On the Performance of Millimeter Wave-based RF-FSO Multi-hop and Mesh Networks

Transcription

1 On the Performance of Mllmeter Wave-based RF-FSO Mult-hop and Mesh Networks Behrooz Makk, Tommy Svensson, Senor Member, IEEE, Mate Brandt-Pearce Senor Member, IEEE, and Mohamed-Slm Aloun, Fellow, IEEE arxv:73.998v [cs.it] 7 Mar 7 Chalmers Unversty of Technology, Gothenburg, Sweden, {behrooz.makk, tommy.svensson}@chalmers.se Unversty of Vrgna, Charlottesvlle, VA, USA, mb-p@vrgna.edu Kng Abdullah Unversty of Scence and Technology KAUST, Thuwal, Saud Araba, slm.aloun@kaust.edu.sa Abstract Ths paper studes the performance of mult-hop and mesh networks composed of mllmeter wave MMW-based rado frequency RF and free-space optcal FSO lnks. The results are obtaned n cases wth and wthout hybrd automatc repeat request HARQ. Takng the MMW characterstcs of the RF lnks nto account, we derve closed-form expressons for the networks outage probablty and ergodc achevable rates. We also evaluate the effect of varous parameters such as power amplfers effcency, number of antennas as well as dfferent coherence tmes of the RF and the FSO lnks on the system performance. Fnally, we determne the mnmum number of the transmt antennas n the RF lnk such that the same rate s supported n the RF- and the FSO-based hops. The results show the effcency of the RF-FSO setups n dfferent condtons. Moreover, HARQ can effectvely mprove the outage probablty/energy effcency, and compensate for the effect of hardware mparments n RF-FSO networks. For common parameter settngs of the RF-FSO dual-hop networks, outage probablty of 4 and code rate of 3 nats-per-channel-use, the mplementaton of HARQ wth a maxmum of and 3 retransmssons reduces the requred power, compared to cases wth open-loop communcaton, by 3 and 7 db, respectvely. Part of ths work has been accepted for presentaton at the IEEE WCNC 7.

2 I. INTRODUCTION The next generaton of wreless networks must provde coverage for everyone everywhere at any tme. To address these demands, a combnaton of dfferent technques s consdered, among whch free-space optcal FSO communcaton s very promsng [] [3]. Coherent FSO systems, made nexpensve by the large fberoptc market, provde fber-lke data rates through the atmosphere usng lasers. Thus, FSO can be used for a wde range of applcatons such as last-mle access, fber back-up, back-haulng and mult-hop networks. In the rado frequency RF doman, on the other hand, mllmeter wave MMW communcaton has emerged as a key enabler to obtan suffcently large bandwdths so that t s possble to acheve data rates comparable to those n the FSO lnks. In ths perspectve, the combnaton of FSO and MMWbased RF lnks s consdered as a powerful canddate for hgh-rate relable communcaton. The RF-FSO related lterature can be dvded nto two groups. The frst group conssts of papers on sngle-hop setups where the lnk relablty s mproved va the jont mplementaton of RF and FSO systems. Here, ether the RF and the FSO lnks are consdered as separate lnks and the RF lnk acts as a backup when the FSO lnk s down, e.g., [3] [], or the lnks are combned to mprove the system performance [] [7]. Also, the mplementaton of hybrd automatc repeat request HARQ n RF-FSO lnks has been consdered n [7] [9]. The second group conssts of the papers analyzng the performance of mult-hop RF-FSO systems. For nstance, [], [] study RF-FSO based relayng schemes wth an RF source-relay lnk and an FSO or RF-FSO relay-destnaton lnk. Also, consderng Raylegh fadng condtons for the RF lnk and amplfy-and-forward relayng technque, [], [3] derve the end-to-end error probablty of the RF-FSO based setups and compare the system performance wth RFbased relay networks, respectvely. The mpact of pontng errors on the performance of dual-hop RF-FSO systems s studed n [4] [6]. Fnally, [7] analyzes decode-and-forward technques n multuser relay networks usng RF-FSO. In ths paper, we study the data transmsson effcency of mult-hop and mesh RF-FSO systems from an nformaton theoretc pont of vew. Consderng the MMW characterstcs of the RF lnks and heterodyne detecton technque n the FSO lnks, we derve closed-form expressons for the system outage probablty Lemmas -6 and ergodc achevable rates Corollary. Our results are obtaned for the decode-and-forward relayng approach n dfferent cases wth and wthout HARQ. Specfcally, we show the HARQ as an effectve technque to compensate for

