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

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, D0, can be made larger than the correlation length, d0, 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 D0 < d0 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 20


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 0 < d 0 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 (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 0 =(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 2 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, ρ l , as follows: which represents the degree of randomness as effect of the dynamic evolution of the turbulence, with being the turbulence correlation time, where λ is the optical wavelength, L is the propagation distance and u ⊥ the component of the wind velocity transverse to the propagation direction.
Finally, τ e is seen as the lifetime of turbulent eddies and it is directly depending on the turbulent kinetic energy dissipation rate, ǫ, 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.

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 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 l → 0, τ e → ∞); and second, when an unrealistic scenario is supposed owing to using only a space-time separable statistics model (ρ l → 1, τ e → 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.

Turbulent atmospheric channel model
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

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Upper Burst Error Bound for Atmospheric Correlated Optical Communications Using an Alternative Matrix Decomposition circumstances, the vector wave equation becomes where E is the vector amplitude of the electric field, k=2π/λ is the wave number of the electromagnetic wave with λ being the optical wavelength; whereas n is the atmospheric refractive index whose time variations have been suppressed, and being a random function of position, r =(x, y, z).T h e∇ operator is the well-known vector derivative (∂/∂x, ∂/∂y, ∂/∂z).Equation (3) can be simplified by imposing certain characteristics of the propagation wave.In particular, since the wavelength λ for optical radiation is much smaller than the smallest scale of turbulence, l 0 , (Strohbehn, 1968) the maximum scattering angle is roughly λ/l 0 ≈ 10 −4 rad.As a consequence, the last term on the left-hand side of Eq. ( 3) is negligible.Such a term is related to the change in polarization of the wave as it propagates (Strohbehn, 1971;Strohbehn & Clifford, 1967).This conclusion permit us to drop the last term and Eq. ( 3) then reduces to Because Eq. ( 4) is easily decomposed into three scalar equations, one for each component of the electric field, E, we may solve one scalar equation and ignore the vector character of the wave until the final solution.Therefore if we let U(r) denote one of the scalar components that is transverse to the direction of propagation along the positive x-axis (Andrews & Phillips, 1998), then Eq. ( 4) may be replaced by the scalar stochastic differential equation The index of refraction, n(r)=n 0 + n 1 (r), fluctuates about the average value whereas n 1 (r) ≪ 1 is the fluctuation of the refractive index from its free space value.Thus For weak fluctuation, it is necessary to obtain an approximate solution of Eq. ( 6) for small n 1 .Most of the literature since 1960 has followed the approach of using the so-called Rytov method, which substitutes U in a series: In Eq. ( 7), ψ 1 , ψ 2 are the first and second order complex phase perturbations, respectively, whereas ψ 0 is the phase of the optical wave in free space.The Rytov solution is widely used in line-of-sight propagation problems because it simplifies the procedure of obtaining both amplitude and phase fluctuations.From the Rytov solution, the wave equation becomes: This is a nonlinear first order differential equation for ∇ψ and is known as the Riccati equation.Consider now a first order perturbation, then

