On the Performance Analysis of Dual-Hop. FSO Fixed Gain Transmission Systems (Technical Report)

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1 On the Performance Analysis of Dual-Hop FSO Fixed Gain Transmission Systems (Technical Report Emna Zedini, Student Member, IEEE, Hamza Soury, Student Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE Abstract Novel exact closed-form results for the end-to-end performance analysis of dual-hop free-space optical (FSO fixed-gain relaying systems under heterodyne detection as well as intensity modulation with direct detection techniques in the presence of atmospheric turbulence as well as pointing errors are presented. By using dual-hop FSO relaying, we demonstrate a better system performance relative to the single FSO link. Numerical and Monte-Carlo simulation results are provided to verify the accuracy of the newly proposed results, and a perfect agreement is observed. Index Terms Free-space optics (FSO, dual-hop relaying, pointing errors, Gamma-Gamma, outage probability, average bit-error rate (BER, ergodic capacity. I. INTRODUCTION Free-space optical (FSO communication systems have gained a growing research interest due to their various features. These include high data rates due to their very high bandwidth, with high security level at the unlicensed optical spectrum, robustness to electromagnetic interference, and E. Zedini, H. Soury and M.-S. Alouini are with the Computer, Electrical, and Mathematical Science and Engineering (CEMSE Division, King Abdullah University of Science and Technology (KAUST Thuwal, Makkah Province, Saudi Arabia ( s:{emna.zedini, soury.hamza, slim.alouini}@kaust.edu.sa.

2 2 ease of deployment among others. This promising technology is able to overcome the spectrum scarcity issue in wireless communications systems [] [8]. However, the performance of this line-of-sight technology can be seriously affected by limited range, turbulence-induced fading, and pointing errors. These effects may result in a significant degradation in the performance of FSO communication systems. Relaying technique has been demonstrated as an efficient solution to mitigate these effects. By using short hops, this technology can broaden the coverage and improve the FSO link performance [9] [2]. As such, considerable efforts have been devoted to study the relay system performance when FSO and radio-frequency (RF are used in series in the so-called mixed RF/FSO systems under both heterodyne detection and intensity modulation with direct detection (IM/DD techniques employing decode-and-forward (DC or amplify-and-forward (AF relaying [3] [23]. In such system models, multiple RF users can be multiplexed and sent over the FSO link. This also has increased the interest to study the performance of dual-hop mixed FSO/RF systems, where the FSO link is used as a broadcast channel that serves different RF users. As per authors best knowledge, the first exact closed-form performance analysis of dual-hop FSO/RF fixed-gain relaying systems has been carried out in [24]. More specifically, the FSO link is considered to be operating over Gamma-Gamma distribution that accounts for pointing errors under heterodyne technique only, and the RF link is modeled by the Nakagami-m fading. As a extension of [24], the authors in [25], have considered both types of detection techniques (i.e. heterodyne detection as well as IM/DD in the FSO link and provided a generalized framework for several fading channels in the RF link including Nakagami-m, Rayleigh, Exponential, Weibull, and Lognormal distributions. Having such interesting results motivates further the analysis of dual-hop FSO fixed-gain relaying communication systems. In [26] closed-form bounds for the performance analysis of dual-hop FSO system with channel-state information (CSI-assisted amplify-andforward (AF relay are presented. An experimental setup of an all-optical 0-Gbps dual-hop FSO system using an AF relay has been presented in [27]. Highly motivated by the experimental verification in [27], this paper presents exact closedform analytical expressions for the outage probability, the average bit-error rate (BER of a variety of binary modulation schemes, and the ergodic capacity of dual-hop FSO systems that experience the Gamma-Gamma fading, which is the most suitable model to characterize moderate to strong turbulence regimes [], under the combined effect of atmospheric turbulence