3 3 the non-deal propertes of the RF-FSO system and mprove the network relablty. We present mappngs between the performance of RF- and FSO-based hops as well as between the HARQbased and open-loop systems, n the sense that wth approprate parameter settngs the same outage probablty s acheved n these setups Corollary, Lemma 6. Also, we determne the mnmum number of transmt antennas n the RF lnks such that the same rate s supported by the RF- and the FSO-based hops Corollary. Fnally, we analyze the effect of varous parameters such as the power amplfers PAs effcency, dfferent coherence tmes of the RF and FSO lnks and number of transmt antennas on the performance of mult-hop and mesh networks. In contrast to [] [9], we consder mult-hop and mesh networks. Moreover, our analytcal/numercal results on the outage probablty, ergodc achevable rate and the requred number of antennas n HARQ-based RF-FSO systems as well as our dscussons on the effect of mperfect PAs/HARQ have not been presented before. The dfferences n the problem formulaton and the channel model makes our analytcal/numercal results and conclusons completely dfferent from the ones n the lterature, e.g., [] [7]. The numercal and the analytcal results show that: Dependng on the codewords length, there are dfferent methods for the analytcal performance evaluaton of the RF-FSO systems Lemmas -6. There are mappngs between the performance of RF- and FSO-based hops, n the sense that wth proper scalng of the channel parameters the same outage probablty s acheved n these hops Corollary. Thus, the performance of RF-FSO based mult-hop/mesh networks can be mapped to ones usng only the RF- or the FSO-based communcaton. Whle the network outage probablty s almost nsenstve to the number of RF-based transmt antennas when ths number s large, the ergodc rate of the mult-hop network s remarkably affected by the number of antennas. The requred number of RF-based antennas to guarantee the same rate as n the FSO-based hops ncreases sgnfcantly wth the sgnal-to-nose rato SNR and, at hgh SNRs, the ergodc rate scales wth the SNR almost lnearly. At low SNRs, the same outage probablty s acheved n HARQ-based RF hops wth N transmt antennas, a maxmum of M retransmssons and C channel realzatons per retransmsson as wth an open-loop system wth MNC transmt antennas and sngle channel realzaton per codeword transmsson Lemma 6. The PAs effcency affects the network outage probablty/ergodc rate consderably. How-

4 4 ever, the HARQ protocols can effectvely compensate for the effect of hardware mparments. Fnally, the HARQ mproves the outage probablty/energy effcency sgnfcantly. For nstance, consder common parameter settngs of the RF-FSO dual-hop networks, outage probablty of 4 and code rate of 3 nats-per-channel-use npcu. Then, compared to cases wth open-loop communcaton, the mplementaton of HARQ wth a maxmum of and 3 retransmssons reduces the requred power by 3 and 7 db, respectvely. II. SYSTEM MODEL In ths secton, we present the system model for a mult-hop setup wth a sngle route from the source to the destnaton. As demonstrated n Secton III.C, the results of the mult-hop networks can be extended to the ones n mesh networks wth multple non-overlappng routes from the source to the destnaton. A. Channel Model Consder a T total -hop RF-FSO system, wth T RF-based hops and T = T total T FSObased hops. As seen n the followng, the outage probablty and the ergodc achevable rate are ndependent of the order of the hops. Thus, we do not need to specfy the order of the RF- and FSO-based hops. The -th, =,...,T, RF-based hop uses a multple-nput-sngleoutput MISO setup wth N transmt antennas. Such a setup s of nterest n, e.g., sde-to-sde communcaton between buldngs/lamp posts [8], as well as n wreless backhaul lnks where the trend s to ntroduce multple antennas and thereby acheve multple parallel streams, e.g., [9]. We defne the channel gans as g j. = h j, =,...,T,j =,...,N, where h j s the complex fadng coeffcents of the channel between the j -th antenna n the -th hop and ts correspondng receve antenna. Whle the modelng of the MMW-based lnks s well known for lne-of-sght wreless backhaul lnks, t s stll an ongong research topc for non-lne-of-sght condtons [3]. Partcularly, dfferent measurement setups have emphaszed the near-lne-of-sght propagaton and the nondeal hardware as two key features of such lnks. Here, we present the analytcal results for the quas-statc Rcan channel model, wth successve ndependent realzatons, whch s an approprate model for near lne-of-sght condtons and has been well establshed for dfferent MMW-based applcatons, e.g., [3] [33].

5 5 Let us denote the probablty densty functon PDF and the cumulatve dstrbuton functon CDF of a random varable X by f X and F X, respectvely. Wth a Rcan model, the channel gan g j,,j, follows the PDF f g j x = K +e K e K +x Ω I Ω K K +x,,j, Ω where K and Ω denote the fadng parameters n the -th hop and I n s the n-th order modfed Bessel functon of the frst knd. Also, defnng the sum channel gan G = N f G x = K +e K N Ω K +x K N Ω N e K +x Ω I N j = gj K K +N x Ω, we have,. Fnally, to take the non-deal hardware nto account, we consder the state-of-the-art model for the PA effcency where the output power at each antenna of the -th hop s determned accordng to [34, eq..4], [35, eq. 3], [36, eq. 3], [37, eq. ] ϑ P P ǫ P cons = ǫ P max P = ϑ P cons Here, P,P max P max ϑ,. 3 and P cons,, are the output, the maxmum output and the consumed power n each antenna of the -th hop, respectvely, ǫ [,] denotes the maxmum power effcency acheved at P = P max and ϑ [,] s a parameter dependng on the PA class. The FSO lnks, on the other hand, are assumed to have sngle transmt/receve termnals. Revewng the lterature and dependng on the channel condton, the FSO lnk may follow dfferent dstrbutons. Here, we present the results for cases wth exponental and Gamma- Gamma dstrbutons of the FSO lnks. For the exponental dstrbuton of the -th FSO hop, the channel gan G follows f G x = λ e λ x,, 4 wth λ beng the long-term channel coeffcent of the -th, =,..., T, hop. Moreover, wth the Gamma-Gamma dstrbuton we have x = a b a+b xa+b K f G a b Γa Γb a b x,. 5 Here, K n denotes the modfed Bessel functon of the second knd of order n and Γx = u x e u du s the Gamma functon. Also, a and b, =,..., T, are the dstrbuton shapng parameters whch can be expressed as functons of the Rytov varance, e.g., [9].

6 6 B. Data Transmsson Model We consder the decode-and-forward technque where at each hop the receved message s decoded and re-encoded, f t s correctly decoded. Therefore, the message s successfully receved by the destnaton f t s correctly decoded n all hops. Otherwse, outage occurs. As the most promsng HARQ approach leadng to hghest throughput/lowest outage probablty [38] [4], we consder the ncremental redundancy INR HARQ wth a maxmum of M retransmssons n the -th, =,...,T total, hop. Usng INR HARQ wth a maxmum of M retransmssons, q nformaton nats are encoded nto a parent codeword of length M L channel uses. The parent codeword s then dvded nto M sub-codewords of length L channel uses whch are sent n the successve transmsson rounds. Thus, the equvalent data rate,.e., the code rate, at the end of round m s q = R npcu where R ml m = q L denotes the ntal code rate n the -th hop. In each round, the recever combnes all receved sub-codewords to decode the message. Also, dfferent ndependent channel realzatons may be experenced n each round of HARQ. The retransmsson contnues untl the message s correctly decoded or the maxmum permtted transmsson round s reached. Note that settng M =,, represents the cases wthout HARQ,.e., open-loop communcaton. III. ANALYTICAL RESULTS Consder the decode-and-forward approach n a mult-hop network consstng of T RF- and T FSO-based hops. Then, because ndependent channel realzatons are experenced n dfferent hops, the system outage probablty s gven by PrOutage = T φ = T = φ, 6 where φ and φ denote the outage probablty n the -th RF- and FSO-based hops, respectvely. Note that the order of the FSO and RF lnks therefore do not matter. To analyze the outage probablty, we need to determne φ and φ,. Followng the same procedure as n, e.g., [38] [4] and usng the propertes of the mperfect PAs 3, the outage probablty of the -th RFand FSO-based hops are found as M mc φ = Pr M C m= c=m C + log + ϑ ǫ P cons P max ϑ G c R M, =,...,T, 7

