Unified Performance Analysis of Mixed Line of Sight RF-FSO Fixed Gain Dual-Hop Transmission Systems

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1 Unified Pefomance Analysis of Mixed Line of Sight RF-FSO Fixed Gain Dual-Hop Tansmission Systems Emna Zedini, Iman Shafique Ansai, and Mohamed-Slim Alouini Compute, Electical, and Mathematical Sciences and Engineeing CEMSE) Division, King Abdullah Univesity of Science and Technology KAUST), Thuwal, Makkah Povince, Saudi Aabia. s: {emna.zedini, iman.ansai, Abstact In this wok, we cay out a unified pefomance analysis of a dual-hop fixed gain elay system ove asymmetic links composed of both adio-fequency RF) and unified feespace optics FSO) unde the effect of pointing eos. The RF link is modeled by the Nakagami-m fading channel and the FSO link by the Gamma-Gamma fading channel subject to both types of detection techniques i.e. heteodyne detection and intensity modulation with diect detection IM/DD)). In paticula, we deive new unified closed-fom expessions fo the cumulative distibution function, the pobability density function, the moment geneation function, and the moments of the end-toend signal-to-noise atio of these systems in tems of the Meije s G function. Based on these fomulas, we offe exact closed-fom expessions fo the outage pobability, the highe-ode amount of fading, and the aveage bit-eo ate of a vaiety of binay modulations in tems of the Meije s G function. Futhe, an exact closed-fom expession fo the end-to-end egodic capacity fo the Nakagami-m-unified FSO elay links is deived in tems of the bivaiate G function. All the given esults ae veified via Compute-based Monte-Calo simulations. Index Tems Asymmetic dual-hop elay system, Fee-space optical FSO) communications, mixed adio fequency RF)/FSO systems, Nakagami-m fading, atmospheic tubulence, pointing eos, outage pobability OP), bit-eo ate BER), amount of fading AF), egodic capacity, Meije s G function. I. INTRODUCTION The demand fo highe data ate aises the question of spectum availability. In many wieless communications systems, data capacity has been enhanced by inceasing the spectal efficiency by means of signal pocessing techniques and advanced modulation schemes. Howeve, eaching ates of Gbps o moe is quite challenging due to the majo limitation facto, scace spectum esouces. Fo futue communication systems, moe spectal esouces ae mandatoy. Of the many popula solutions, fee-space optical FSO) communication systems have gained significant eseach attention as effective means of tansfeing data at high ates ove shot distances mainly because they can povide not only highe capacity but also wide bandwidth elative to the taditional adio fequency RF) tansmission. Additionally, FSO links offe This wok was suppoted in pat by a gant fom King Abdulaziz City of Sciences and Technology KACST). a high secuity level, a license-fee spectum, and hence a cost effective solution compaed to the RF communication systems. These advantages of FSO communication systems potentially help solving the issues that the RF communication systems face due to the expensive and scace spectum 6. Howeve, a numbe of technical challenges need to be ovecome o bette undestood. In fact, contay to RF links, the majo sevee limiting facto in FSO communications is its high vulneability to the atmospheic tubulence conditions 2. Fog, snow, and ain may cause a sevee degadation in the oveall pefomance. Anothe dominant facto affecting the eliability of FSO channels is building sway caused by themal expansion, wind loads, and weak eathquakes 7, 8. This phenomenon leads to a