in densely populated urban areas. However, even in clear sky conditions, wireless optical links may experience fading due to the turbulent atmosphere. In this respect, inhomogeneities in the temperature and pressure of the atmosphere lead to variations of the refractive index along the transmission path. These random refractive index variations produce fluctuations in both the intensity and the phase of an optical wave propagating through this medium. Such fluctuations can lead to an increase in the link error probability limiting the performance of communication systems. In this particular scenario, the turbulence-induced fading is called scintillation.

If the receiving aperture size in these optical systems, *D*0, can be made larger than the correlation length, *d*0, then the received irradiance becomes a spatial average over the aperture area and the scintillation level measured by the detector begins to decrease. This effect is known as aperture averaging (Andrews & Phillips, 1998). Unfortunately, it could be neither practical nor desirable to satisfy this condition, especially in diversity receivers, so we will assume that *D*<sup>0</sup> < *d*<sup>0</sup> throughout this chapter.

Finally, weather-induced attenuation caused by rain, snow and fog can also degrade the performance of atmospheric optical communication systems in the way shown in (Al Naboulsi & Sizun, 2004; Muhammad et al., 2005), but are not considered in this chapter.

Spatial diversity reception is a good proposal in order to mitigate the adverse effect of the scintillation on the transmitted signal. Nevertheless, many researchers assume in a first approach that turbulence-induced fading is uncorrelated at each of the optical receivers

from (Jurado-Navas & Puerta-Notario, 2009) with different cross-correlation (CC) coefficient in terms of burst error rate, for an atmospheric optical link in the following two extreme situations: first, when a full accomplishment of the frozen turbulence hypothesis is assumed (*ρ<sup>l</sup>* → 0, *τ<sup>e</sup>* → ∞); and second, when an unrealistic scenario is supposed owing to using only a space-time separable statistics model (*ρ<sup>l</sup>* → 1, *τ<sup>e</sup>* → *τ*0), so that it can be assumed that the frozen-in turbulence is not incorporated to the system. Naturally, this latter situation is not corresponding to a real scenario, but it is presented in this paper as a benchmark in order to compare the obtained performance in burst error rate. In view of the results obtained by this latter scenario (shown through section 6 in this proposal of chapter) in comparison to the ones derived by using the first model (incorporating the frozen turbulence), it is concluded that a realistic upper error limit can be easily achieved with a high simplicity by using a space-time

<sup>431</sup> Upper Burst Error Bound for Atmospheric Correlated

Optical Communications Using an Alternative Matrix Decomposition

We must remark, however, that the space-time separable statistics model proposed here is an efficient approach that accomplishes more realistic performances when higher wind velocities are considered. In fact, this approach is based on coloring independent Gaussian sequences first between them and then in time in order to generate *m* log-normal random processes of scintillation. Evidently, such method is restricted to have cross-correlation functions that have the same time-dependencies as the autocorrelation functions, i.e., the obtained sequences have statistics that are space-time separable. Due to this fact, Taylor's hypothesis (Tatarskii, 1971) is not fully satisfied. For such a case, we must redirect readers to (Jurado-Navas & Puerta-Notario, 2009), where Taylor's frozen turbulence hypothesis is properly taken into account. Conversely, the frozen-in hypothesis is unquestionably an approximation and must fail as distance between receivers becomes large or in especial situations when there are both strong velocity fluctuations of the wind or long-range spatial correlations (Burghelea et al., 2005; Moore et al., 2005), or even in urban atmospheres, especially near or among roughness elements, where strong wind shear is expected to create high turbulent kinetic energy (Christen et al., 2007). Furthermore, in urban canopies and cloud streets up to 2 − 5 times the average building height of such particular streets (Christen et al., 2007), the strong wind shear creates turbulence intensities that are tipically near the threshold where the hypothesis of frozen turbulence becomes inapplicable (Christen et al., 2007; Willis & Deardorff, 1976). In this fashion, difference in performance obtained from the realistic model presented in (Jurado-Navas & Puerta-Notario, 2009) and the upper bound performance proposed in this chapter are even closer to each other. Thus, the separable statistics model proposed here may be seen as a highly accurate upper error bound of the complete model detailed in (Jurado-Navas & Puerta-Notario, 2009), with the advantage

of a reduced computational complexity in comparison to an AR method.

There is an extensive literature on the subject of the theory of line-of-sight propagation through the atmosphere (Andrews & Phillips, 1998; Andrews et al., 2000; Fante, 1975; Ishimaru, 1997; Strohbehn, 1978; Tatarskii, 1971). One of the most important works was developed by Tatarskii (Tatarskii, 1971). He supposed a plane wave that is incident upon the random medium (the atmosphere in this particular case). It is assumed an atmosphere having no free charges with a constant magnetic permeability. In addition, it is suppossed that the electromagnetic field has a sinusoidal time dependence (a monochromatic wave). Under these

**3. Turbulent atmospheric channel model**

separable statistics model.