7 7 and φ = Pr M C M m C m= c=m C + log + P G c R, =,..., M T, 8 respectvely. Here, 7-8 come from the maxmum achevable rates of Gaussan channels where, n harmony wth, e.g., [], [7], [9], [4], [5], [38], [4], we have used the Shannon s capacty formula. Thus, our results provde a lower bound of outage probablty whch s tght for moderate/large codewords lengths. We assume that the FSO system s well-modeled as an addtve whte Gaussan nose channel, wth nsgnfcant sgnal-dependent shot nose contrbuton. Moreover, P denotes the transmsson power n the -th, =,..., T, FSO-based hop. We have consdered a heterodyne detecton technque n 8. Also, wth no loss of generalty, we have normalzed the recevers nose varances. Hence, P, P n db, log P,log P represent the SNR as well. Then, C, =,...,T, and C, =,..., T, represent the number of channel realzatons experenced n each HARQ-based transmsson round of the -th RFand FSO-based hops, respectvely. The number of channel realzatons experenced wthn a codeword transmsson s determned by the channel coherence tmes of the lnks, the codewords lengths, consdered frequency as well as f dversty ganng technques such as frequency hoppng are utlzed. Fnally, G c and G c are the sum channel gans for the channel fadng realzaton c n the -th RF- and FSO-based hop, respectvely. In the followng, we present near-closed-form expressons for 7-8, and, consequently, 6. Then, Corollary determnes the ergodc achevable rate of mult-hop networks as well as the mnmum number of requred antennas n the RF-based hops to guarantee the ergodc achevable rate. Fnally, Secton III.C extends the results to mesh network. Snce there s no closed-form expresson for the outage probabltes, we need to use dfferent approxmaton technques see Table I for a summary of developed approxmaton schemes. In the frst method, we concentrate on cases wth long codewords where multple channel realzatons are experenced durng data transmsson n each hop,.e., C and C,, are assumed to be large. Here, we use the central lmt theorem CLT to approxmate the contrbuton of the RF- and FSO-based hops by equvalent Gaussan random varables. Usng the CLT, we fnd dfferent approxmaton results for the network outage probablty/ergodc rate Lemmas -5, For smplcty, we present the results for cases wth normalzed symbol rates. However, usng the same approach as n [], t s straghtforward to represent the results wth dfferent symbol rates of the lnks

8 8 Table I SUMMARY OF DEVELOPED APPROXIMATION TECHNIQUES. Hop type Metrc Tghtness condton Lemma RF Outage probablty Long codeword, low SNR Lemmas -4 RF Outage probablty Long codeword Lemma 5 FSO Outage probablty Long codeword Corollary RF, FSO Outage probablty Long codeword Corollary RF, FSO Ergodc rate/requred number of antennas Long codeword Lemma 6 RF, FSO Outage probablty Short codeword Corollary. Then, Secton III.B studes the system performance n cases wth short codewords,.e., small values of C, C, Lemma 6. It s mportant to note that the dfference between the analytcal schemes of Sectons III.A and B comes from the values of the products M C and M C,. Therefore, from 7-8, the long-codeword results of Secton III.A can be also mapped to cases wth short codewords, large number of retransmssons and scaled code rates. As shown n Secton IV, our derved analytcal results are n hgh agreement wth the numercal smulatons. A. Performance Analyss wth Long Codewords Lemma : At low SNRs, the outage probablty 7 s approxmately gven by 5 wth µ and σ defned n and, respectvely. Proof. Usng log+x x for small values of x, 7 s rephrased as M mc R φ Pr G c, 9 M C ϑ ǫ P M cons M C M m= m= c=m C + P max ϑ where for long codewords/large number of retransmssons, we can use the CLT to replace the random varable mc c=m C + G c by an equvalent Gaussan varable V Nµ, M C σ wth and µ = σ = γ µ, γ = xf G x a = Ω e K N N K + F N +;N ;K N, x f G x b = Ω e K N N + N K + F N +;N ;K N.

