Performance Analysis of Space-Diversity FSO Systems over the Correlated Gamma-Gamma Fading Channel Using Padé Approximation Method

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1 Performance Analysis of Space-Diversity FSO Systems over the Correlated Gamma-Gamma Fading Channel Using Padé Approximation Method G. Yang 1, M. A. Khalighi 2, Z. Ghassemlooy 3, and S. Bourennane 2 1 College of Communication Engineering, Hangzhou Dianzi University, Hangzhou, P. R. China 2 Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, France 3 Optical Communications Research Group, Faculty of Engineering and Environment, Northumbria University, Newcastle Upon Tyne, UK 1 Abstract We propose an analytical approach to evaluate the performance of space-diversity free-space optical (FSO) systems over correlated Gamma-Gamma (ΓΓ) fading channels. To do this, we firstly derive an infinite series representation for the moment generating function (MGF) of the sum of arbitrarily-correlated ΓΓ random variables (RVs). Then, a closed-form approximate expression for the probability density function (PDF) of the sum of ΓΓ RVs is obtained from the MGF with the aid of Padé approximation. The performance of the FSO system is then evaluated based on this PDF. Through some numerical results we illustrate the accuracy of the proposed method and further discuss its limitations. I. INTRODUCTION FSO communication is well known as a cost-effective solution for very high-rate data transmissions [1] [3]. One of the main impairments that affects the performance of an FSO link under clear weather conditions is the atmospheric turbulence. This is due to the inhomogeneities in the temperature and the pressure of the atmosphere, resulting in variations of the air refractive index.

2 2 Here, we consider the conditions of moderate-to-strong turbulence that are usually valid when the link distance is on the order of a few kilometers. In such situations, the required collecting optical lens for efficient aperture averaging is too large and inappropriate for a practical compact design [4]. More efficient fading reduction is achieved through the spatial diversity that can be realized by using multiple apertures at the receiver and/or multiple beams at the transmitter [4], [5]. However, the efficacy of this solution is reduced under the conditions of correlated fading on the underlying sub-channels [6] [10]. In fact, it is not practically feasible to comply with the required spacing between the apertures at the receiver and/or between laser beams at the transmitter to safeguard the conditions of independent fading [9] [11]. A number of previous works have investigated the correlated fading problem by considering simplified statistical models. For instance, in [2] and [12], the authors investigated the effect of fading correlation on the system bit-error-rate (BER) performance by considering the log-normal distribution for intensity fluctuations. This model, however, is applicable only to the weak turbulence regime. Also, the K distribution was considered in [13] for modeling the turbulence but this model is suitable only for the very strong turbulence regime. Here, we consider the ΓΓ distribution that has widely been adopted for modeling the FSO channel due to its excellent agreement with the experimental data over a wide range of turbulence conditions [14], [15]. This model has been applied in [4], [14], for instance, to the case of multiple-aperture systems assuming independent fading conditions. Also, in [16], [17], two approximations to the sum of independent ΓΓ random variables (RVs) were proposed in order to evaluate the performance of space-diversity FSO systems. Our aim here is to treat the case of arbitrarily-correlated fading conditions [18] under the ΓΓ statistical model. In some previous research works this problem is investigated by considering a number of specific assumptions. For instance, in [19], a multivariate ΓΓ model with exponential correlation was proposed. However, this correlation model is not appropriate for most FSO system configurations. In [11], multiple ΓΓ channels were modeled by a single ΓΓ channel whose parameters were set by approximating it by a Gaussian RV. This solution would not guarantee enough accuracy for the system performance evaluation in general. In [20], the sum of correlated ΓΓ RVs is approximated by a ΓΓ RV whose parameters were determined based on numerical