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Computational Simulations and Applications www.intechopen.com Operating, assuming that |∇ψ 1 |≪|∇ψ 0 |,dueton 1 (r) ≪ 1, neglecting n 2 1 (r) in comparison to 2n 1 (r), and equating the terms with the same order of perturbation, then the following expressions are obtained: The first one is the differential equation for ∇ψ in the absence of the fluctuation whereas turbulent atmosphere induced perturbation are found in the second expression.The resolution of Eq. ( 10b) is detailed in (Fante, 1975;Ishimaru, 1997).For the particular case of a monochromatic optical plane wave propagating along the positive x-axis, i.e., U 0 (L, r)= exp (jkx), this solution can be written as: where the position (L, r) denotes a position in the receiver plane (at x = L)w h e r e a s(x ′ , r ′ ) represents any position at an arbitrary plane along the propagation path.The mathematical development needed to solve Eq. ( 11) can be consulted in (Andrews & Phillips, 1998;Ishimaru, 1997).Furthermore, the statistical nature of ψ 1 (L, r) can be deduced in an easy way.Equation ( 11) has the physical interpretation that the first-order Rytov perturbation, ψ 1 (L, r) is a sum of spherical waves generated at various points r ′ throughout the scattering volume V, the strength of each sum wave being proportional to the product of the unperturbed field term U 0 and the refractive-index perturbation, n 1 ,atthepointr ′ (Andrews & Phillips, 1998).Thus it is possible to apply the central limit theorem.According to such a theorem, the distribution of a random variable which is a sum of N independent random variables approaches normal as N → ∞ regardless of the distribution of each random variable.Application of the central limit theorem to this integral equation leads to the prediction of a normal probability distribution for ψ.Since we can substitute Ψ = χ + jS,w h e r eχ and S are called the log-amplitude and phase, respectively, of the field, then application of the central limit theorem also leads to the prediction of a Gaussian (normal) probability distribution for both χ and S, at least up to first order corrections (χ 1 and S 1 ).Accordingly, under this first-order Rytov approximation, the field of a propagating optical wave at distance L from the source is represented by: with U 0 (L, r) being the unperturbed portion of the field in the absence of turbulence.Hence, the irradiance of the random field shown in Eq. ( 12) takes the form: where, from now onwards, we denote χ 1 as χ for simplicity in the notation.Hence, In Eq. ( 13), operator * denotes the complex conjugate, |U 0 | is the amplitude of the unperturbed field and I 0 is the level of irradiance fluctuation in the absence of air turbulence that ensures that the fading does not attenuate or amplify the average power, i.e., E[I]=|U 0 | 2 .Th isma yb e thought of as a conservation of energy consideration and requires the choice of E[χ]=−σ 2 χ , as was explained in (Fried, 1967;Strohbehn, 1978), where E[χ] is the ensemble average of log-amplitude, whereas σ 2 χ is its variance depending on the structure parameter, C 2 n .With all of these expressions, we have modeled the irradiance of the random field, I,inthespaceata single instant in time.Now, because the state of the atmospheric turbulence varies with time, the intensity fluctuations is also temporally correlated.Then, Eq. ( 14) can be expressed as: whereas α sc (t)=exp (2χ(t)) is the temporal behavior of the scintillation sequence and represents the effect of the intensity fluctuations on the transmitted signal.In Eq. ( 15), a space-to-time statistical conversion has been assumed by employing the well-known Taylor's hypothesis of frozen turbulence (Tatarskii, 1971;Taylor, 1938).
As analyzed before, and by the central limit theorem, the marginal distribution of the logamplitude, χ, is Gaussian.Thus, Hence, from the Jacobian statistical transformation (Papoulis, 1991), the probability density function of the intensity, I, can be identified to have a lognormal distribution typical of weak turbulence regime.Then: Theoretical and experimental studies of irradiance fluctuations generally center around the scintillation index.It was evaluated in (Mercier, 1962) and it is defined as the normalized variance of irradiance fluctuations: Hence it is possible to define the weak turbulence regimes as those regimes for which the scintillation index given in Eq. ( 19) is less than unity.With all these considerations taken into account, an efficient channel model for FSO communications using intensity modulation and direct detection (IM/DD) was presented in (Jurado- Navas et al., 2007;2011a) under the assumption of weak turbulence regime.For these systems, the received optical power, Y(t), can be written as being X(t) the received optical power without scintillation; whereas α sc (t)=exp 2χ(t) is the temporal behavior of the scintillation sequence and represent the effect of the intensity fluctuations on the transmitted signal.To generate α sc (t), a scheme based on a lowpass filtering of a random Gaussian signal, z(t), is implemented as in (Jurado- Navas et al., 2007;2011a).χ(t) is, as was explained above, the log-amplitude of the optical wave governed by Gaussian statistics with ensemble average E[χ] and variance σ 2 χ .Finally, the additive 434 Computational Simulations and Applications www.intechopen.comwhite Gaussian noise, N(t), is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal.In the following section, we complete the scheme presented in (Jurado- Navas et al., 2007) to afford the inclusion of m correlated scintillation sequences approximating to the model introduced in (Jurado- Navas & Puerta-Notario, 2009).

Proposed approximation: space-time separable statistics channel model
As indicated in the last section, we can simplify the model proposed in (Jurado- Navas & Puerta-Notario, 2009) by an unrealistic but reasonably accurate space-time separable statistics model with a reduced computational load if it is compared with the complete model.Through this section, we develop the space-time separable statistics model.