9 Here, s F t a,...,a s ;b,...,b t ;x = a j...a s j j= x j b j...b t j j!,a =,a j = aa +...a + j,j >, denotes the generalzed hypergeometrc functon. Also, to fnd a-b we have frst used the property [43, eq ] to represent the PDF as I n x = f G x = K + N e K N Ω N ΓN Γn+ x n F n+; x, 4 x N e K +x Ω F N ; K K +N x, 3 Ω and then derved a-b based on the followng ntegral dentty [44, eq ] e x x ν s F ta,...,a s ;b,...,b t ;αxdx = Γν s+ F t ν,a,...,a s ;b,...,b t ;α. 4 In ths way, usng the CDF of Gaussan random varables and the error functon erfx = x π dt, 7 s obtaned by e t φ Pr V R ϑ ǫ P M cons P max ϑ = M C +erf R ǫ P M ϑ cons P max ϑ σ 9 µ, 5 as stated n the lemma. To present the second approxmaton method for 7, we frst represent an approxmate expresson for the PDF of the sum channel gan G,, as follows. Lemma : For moderate/large number of antennas, whch s of nterest n MMW communcaton, the sum gan G,, s approxmated by an equvalent Gaussan random varable Z NN ζ,n ν wth ζ = S, ν = S 4 S and S n =. n Ω K Γ + n + L n K. Here, L n x = ex d n e x x n denotes the Laguerre polynomal of the n-th order and K n! dx n,ω are the fadng parameters as defned n. Proof. Usng the CLT for moderate/large number of antennas, the random varableg = N j = gj s approxmated by the Gaussan random varable Z NN ζ,n ν. Here, from, ζ and ν are, respectvely, determned by ζ = xf j g xdx = K +e K Ω xe K +x K Ω K +x I dx, 6 Ω

10 and ν = ρ ζ, ρ = x f j g xdx = K +e K Ω x e K +x Ω I K K +x dx, 7 Ω whch, usng the varable transform t = x, some manpulatons and the propertes of the Bessel functon x n+ e x +c b b I cx b dx = b n n Γ + n L c, c,b,n, are determned b as stated n the lemma. Lemma 3: The outage probablty 7 s approxmated by wth ˆµ and ˆσ gven n 9-, respectvely. Proof. Replacng the random varable M C M ts equvalent Gaussan random varable U N as where ˆµ = = Q log ϑ mc m= c=m C + log + ϑ ǫ P cons G P max ϑ c by, the probablty 7 s rephrased ˆµ, M C ˆσ φ Pr U R,U N ˆµ, M M C ˆσ, 8 + ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ x,,n ζ,n ν,s f G xdx c Q ϑ Y xf Z xdx ǫ P cons P max +Q r,θ r d,n ζ,n ν, Q r,θ r d,n ζ,n ν,s, Qa,a,a 3,a 4,x =. a a 3 +a erf ϑ,,n ζ,n ν, a3 x a 4 a4 π a e a 3 x a 4, 9