3 3 fitting applied to the simulated data. However, this approach depends on the underlying system structure and cannot directly be used to accurately evaluate the performance of a general FSO system. In a recent work [21], we proposed a general model for correlated ΓΓ fading channels in order to evaluate the performance of space-diversity FSO systems via Monte-Carlo simulation. In order to evaluate the system performance analytically, we recently proposed in [10], [22], [23] to approximate the sum of correlated ΓΓ RVs by an α-µ distribution. By this method, we set equal the first three moments of the sum of correlated ΓΓ RVs and the approximate α-µ distribution. However, this method loses its accuracy for increased diversity order and in the case of relatively large correlation coefficients [10], [23]. In this paper, we propose an analytical approach based on the theory of Padé approximants to evaluate the performance of space-diversity FSO systems over correlated ΓΓ fading channels. Compared to the α-µ approximation method, this approach involves more moments and can lead to a more accurate performance prediction for any diversity order and whatever the correlation coefficient. The Padé approximation has been used in several previous works for the performance analysis of space-diversity schemes over different fading channel models including Nakagamim, Weibull, and Gamma [24] [27]. Thereby, the Padé approximants of the moment generating function (MGF) of the received signal-to-noise ratio (SNR) were employed to analyze the system performance. However, we were unable to obtain satisfying results by this approach for the case of ΓΓ fading channels (this is explained in detail in Appendix B). Instead, we apply here the Padé approximation to the sum of fading coefficients. As we will see, this is beneficial when considering the equal gain combining (EGC) technique [12] at the receiver for the case of multiple-aperture systems, and/or repetition coding (RC) [5] at the transmitter for the case of multiple-beam systems. Note that EGC and RC are quasi-optimal schemes for the case of intensity modulation with direct detection [12], [28] [30]. Furthermore, we consider the case of arbitrarily-correlated fading channels. To treat correlated ΓΓ fading, we use the conclusions of one of our recent works [21] according to which the fading correlation can practically be attributed solely to the large-scale turbulence, and the small-scale turbulence can effectively be considered as uncorrelated. Using this, we firstly derive a general expression for the calculation of the n th -order joint moments of arbitrarily-correlated large-scale turbulence Gamma RVs from

4 4 the corresponding MGF. To the best of our knowledge this has not been done before. Next, an infinite series representation for the MGF of the sum of correlated ΓΓ RVs is obtained and approximated by Padé approximants in terms of a simple rational function [31], [32]. Then, the probability density function (PDF) of the sum of ΓΓ RVs is obtained by taking the inverse Laplace transform of the Padé approximant via the residue theory [32]. The system performance in terms of the average BER and the outage probability, is then evaluated based on the resulting PDF. The accuracy of the proposed approach is illustrated through some numerical results for the case study of a dual-diversity FSO system by contrasting them with Monte-Carlo simulation results. We further discuss the limitation of the Padé approximation method in predicting the system perfromance. The remainder of this paper is organized as follows. The channel model and general assumptions are provided in Section II. In Section III, we present MGF of the sum of correlated ΓΓ RVs and describe the Padé approximation to MGF. Performance analysis using the Padé approximation method is explained in Section IV. We next provide in Section V some numerical results to validate our proposed approach before concluding the paper in Section VI. II. CHANNEL MODEL AND ASSUMPTIONS Let us consider an L-branch space-diversity FSO system over correlated ΓΓ fading channels. We reasonably assume that the parameters of the corresponding ΓΓ distributions are the same for all the L underlying sub-channels. Let us denote the channel fading coefficient of the k th sub-channel by I k. By the ΓΓ model, I k is considered as the product of two independent Gamma RVs, X k and Y k, which represent the irradiance fluctuations arising from large- and small-scale turbulence, respectively. The PDF of I k is given by [14]: f(i k ) = 2(αβ)(α+β)/2 Γ(α)Γ(β) I (α+β) 2 1 k K α β ( 2 α β I k ), (1) where α and β denote the effective numbers of large- and small-scale turbulence eddies, respectively. Also, Γ( ) is the Gamma function and K υ ( ) is the modified Bessel function of the second kind and order υ. For the general case of an M-beam N-aperture FSO system employing RC at the transmitter and EGC at the receiver, the sum I = L k=1 I k will correspond to the total received signal intensity where L = MN.