Spatial diversity reception
When multiple receivers are considered, then, as shown in (Zhu & Kahn, 2002), the real symmetric auto-covariance matrix of the log-amplitude, C χ ={c χ (i,j)} m i,j=1 ,a tm receivers in a plane transverse to the direction of propagation is given by: where d ij and ρ d ij are the distance and its normalized CC coefficient respectively between points i and j in the receiver plane.In (Jurado- Navas & Puerta-Notario, 2009;Zhu & Kahn, 2002), a Gaussian spatial covariance function for the log-amplitude fluctuations is employed that approximates the theoretical covariance function resulting from Rytov theory.
In order to satisfy the hypothesis of frozen turbulence, we can adopt an AR model as in (Baddour & Beaulieu, 2002;Jurado-Navas & Puerta-Notario, 2009) as a possible solution.
By doing so, the multichannel Yule-Walker equations are expressed as (Jurado- Navas & Puerta-Notario, 2009): where A H [k], k = 1, 2, ...p are m × m matrices containing the multichannel AR model coefficients; whereas C χ [j] is the covariance matrix evaluated in the 'j'-time instant.Then, the system of equations in Eq. ( 22) can be solved efficiently via the Levinson-Wiggins-Robinson algorithm (Kay, 1988).Once the A H [k] coefficient matrices have been determined, we can obtain the m × m covariance matrix of the driving noise vector process of the AR model from: Thus, after obtaining C w = E{ω[n]ω[n] T }, where the hermitian operator has been substituted by a transpose operator due to all the samples are real; the driving noise process, ω[n], can be accomplished as was indicated in (Baddour & Beaulieu, 2002).But first we need to compute the factorization C w = LL T .I fC w is positive definite, a Cholesky decomposition is performable using the interval Cholesky method proposed in (Alefeld & Mayer, 1993) in order to include the desired correlation among scintillation sequences, with L being a lower triangular matrix (Beaulieu & Merani, 2000;Ertel & Reed, 1998).So, the driving process is then generated by the product ω[n]=Lz[n],w h e r ez[n] is an m × 1 vector of independent zero mean Gaussian variates with unit variance and a autocorrelation function expressed as: where I m is the 'm'-element identity matrix.Finally, But if C w is not positive definite, then an alternative efficient decomposition algorithm that let the matrix be factorized was proposed in (Jurado-Navas & Puerta-Notario, 2009).
In addition, as said in the introduction, we can simplify the model proposed in (Jurado-Navas & Puerta-Notario, 2009) by an unrealistic but reasonably accurate space-time separable statistics model with a reduced computational load if it is compared with the complete model.Hence, we work directly with the covariance matrix, C χ , unlike the work proposed in (Jurado- Navas & Puerta-Notario, 2009), where the matrix to be decomposed is C w .Thus, when C χ is positive definite, as in the Gaussian approximation employed in (Zhu & Kahn, 2002), a Cholesky decomposition is performable using the interval Cholesky method proposed in (Alefeld & Mayer, 1993) in order to include the desired correlation among scintillation sequences.Hence we find a lower triangular matrix L such that C χ = LL H in a similar way as in (Beaulieu & Merani, 2000;Ertel & Reed, 1998).