misalignment between the tansmitte and the eceive defined as pointing eo, which is a seious poblem that degades the channel pefomance, 9. It is wothy to mention that the main type of detection technique in FSO systems is IM/DD. Coheent modulation is ecently employed as an altenative detection appoach. Despite of the complexity of implementing coheent eceives elative to IM/DD systems, heteodyne detection that belongs to coheent mode offes bette pefomance in ovecoming themal noise effects,. Relaying technique has gained an enomous inteest due to its advantages including not only wide and enegy-efficient coveage but also inceased capacity in the wieless communication systems. Recently, seveal effots have been conducted to investigate the elay system pefomance unde vaious fading conditions 2 5. Moeove, liteatue egading the asymmetic elay netwoks based on both RF as well as FSO chaacteistics includes 6 9. In 6, the pefomance analysis of asymmetic dual-hop RF-FSO elay system is pesented. In 8, the pefomance analysis of a dual-hop vaiable gain elay RF-unified FSO tansmission system subject to pointing eos is pesented. 7 investigates the pefomance of a dual-banch fixed gain elay RF-FSO tansmission system unde the effect of pointing eos and subject to both types of detection techniques i.e. IM/DD as well as heteodyne detection). Howeve, the esults pesented in 7 wee deived unde the assumption of a non light of sight NLOS) Rayleigh fading in the RF link, and as such does not cove the case

2 when a line of sight LOS) component is pesent between the souce and elay. Since the Rician and Nakagami-m fading models ae moe appopiate fo popagation envionments in LOS communications 2, in this wok, we extend the model pesented in 7 to study fo the fist time the pefomance of asymmetic LOS RF-FSO dual-hop fixed gain elay tansmission systems with mixed Nakagami-m RF-unified FSO links, which is a non-tivial contibution. Moe specifically, the FSO link is assumed to be opeating ove unified Gamma-Gamma fading envionment 3, 4 unde the effect of pointing eos, and the RF link ove Nakagami-m fading that includes the Rayleigh fading as a special case. In this context and in ou pefomance analysis study, we used the finite seies epesentation of the incomplete Gamma function along with the binomial expansion to deive unified exact closed-fom expessions fo the cumulative distibution function CDF), the pobability density function PDF), the moment geneating function MGF), the moments, the highe-ode amount of fading AF), the outage pobability OP), the bit-eo ate BER) of a binay modulation schemes in tems of the Meije s G function. Additionally, we pesent the egodic capacity in tems of the bivaiate G function. Futhe, we intoduce the asymptotic expessions fo all the expessions deived ealie in tems of the Meije s G function at high signal-to-noise atio SNR) egime in tems of simple elementay functions by utilizing Meije s G function expansion. The emainde of the pape is oganized as follows. In Section II, the system model and channel model fo fixed Nakagami-m-unified FSO elay scheme is intoduced. Exact closed-fom esults to chaacteize Nakagami-m-unified FSO elay including the PDF, the CDF, the MGF, the moments, the AF, the OP, the BER, and the egodic capacity followed by the asymptotic expessions ae pesented in Sections III and IV. The deived analytical expessions in the pevious sections ae numeically evaluated, illustated, and intepeted in Section V. Finally, we eview ou main esults and we daw some conclusions in Section VI. II. CHANNEL AND SYSTEM MODELS We conside an asymmetic