5 5 A. Infinite series representation of MGF The MGF of I is defined as: III. MGF OF SUM OF ΓΓ RANDOM VARIABLES M I (s) = E { e si}, (2) where E{ } stands for expectation. The MGF can be written in the form of Maclaurin series as follows. M I (s) = The n-th moment of I can be calculated as: {( L ) n } n v 1 E {I n } = E X i Y i = i=1 v 1 =0 v 2 =0 n=0 v L 2 v L 1 =0 E {I n } s n (3) n! ( n v 1 )( v1 v 2 ) ( vl 2 v L 1 E { X n v 1 1 X v 1 v 2 2 X v } { L 1 L E Y n v 1 1 Y v 1 v 2 2 Y v } L 1 L, (4) where v 1, v 2,, v L, n are non-negative integers [33]. For (4), we require the n-th joint moments of arbitrarily-correlated Gamma RVs, which are derived in Appendix A. In addition, we need the normalized auto-correlation matrices of the two Gamma RV sets {X k } and {Y k }, that we denote by R X and R Y, respectively. Defining R I as the normalized correlation matrix of the L ΓΓ sub-channel fading coefficients, matrices R X and R Y are related to R I through the following relationship [21]: R I (i, j) = αr Y(i, j) + βr X (i, j) + R X (i, j)r Y (i, j), (5) α + β + 1 where A(i, j) denotes the (i, j) th entry of matrix A. Matrix R I can be obtained through waveoptics simulations or experimental measurements [6] [9], [11]. According to the conclusions of [21], we set R Y to an Identity matrix. ) B. Padé approximants of MGF To evaluate the system performance via MGF, the idea is to take a finite number of terms in (3). One powerful approach to truncate the infinite series of (3) is to use the Padé approximation method, as described in [24], [25], [32]. In fact, the resulting Padé approximants often give a

6 6 better approximation of MGF, compared to the direct truncation of the power series. Furthermore, this method is particularly interesting when dealing with series that converge very slowly or when only a small number of their coefficients are known [24], [32]. Due to this reason, the Padé approximation method is widely used in numerical calculations. Following this method, the infinite series of MGF M I (s) is estimated by the Padé approximant that is a rational function of order D for the denominator and order C for the nominator: C i=0 R [C/D] (s) = c is i 1 + D i=1 d is. (6) i Here, the coefficients {c i } and {d i } are determined via the following relationship: C i=0 c is i C+D 1 + D i=1 d is = E {I n } s n + O(C + D + 1), (7) i n! n=0 where O(C + D + 1) represents the truncated terms of order higher than (C + D) in (3). Here, we approximate M I (s) by using the subdiagonal Padé approximants R [U 1/U] (i.e., taking C = U 1 and D = U), because it is only for such approximants that the convergence rate and the uniqueness can be assured [24]. Note that the order of Padé approximants U should be chosen appropriately to achieve a good estimation (see [32, Section II]). Methods for calculating the two sets of {c i } and {d i } have been widely developed in the literature, e.g., in [26], [32]. These methods are also available in several mathematical softwares, such as MATHEMATICA R, MAPLE R, and MATLAB R. IV. PERFORMANCE ANALYSIS USING PADÉ APPROXIMATION METHOD In the following, we analyze the performance of space-diversity FSO systems over arbitrarilycorrelated ΓΓ fading channels based on the Padé approximants of MGF. In fact, in a first attempt, we tried to use the Padé approximants of MGF of the received SNR (after EGC), but we found that the predicted performances do not match adequately with the Monte-Carlo simulation results (a detailed analysis is provided in Appendix B). Instead, we applied the Padé approximation to the received intensity after EGC. This way, we obtained the approximate PDF from the Padé approximants of MGF via the inverse Laplace transform [32], and averaged the conditional BER over the approximate PDF. We noticed that, as we will show in Section V, the obtained average