Constructing the coloring matrix
As we have already said, the AR model presented in (Jurado-Navas & Puerta-Notario, 2009) can be simplified by directly coloring independent Gaussian sequences first between them and then in time (space-time separable statistical model), avoiding the realization of the AR model.Obviously, this procedure is unrealistic because it implies that Taylor's hypothesis is not considered.However, numerical results obtained (and included through this chapter) show that this simplification behaves as a good approximation in terms of burst error rate.Thus, we can directly consider the auto-covariance matrix of the log-amplitude, C χ .A s commented before, when C χ is positive definite, a Cholesky decomposition is performable by using the interval Cholesky method proposed (Alefeld & Mayer, 1993) in order to include the desired correlation among scintillation sequences.Hence we find a lower triangular matrix, L, as a coloring matrix such that C χ = LL H in a similar way as in (Ertel & Reed, 1998).Nevertheless, C χ may not be positive definite so the Cholesky decomposition is not always feasible.Although any theoretical covariance matrix must be positive or, at least, semipositive definite, when such matrices are processed on a computer, small deviations from theory are introduced by the limits of floating point computations, inducing numeric inconsistencies.In this respect, there are no guarantees that the finite precision representation of the matrix can ensure positive definiteness, even more when the Cholesky algorithm is unstable for positive definite matrices that have one or more eigenvalues close to 0, as it is explained in (Holton, 2004).In (Abramovich et al., 2001), the particular problem of non-positive definite matrices that arise from characterizing a realistic system is investigated in detail, showing that in 988 outcomes of their 1000 Monte Carlo trials, gave rise to a non-positive definite covariance matrix, although the corresponding "contracted" matrix may well be positive definite.In addition, in most practical situations the covariance matrix must be estimated from scintillation measurements (Monserrat et al., 2007;Moore et al., 2005) and outliers and other special situations as urban canopies may induce that correlation matrices derived from measurements are not always guaranteed to be positive definite, although a theoretical correlation matrix must always be.
In such cases, a real Schur decomposition is proposed (Golub & van Loan, 1996).Since the symmetric matrix C χ ∈ℜ mxm , then there exists an orthogonal matrix, Q ∈ℜ mxm ,suchthat as was shown in (Golub & van Loan, 1996), where superscript T denotes the transpose and λ i , ∀i = 1...m, are the eigenvalues of C χ .In this case, we allow C χ to have zero-eigenvalues.But however, as we will explain later, all the eigenvalues must be either zeros or positives.
Unluckily, there will exist particular cases where C χ might have any negative eigenvalue, as was indicated above.When this circumstance occurs, an adjustment of the Schur decomposition will be accomplished in order to obtain a positive semi-definite approximation, C χ ={ c χ (i,j)}, to the original auto-covariance matrix of the log-amplitude, C χ , by minimizing the distance (Halmos, 1972): where • represents the norm of the matrix C χ − C χ .We distinguish two main situations (Jurado- Navas & Puerta-Notario, 2009) corresponding to minimize the Frobenius norm (Golub & van Loan, 1996) or the the p-norms (with p = 1, 2, ∞), respectively.In both cases, we can substitute the diagonal matrix Λ by a resulting matrix written as R = Λ + Υ,w h e r eΥ is a perturbation matrix obtained after minimizing Eq. ( 27) in any of the two situations mentioned above.After that, and for any of these two scenarios, we can form the coloring matrix, K,a sK = Q(R) 1/2 in order to generate the correlated log-normal scintillation samples.Here it is shown why we need that C χ were approximated by a positive semi-definite matrix if any eigenvalue of C χ is negative.In this case, however, if all eigenvalues of C χ are positive, then Let us see these two scenarios in detail:

General scenario
If any λ i in Λ is negative, then we can approximate the matrix Λ substituting every negative eigenvalue for 0. The resulting matrix may be written as where Υ is a diagonal perturbation matrix with diagonal elements, {υ i } m i=1 ,of In this case, the approximation C χ = QRQ T is proven to optimize the Frobenius norm if (v i + λ i ) → 0, and with the constraint of having a positive semi-definite solution matrix, C χ ; where the Frobenius norm is detailed in (Golub & van Loan, 1996) and represented by

Specific scenario
If instead of employing the Frobenius norm, the expression ( 27) is optimized for the relevant 2−norm matrix defined as: then we can obtain a new and different solution for the same problem.In equation ( 33), defined as the supremum among the absolute values of its eigenvalues.Obviously, the hermitian operator can be substituted by a transpose operator since the samples are all real.As a characteristic feature, optimizing the 2−norm entails, in general, multiple solutions for C χ , as a main difference with respect to the previous situation.Again, we define R = Λ + Υ,beingΥ the perturbation matrix.One of the possible solutions for Υ that let R be a semi-definite positive matrix is obtained by constructing Υ as a diagonal matrix whose elements are defined as {υ i = −λ ∀ i = 1...m},beingλ the highest among the absolute values of the negative eigenvalues of the matrix C χ .Obviously, as in the above scenario, we obtain greater accuracies when smaller magnitudes were summed to Λ.T h e limit, again, is the necessary magnitude that let the negative eigenvalue be canceled and so, C χ were approximated to the closest positive semi-definite matrix.In this way, this second scenario is proven to minimize the p-norms (with p = 1, 2, ∞)sinceΥ is a diagonal matrix in which all its entries have the same value.These norms can be consulted in (Golub & van Loan, 1996).
As final comment to this second scenario, we can remark that the great resemblance obtained between C χ and C χ is based on the orthogonality of the rows of the matrix Q.For example, suppose that the covariance matrix is non-positive definite expressed as 0.100 0.079 0.010 0.079 0.100 0.079 0.010 0.079 0.100 Hence, applying the decomposition shown in scenario 2, we obtain that C χ = C χ ,exceptfor the principal diagonal, that now is c χ (i, i)=0.106∀i = 1...3, i.e.: 0.106 0.079 0.010 0.079 0.106 0.079 0.010 0.079 0.106 438 Computational Simulations and Applications www.intechopen.com Finally, if the original auto-covariance matrix is not symmetric, then the Schur decomposition is written as where U ∈ℜ nxn is upper triangular as was shown in (Golub & van Loan, 1996).In these circumstances, we can form the matrix Λ taking into account solely the diagonal elements of U before applying any of the two approximations proposed in this paper.

