Performance Analysis of Hybrid FSO Systems Using FSO/RF-FSO Link Adaptation

Open Access Performance Analysis of Hybrid FSO Systems Using FSO/RF-FSO Link Adaptation Volume 10, Number 3, June 2018 Banibrata Bag Akinchan Das Imran Shafique Ansari Aleˇ s Prokeˇ s Chayanika Bose Aniruddha Chandra DOI: 10.1109/JPHOT.2018.2837356 1943-0655 © 2018 CCBY IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems Performance Analysis of Hybrid FSO Systems Using FSO/RF-FSO Link Adaptation Banibrata Bag ,1Akinchan Das ,1Imran Shafique Ansari ,2 Aleˇ sProke ˇ s,3Chayanika Bose,4and Aniruddha Chandra 5 1Haldia Institute of Technology, Haldia 721631, India 2Global College of Engineering and Technology, Muscat 2546, Oman 3Brno University of Technology, Brno 601 90, Czech Republic 4Jadavpur University, Kolkata 700032, India 5National Institute of Technology, Durgapur 713209, India DOI:10.1109/JPHOT.2018.2837356 This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/. Manuscript received February 19, 2018; revised May 4, 2018; accepted May 12, 2018. Date of publication May 16, 2018; date of current version June 12, 2018. This work was supported by the Czech Science Foundation, Project No. 17-27068S Mobile channel analysis and modelling in millimeter wave band, and by the National Sustainability Program under grant LO1401 Interdisciplinary Research of Wireless Technologies. For the research, the infrastructure of the SIX Center was used. Corresponding authors: A. Chandra and A. Prokeˇ s (e-mail: [email protected]; prok[email protected].cz). Abstract: Free-space optical (FSO) links are considered as cost-effective, noninvasive alternative to fiber optic cables for 5G cellular backhaul networking. For FSO-based backhaul networks, we propose an additional millimeter-wavelength (MMW) radio-frequency (RF)- FSO link, used as a backup. Uninterrupted and reliable network connection is possible by switching between primary FSO link and the secondary RF-FSO link; when the primary link is under atmospheric turbulence, the secondary link maintains connectivity as the MMW RF link exhibits complementary characteristics to atmospheric effects. In order to analytically assess the improvement, we also derive concise mathematical expressions for different performance metrics, such as outage probability, average bit error rate (BER), and capacity. Our results demonstrate that the FSO/RF-FSO topology performs better than a single FSO link in terms of outage probability and BER. The dual-hop mixed RF-FSO link is realized with an amplify and forward (AF) relay that adapts an average power scaling strategy. The irradiance fluctuations in the FSO links are modeled by gamma–gamma distribution, assuming strong atmospheric turbulence while it is assumed that the RF link experiences multipath Rayleigh fading. For switching between links, a single FSO threshold is considered first, followed by a dual FSO threshold to prevent unnecessary switching. Index Terms: Free-space optics, 5G cellular backhaul, gamma-gamma fading, backup RFFSO link, amplify-and-forward relay, link switching probability. 