7 7 BER has an excellent agreement with the simulation results. Moreover, the outage probability can also be accurately obtained from the corresponding cumulative distribution function (CDF). This approach is detailed in the following. A. Approximate PDF and CDF To calculate the approximate PDF, we perform the partial fraction decomposition on the Padé approximant of MGF, that is: M I (s) = R [U 1/U] (s) = U 1 i=0 c i(s) i 1 + U i=1 d i(s) i = U i=1 λ i s + p i, (8) where {p i } and {λ i } are the poles and residues of the Padé approximant, respectively. Then, we take the inverse Laplace transform of the Padé approximant via the residue theory, according to which the approximate PDF is expressed as a sum of decaying exponentials [32]: f PA (I) = Also, the corresponding CDF is given by: U λ i e pii. (9) i=1 F PA (I) = 1 + U i=1 λ i p i e p ii. (10) B. BER and outage probability To verify the usefulness of the Padé approximation method in predicting the system performance, we consider an FSO system employing intensity modulation with direct detection using uncoded on-off keying (OOK) modulation. Remember from Section II that we consider the general case of an M-beam N-aperture (M N) FSO system, where RC and EGC are performed at the transmitter and the receiver, respectively. We consider optimal signal demodulation assuming a perfect available channel knowledge at the receiver. Without loss of generality, we assume that the dominant noise source at the receiver is the thermal noise and model it by as additive white Gaussian with zero mean and variance σn. 2 Note that the background (ambiant) noise effect can be effectively reduced by using narrow spatial and spectral bandpass filtering

8 8 prior to photo-detection [34]. We also set the transmit signal intensity (in on slots) and the optical-to-electrical conversion efficiency to one. The electrical received SNR is given by [2]: γ = I2. (11) 4Nσn 2 Then, the average system BER is calculated by averaging the conditional BER over the PDF f PA (I) [14]: P e = ( ) I f PA (I) erfc 2 di, (12) 2Nσ n where erfc( ) is the complementary error function defined as erfc(z) = 2 π z e t2 dt. Also, the outage probability, which is defined as the probability that the received SNR γ falls below a specified threshold γ th is given in closed-form as follows: P out (γ th ) = F PA (2σ n Nγth ) = 1 + U i=1 V. NUMERICAL RESULTS λ i e 2p iσ n Nγth. (13) p i Here, we provide some numerical results to illustrate the accuracy of the proposed approach. We consider as case study a dual-aperture FSO system that we will denote by (1 2), where each aperture has the diameter D R. We consider two conditions of L = 2 km and D R = 50 mm and L = 5 km and D R = 100 mm that we will refer to as Cases (1) and (2), respectively, in the following. We consider the sets of correlation coefficients ρ = 0.2, 0.4, 0.6 and ρ = 0.3, 0.5, 0.7 for Cases (1) and (2), respectively. Notice that the largest ρ values could not hold in practice [7] [9], [11], and our aim by considering them is to see the accuracy of the proposed approach. Concerning the numerical simulation parameters, we consider a diverging Gaussian beam at the transmitter at λ = 1550 nm with the beam waist W 0 = 1.59 cm and the curvature radius of the phase front of F 0 = 69.9 m, corresponding to a beam divergence of θ div = 0.46 mrad. Also, we consider the index of structure parameter Cn 2 = m 2/3, and the inner and outer scales of turbulence of l 0 = 6.1 mm and L 0 = 1.3 m, respectively. These parameters correspond to the experimental works reported in [35]. This way, Cases (1) and (2) correspond to the strong turbulence regime with Rytov variances [14] of σ 2 R = 4.61 and 24.74, respectively. To set the