Generation of turbulent spatially-correlated channels
Now we can form the coloring matrix, K ∈ℜ mxm in order to generate the correlated log-normal scintillation samples.Then, it follows that (Jurado-Navas & Puerta-Notario, 2009) as was shown in Eq. ( 28), where C χ = C χ if there does not exist any negative element in diagonal matrix Λ written in (26).Equation ( 37) shows why we need that C χ were approximated by a positive semi-definite matrix if any eigenvalue of C χ is negative even when a Schur decomposition could be always accomplished if the starting matrix is squared.
Next, define Z = z (1) [n], z (2) [n], ... z (m) [n] T as a set of uncorrelated white Gaussian signals.Its correlation matrix, R ZZ ,isgivenby where I m is the 'm'-element identity matrix.By calculating w[n]=Kz[n],being w[n]= w (1) [n], w (2) [n], ... w (m) [n] the desired auto-covariance matrix of the log-amplitude fluctuations is obtained since Furthermore, every w (i) [n], ∀i = 1...m in Eq. ( 40) remains statistically Gaussian so it is possible to filter them in the way proposed in (Jurado- Navas et al., 2007) in order to obtain α (i) sc (t) for each of the m receivers.The filter employed in each branch is the Gaussian filter shown in (Jurado-Navas et al., 2007).Therefore, the output signal of these filters is the log-amplitude fluctuations, χ (i) (t) ∀i = 1...m, for each receiver.These outputs exhibit a Gaussian probability density function and have the desired covariance matrix, C χ ,a m o n gs e q u e n c e sb u tw i t h o u t its inherent time-dependencies that would be enforced by the frozen-in hypothesis, according to the space-time separable statistics approximation imposed in this chapter.Next, its probability density function is converted from Gaussian to lognormally distributed, generally accepted for the irradiance fluctuations under weak turbulence conditions.Figure 1 shows clearly the overall process.In this respect, it is possible to employ the same procedure explained in this paper but knowing that, instead of the log-normal employed here, a Beckman probability density much more accurately reflects the statistics of the intensity scintillations (Hill & Frehlich, 1997) if Rytov variance increases even beyond the limits of the weak turbulence regime; or even the recently discovered Málaga probability density function (Jurado-Navas et al., 2011b), also very accurate with the statistics of the intensity scintillations.

System model
To study the performance of different values of spatial CC coefficients, ρ ij , between points i and j in the receiver plane, intensity modulation and direct detection (IM/DD) links are assumed operating at a laser wavelength of 830 nm through a 250 m horizontal path at a bit rate of 50 Mbps.Assume three receivers in the system, where a conventional equal-gain combining (EGC) of diversity branches is implemented.Figure 2 shows the temporal behavior of correlated scintillation sequences for these three different receivers with a CC coefficient of ρ 12 = 0.56 and ρ 13 = 0.1 respectively, for this two extreme scenarios: an AR model, as in (Jurado- Navas & Puerta-Notario, 2009), and the space-time separable statistics approximation employed in this chapter.The component of the wind velocity, u ⊥ , transverse to the propagation direction is fixed to u ⊥ = 20 m/s, where ρ ij is the normalized CC coefficient between points i and j in the receiver plane.Clearly, the election of a 20 m/s wind speed in this chapter will not be by far the typical operational scenario unless one of the terminals is in motion or the optical link is settled in particular geographical locations specifically affected by strong wind (Campins et al., 2007); but, however, such value is selected for numerical convenience so that we can work with a lower computational cost, because the total amount of samples needed to obtain a scintillation sequence with the desired lognormal statistics is smaller than using more conventional values for u ⊥ , but letting us extract the same conclusions than employing a standard velocity.Hence, a different temporal variability is shown in Figure 2 (a) and (b) for the EGC-combined scintillation sequence, being faster for the obtained sequence from a space-time separable statistics model.This different variability must have a different repercussion in the associated burst error rate curves, as shown below, where we followed Fig. 2. Scintillation sequences generated from (a) an AR model; (b) a space-time separable statistics model, for a 3-receivers system.The EGC sequence is displayed in thicker solid line.(Deutsch & Miller, 1981) definition of a burst error with lengths of 192, and 64 bits, not containing more than 4 consecutive correct bits (L b = 5, as explained in (Deutsch & Miller, 1981)) any sequence of burst error.Then, pulses with on-off keying format (OOK) and Gaussian shape (OOK-GS) with a duty cycle (d.c.) of 100% are adopted, where identical average optical power is transmitted for every simulated case representing the reference condition to establish the comparative analysis (Jurado- Navas et al., 2010).All these features are included in the system model, where its remarkable elements are: first, the channel model presented in (Jurado- Navas et al., 2007) corresponding to a turbulent atmospheric environment, but included in a m-branch reception as in (Jurado- Navas & Puerta-Notario, 2009), which represent the m different correlated turbulence-induced fadings at each of the optical receivers; secondly, a 500 kHz three-pole Bessel highpass filter for natural and artificial light adverse effects suppression; and, thirdly, a five-pole, Bessel, lowpass filter employed as a matched filter.As said above, a conventional EGC is implemented whereas the detection procedure considered is a maximum likelihood (ML) detection.The receivers employed in this paper are point receivers whereas the weather-induced attenuation is neglected so that we concentrate our attention on turbulence effects.Furthermore, the atmospheric-induced beam spreading that causes a power reduction at the receiver is also neglected because, in our specific case, we are considering a terrestrial link where beam divergence is typically on the order of 10 μRad.This temporal spreading may be considered at high data rate, as in (Jurado- Navas et al., 2009), particularly when operating in special scenarios where dust particles are likely present.