3 3 and pointing errors using both heterodyne detection and IM/DD. To the best of the authors knowledge, this represents the first exact closed-form performance study of such systems using fixed gain relaying. The results show a significant improvement in the performance of the dualhop FSO system over the single FSO link and are as such in a perfect agreement with what was observed experimentally in [27]. The remainder of this paper is organized as follows. In Section II, we introduce the channel and communication system model. We derive the cumulative distribution function (CDF, the probability density function (PDF, and the moments of the end-to-end signal-to-noise ratio (SNR of dual-hop FSO systems in Section III. Capitalizing on these results, the outage probability, the average BER, and the ergodic capacity are obtained in exact closed-form in Section IV. Section V presents some numerical and simulation results to illustrate the mathematical formalism presented in the previous section. Finally, some concluding remarks are drawn in Section VI. II. CHANNEL AND SYSTEM MODELS We consider a dual-hop optical communication system where the source terminal S is communicating with the destination terminal terminal D through the intermediate terminal R which acts as a relay. The two FSO hops (i.e. S-R and R-D are assumed to be subject to independent but not necessarily identically distributed Gamma-Gamma fading that accounts for pointing errors and both types of detection techniques (i.e. IM/DD as well as heterodyne detection. The overall instantaneous SNR of a dual-hop FSO system employing AF fixed gain relay can be written as γ = γ γ 2 γ 2 +C [22, Eq.(], where C is a fixed relay gain, and γ i represents the instantaneous SNR of the ith hop for i (, 2 with the PDF given in [22, Eq.(3] as ( ξi 2 γ r i ξ f γi (γ = r i Γ(α i Γ(β i γ G3,0 i,3 α i β i h i µ ri 2 +, ( ξi 2, α i, β i where h i = ξ2 i ξi 2+, r i is the parameter that represents the type of detection being used (i.e. r i = is associated with heterodyne detection and r i = 2 associated with IM/DD, ξ i denotes the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter at the receiver given as ξ i = w zeq,i 2 σ s,i, with σ 2 s,i is the jitter variance at the receiver and w zeq,i is the equivalent beam radius at the receiver [28], G,, ( is the Meijer s G function, and

4 4 µ ri refers to the the electrical SNR of the ith hop. In particular, for r i =, µ i = µ heterodynei = E[γ i ] = γ i and for r i = 2, µ 2i = µ IM/DDi = γ i α i β i ξ 2 i (ξ 2 i + 2/[(α i + (β i + (ξ 2 i + 2 ], with α i and β i the fading parameters related to the atmospheric turbulence conditions [3]. More specifically, assuming a plane wave propagation in the absence of inner scale, α i and β i can be determined from the Rytov variance as [3] where σ 2 R,i =.23 C2 n,i 0.49 σr,i 2 α i = exp ( +. σ 2/5 R,i 0.5 σr,i 2 β i = exp ( σ 2/5 R,i ( 7 2π λ i 6 L 6 i 7/6 5/6 (2, (3 is the Rytov variance, C 2 n,i denotes the refractive-index structure parameter, λ i is the wavelength, and L i represents the propagation distance of the ith hop for i (, 2. Moreover, by using [29, Eq.(2.24.2/3] then [30, Eq.(2.4.5], we can obtain the CDF of γ i as ξ 2 i F γi (γ = Γ(α i Γ(β i G4,0 2,4 A. Cumulative Distribution Function α i β i h i ( γ µ ri III. END-TO-END SNR STATISTICS r i, ξ 2 i + 0, ξ 2 i, α i, β i. (4 The CDF of the overall SNR, γ, for a dual-hop FSO system under both types of detection techniques (i.e. heterodyne detection as well as IM/DD with pointing errors taken into account can be given in exact closed form in terms as ξ 2 ξ2 2 F γ (γ = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 (, r 2, ( H 0,:4,0:0,3 ( + ξ2, 2 ( µr C r 2 γ,0:,4:3,2 α (ξ2, 2, (α 2,, (β 2,, (0, r 2 2 β 2 h 2,, (5 µ r2 α β h ( ξ, 2 ( α, ( β, ( ξ, 2 (0,

5 5 where H.,.:.,.:.,..,.:.,.:.,. is the bivariate H-Fox function, known also as the H-Fox function of two variables [3] whose MATLAB implementation is presented in [32]. Proof: See Appendix A. It is important to mention that for the special case where the two FSO hops operate under heterodyne detection, i.e. = and r 2 =, the unified expression given by (5 can be simplified in terms of the extended generalized bivariate Meijer s G function, G.,.:.,.:.,..,.:.,.:.,. with the help of [30, Eq.(2.9.] as ξ 2 ξ2 2 F γ (γ = Γ(α Γ(α 2 Γ(β Γ(β 2 G 0,:4,0:0,3 + ξ 2 2 ξ, 2 α, β,0:,4:3,2 ξ2, 2 α 2, β 2, 0 ξ, 2 0 α 2 β 2 h 2 C µ r2, µ r α β h γ. (6 Note that an efficient MATHEMATICA R implementation for G.,.:.,.:.,..,.:.,.:.,. is presented in [33, Table II]. B. Probability Density Function Differentiating (5 with respect to γ results in the exact closed-form expression of the PDF of γ in terms of the bivariate H-Fox function, that is, ξ 2 ξ2 2 f γ (γ = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 γ (, r 2, ( H 0,:4,0:0,3 ( + ξ2, 2 ( µr C r 2 γ,0:,4:3,2 α (ξ2, 2, (α 2,, (β 2,, (0, r 2 2 β 2 h 2,. (7 µ r2 α β h ( ξ, 2 ( α, ( β, ( ξ, 2 (, Proof: See Appendix B. C. Moments The nth moments of the end-to-end SNR of a dual-hop FSO system using both types of detection techniques, defined as E[γ n ] = 0 γ n f γ (γ dγ, can be shown to be given in terms of