1. Introduction The digital society of new generation is being accustomed to machine-to-machine (M2M) communication with high-speed Internet applications, and a demand for 1 Gbps connectivity per user is required to fulfil the dream of Internet-of-Things (IoT) in 5G networks. M2M communication in IoT would require communication between a huge number of connected devices. The challenge is to realize a backhaul infrastructure that supports a large node density and can carry an overwhelming amount of aggregated data. To extend the capacity, network operators are constantly diminishing Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems the size of the cells, but with every added base station the design of the backhaul network is becoming complex as well as expensive. Free-space optical (FSO) communication is expected to play a vital role in 5G wireless networks. FSO links serve as a promising alternative to the conventional fiber optic cables utilized for backhaul links due to the ease of deployment, rapid setup time, and low maintenance cost [1], [2]. The downside of FSO is that it requires a clear line-of-sight (LOS) path and the propagation is highly influenced by the atmospheric turbulence [3]. Some hybrid paradigms incorporating both radiofrequency (RF) and FSO links have been proposed to combine the advantages of both links. In particular, FSO links offer much better data rates than RF links but suffer from atmospheric loss due to fog and scintillation whereas the RF link is a very good complement to FSO as RF is relatively insensitive to weather and it can penetrate fog easily. In our proposed FSO/RF-FSO scheme (see Fig. 1), the primary FSO link is utilized for better data rate when a clear LOS path exists, else a backup RF-FSO link is activated to sustain the connectivity. Under primary FSO link failure, this scheme provides a backup link to cope up with the weather conditions in hilly regions and industrial belts where dense fog, cloud, or smog often engulfs a part of the FSO transmission path. 1.1 Prior Work While the idea of transmitting RF signals over FSO links in cellular mobile backhauls has been around the corner for quite some time [4], the network architectures proposing a combination of RF and FSO links are relatively new [5]. The proposed combinations are either serial where the middle node (relay node) of the cascaded link is utilized for RF-optical conversion [6] or parallel where a pair of RF and FSO links connects two network nodes to improve reliability [7]. The serial RF-FSO combination have been addressed in [8]–[10] where analytical expressions for amount of fading, outage probability, bit/ symbol error rate, and ergodic capacity have been derived. On the other hand, the parallel RF/FSO combination are considered in [11]–[13] where novel coding schemes are proposed for switching between RF and FSO paths. Again, for channel characterization, different statistical models have been prescribed for both part of the links as RF and FSO links experience different atmospheric perturbations. For the RF link, Rayleigh [14], Ricean [15], Nakagami-m[16], and generalized distributions [17] are proposed to model the multipath fading. For characterizing atmospheric turbulence induced fading in FSO links, log-normal distribution has been utilized for long [18], although the distribution is suitable for modeling only weak turbulence. Recent experimental studies indicate that for FSO channel modeling gamma-gamma distribution is the most preferable candidate as it can model weak, moderate, and strong atmospheric turbulence conditions [19], [20]. 