9 Pe 10 3 Pe Padé, ρ = 0 Padé, ρ = 0.2 Padé, ρ = 0.4 Padé, ρ = 0.6 Simulation γ1 (db) Padé, ρ = 0 Padé, ρ = 0.3 Padé, ρ = 0.5 Padé, ρ = 0.7 Simulation γ1 (db) (a) (b) Fig. 1. Contrasting BER performances of the (1 2) system obtained by Monte-Carlo simulations and the Padé approximation method. (a) Case (1): L = 2 km and D R = 50 mm. (b) Case (2): L = 5 km and D R = 100 mm Pout 10 3 Pout Padé, ρ = 0 Padé, ρ = 0.2 Padé, ρ = 0.4 Padé, ρ = 0.6 Simulation γth/γ1 (db) Padé, ρ = 0 Padé, ρ = 0.3 Padé, ρ = 0.5 Padé, ρ = 0.7 Simulation γth/γ1 (db) (a) (b) Fig. 2. Contrasting outage probabilities of the (1 2) system obtained by Monte-Carlo simulations and the Padé approximation method. (a) Case (1): L = 2 km and D R = 50 mm. (b) Case (2): L = 5 km and D R = 100 mm. SNR, we take a single-aperture FSO system as reference and denote the corresponding SNR by γ 1 = I2 1 4σ 2 n and its average by γ 1 = E{I 1} 2 4σ 2 n [2], [16]. Figure 1 shows plots of analytical BER obtained using the Padé approximation method, as well as those obtained via Monte-Carlo simulations. Interestingly, we notice that the proposed analytical approach allows highly accurate performance predication, even for the case of relatively large fading correlation coefficients. We have also presented the curves of the outage probability

10 CDF Simulation Padé, U=9 Padé, U=17 Padé, U=25 Padé, U=35 Padé, U= Intensity I Fig. 3. CDFs of the received intensity I for (1 2) system, Case (1) with ρ = 0.2. CDF is shown in logarithmic scale to better show the difference between different approximation orders. P out versus the normalized threshold γ th /γ 1 for the two cases in Fig. 2. Again, we notice an excellent agreement between the analytical and simulation results. A. Limitation of Padé approximation method In order to minimize the difference between the exact MGF and the corresponding Padé approximant, we should find the optimal order U, as explained in [32]. In other words, the difference between these functions do not decrease constantly by increasing the order U. For the sake of completeness, we have provided the optimal orders for the considered case studies in Table I. To illustrate the impact of U on the accuracy of the Padé approximation method, we have shown in Fig. 3 plots of the CDF of the received intensity (after EGC) obtained from Monte- Carlo simulation and by Padé approximation using different orders for Case (1) with ρ = 0.2. The optimal order of Padé approximation is 25. We have a better fit between the CDFs of Padé approximation and Monte Carlo approaches when U increases from 5 to the optimal value,

11 11 TABLE I OPTIMAL PADÉ APPROXIMATION ORDERS FOR DIFFERENT CASE STUDIES. Case (1) Case (2) ρ = ρ 1 ρ 2 ρ ρ 1, ρ 2, ρ 3 are 0.2, 0.4, 0.6 and 0.3, 0.5, 0.7 in Cases (1) and (2), respectively. whereas the fit worsens for U > 25. We also notice that for lower values of I (i.e., when I 0), the obtained CDF by Padé approximation suddenly deviates from that obtained via Monte-Carlo simulation (even, the former oscillates around the latter). This also occurs for the optimal U (although this does not appear in the figure). It is difficult to foresee where the deviation occurs (i.e. at which I) and how much is the deviation from the Monte Carlo simulated data. In particular, we found that the accuracy of this method is relatively poor for very low BERs (< 10 8 ). In fact, all previous works using the Padé approximants method have considered not-very-low error probabilities and, hence, have not realized this limitation of this method. B. PDF approximation error In order to study in more detail the approximation error of the proposed method, we consider the relative error between the PDFs, defined as [33]: relative error = log 10 f(i) f PA (I) f(i) (14) where f(i) and f PA (I) are the PDF of received intensity and that obtained by Padé approximation, respectively. Since we do not have an analytical expression for f(i), we cannot directly calculate the exact expression of the relative error. Instead, we use the PDF obtained from Monte-Carlo simulations and consider it as the exact f(i). As an example, we have shown in Fig. 4 the relative error of the approximation for Case (1) with ρ = 0.2 (for U = 25). We notice that for