11 and ˆσ = ˆγ ˆµ, ˆγ = = T log + ϑ ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ x,,n ζ,n ν,s f G xdx d T ϑ Y xf Z xdx ǫ P cons P max +T r,θ r d,n ζ,n ν, T r,θ r d,n ζ,n ν,s, ϑ,,n ζ,n ν, erf T a,a,a 3,a 4,x =. x +a 3 x a3 π e a 4 a 4 a e a 3 x a 4 a a 3 +x+a a4 + πe x +a 3 a 4 a a 3 +a 4 +a a a 3 +a. Here, c and d n 9 and come from approxmatng f G x by f Z x defned n Lemma and the approxmaton log + ϑ ǫ P cons x Y x where P max ϑ ǫ P cons x, x [,s P Y x = max ϑ ] θ+r x d, x > s, s = r = ϑ ϑ ǫ P cons P max ϑ ǫ P cons P max θ e θ ϑ ϑ ϑ ǫ P cons P max ϑ e θ,d = eθ ϑ ǫ P cons P max ϑ,. Then, followng the same procedure as n 5, 8 s obtaned as φ R M C M +erf ˆµ, θ >. ˆσ, Note that, n 9-, θ > s an arbtrary parameter and, based on our smulatons, accurate approxmatons are obtaned for a broad range of θ >. Lemma 4: The outage probablty of the RF-hop,.e., 7, s approxmately gven by φ R M C M +erf µ, 3 σ wth µ and σ defned n 4 and 6, respectvely.

12 Proof. To prove the lemma, we agan use the CLT where the achevable rate random varable M mc M C m= c=m C + log + ϑ ǫ P cons G P max ϑ c s replaced by Ŭ N µ, M C σ wth ǫ µ = log + ϑ P cons x f P max ϑ G xdx e ǫ ϑ P cons F Z x dx P max ϑ x f ϑ +A A ǫ P cons P max ϑ ϑ ϑ W x + ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ dx = log x + ϑ ǫ P cons P max,, πn ν + N ζ, πn ν +N πn ν ζ,, πn ν + N ζ πn ν, +N ζ, πn ν Aa,a,a 3,x =. a a x log+a x a a x 4 +a 3 log+a x+ a x. + ϑ ǫ P cons P max ϑ ϑ πn ν +N ζ a log+a x a a a 3 x+a a 3 xlog+a x Here, e comes from Lemma and partal ntegraton. Then, f s obtaned by the lnearzaton x N technque Q ζ W x wth N ν f x πn ν +N ζ, W x =. x N ζ πn ν [ ] πn ν f x πn 5 ν +N ζ, +N ζ, πn ν f x > +N ζ, x N whch s found by lnearly approxmatng Q ζ near the pont x = N ζ. Fnally, the last N ν equalty s obtaned by partal ntegraton and some manpulatons. Also, followng the same 4

13 3 procedure, we have σ = ρ µ ρ = log + ϑ = log + ϑ +B B ϑ ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ ǫ P cons P max ǫ P cons P max ϑ ϑ x πn ν f G xdx ϑ +N ζ ǫ P cons P max ϑ log,, πn ν + N ζ, πn ν +N ζ πn ν,, πn ν + N ζ πn ν, +N ζ, πn ν + ϑ ǫ P cons P max ϑ + ϑ ǫ P cons P max ϑ Ba,a,a 3,x. = a a 3 a a log +a x a x+ a a x+a a log+a x. 6 In ths way, the outage probablty s gven by 3. Fnally, Lemma 5 represents the outage probablty of the FSO-based hops as follows. Lemma 5: The outage probablty of the FSO-based hop,.e., 8, s approxmately gven by φ R M C M +erf µ, 7 σ x W x dx x where µ and σ are gven by 8-9 and [9, eq ] for the exponental and the Gamma- Gamma dstrbutons of the FSO lnks, respectvely. Proof. Usng the CLT, the random varable M mc M C m= c=m C + log P + G c s approxmated by ts equvalent Gaussan random varable R N µ, M C σ, where for the exponental dstrbuton of the FSO lnk we have µ = xlog + P f G x dx g x λ F G = P + P x dx = e P E λ, 8 P and σ = ρ µ, ρ = xlog + P f G x dx h = P H x = e λ P λ P x 3 F 3,,;,,; λ x + logx log λ x +E Γ P e λ x + P x log + P x dx =H H, P, λ x P +logx. 9