dual-hop elaying system whee the souce node S and the destination node D ae communicating though an intemediate elay node R. The RF point-topoint popagation link i.e. S-R link) is assumed to follow a Nakagami-m distibution. On the othe hand, we assume the second FSO link i.e. R-D link) expeiences unified Gamma- Gamma fading with pointing eo impaiments. In the fixed gain elaying scheme, the end-to-end SNR can be expessed as 2 γ = γ γ 2 γ 2 + C, ) whee γ denotes the SNR of the S-R hop, γ 2 epesents the SNR of the R-D hop, and C stands fo a fixed elay gain 2, 6. In this pape, we assume that the RF S-R link expeiences Nakagami-m fading distibution with the PDF in 2 m ) m γ m f γ γ ) = Γm) exp m ) γ, 2) whee m is the Nakagami-m fading paamete m 2 ), Γ ) is the Gamma function as defined in 2, Eq.8.3), and epesents the aveage fading powe, i.e. = E γ γ with E denoting the expectation opeato. It is impotant to note that the PDF in 2) includes the Rayleigh distibution m = ) as a special case. The FSO R-D link is assumed to follow a unified Gamma Gamma fading distibution with pointing eo impaiments fo which the PDF of the SNR is given by 22 f γ2 γ 2 ) = ξ 2 γ 2 Γα)Γβ) G3,,3 h α β γ2 µ ) ξ 2 + ξ 2, α, β 3) whee is the paamete specifying the detection technique type i.e. = accounts fo heteodyne detection and = 2 epesents IM/DD), h = ξ2 ξ 2 +, ξ denotes the atio between the equivalent beam adius at the eceive and the pointing eo displacement standad deviation jitte) at the eceive, 23 i.e. fo negligible pointing eos, ξ ), α and β ae the the fading/scintillation paametes elated to the atmospheic tubulence conditions with small values of these two paametes pointing to sevee fading conditions 4, 9, G,, ) is the Meije s G function as defined in 2, Eq.9.3), and µ standing fo the aveage electical SNR. Moe specifically, fo µ, when =, µ = µ heteodyne = Eγ 2 = γ 2 and when = 2, µ 2 = µ IM/DD = γ 2 α β ξ 2 ξ 2 + 2)/α + )β + )ξ 2 + ) III. STATISTICAL CHARACTERISTICS A. Cumulative Distibution Function The CDF of γ is given by F γ γ) = P µ γ γ 2 γ 2 + C < γ, 4) which can be expessed as γ γ 2 F γ γ) = P γ 2 + C < γ γ 2 f γ2 γ 2 ) dγ 2 ξ 2 = Γ m, m γ γ ) 2 + C) Γα) Γβ) Γm) γ 2 γ 2 ) G 3, γ2,3 h α β ξ 2 + ξ 2 dγ, α, β 2. 5) To the best of the authos knowledge, the solution to the integal in 5) is not available in exact closed-fom no in tems of the extended genealized bivaiate Meije s G function EGBMGF) because of the shift in the incomplete Gamma function. Theefoe, we utilize the finite seies epesentation of the incomplete Gamma function in 2, Eq ) to ewite Γ ) m, m γ γ2+c) γ 2 ) m k= k! as m γ ) k + C γ 2 ) k. m )! exp m γ ) exp m C γ γ 2 Since the summation is uppe limited by m, ou esults ae limited to the case of Nakagami-m with intege values of m. Futhe using the binomial expansion in 2, ) k Eq..), + C k γ 2 can be expessed as k j= j) ) j. C γ 2 Now, along with the above modifications, we apply 24,,

3 Eq ) and some mathematical manipulations to get the CDF of γ as F γ γ) = A exp m γ G 3+, γ,3+ µ ) m k κ j! k j)! m γ, 6) whee A = α+β 2 ξ 2 2π) Γα)Γβ), B = h α β), 2 ξ κ = 2 +,..., ξ2 + compises tems, and ξ = 2,..., ξ2 +, α,..., α+, β,..., β+, j compises 3 + tems. Fo m =, as a special case, the CDF in 6) is in ageement with the CDF of the hybid Rayleigh/FSO fixed gain dual hop tansmission systems with pointing eos pesented in 7, Eq.2). The aguments of the Meije s G function in 6) can be inveted using 25, Eq.6.2.2). Then, by applying 22, Eq.26), the asymptotic expession of the CDF at high SNR can be deived in tems of basic elementay functions as F γ γ) µ A exp 3+ i= µ γ m γ ) m k m γ j! k j)! ) κ2,i 3+ l=;l i Γ,l,i ) l= Γκ,,l,i ) 7) whee κ u,v stands fo the v th -tem of κ u. This asymptotic expession fo the CDF in 7) can be futhe expessed via only one dominant tem, j, that epesents the 3 + ) th -tem in. B. Pobability Density Function The PDF of γ can be obtained by diffeentiating 6) with espect to γ. Theefoe, utilizing the poduct ule then applying 24, Eq ), we get afte some algebaic manipulations the PDF in exact closed-fom in tems of the Meije s G functions as f γ γ) =A exp m γ ) m k m γ j! k j)! { m k j ) G 3+, γ,3+ γ µ κ γ γ G3+, +,3+2 µ, κ }. 8), Fo m =, as a special case, the PDF in 8) is in a pefect ageement with the PDF in 7, Eq.3). C. Moment Geneating Function It is well known that the MGF is defined as M γ s) = Ee γs. Using integation by pats, the MGF can be expessed in tems of CDF as M γ s) = s e γs F γ γ) dγ. 9) Placing 6) into 9) and utilizing 24, Eq ), the MGF of γ can be pesented as m k m M γ s) = s A s + m ) j k j! k j)! G 3+, +,3+ µ s + m) j k, κ. ) When m =, as a special case, the MGF in ) can be easily shown to be equal to 7, Eq.6). Simila to the CDF, the asymptotic expansion of the MGF high SNR can be detemined as m M γ s) s A µ 3+ i= 3+ k µ s + m) m j! k j)! ) κ2,i s + m ) j k l=;l i Γ,l,i )Γ +,i j + k) + l=2 Γκ,l,i ), ) and can be futhe expessed via only the dominant tem, j, which is the 3 + ) th -tem in. D. Moments The moments specified as Eγ n can be deived in tems of the complementay CDF CCDF) Fγ c γ) = F γ γ), via integation by pats, as Eγ n = n γ n F c γ γ) dγ. 2) Placing 6) into 2) and applying 24, Eq ), the moments educe to ) n m k Eγ n =n A m j! k j)! G 3+, B C +,3+ k + j n, κ. 3) µ Fo m =, as a special case, the moments in 3) can be easily shown to agee with 7, Eq.8). It is impotant to mention that the moments ae exploited to deive the expessions of the highe-ode amount of fading in the next section. IV. APPLICATIONS TO THE PERFORMANCE OF ASYMMETRIC NAKAGAMI-m-UNIFIED FSO RELAY TRANSMISSION SYSTEMS WITH FIXED GAIN RELAY A. Outage Pobability The OP is an impotant measue fo the pefomance of a wieless communication system. An outage of the communication system is encounteed when the instantaneous output SNR γ falls below a pedetemined theshold γ th. Setting γ = γ th in 6), we obtain the OP as P out γ th ) = F γ γ th ). 4)

4 B. Highe-Ode Amount of Fading Fo the instantaneous SNR γ, the n th -ode amount of fading is defined as 26 AF n) γ = Eγn. 5) Eγ n Substituting 3) in 5) yields to the n th -ode AF. C. Aveage BER The aveage BER fo a vaiety of binay modulations is intoduced as 27, Eq.2) P b = qp 2 Γp) exp q γ) γ p F γ γ) dγ, 6) whee p and q ae paametes that change fo diffeent modulation schemes 28. Replacing F γ γ) by its expession in 6) and utilizing 24, Eq ) with some algebaic manipulations, we obtain the BER as P b = 2 A qp 2Γp) G 3+, +,3+ m k µ q + m) m q + m ) j k p j! k j)! p k j, κ. 7) Fo m =, as a special case, we get the BER of the mixed Rayleigh/FSO fixed gain dual hop tansmission systems with pointing eos given in 7, Eq.). At high SNR and simila to the CDF, the BER can be expessed asymptotically as P b µ 2 A qp m k m 2Γp) j! k j)! 