6 6 the H-Fox function by Proof: See Appendix C. E[γ n ξ 2 ξ2 2 Γ( n + α Γ( n + β µ n r ] = Γ(α Γ(α 2 Γ(β Γ(β 2 Γ(n( n + ξ(α 2 β h n H 4, C (α 2β 2 h 2 r 2 ( n, ( + ξ 2, 2 r 2 2,4 µ r2. (8 (ξ2, 2 r 2 (α 2, r 2 (β 2, r 2 (0, Note that an efficient MATHEMATICA R implementation for evaluating the H-Fox function H,, ( is presented in [34]. Furthermore, in the special case of a dual-hop FSO system operating under heterodyne detection, (8 simplifies to E[γ n ] = ξ 2 ξ2 2 Γ(n + α Γ(n + β µ n Γ(α Γ(α 2 Γ(β Γ(β 2 Γ(n(n + ξ(α 2 β h n G4, 2,4 Cα 2β 2 h 2 µ r2 n, + ξ 2 2. (9 ξ2, 2 α 2, β 2, 0 IV. END-TO-END PERFORMANCE METRICS A. Outage Probability The outage probability is a standard performance metric of an FSO communication system. It is defined as the probability that the end-to-end SNR, γ, falls below a certain specified threshold, γ th. An exact closed-form expression for the outage probability of dual-hop fixed-gain relaying FSO systems in operation under both heterodyne detection as well as IM/DD with pointing error impairments can be easily obtained by setting γ = γ th in (5. B. Average Bit-Error Rate The average BER for different binary modulation schemes can be written in terms of the CDF of γ as P b = ba 2Γ(a 0 γ a e bγ F γ (γ dγ, (0 where a and b represent different binary modulations schemes. For instance, a = /2 and b = are for coherent binary phase shift keying (CBPSK, a = /2 and b = /2 are for coherent binary frequency shift keying (CBFSK, a = and b = /2 are for non-coherent BFSK (NCFSK, and a = and b = for differential BPSK (DBPSK. Then, referring to Appendix D, an exact closed-form expression for the average BER of dual-hop FSO systems using fixed-gain relaying

7 7 and operating under both types of detection techniques can be derived in terms of the bivariate H-Fox function, that is, P b = 2 ξ 2 ξ2 2 2 r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 Γ(a (, r 2, H 0,:4,0:,3 ( + ξ2, 2 ( C r 2 (b µ r,0:,4:3,3 α 2 β 2 h 2,. ( (ξ2, 2 (α 2, (β 2, (0, r 2 µ r2 α β h ( ξ, 2 ( α, ( β, (a, ( ξ, 2 (0, Proof: See Appendix D. It is worth mentioning that, when the two FSO links employ heterodyne detection technique (i.e. = r 2 =, the average BER expression presented in terms of the bivariate H-Fox function in ( simplifies to P b = 2 ξ 2 ξ2 2 2 r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 Γ(a G 0,:4,0:,3 + ξ 2 2 ξ, 2 α, β,0:,4:3,3 ξ2, 2 α 2, β 2, 0 a, ξ, 2 0 α 2 β 2 h 2 C µ r2, bµ r α β h. (2 C. Ergodic Capacity The ergodic capacity of dual-hop FSO communication systems in operation under both heterodyne technique and IM/DD can be given by [35, Eq.(26], [36, Eq.(7.43] C E[ln( + c γ] = 0 ln( + c γf γ (γ dγ, (3 where c is a constant such that c = e/(2π for IM/DD technique (i.e. r = 2 and c = for heterodyne technique (i.e. r =. Note that the expression in (3 is exact for r = while it is a lower-bound for r = 2, and can be derived in closed-form in terms of the bivariate H-Fox