12 Relative error Intensity I Fig. 4. The relative error of Padé approximation for Case (1) with ρ = 0.2. I 0.6, we have a very good fit 1 between the two PDFs as the relative error is less than However, low BERs correspond to lower values of I. We notice that the relative error increases quickly when I decreases from 0.6, which means that we have a worse fit. This observation is in accordance with the explanations of the above paragraph on the limitation of the Padé approximation method to predict very low BERs. VI. CONCLUSION AND FUTURE WORK We presented an analytical approach to evaluate the performance of space-diversity FSO systems over correlated ΓΓ fading channels based on the Padé approximation method. We derived the general expressions of the n th order moment of a set of arbitrarily-correlated Gamma RVs, and used it to derive an infinite series representation for the MGF of the sum of correlated ΓΓ RVs. Performing the Padé approximation and the inverse Laplace transform on the resulting MGF, the approximate PDF and CDF of the sum of ΓΓ RVs were obtained and used to calculate the system average BER and outage probability. The accuracy of the proposed approach was 1 Note that the sharp decrease in some I values (e.g. around 0.8) is specific to the Padé approximation method; see [32] for details.

13 13 illustrated by considering the case study of a dual-diversity FSO system. We further show that, due to the limitation of the Padé approximation method, this method cannot be used for very low BERs, i.e., lower than We are currently investigating a modified approach to overcome this problem. APPENDIX A CALCULATION OF THE n TH MOMENT OF CORRELATED GAMMA RVS Using the characteristic function of a set of Gamma (squared Nakagami) RVs provided in [36], we explain here how to obtain the MGF of multiple correlated Gamma RVs that we use to calculate their n th moment. Let us consider the vector W = [W 1, W 2,, W L ], whose L elements are arbitrarily-correlated Gamma RVs with equal shape and inverse scale parameters, denoted by m 1, m 2,..., m L, respectively. We also denote the auto-correlation matrix of W by R W. Without loss of generality, we assume that the elements W i are arranged in ascending order of their fading parameters, i.e. m 1 m 2 m L. The MGF of W, denoted by M W (s), can be expressed as [36]: L M W (s) = M W (s 1, s 2,, s L ) = det(i S i A i ) n i, (15) where I represents an (L L) Identity matrix, det(.) denotes matrix determinant, and n i denotes the difference of the fading parameters, defined as follows: m 1, i = 1 n i = m i m i 1, i = 2, 3,, L. Also, S 1 = S is a diagonal matrix of diagonal entries s 1, s 2,, s L, denoted by diag(s 1, s 2,, s L ). In addition, A 1 = A is a positive-definite symmetric matrix that can be determined given m i and R W. Having S 1 and A 1, matrices S i and A i correspond to their lower (L i + 1) (L i + 1) sub-matrices. In other words: A(i, i) A(i, i + 1) A(i, L) A i =..., (17) A(L, i) A(L, i + 1) A(L, L) i=1 (16) S i = diag(s i, s i+1,, s L ). (18)

14 14 The first and the second moments of W are calculated in [36] and are given below: E{W j } = m j A(j, j), (19) E{W j W k } = m j m k A(j, j) A(k, k) + min(m j, m k ) A(j, k) A(k, j). (20) To determine the MGF M W (s), we should first calculate the matrix A (and A i ). Using (19) and (20), we can show that the correlation coefficient ρ jk W entry of R W, can be written as: ρ jk between W j and W k, i.e. the (j, k)-th = min(m j, m k ) A 2 (j, k) W mj m k A(j, j) A(k, k). (21) Note that A(j, k) = A(k, j) due to the symmetry of correlation matrix. Then, the diagonal entries of A can be determined from the mean values of W j from (19). Then, the entries A(j, k) can be calculated from (21). The joint moments of W can be calculated by taking the derivatives and partial derivatives of the MGF [37, Theorem 11.7]. The third moment is defined as: E {W j W k W l } = 3 M W (s) s j s k s l. (22) s=0 Calculating the third-order derivative, we have: 3 M W (s) = 2 M W (s) s j s k s l s k s l where h i (s) is defined as: j i=1 n i h i (s) + M W (s) s k + M W (s) s l min(j,k) i=1 min(j,l) i=1 n i h i (s) s k n i h i (s) s l min(j,k,l) 2 h i (s) + M W (s) n i, (23) s k s l { (I ) } h i (s) = tr S T i A T 1 i (Ei (j, j)a i ) T. (24) Here, tr{ } denotes the trace of matrix, ( ) T stands for transposition, and E i (j, j) represents an (L i + 1) (L i + 1) matrix specified below: E i (j, j) = diag(0,, 0, 1, 0,, 0), j i. (25) }{{}}{{} j i L j i=1