14 4 Here, Ex = x e t dt t denotes the exponental ntegral functon. Also, g and h are obtaned by partal ntegraton. Then, denotng the Euler constant by E, s gven by the varable transformaton + P x = t, some manpulatons, as well as the defnton of the Gamma ncomplete functon Γs,x = x ts e t dt and the generalzed hypergeometrc functon a F a. For the Gamma-Gamma dstrbuton, on the other hand, the PDF f G n 8-9 s replaced by 5 and the mean and varance are calculated by [9, eq. 43] and [9, eq. 44], respectvely. In ths way, followng the same arguments as n Lemmas, 3-4, the outage probablty of the FSO-based hops s gven by 7. Lemmas -5 lead to dfferent corollary statements about the performance of mult-hop RF-FSO systems, as stated n the followng. Corollary : Wth long codewords, there are mappngs between the performance of FSOand RF-based hops, n the sense that the outage probablty acheved n an RF-based hop s the same as the outage probablty n an FSO-based hop experencng specfc long-term channel characterstcs. Proof. The proof comes from Lemmas -5 where for dfferent hops the outage probablty s gven by the CDF of Gaussan random varables. Thus, wth approprate long-term channel characterstcs, µ,σ, ˆµ,ˆσ, µ, σ and µ, σ n Lemmas and 3-5 can be equal leadng to the same outage probablty n these hops. In ths way, the performance of RF-FSO based mult-hop/mesh networks can be mapped to ones usng only the RF- or the FSO-based communcaton. Corollary : Wth asymptotcally long codewords, the mnmum number of transmt antennas n an RF-based hop, such that the same rate s supported n all hops, s found by the soluton of { ǫ P cons N = arg x +A A ϑ ϑ log ǫ P cons P max + ϑ ϑ ǫ P cons P max, P max whch can be calculated numercally. ϑ, ϑ πxν +xζ, πxν + xζ, πxν πxν, πxν + xζ πxν, πxν +xζ +xζ = mn { µ j } j=,..., T },, 3

15 5 Also, the ergodc achevable rate of the mult-hop network s approxmately gven by CT, T = mn mn { µ j}, mn { µ j }, 3 j=,...,t j=,..., T wth µ defned n 4 and µ gven by 8 and [9, eq. 43] for the exponental and Gamma-Gamma dstrbutons of the FSO lnk, respectvely. Proof. Wth asymptotcally long codewords,.e., very large C, C, the achevable rates n the RF- and FSO-based hops converge to ther correspondng ergodc capacty, and there s no need for HARQ because the data s always correctly decoded f t s transmtted wth rates less than or equal to the ergodc capacty. Denotng the expectaton operaton by E{ }, the ergodc capacty of an FSO-hop s gven by µ = E{log + P G } whch s determned by 8 and [9, eq. 43] for the exponental and Gamma-Gamma dstrbutons of the FSO-based hop, respectvely. For the RF-based hop, on the other hand, the ergodc capacty s found as { } E log + ϑ ǫ P cons G µ wth µ gven n 4. In ths way, the maxmum achevable P max ϑ rate of the FSO-based hops s R = mn { µ j }. Also, the mnmum number of requred antennas j=,..., T n the -th RF-based hop s found by solvng µ = R whch, from 4, leads to 3. Note that 3 s a sngle-varable equaton and can be effectvely solved by dfferent numercal technques. Fnally, followng the same argument, the ergodc achevable rate of the RF-FSO network s gven by 3,.e., the maxmum rate n whch the data s correctly decoded n all hops. B. Performance Analyss wth Short Codewords Up to now, we consdered the long-codeword scenaro such that the CLT provdes accurate approxmaton for the sum of ndependent and dentcally dstrbuted IID random varables. However, t s nterestng to analyze the system performance n cases wth short codewords,.e., when C and C,, are small. Ths case s especally mportant for FSO lnks snce the coherence tme can be qute long mllseconds. Here, we manly concentrate on the Gamma- Gamma dstrbuton of the FSO-based hops. The same results as n [45] can be appled to derve the outage probablty of the FSO-based hops n the cases wth exponental dstrbuton. Lemma 6: For arbtrary numbers of M,C and C, The outage probabltes of the FSO- and RF-based hops are bounded by 33 and 36, respectvely.