8 8 function as ξ 2 ξ2 2 C = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 (, r 2, H 0,:4,0:,4 ( + ξ2, 2 ( C r 2 (c µ r,0:,4:4,3 α 2 β 2 h 2,. (4 (ξ2, 2 (α 2, (β 2, (0, r 2 µ r2 α β h ( ξ, 2 ( α, ( β, (, (, ( ξ, 2 (0, Proof: See Appendix E. In the special case when = and r 2 =, (4 further simplifies to ξ 2 ξ2 2 C = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 G 0,:4,0:,4 + ξ 2 2 ξ, 2 α, β,,0:,4:4,3 ξ2, 2 α 2, β 2, 0, ξ, 2 0 α 2 β 2 h 2 C µ r2, c µ r α β h. (5 V. NUMERICAL RESULTS In this section, we provide some numerical results to illustrate the mathematical formalism presented above and prove its correctness by means of Monte Carlo simulations. Without loss of generality, we assume equal average SNRs of both the links, γ = γ 2 = γ with turbulence parameters for S-R and R-D FSO links α = α 2 = α and β = β 2 = β and pointing errors ξ = ξ 2 = ξ. The wavelength is assumed to be λ = 550 nm. A fixed relay gain C =. is considered. Moderate turbulence is characterized by C 2 n turbulence is associated with C 2 n = 0 3 m 2 3 [3]. = m 2 3, whereas strong Fig. demonstrates the impact of the pointing error on the outage probability of a dual-hop FSO link with L S R = L R D = 000 m under moderate turbulence with turbulence parameters α = 5.42 and β = 3.79 calculated from (2 and (3, respectively. Results for a single 2000 m long FSO link with turbulence parameters α = 4 and β =.65, are also included for comparison purposes (we divide the single link into two 000 m long links. The exact closed-form expression for the outage probability of a single FSO link under both heterodyne detection and IM/DD is given by [37, Eq.(5]. Clearly, we observe from Fig. that the analytical results provide a perfect

9 9 match to the MATLAB simulated results proving the accuracy of our derivation. As expected, it can also be observed from this figure that for both dual-hop FSO and single FSO links, the outage probability performance degrades in the case of strong pointing errors. Furthermore, it can be seen that connecting two FSO links in series can significantly mitigate the pointing error impairments and as such improve the system performance, compared to the single FSO link. This result is in a perfect agreement with what was experimentally demonstrated in [27]. 0 0 Outage Probability Single Link ξ =. Dual-Hop ξ =. Single Link ξ = 6.7 Dual-Hop ξ = 6.7 Simulation Average SNR (db Fig. : Outage probability of single FSO and dual-hop FSO links for various values of ξ under moderate (C 2 n = m 2 3 turbulence conditions using IM/DD with a total length of 2000 m. Fig. 2 depicts the outage probability performance of a dual-hop FSO link in the presence of moderate (α = 5.42, β = 3.79 and strong (α = 4, β =.7 turbulence conditions under both IM/DD and heterodyne detection for negligible effect of the pointing error (ξ = 6.7. We observe that for a given type of detection, P out increases with an increase in the turbulence severity leading to a performance deterioration. It can also be shown that implementing heterodyne detection results in a considerable improvement in the dual-hop system performance compared to IM/DD, as expected. In fact, despite its complexity, heterodyne detection has been proposed as an alternative type of detection in FSO communication systems being able to better overcome

10 0 the turbulence effects, relative to IM/DD technique [38] IM/DD Outage Probability Heterodyne Cn 2 = 0 3 m 3 2 Cn 2 = m 3 2 Simulation Average SNR (db Fig. 2: Outage probability of a dual-hop FSO/FSO system for negligible pointing errors (ξ = ξ 2 = 6.7 with strong (C 2 n = 0 3 m 2 3 and moderate (C 2 n = m 2 3 turbulence conditions using both IM/DD and heterodyne detection. In Fig. 3, the average BER for various binary modulation schemes of a dual-hop FSO system derived in ( as well as a single FSO link given by [37, Eq.(0] in operation under IM/DD is plotted. It can be observed from Fig. 3 that the dual-hop FSO system offers a better performance than the single FSO link. This result, being experimentally verified in [27], emphasizes the importance of the dual-hop FSO system in mitigating the turbulence conditions as well as the pointing errors. In addition, it can be shown from Fig. 3 that CBPSK modulation outperforms DBPSK and NCFSK, as expected. The ergodic capacity for both dual-hop FSO and single FSO links in operation under heterodyne detection as well as IM/DD is presented in Fig. 4. We can see from this figure that the analytical results of the ergodic capacity given by (4 for the dual-hop system and [37, Eq.(3] for the single FSO link are in a good match with the simulated results. One of the most important outcomes of Fig. 4 is the capacity gain achieved by cascading two FSO links in series. In addition, it can be seen from this figure that heterodyne detection outperforms the