15 15 Using [36] and some matrix derivation properties from [38], [39], the first- and second-order partial derivatives of h i (s) are calculated as follows: h i (s) s k { = tr (E i (j, j)a i ) [ (I S i A i ) 1 (E i (k, k)a i ) (I S i A i ) 1]} (26) 2 h i (s) s k s l { = tr (E i (j, j)a i ) (I S i A i ) 1 (E i (l, l)a i ) (I S i A i ) 1 (E i (k, k)a i ) (I S i A i ) 1 + (E i (j, j)a i ) (I S i A i ) 1 (E i (k, k)a i ) (I S i A i ) 1 (E i (l, l)a i ) (I S i A i ) 1 }. (27) In addition, we have: h i (s) s=0 = A(j, j), h i (s) s=0 s k = A(j, k) A(k, j), 2 h i (s) s k s l s=0 = A(j, l) A(l, k) A(k, j) + A(j, k) A(k, l) A(l, j). Using (22)-(28), the general form of third moment is obtained as: E {W j W k W l } = 3 M W (s) s j s k s l s=0 (28) = m j m k m l A(j, j) A(k, k) A(l, l) + m j A(j, j) min(m k, m l ) A 2 (k, l) + m k A(k, k) min(m l, m j ) A 2 (l, j) + m l A(l, l) min(m j, m k ) A 2 (j, k) + 2 min(m j, m k, m l ) A(j, k) A(k, l) A(l, j). (29)

16 16 Similarly, the fourth moment can be calculated as follows: E {W j W k W l W q } = 4 M W (s) s j s k s l s q s=0 = E {W k W l W q } m j A(j, j) + E {W k W l } min(m j, m q ) A 2 (j, q) + E {W l W q } min(m j, m k ) A 2 (j, k) + E {W q W k } min(m j, m l ) A 2 (j, l) + 2 E {W k } min(m j, m l, m q ) A(j, l) A(l, q) A(q, j) + 2 E {W l } min(m j, m k, m q ) A(j, k) A(k, q) A(q, j) + 2 E {W q } min(m j, m k, m l ) A(j, k) A(k, l) A(l, j) + 3 min(m j, m k, m l, m q ) [A(j, k) A(k, l) A(l, q) A(q, j) + A(j, k) A(k, q) A(q, l) A(l, j) + A(j, q) A(q, k) A(k, l) A(l, j)] (30) Following the expressions of the first four moments as given in (19), (20), (29) and (30), we can deduce the general expression of the n th moment of W, i.e., E {W r1 W r2 W rn } through a recursive formula, where r 1, r 2,, r n {1, 2,, n} represent the indices of Gamma RVs. For the sake of notation simplicity, let us use the notation E (r 1, r 2,, r n ) instead of E {W r1 W r2 W rn }. We have rewritten the expressions of the first four moments using this notation in (31). Then, the recursive formulae for calculating the n th moment are given in (32)-(36). Note that all possible n th -order joint moments can be calculated from these equations. For instance, E {W 3 1 W n 3 n } is obtained by setting the indices r 1 = r 2 = r 3 = 1 and r 4 = = r n = n. APPENDIX B JUSTIFICATION OF APPLYING PADÉ APPROXIMATION TO THE PDF OF INTENSITY I In the previous works [24] [27], the Padé approximation was applied to the MGF of the