3+ ) κ2,i µ q + m) i= 3+ q + m ) j k p l=;l i Γ,l,i )Γ,i + p + k j) + l=2 Γκ,l,i ), 8) and can be futhe expessed via only the dominant tem j. D. Egodic Capacity The egodic capacity defined as C = Elog 2 + γ) can be witten in tems of the CCDF of γ as 29, Eq.5) C = / ln2) F c γ γ)/ + γ) dγ. 9) Using 3 to epesent + γ) as G,, γ, and utilizing the integal identity 27, Eq.2), we obtain the egodic capacity in tems of the EGBMGF as C = A ln2) m m G,:,:3+,,:,:,3+ k j! k j)! k j + κ m, B C µ. 2) An efficient Mathematica implementation of the EGBMGF is given in 27, Table II. Fo m =, as a special case, the egodic capacity in 2) is in ageement with 7, Eq.3). V. NUMERICAL RESULTS In this section, we pesent simulation and numeical esults fo diffeent pefomance metics of asymmetic dualhop Nakagami-m-unified FSO elay tansmission system with fixed gain elay, as an illustation of the analytical expessions given in the pevious sections. The FSO link i.e. the R-D link) is modeled as a unified Gamma-Gamma fading channel fo weak α = 2.92 and β = 2.5), modeate α = and β =.822), and stong α = 2.64 and β =.342) tubulent FSO channel conditions. In this section, the aveage SNR between the elay and the destination R-D link) is set such that γ 2 = db except fo the figues showing the asymptotic esults whee γ 2 is vaying. Fo the fixed gain scheme, the elay is set such us C =. The outage pobability pefomance fo both heteodyne and IM/DD detection techniques vesus the nomalized aveage fading powe of the RF link i.e S-R link) is pesented in Fig.. The effect of pointing eo is fixed at ξ =.. We can see fom Fig. that the analytical esults povide a pefect match to the simulation esults pesented in this pape. It can also be obseved that the heteodyne detection technique = ) povides bette pefomance than the IM/DD technique = 2). Moeove, it can be shown that the pefomance deteioates as the atmospheic tubulence conditions get sevee i.e. the highe the values of α and β, the lowe will be the OP) and vice vesa. Outage Pobability OP), Pout α=2.92 and β=2.5 Weak Tubulence) α=2.296 and β=.822 Modeate Tubulence) α=2.64 and β=.342 Stong Tubulence) =; Heteodyne Detection =2; IM/DD Aveage fading powe of the RF S-R) link db), Fig.. OP showing the pefomance of both the detection techniques heteodyne and IM/DD) unde stong, modeate, and weak tubulent FSO channels fo stong pointing eo ξ =.. Fig. 2 pesents the OP unde both the detection techniques heteodyne and IM/DD) fo stong pointing eo ξ =. along with the asymptotic esults in high SNR egime. It can be shown that at high SNR, the asymptotic expession utilizing the Meije s G function expansion and all the tems ae consideed in the summation in 7) conveges quite fast to the exact esult poving this asymptotic expession to be tight enough. Moeove, if we select the appopiate single dominant tem, we get also a convegence to the exact esult though elatively slowe. In Fig. 3, we illustate the OP unde IM/DD technique with vaying effects of pointing eo ξ = and 6.7). As expected, the OP inceases as the pointing eo gets sevee i.e. the lowe the values of ξ, the highe will be the OP). Additionally, it can

5 Outage Pobability OP), Pout Aveage electical SNR of the FSO R-D) link db), µ =2; IM/DD =; Heteodyne Detection Asymptotic Result All Tems) Asymptotic Result Single Dominant Tem) α=2.64 and β=.342 Stong Tubulence) Fig. 2. OP showing the pefomance of both the detection techniques heteodyne and IM/DD) unde stong tubulence conditions fo stong pointing eo ξ =. along with the asymptotic esults in high SNR egime fo = 2 db. Aveage Bit Eo Rate BER), Pb α=2.92 and β=2.5 Weak Tubulence) α=2.296 and β=.822 Modeate Tubulence) α=2.64 and β=.342 Stong Tubulence) =; Heteodyne Detection Aveage fading powe of the RF S-R) link db), =2; IM/DD Fig. 4. Aveage BER of BDPSK binay modulation scheme showing the pefomance of both the detection techniques heteodyne and IM/DD) unde stong, modeate, and weak tubulent FSO channels fo stong pointing eo ξ =.. be obseved that fo lowe effect of the atmospheic tubulence, the