(Ibrahim & Ibrahim, 1996; Lee & Chan, 2004; Razavi & Shapiro, 2005). In order for this assumption to hold true, the spacing between receivers should be greater than the fading correlation length, what may be difficult to satisfy in practice because of the available physical space or due to the fact that the receiver spacing required for uncorrelated fading may exceed the beam diameter in power-limited links with well-collimated beams. For instance, with a propagation path length, *L*, of 1 km and an optical wavelength, *λ*, of 830 nm, the fading correlation length, approximated by *d*<sup>0</sup> = (*λL*)1/2 (Zhu & Kahn, 2002), would be of 2.89 cm. But, if *λ* = 1550 nm and *L* = 10 km, then the receiver spacing required for uncorrelated scintillation should be greater than 12.45 cm. In this respect, the spatial correlation is studied in detail in (Anguita et al., 2007), presenting a dependence on the turbulence parameter *C*<sup>2</sup> *n* and, above all, a more remarkable dependence on the propagation distance and on the receiver aperture.

Thus, in (Jurado-Navas & Puerta-Notario, 2009), a complete model using an autorregresive (AR) model was presented to include correlated scintillations in simulations of free space optical links using multiple receivers. Obtained results showed a diversity gain penalty due to the impact of the spatial coherence which should not be ignored in many practical scenarios. Hence, the method proposed in (Jurado-Navas & Puerta-Notario, 2009) extended the applicability of the existing techniques (Beaulieu, 1999; Ertel & Reed, 1998), including the effect of the atmospheric dynamics in order to break the uniformity of the frozen-in hypothesis (Zhu & Kahn, 2002). This latter effect was incorporated by defining a factor, *ρl*, as follows:

$$
\rho\_l = \tau\_0 / \tau\_\varepsilon \tag{1}
$$

which represents the degree of randomness as effect of the dynamic evolution of the turbulence, with

$$
\pi\_0 = \frac{\sqrt{\lambda L}}{\mu\_\perp} \tag{2}
$$

being the turbulence correlation time, where *λ* is the optical wavelength, *L* is the propagation distance and *<sup>u</sup>*<sup>⊥</sup> the component of the wind velocity transverse to the propagation direction. Finally, *τe* is seen as the lifetime of turbulent eddies and it is directly depending on the turbulent kinetic energy dissipation rate, *�*, that represents the atmospheric dynamics. as the rate of energy cascading from larger eddies to smaller ones.

The method is focused on a multichannel generalization of the autoregressive (AR) variate generation method in a way similar to (Baddour & Beaulieu, 2002) in order to satisfy Taylor's hypothesis of frozen turbulence. Therefore, *m* lognormal scintillation sequences are generated with specified second-order statistics: concretely, the cross-correlation function and the autocorrelation function between different sequences that let spatial and temporal correlations be interrelated.

#### **2. Upper error bound in a simpler channel model**

The AR model presented in (Jurado-Navas & Puerta-Notario, 2009) and commented above is computationally complex due to its inherent numerically ill-conditioned covariance matrix (Baddour & Beaulieu, 2002). In this chapter, we propose a space-time separable statistics model, extremely simple, to avoid such a problem, providing an excellent accurate upper burst error bound and with the advantage of a reduced computational time, for correlated atmospheric terrestrial links operating at optical wavelengths. This limit is heuristically corroborated after comparing the obtained performance using scintillation sequences derived 2 Numerical Simulations / Book 1

(Ibrahim & Ibrahim, 1996; Lee & Chan, 2004; Razavi & Shapiro, 2005). In order for this assumption to hold true, the spacing between receivers should be greater than the fading correlation length, what may be difficult to satisfy in practice because of the available physical space or due to the fact that the receiver spacing required for uncorrelated fading may exceed the beam diameter in power-limited links with well-collimated beams. For instance, with a propagation path length, *L*, of 1 km and an optical wavelength, *λ*, of 830 nm, the fading correlation length, approximated by *d*<sup>0</sup> = (*λL*)1/2 (Zhu & Kahn, 2002), would be of 2.89 cm. But, if *λ* = 1550 nm and *L* = 10 km, then the receiver spacing required for uncorrelated scintillation should be greater than 12.45 cm. In this respect, the spatial correlation is studied in detail in (Anguita et al., 2007), presenting a dependence on the turbulence parameter *C*<sup>2</sup>

and, above all, a more remarkable dependence on the propagation distance and on the receiver