1.2 Contributions In this paper: rA hybrid FSO/RF-FSO transmission scheme is presented to increase the availability and reliability of next generation cellular backhaul networks. To the best of our knowledge, analysis of performance metrics for a backhaul system where the primary FSO backhaul link is augmented with a serial RF-FSO backup link, has not been reported in the open literature so far. The proposed system is different from a hybrid RF/RF-FSO implementation [21]1where the mobile users communicate with the respective base station via a RF or a RF-FSO link. rBy modeling the RF fading and atmospheric turbulence induced FSO fading with Rayleigh and gamma-gamma statistics, respectively, we derive analytical expressions for outage probability, average bit-error rate (BER), and ergodic capacity. rThe derived mathematical expressions for different performance metrics are presented in terms of Meijer’s G-functions that can be accurately and easily computed using MATLAB or Wolfram Mathematica. 1This paper was co-authored by one of the current authors. Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems Fig. 1. Dual-hop mixed RF-FSO backup link with source-to-relay RF link and relay-to-destination FSO link. rTwo different link switching strategies are examined - a single FSO threshold scheme which offers simplicity in transceiver design, and a dual FSO threshold scheme that prevents unnecessary switching between primary FSO link and backup mixed RF-FSO link. rAll the mathematical analyses are verified through Monte Carlo simulations. 1.3 Organisation The rest of this paper is structured as follows. Section II describes the proposed system model followed by statistical channel modeling of the primary FSO link and the backup RF-FSO link. Section III describes the link switching operation in Algorithm-1 under single FSO threshold scheme and presents analytical framework for calculation of outage probability, average BER, and ergodic capacity. The link switching operation for dual FSO threshold scheme is given in Algorithm-2 in Section IV followed by calculation of all the above-mentioned performance metrics under the dual threshold scheme. Both these sections contain the respective plots of numerical results as well. Finally, Section V concludes the paper with a brief summary and mentions possible directions to extend the current work. 2. System Configuration and Channel Model In our proposed hybrid FSO/RF-FSO system, the FSO link works in parallel with a mixed RF-FSO link,2as the one depicted in Fig. 1. The source (S) contains an RF transmitter in addition to the regular FSO transmitter. The relay (R) is capable of receiving RF signal and subsequent RF-FSO conversion. At destination (D), there are two distinct optical receivers present; one for receiving data via S-Dlink and another for receiving the data through the relay, i.e. via S-R-Dlink. The channel state information (CSI) about the primary FSO link is sent from Dto Svia a feedback path. If the primary FSO link is obscured due to atmospheric turbulence, Sswitches from FSO to RF transmission and notify Dto switch to the receiver aligned with R. At regular intervals, S transmits a pilot signal through primary FSO link to gauge the turbulence condition. If the link quality meets the desired service level, Dconfirms it by sending a feedback, and the primary FSO link is re-activated. 