espective pefomance gets bette. Outage Pobability OP), Pout α=2.92 andβ=2.5 Weak Tubulence) α=2.296 and β=.822 Modeate Tubulence) α=2.64 and β=.342 Stong Tubulence) ξ=6.7 Aveage fading powe of the RF S-R) link db), Fig. 3. OP showing the pefomance of IM/DD technique unde stong, modeate, and weak tubulent FSO channels with vaying effects of pointing eo. Fig. 4 demonstates the aveage BER pefomance fo diffeential binay phase shift keying DBPSK) binay modulation scheme whee p = and q = ae the paametes of DBPSK, fo both types of detection techniques i.e. IM/DD and heteodyne) with fixed effect of the pointing eo ξ =.). As clealy seen in the figue, the analytical esults and the simulation esults coincide. We can also see fom this figue that the heteodyne detection technique outpefoms the IM/DD technique. Moeove, it can be obseved that the pefomance impoves as the effect of the atmospheic tubulence dops. Fig. 5 pesents the aveage BER fo DBPSK binay modulation scheme unde IM/DD technique fo vaying effects of the pointing eo ξ = and 6.7) along with the asymptotic esults in high SNR egime. It can be obseved that at high SNR, the asymptotic expession utilizing the Meije s G function expansion and consideing all the tems in the summation in 8) conveges quite fast to the exact esult poving the tightness of this asymptotic appoximation. Additionally, when we select the elevant single dominant tem of 8) deived via Meije s G function expansion, a slowe convegence is clealy obseved. ξ= Aveage Bit Eo Rate BER), Pb Aveage electical SNR of the FSO R-D) link db), µ =2; IM/DD =; Heteodyne Detection Asymptotic Result All Tems) Asymptotic Result Single Dominant Tem) α=2.64 and β=.342 Stong Tubulence) Fig. 5. Aveage BER of BDPSK binay modulation scheme showing the pefomance of both the detection techniques heteodyne and IM/DD) unde stong tubulence conditions fo stong pointing eo ξ =. along with the asymptotic esults in high SNR egime fo = 2 db. Fig. 6 depicts the aveage BER fo DBPSK binay modulation scheme unde IM/DD technique fo vaying effects of the pointing eo ξ = and 6.7). Expectedly, as the pointing eo deceases ξ ), the espective system pefomance gets bette. Aveage Bit Eo Rate BER), Pb α=2.92 and β=2.5 Weak Tubulence) α=2.296 and β=.822 Modeate Tubulence) α=2.64 and β=.342 Stong Tubulence) ξ=6.7 Aveage fading powe of the RF S-R) link db), Fig. 6. Aveage BER of BDPSK binay modulation scheme showing the pefomance of IM/DD technique unde stong, modeate, and weak tubulent FSO channels with vaying effects of pointing eo. In Fig. 7, the egodic capacity unde both heteodyne ξ=

6 and IM/DD detection techniques fo vaying effects of the pointing eo ξ = and 6.7) fo stong tubulence conditions is pesented. It can be obseved that heteodyne detection pefoms much bette than the IM/DD technique. Additionally, it can be shown that as the pointing eo gets sevee, the egodic capacity deceases i.e. the highe values of ξ, the highe will be the egodic capacity). Egodic Capacity, C bits/sec/hz) ξ=. ξ=6.7 α=2.64 and β=.342 Stong Tubulence) =2; IM/DD Aveage fading powe of the RF S-R) link db), =; Heteodyne Detection Fig. 7. Egodic capacity esults showing the pefomance of both heteodyne and IM/DD techniques unde stong tubulence conditions fo vaying pointing eos. VI. CONCLUSION In this wok, we, fo the fist time, povided unified exact closed-fom expessions fo the PDF, the CDF, the MGF, and the moments of a dual-hop fixed gain elay system ove the asymmetic links composed of both Nakagami-m and unified Gamma-Gamma fading envionments. 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