Numerical results
The first set of results are displayed in Figure 3 for a three-receivers system and for σ 2 χ = 0.25 and σ 2 χ = 0.1, where the CC coefficients between scintillation sequences are: Hence, the AR-model proposed in (Jurado-Navas & Puerta-Notario, 2009) and its inherent numerically ill-conditioned covariance matrix (Baddour & Beaulieu, 2002) may be avoided in many cases when calculating burst error rate curves due to the difference between the two extreme scenarios studied in this chapter is usually limited to approximately 2 dB at a burst rate of 10 −6 .In this sense, the space-time separable statistics model proposed here can be used to consider spatial correlations among scintillation sequences without fear of making big mistakes and with the advantage of a reduced computational time.Thus, such separable statistics model may be seen as a highly accurate upper error bound of the whole model detailed in (Jurado-Navas & Puerta-Notario, 2009).

Acknowledgment
This work was supported by the Spanish Ministerio de Ciencia e Innovación, Project TEC2008-06598.

A[k]
m × m matrices containing the multichannel AR model coefficients.C 2 n Refractive-index structure parameter.

C w
Covariance matrix of the driving noise vector process of an AR model.

C χ
Covariance matrix of the log-amplitude scintillation.C χ [j] Covariance matrix of the log-amplitude scintillation evaluated in the j-time instant.

C χ
Positive semi-definite approximation of C χ .d 0 Correlation length of intensity fluctuations.
Distance between points i and j in the receiver plane (m).

E
Vector amplitude of the electric field.f χ (χ) Probability density function of random log-amplitude scintillation.f I (I) Probability density function of intensity fluctuations (= f α sc (α sc )).

) 437 Upper
Burst Error Bound for Atmospheric Correlated Optical Communications Using an Alternative Matrix Decomposition www.intechopen.com Fig. 1.Block diagram representing the generation of 'm' equal power log-normal scintillation sequences with a space-time separable statistics approximation.

I
Irradiance of the random field.I 0 Level of irradiance fluctuation in the absence of air turbulence.Vector of independent zero mean Gaussian variates with unit variance.αsc(t)Time-varying atmospheric scintillation sequence.χ(t)Log-amplitudefluctuation of scintillation.Λ Diagonal matrix containing the eigenvalues of C χ .ǫTurbulentkineticenergy dissipation rate (m 2 /s 3 ).Phase perturbations of Rytov approximation.ρijNormalizedcross-correlationcoefficient between points i and j in the receiver plane (m).ρ l Degree of randomness as effect of the dynamic evolution of the turbulence (=ρ l = τ 0 /τ e ).Driving noise vector process of an AR model.ΥPerturbation matrix employed to obtain C χ .
⊥Component of the wind velocity transverse to the propagation direction.w[n]ColoringGaussian vector (= Kz[n]).