17 17 E (r 1 ) = C min (r 1 ) = m r1 A (r 1, r 1 ), E (r 1, r 2 ) = E (r 2 ) C min (r 1 ) + C min (r 1, r 2 ), E (r 1, r 2, r 3 ) = E (r 2, r 3 ) C min (r 1 ) + E (r 3 ) C min (r 1, r 2 ) + E (r 2 ) C min (r 1, r 3 ) + C min (r 1, r 2, r 3 ), E (r 1, r 2, r 3, r 4 ) = E (r 2, r 3, r 4 ) C min (r 1 ) + E (r 3, r 4 ) C min (r 1, r 2 ) + E (r 2, r 4 ) C min (r 1, r 3 ) + E (r 2, r 3 ) C min (r 1, r 4 ) + E (r 4 ) C min (r 1, r 2, r 3 ) + E (r 3 ) C min (r 1, r 2, r 4 ) + E (r 2 ) C min (r 1, r 3, r 4 ) + C min (r 1, r 2, r 3, r 4 ). E (r 1, r 2,, r n ) = E (r 2,, r n ) C min (r 1 ) n n n + + i 2 =1 (i 2 i 1 ) n i 2 =1 (i 2 i 1 ) i 3 =1 (i 3 {i 1,i 2 }) n i 3 =1 (i 3 {i 1,i 2 }) + i n 1 =1 (i n 1 {i 1,i 2,,i n 2 }) n i n 2 =1 (i n 2 {i 1,i 2,,i n 3 }) n i 2 =1 (i 2 i 1 ). E ( r i2,, r in 1 ) Cmin (r i1, r in ) E ( r i2,, r in 2 ) Cmin ( ri1, r in 1, r in ) E (r i2 ) C min ( ri1, r i3,, r in 1, r in ) + C min ( ri1, r i2,, r in 1, r in ) (31) i 1 = 1, i 2 i 1, i 3 {i 1, i 2 },, i n {i 1, i 2,, i n 1 }, (32) C min (r 1, r 2,, r n ) = min (m r1, m r2,, m rn ) n n n... A (r i1, r i2,, r in ), i 1 {1, 2,, n}, (33) i 2 =1 (i 2 i 1 ) i 3 =1 (i 3 {i 1,i 2 }) in=1 (in {i 1,i 2,,i n 1 }) min (m ri ) = m ri, (34) A (r i ) = A (r i, r i ), (35) A (r i1, r i2,, r in ) = A (r i1, r i2 ) A (r i2, r i3 ) A ( r in 1, r in ) A (rin, r i1 ). (36)

18 18 received SNR. Instead of this classical approach, we consider here the application of Padé approximation to the received intensity I. Note that, given (11), the received SNR is proportional to I 2 in our case. We explain here the reason behind this modification. Let us consider the (1 2) FSO system under the two cases of weak and strong turbulence conditions, with the corresponding ΓΓ model parameters of (α = 20.0, β = 25.0) and (α = 2.0, β = 3.0), respectively. Without loss of generality, let us assume the conditions of independent fading. We have shown plots of the PDFs of the total received intensity I and the squared intensity I 2 in Fig. 5 for the two cases of Monte Carlo simulations (f(.)) and Padé approximation (f PA (.)). We notice from Fig. 5(a) that, concerning the PDF of I, the Padé approximation has a very good agreement with the Monte-Carlo simulation results. However, for the PDF of I 2, presented in Fig. 5(b), we have a good fit for the weak turbulence regime but the fit is not satisfying for the strong turbulence case and it worsens when I 2 approaches zero. In particular, we should reasonably have f(i 2 ) 0 for I 2 0 but this is not the case for f PA (I 2 ). Since we consider in this work the conditions of moderate to strong turbulence that are mostly valid in practice for link distances on the order of a few kilometers, and we are particularly interested in relatively low BERs, we should not apply the Padé approximation to the SNR after EGC; otherwise, we cannot provide an accurate performance prediction for the system. As a result, we apply the Padé approximation to the PDF of I. As a matter of fact, after carrying out a number of trials, we concluded that the Padé approximation does not provide a good fit when we should deal with the specific situations where we have a relatively fast decrease in the PDF close to I = 0, which is the case in Fig. 5(b), for example. ACKNOWLEDGMENT The authors would like to acknowledge the support by EU Opticwise COST Action IC1101. They are also grateful to Prof. George K. Karagiannidis for the fruitful discussions on the Padé approximation method.

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