Thus, in (Jurado-Navas & Puerta-Notario, 2009), a complete model using an autorregresive (AR) model was presented to include correlated scintillations in simulations of free space optical links using multiple receivers. Obtained results showed a diversity gain penalty due to the impact of the spatial coherence which should not be ignored in many practical scenarios. Hence, the method proposed in (Jurado-Navas & Puerta-Notario, 2009) extended the applicability of the existing techniques (Beaulieu, 1999; Ertel & Reed, 1998), including the effect of the atmospheric dynamics in order to break the uniformity of the frozen-in hypothesis (Zhu & Kahn, 2002). This latter effect was incorporated by defining a factor, *ρl*, as follows:

which represents the degree of randomness as effect of the dynamic evolution of the

being the turbulence correlation time, where *λ* is the optical wavelength, *L* is the propagation distance and *<sup>u</sup>*<sup>⊥</sup> the component of the wind velocity transverse to the propagation direction. Finally, *τe* is seen as the lifetime of turbulent eddies and it is directly depending on the turbulent kinetic energy dissipation rate, *�*, that represents the atmospheric dynamics. as

The method is focused on a multichannel generalization of the autoregressive (AR) variate generation method in a way similar to (Baddour & Beaulieu, 2002) in order to satisfy Taylor's hypothesis of frozen turbulence. Therefore, *m* lognormal scintillation sequences are generated with specified second-order statistics: concretely, the cross-correlation function and the autocorrelation function between different sequences that let spatial and temporal correlations

The AR model presented in (Jurado-Navas & Puerta-Notario, 2009) and commented above is computationally complex due to its inherent numerically ill-conditioned covariance matrix (Baddour & Beaulieu, 2002). In this chapter, we propose a space-time separable statistics model, extremely simple, to avoid such a problem, providing an excellent accurate upper burst error bound and with the advantage of a reduced computational time, for correlated atmospheric terrestrial links operating at optical wavelengths. This limit is heuristically corroborated after comparing the obtained performance using scintillation sequences derived

<sup>√</sup>*λ<sup>L</sup> u*⊥

*τ*<sup>0</sup> =

the rate of energy cascading from larger eddies to smaller ones.

**2. Upper error bound in a simpler channel model**

*ρ<sup>l</sup>* = *τ*0/*τ<sup>e</sup>* (1)

aperture.

turbulence, with

be interrelated.

*n*

(2)

from (Jurado-Navas & Puerta-Notario, 2009) with different cross-correlation (CC) coefficient in terms of burst error rate, for an atmospheric optical link in the following two extreme situations: first, when a full accomplishment of the frozen turbulence hypothesis is assumed (*ρ<sup>l</sup>* → 0, *τ<sup>e</sup>* → ∞); and second, when an unrealistic scenario is supposed owing to using only a space-time separable statistics model (*ρ<sup>l</sup>* → 1, *τ<sup>e</sup>* → *τ*0), so that it can be assumed that the frozen-in turbulence is not incorporated to the system. Naturally, this latter situation is not corresponding to a real scenario, but it is presented in this paper as a benchmark in order to compare the obtained performance in burst error rate. In view of the results obtained by this latter scenario (shown through section 6 in this proposal of chapter) in comparison to the ones derived by using the first model (incorporating the frozen turbulence), it is concluded that a realistic upper error limit can be easily achieved with a high simplicity by using a space-time separable statistics model.

We must remark, however, that the space-time separable statistics model proposed here is an efficient approach that accomplishes more realistic performances when higher wind velocities are considered. In fact, this approach is based on coloring independent Gaussian sequences first between them and then in time in order to generate *m* log-normal random processes of scintillation. Evidently, such method is restricted to have cross-correlation functions that have the same time-dependencies as the autocorrelation functions, i.e., the obtained sequences have statistics that are space-time separable. Due to this fact, Taylor's hypothesis (Tatarskii, 1971) is not fully satisfied. For such a case, we must redirect readers to (Jurado-Navas & Puerta-Notario, 2009), where Taylor's frozen turbulence hypothesis is properly taken into account. Conversely, the frozen-in hypothesis is unquestionably an approximation and must fail as distance between receivers becomes large or in especial situations when there are both strong velocity fluctuations of the wind or long-range spatial correlations (Burghelea et al., 2005; Moore et al., 2005), or even in urban atmospheres, especially near or among roughness elements, where strong wind shear is expected to create high turbulent kinetic energy (Christen et al., 2007). Furthermore, in urban canopies and cloud streets up to 2 − 5 times the average building height of such particular streets (Christen et al., 2007), the strong wind shear creates turbulence intensities that are tipically near the threshold where the hypothesis of frozen turbulence becomes inapplicable (Christen et al., 2007; Willis & Deardorff, 1976). In this fashion, difference in performance obtained from the realistic model presented in (Jurado-Navas & Puerta-Notario, 2009) and the upper bound performance proposed in this chapter are even closer to each other. Thus, the separable statistics model proposed here may be seen as a highly accurate upper error bound of the complete model detailed in (Jurado-Navas & Puerta-Notario, 2009), with the advantage of a reduced computational complexity in comparison to an AR method.