11 IM/DD technique for both dual-hop and single FSO links. 0 0 Average Bit-Error Rate NCFSK DBPSK CBPSK Simulation Single Link Dual-Hop Average SNR (db Fig. 3: Average BER for a variety of binary modulation schemes of single FSO and dual-hop FSO links under moderate turbulence conditions using IM/DD with negligible pointing errors for a total length of 2000 m. VI. CONCLUSION In this paper, we have investigated analytically for the first time the outage performance, the average BER, and the ergodic capacity of a dual-hop FSO system using AF fixed-gain relaying in operation under both heterodyne detection as well as IM/DD including pointing error effects. We demonstrated that the dual-hop FSO system outperforms the single FSO link and is capable of mitigating turbulence-induced fading and pointing errors. The effect of atmospheric turbulence and pointing errors on the dual-hop FSO link performance has also been studied and, as expected, severe pointing errors and strong turbulence can severely degrade the overall system performance of both single FSO and dual-hop FSO links.

12 2 Ergodic Capacity (Nats/Sec/Hz Dual-Hop Single Link Simulation Heterodyne IM/DD Average SNR (db Fig. 4: Ergodic Capacity of single FSO and dual-hop FSO links under moderate turbulence conditions using both heterodyne detection and IM/DD with negligible pointing errors for a total propagation distance of 2000 m. APPENDIX A CDF OF THE END-TO-END SNR In this appendix, we derive the CDF of the end-to-end SNR γ starting with [ ] γ γ 2 F γ (γ = Pr γ 2 + C < γ, (A. which can be expressed as [ ] γ γ 2 F γ (γ = Pr 0 γ 2 + C < γ γ 2 f γ2 (γ 2 dγ 2 = F γ (γ ( + Cγ2 f γ2 (γ 2 dγ 2 = 0 γ x=0 t=0 f γ (tf γ2 (x dt dx + x=0 γ+ Cγ x Integrating over the same area and interchanging the integrals yields F γ (γ = γ t=0 = F γ (γ + f γ2 (xf γ (t dx dt + x=0 γ t=γ t=γ ( Cγ F γ2 f γ (t dt. t γ Cγ t γ x=0 f γ (tf γ2 (x dt dx. f γ2 (xf γ (t dx dt (A.2 (A.3

13 3 Substituting ( and (4 in (A.3 we obtain ξ 2 ξ2 2 F γ (γ = Γ(α Γ(α 2 Γ(β Γ(β 2 γ t G4,0 α ( 2β 2 h 2 Cγ r 2, ξ2 2,4 r µ 2 t γ 2 + G 3,0 α β h ξ r 0, ξ2, 2,3 t α 2, β r 2 µ 2 + dt. ξ, 2 α, β r 2 (A.4 Using the change of variable x = t γ and the primary definition of the Meijer s G function in [39, Eq.(9.30] with the integral identity [39, Eq.(3.94/3], the CDf can be written as ξ 2 ξ 2 2 F γ (γ = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 (2πi 2 r 2 C C 2 ( s Γ + t r 2 Γ(ξ2 2 sγ(α 2 sγ(β 2 sγ( s Γ(ξ 2 + tγ(α + tγ(β + t Γ( + ξ2 2 s Γ(ξ tγ( + t ( s ( ( t r 2 µr (α 2 β 2 h 2 ds dt, Cµr2 α β h γ (A.5 (A.6 where C and C 2 are the s-plane and the t-plane contours, respectively. Now, by utilizing [3, Eq.(.], we obtain the CDF expression of the end-to-end SNR given in (5. APPENDIX B PROBABILITY DENSITY FUNCTION Taking the derivative of (A.5 with respect to γ yileds ξ 2 ξ 2 2 f γ (γ = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 (2πi 2 Γ(ξ2 2 sγ(α 2 sγ(β 2 sγ( s Γ( + ξ2 2 s ( s t r 2 (α 2 β 2 h 2 µ Cµr2 α β h r 2 C C 2 ( s Γ + t r 2 Γ(ξ 2 + tγ(α + tγ(β + t Γ(ξ tγ( + t dγ t dγ ds dt. (B.