2For a typical urban cellular backhaul application, the relay may be put on rooftops or it can be an unmanned aerial vehicle (UAV) node [2] placed at a suitable position to avoid the turbulence effect near the transmitter, to optimize energy consumption and to improve quality of service (QoS). Alternatively, the relay node may be just another regular transmitter with added RF-FSO conversion capability, i.e. all the transmitters can have an additional RF transceiver and an RF to FSO converter to realize this backup link concept. Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems Fig. 2. (a) Dual-hop backup RF-FSO link with S-RRF hop and R-DFSO hop. (b) Block diagram of RF to optical converter implemented at R[14]. 2.1 Modeling the Primary FSO Link The probability density function (PDF) of the received instantaneous electrical signal to noise ratio (SNR) of the S-Dlink, ξfso =(ηI)2/N0, follows gamma-gamma distribution [22] fξfso (ξfso)=(αtβt)αt+βt 2 (αt)(βt)ξfsoξfso ξfso ξfso αt+βt 2−1 Kαt−βt⎛ ⎝2   αtβtξfso ξfso ⎞ ⎠;ξfso ≥0,(1) having an average electrical SNR of ξfso =η¯ I2/N0, where ηis the photo-current conversion ratio of the receiver, Iis the received light intensity, N0is the one-sided power spectral density of additive white Gaussian noise (AWGN), and the operator ¯ (·) denotes mean value. The coefficients, αtand βt, are the effective number of smalland large-scale eddies of scattering environment [23], and together define the scintillation index, SI =1/αt+1/βt+1/(αtβt), utilized for measuring the optical intensity variation by atmospheric turbulence. Further, in (1), (·) is the Gamma function and Kv(·) is the modified Bessel function of second kind of order v. The cumulative distribution function (CDF) of electrical SNR, Fξfso (ξfso), is found by using [24, eq. (8.4.23.1)] and integrating the PDF in (1) as Fξfso (ξfso)=(αtβt)αt+βt 2 2(αt)(βt)ξτ1 fso ξfso th 0 ξτ1−1 fso G20 02αtβtξfso ξfso αt−βt 2 βt−αt 2dξfso.(2) where τ1=(αt+βt)/4. Now, utilizing [25, eq. (26)], the above equation may be expressed as Fξfso (ξfso)=(αtβt)αt+βt 2 (αt)(βt)ξτ1 fso ξ βt+αt 4 fso G21 13αtβtξfso ξfso  1−αt+βt 2 αt−βt 2 βt−αt 2−αt+βt 2.(3) 2.2 Modeling the Backup RF-FSO Link During primary link failure, Scommunicates with Dvia intermediate relay, R, using the backup S-R-D link. We have considered an average power scaling (APS) based fixed-gain amplify-and-forward (AF) type relay, as it is suitable for low budget relay based applications [26]. The S-Rlink can be characterized with Rayleigh fading whereas, assuming moderate to strong atmospheric turbulence, the R-DFSO link perturbations may be described by gamma-gamma distribution, as demonstrated in Fig. 2. The RF to FSO conversion is realized with a Mach-Zehnder modulator (MZM). The converter at the relay accepts RF signals from the source antenna and after conversion, the relay’s optical transmitter sends an optical signal to the photo detector at the destination node lens. Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems If the instantaneous received electrical SNR for S-Rand R-Dlinks are denoted with ξrf and ξfso, respectively, the equivalent end-to-end instantaneous electrical SNR for an APS-AF relay is given by ξmix =ξrf ξfso/(gr+ξfso) [27, eq. (6)], where the relay gain is gr=(1+1/N0). As per our model, the S-Rlink experiences Rayleigh fading and the PDF of immediate SNR can be expressed as [28] fξrf (ξrf )=1/ξrf exp −ξrf /ξrf ;ξrf ≥0,(4) where ξrf is the average SNR. The PDF of instantaneous electrical SNR of the R-Dlink follows gamma-gamma distribution as expressed in (1). The CDF of end-to-end electrical SNR, Fξmix (ξmix), is derived by integrating the conditional density over the whole range of ξfso using [24, eq. (2.24.3.1)], resulting in a closed-form expression Fξmix (ξmix)=1−K1exp −ξmix/ξmixξτ1 mixG50 05ωξmix − P,(5) where K1=(αtβt)αt+βt 2(gr/ξrf )τ1 4π(αt)(βt)(ξfso)τ1,ω=α2 tβ2 tgr 16ξrf ξfso ,P∈αt−βt 4,αt−βt+2 4,βt−αt 4,βt−αt+2 4,−τ1, and Gmn pqz (ap) (bq) is the Meijer’s G-function [24, eq. (8.2.1)]. Further, differentiating (5) using [24, eq. (8.2.2.30)], the corresponding PDF is obtained as fξmix (ξmix)=K1exp −ξmix/ξmixξτ1−1 mix G60 16ωξmix −τ1 1−τ1,P +K1/ξmixexp −ξmix/ξmixξτ1 mixG50 05ωξmix − P.(6) 3. Single FSO Threshold Scheme Quality of the high speed primary FSO link (PL) is estimated by checking the signal level at D at frequent intervals. When the received signal level falls below a certain threshold, Dsends this feedback to Sso that an adaptive algorithm can determine the appropriate transmission path for further data transmission. Under the single threshold scheme, Sautomatically switches over to the RF-FSO secondary link (SL) if SNR falls below a certain fixed FSO threshold, ξfso th , or falls into outage if the qualities of either of the RF or FSO link of mixed RF-FSO transmission path also falls below a certain common threshold, ξmix th [11]. 3.1 Link Switching Operation The transmission path variable Xat time Tis selected as per Algorithm 1 described next. 3.2 Outage Probability Analysis An outage occurs when both primary S-Dlink and secondary S-R-Dlink are down as the SNRs do not meet the respective threshold levels. The outage probability is thus P(1) out =Pfso(PL ) out ξfso th Pmix out ξmix th ,(7) where Pfso(PL ) out is the outage probability of the primary FSO S-Dlink and Pmix out is the outage probability of the secondary RF-FSO S-R-Dlink. The thresholds act as minimum SNR values above which the links can guarantee a specific QoS. From (3) it is easy to find that Pfso(PL ) out (ξfso th )=Fξfso (ξfso th ), i.e. Pfso(PL ) out ξfso th =(αtβt)αt+βt 2 (αt)(βt)ξτ1 fso ξfso th βt+αt 4G21 13αtβtξfso th /ξfso 1−αt+βt 2 αt−βt 2 βt−αt 2−αt+βt 2.(8) Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems Algorithm 1: Link Switching for Single FSO Threshold. if ξPL fso ≥ξfso th then XT←FSO else if ξrf ≥ξmix th ∧ξSL fso ≥ξmix th then XT←RF-FSO else Transmission suspended / Outage occurred end if end if Fig. 3. Outage probability with single FSO threshold. Similarly, the outage probability of the secondary RF-FSO link, Pout(ξmix th )=Fξmix (ξmix th ), is derived from (5) as Pmix out (ξmix th )=1−K1exp −ξmix th /ξmixξmix th τ1G50 05ωξmix th  − P.(9) In Fig. 3, the numerical values obtained from the expressions developed for the outage probability are plotted and validated by Monte-Carlo simulations.3It may be seen that for an outage threshold of 5 dB, the outage probability is reduced by an order for an average electrical SNR of 30 dB, when we replace a single FSO link (blue line with square markers) with the proposed hybrid FSO/RF-FSO setup (black line with circle markers). 3.3 Average BER Analysis During non-outage period, any one link can be active at any given instance. Therefore, three distinct scenarios should be considered for calculating the average BER. Extending the basic formulation 3We consider a fixed set of turbulence parameters, (αt,β t)=(5.07,1.53), for the plot. Also, the average SNR per hop and the threshold SNR values are considered to be identical across all the branches, i.e. ¯ ξrf =¯ ξ(SL ) fso =¯ ξmix =¯ ξ(PL ) fso and ξrf th =ξfso(SL ) th =ξmix th =ξfso(PL ) th . These system parameters are used throughout for all the subsequent plots unless otherwise stated. Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems presented in [11], the average BER during the non-outage period can be expressed as BER(1) = Bfso ξfso th  1−P(1) out +Prf out ξmix th Bmix ξmix th  1−Pfso(PL ) out ξfso th +Pfso(SL ) out ξmix th Bmix ξmix th  1−P(1) out1−Prf out ξmix th ,(10) where Bfso and Bmix are the average BER when PL and SL are active, respectively, while Prf out ξmix th  and Pfso(SL ) out are the outage probabilities of the S-Rlink and R-Dlink. The first term in (10) accounts for the situation when ξPL fso >ξ fso th rendering the PL active. The second and third terms denote BER when SL is active and when ξrf <ξ mix th or when ξrf >ξ mix th but ξSL fso <ξ mix th . If we assume on-off keying (OOK) modulation, the conditional error probability is given by P(e|ξ)= (1/2)erfc ξ/2. The average BER when primary FSO link is active Bfso ξfso th =∞ ξfso th PeξPL fso fξfso ξPL fso dξPL fso .(11) under OOK modulation is thus found by substituting (1) in (11) as Bfso ξfso th =K2∞ ξfso th erfc ξPL fso /2ξPL fso τ1−1G20 02αtβtξPL fso /ξPL fso αt−βt 2 βt−αt 2dξPL fso ,(12) where K2=(αtβt)αt+βt 2[4(αt)(βt)ξPL fso τ1]. After some mathematical manipulations, we may express the integral as, Bfso ξfso th =I1−(I2a+I2b), where (see Appendix A for derivations) I1=K2·2τ1−1 π3/2G42 25⎡ ⎣ (αtβt)2 8ξPL fso  1−τ1,1/2−τ1 P⎤ ⎦,(13) I2a=K2(ξfso th )τ1 3 ∞  k=0 (−ξfso th /2)k k!G21 13αtβtξfso th /ξPL fso  1−αt+βt 2−2k αt−βt 2 βt−αt 2−αt+βt 2−2k,(14) and I2b=K2(ξfso th )τ1∞  k=0 (−2ξfso th /3)k k!G21 13αtβtξfso th /ξPL fso  1−αt+βt 2−2k αt−βt 2 βt−αt 2−αt+βt 2−2k.(15) The average BER when RF-FSO link is active Bmix ξmix th =∞ ξmix th Peξmixfξmix (ξmix)dξmix.(16) Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems can be expressed as a sum of four individual integrals, Bmix ξmix th =(I3a+I3b)+(I4a+I4b), where (see Appendix B for derivations) I3a=K1 12 1 ξmix +1 2 −τ1 G61 26⎡ ⎣ ω 1 ξmix +1 2 −τ1−τ1 1−τ1P⎤ ⎦ −K1 12 ∞  k=0 (−1)k k!1 ξmix +1 2 k ξτ1+k mix G61 27ωξmix 1−τ1−k−τ1 1−τ1,P−τ1−k,(17) I3b=K1 41 ξmix +2 3 −τ1 G61 26⎡ ⎣ ω 1 ξmix +2 3 −τ1−τ1 1−τ1P⎤ ⎦ −K1 4 ∞  k=0 (−1)k k!1 ξmix +2 3 k ξτ1+k mix G61 27ωξmix 1−τ1−k−τ1 1−τ1,P−τ1−k,(18) I4a=K1 12ξmix 1 ξmix +1 2 −τ1−1 G51 15⎡ ⎣ ω 1 ξmix +1 2 −τ1 P⎤ ⎦ −K1 12ξmix ∞  k=0 (−1)k k!1 ξmix +1 2 k ξτ1+k+1 mix G51 16ωξmix −τ1−k P−τ1−k−1,(19) and I4b=K1 4ξmix 1 ξmix +2 3 −τ1−1 G51 15⎡ ⎣ ω 1 ξmix +2 3 −τ1 P⎤ ⎦ −K1 4ξmix ∞  k=0 (−1)k k!1 ξmix +2 3 k ξτ1+k+1 mix G51 16ωξmix −τ1−k P−τ1−k−1.(20) We have computed all the terms in (10) except Prf out, which, by definition, is Prf out ξrf th=ξrf th 0 fξrf (ξrf )dξrf .(21) Placing (4) in (21) and utilizing the lower incomplete gamma function [29, eq. (8.350.1)] and [24, eq. (8.4.16.1)], we may write Prf out ξrf th=γ1,ξrf th ξrf =G11 12ξrf th ξrf  1 10.(22) BER performance with single FSO threshold is demonstrated in Fig. 4. The improvement is clearly visible, for example, at a target BER of 10−2and for a threshold value of 5 dB, the proposed setup (black line with square markers) achieves an electrical SNR gain of 15 dB over the single FSO link (blue line with diamond markers). Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems may be solved utilizing [24, eq. (8.4.14.2)] and [25, eq. (21)] to obtain (13). On the other hand, with a simple change of variable, x=ξfso/2, the second integral, I2=K2ξfso th 0 erfc ξPL fso /2ξPL fso τ1−1G20 02αtβtξPL fso /ξPL fso αt−βt 2 βt−αt 2dξPL fso ,(36) may be expressed with new limits as I2=K22τ1+1√ξfso th /2 0 x2τ1−1erfc (x)G20 02αtβtx2/ξPL fso αt−βt 2 βt−αt 2dx.(37) Again, using the exponential series expansion of erfc(·), erfc(x)=1 6exp(−x2)+1 2exp −4 3x2,we may write, I2≈I2a+I2b, where I2a=K22τ 1/3√ξfso th /2 0 exp(−x2)x2τ1−1G20 02αtβtx2/ξPL fso αt−βt 2 βt−αt 2dx,(38) and I2b=K22τ 1√ξfso th /2 0 exp −4 3x2 x2τ1−1G20 02αtβtx2/ξPL fso αt−βt 2 βt−αt 2dx.(39) Using [29, eq. (1.211.3)] and [25, eq. (27)] in the above equations, we get the simplified expressions as provided in (14) and (15) from (38) and (39), respectively. Appendix B. Derivation of (17)–(20) We begin with breaking the term Bmix ξmix th into two separate integrals, Bmix ξmix th =∞ ξmix th Peξmixfξmix (ξmix)dξmix =I3+I4,(40) where I3=(K1/2) ∞ ξmix th erfc ξmix/2ξτ1−1 mix exp −ξmix/ξmixG60 16ωξmix −τ1 1−τ1,Pdξmix,(41) and I4=K1/(2ξmix)∞ ξmix th erfc ξmix/2ξτ1 mix exp −ξmix/ξmixG50 05ωξmix − Pdξmix.(42) Now, using the same exponential series expansion of erfc(·) as used in Appendix A, we may write I3as I3≈(K1/2) ∞ ξmix th %(1/6) exp −ξ2 mix/2+(1/2) exp −4ξ2 mix/3&ξτ1−1 mix ×exp −ξmix/ξmixG60 16ωξmix −τ1 1−τ1,Pdξmix =I3a+I3b,(43) where I3a=(K1/12) ∞ ξmix th ξτ1−1 mix exp '−ξmix 1/ξmix +1/2(G60 16ωξmix −τ1 1−τ1,Pdξmix,(44) and I3b=(K1/4) ∞ ξmix th ξτ1−1 mix exp '−ξmix 1/ξmix +2/3(G60 16ωξmix −τ1 1−τ1,Pdξmix.(45) Now, changing the limit of the above equation and after some mathematical manipulations using [24, eq. (2.24.3.1)] and [25, eq. (26)], we obtain the solutions given in (17) and (18). Following the Vol. 10, No. 3, June 2018 7904417 IEEE Photonics Journal Performance Analysis of Hybrid FSO Systems same approach for I4in (42), we end up with two integrals, i.e. I4=I4a+I4b, which results in the expressions as given in (19) and (20), respectively. Appendix C. Derivation of (25)–(26) The two components of CPL fso ξfso th are I5and I6, and the first component I5=K1 ln (2) ∞ 0 ln 1+ξPL fso ξPL fso τ1−1G20 02(αtβt)ξPL fso /ξPL fso βt−αt 2−βt−αt 2dξPL fso ,(46) can be deduced to (25) using [25, eq. (11)] and [24, eq. (2.24.1.1)]. On the other hand, the second component I6=K1 ln (2) ξfso th 0 ln 1+ξPL fso ξPL fso τ1−1G20 02(αtβt)ξPL fso /ξPL fso βt−αt 2−βt−αt 2dξPL fso ,(47) can be expressed as in (26) using the Taylor series expansion [24, eq. (45)], ln (1+x)=)∞ n=1 (−1)n+1xn/n!. Appendix D. Derivation of (28)–(31) The Taylor series expansion, ln (1+x)=)∞ n=1(−1)n+1xn/n!, can be used to obtain Cmix ξmix th =K1 ln (2) ∞  n=1 (−1)n+1 n∞ ξmix th ξn+τ1−1 mix exp −ξmix ξmix G60 16ωξmix −τ1 1−τ1Pdξmix +K1 ln (2)ξmix ∞  n=1 (−1)n+1 n∞ ξmix th ξn+τ1 mix exp −ξmix ξmix G50 05!ωξmixP"dξmix,(48) from (27). Now, expressing the integral interval as a difference, the above equation can be expressed as Cmix ξmix th =(C1−C2)+(C3−C4). The first integral, C1=K1 ln (2) ∞  n=1 (−1)n+1 n∞ 0 ξn+τ1−1 mix exp −ξmix ξmix G60 16ωξmix −τ1 1−τ1Pdξmix.(49) is solved with the help of [39, eq. (2.24.3.1)] to obtain (28). Next, using the series, exp (−ξmix/ξmix)= )∞ n=1(−1)k(ξmix/ξmix)k/k! [29, eq. (1.211.3)], the second term, C2=K1 ln (2) ∞  n=1 (−1)n+1 nξmix th 0 ξn+τ1−1 mix exp −ξmix ξmix G60 16ωξmix −τ1 1−τ1Pdξmix.(50) may be resolved into (29). 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