14 4 Using Γ( + t = t Γ( t with some algebraic manipulations, (B. becomes f γ (γ = ξ 2 ξ2 2 r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 γ (2πi 2 r 2 C C 2 Γ ( s + t r 2 Γ(ξ2 2 sγ(α 2 sγ(β 2 sγ( s Γ(ξ 2 + tγ(α + tγ(β + t Γ( + ξ2 2 s Γ(ξ tγ( t ( s ( ( t r 2 µr (α 2 β 2 h 2 ds dt, Cµr2 α β h γ (B.2 Applying [3, Eq.(.], we get the desired PDF expression given by (7. The moments can be written as APPENDIX C MOMENTS E[γ n ξ 2 ξ2 2 ] = r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 ( 0 x H,0 0, x Cx H 4,0 r 2 ( + ξ2,,4 α 2 β 2 h 2 (0, µ r2 2 (ξ2, 2 (α 2, (β 2, (0, r 2 ( µr γ n H 0,3 x γ ( ξ 3,2, 2 ( α, ( β, 0 α β h dγ dx, ( ξ, 2 (, (C. by means of substituting (7 into the definition of the moments then applying [3, Eq.(2.3] to represent the the bivariate H-Fox function in terms of an integral involving the product of three H-Fox functions. Using [40, Eq.(.59] along with the Mellin transform of the H-Fox function given by [40, Eq.(2.8], (C. reduces to E[γ n ξ 2 ξ2 2 Γ( n + α Γ( n + β µ n r ] = Γ(α Γ(α 2 Γ(β Γ(β 2 Γ(n( n + ξ(α 2 β h n ( x n H,0 0, x Cx H 4,0 r 2 ( + ξ2,,4 α 2 β 2 h 2 0 (0, µ r2 2 dx. (ξ2, 2 (α 2, (β 2, (0, r 2 (C.2 Finally, employing [30, Eq.(2.8.4] together with [40, Eq.(.59], the moments can easily simplify into (8 by means of some algebraic manipulations.

15 5 APPENDIX D AVERAGE BIT-ERROR RATE Substituting (A.5 into (0 and then utilizing [39, Eq.(3.38/4], the average BER may be written as P b = 2 ξ 2 ξ2 2 2 r 2 Γ(α Γ(α 2 Γ(β Γ(β 2 (2πi 2 C C 2 ( s Γ + t r 2 Γ(ξ2 2 sγ(α 2 sγ(β 2 sγ( s r 2 Γ(ξ 2 + tγ(α + tγ(β + tγ(a t Γ( + ξ2 2 s Γ(ξ tγ( + t ( s ( r t 2 (α 2 β 2 h 2 (bµ r Cµr2 ds dt. α β h Using [3, Eq.(.] results in the unified average BER expression in (. (D. APPENDIX E ERGODIC CAPACITY By substituting (B.2 into (3, the ergodic capacity can be written as p ξ 2 ( C = Γ ps + t r Γ(mΓ(αΓ(β (2πi 2 r Γ(m sγ( ps Γ(ξ2 + tγ(α + tγ(β + t Γ(ξ t Γ( + t r ( ( ps C d µ t r γ t r ln( + c γ dγ ds dt. αβh γ 2 0 C C 2 (E. Now, using [39, Eq.(4.293/0] and [3, Eq.(.], the ergodic capacity can be obtained in closed-form as in (4. REFERENCES [] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications. SPIE Press, 200. [2] K. Peppas and C. Datsikas, Average symbol error probability of general-order rectangular quadrature amplitude modulation of optical wireless communication systems over atmospheric turbulence channels, IEEE/OSA Journal of Optical Communications and Networking, vol. 2, no. 2, pp. 02 0, Feb [3] W. Popoola and Z. Ghassemlooy, BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence, IEEE/OSA Journal of Lightwave Technology, vol. 27, no. 8, pp , Apr [4] J. Park, E. Lee, and G. Yoon, Average bit-error rate of the Alamouti scheme in Gamma-Gamma fading channels, IEEE Photonics Technology Letters, vol. 23, no. 4, pp , 20.

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