16 6 At low SNRs, a MISO-HARQ RF-based hop wth M retransmssons, N transmt antennas and C channel realzatons per sub-codeword transmsson can be mapped to an open-loop MISO setup wth M N C transmt antennas and sngle channel realzaton per codeword transmsson. Proof. Consderng the FSO-based hops, one can use the Mnkowsk nequalty [46, Theorem 7.8.8] to wrte φ = Pr M C M = Pr M m= c=m C + + m C m= c=m C + m C Pr + P M n = + P G c m C m= c=m C + n x n n +x, 3 = log+ P G c R e C R M M C G c e R M = F J e R P M M C, 33 where, usng the results of [4, Lemma 3] and for the Gamma-Gamma dstrbuton of the m C varables G, the random varable J = M G m= c=m C + c follows the CDF F J x = Γ M C a Γ M C b GM C, a,m C + b M C x, 34 a,a,...,a }{{},b,b,...,b, }{{} M C tmes M C tmes wth G. denotng the Mejer G-functon. Note that, the results of 33 are mathematcally applcable for every values of M, C. However, for, say M C 6, the mplementaton of the Mejer G-functon n MATLAB s very tme-consumng and the tghtness of the approxmaton decreases wth M, C. Therefore, 33 s useful for the performance analyss n cases wth small M, C,, whle the CLT-based approach of Secton III.A provdes accurate performance evaluaton for cases wth long codewords. For the RF-based hop, on the other hand, we use n n log + x j n log+x j log + n n j= j= n x j, n,x j, 35 j=

17 7 to lower- and upper-bound the outage probablty by Pr log + φ Pr F G ϑ ǫ P cons P max ϑ M C M C log M C e R M ϑ ǫ P cons P max ϑ M mc m= c=m C + + ϑ Here, G = M mc N m= c=m C + ǫ P cons P max ϑ φ F G j = gj G c R M mc M m= c=m C + er C ϑ ǫ P cons P max ϑ G c R M. 36 c s an equvalent sum channel gan varable wth M C N antennas at the transmtter whose PDF s obtaned by replacng N wth M C N n. Also, F G denotes the CDF of the equvalent sum channel gan varable. To prove Lemma 6 part, we note that lettng x,, the nequaltes n 35 are changed to equalty. Thus, as a corollary result, at low SNRs a MISO-HARQ RF-lnk wth N transmt antennas, M retransmssons and C channel realzatons wthn each retransmsson round can be mapped to an open-loop MISO setup wth M C N transmt antennas, n the sense that the same outage probablty s acheved n these setups. Note that the boundng schemes of 36 are mathematcally applcable for every values of M,C. However, whle the results of 36 tghtly match the exact numercal results for small values of M,C, the tghtness decreases for large M,C s. Thus, the results of Lemmas -4 and Lemma 6 can be effectvely appled for the performance analyss of the RF-based hops n the cases wth long and short codewords, respectvely. Fnally, as another approxmaton for the cases wth M =,C =, we have φ m =,C = +erf er ǫ P ϑ cons P max ϑ N ζ N ν, 37 whch comes from Lemma. C. Performance Analyss n Mesh Networks Consder a mesh network consstng of X non-overlappng routes from the source to the destnaton wth ndependent channel realzatons for the hops. The -th, =,...,X, route s

18 8 made of T RF- and T FSO-based hops and the routes can have dfferent total number of hops T total = T + T, =,...,X. In ths case, the network outage probablty s gven by PrOutage mesh = X Pr Outage, 38 = where PrOutage s the outage probablty n the -th route as gven n 6. In 38, we have used the fact that n a mesh network an outage occurs f the data s correctly transferred to the destnaton through none of the routes. Wth the same arguments, the ergodc achevable rate of the mesh network s obtaned by } C mesh = max { C, 39 =,...,X wth C derved n 3. Ths s based on the fact that, knowng the long-term channel characterstcs, one can set the data rate equal to the maxmum achevable rate of the best route and the message s always correctly decoded by the destnaton, f the codewords are asymptotcally long. The performance of mesh networks s studed n Fg. 9. IV. NUMERICAL RESULTS Throughout the paper, we presented dfferent approxmaton technques. The verfcaton of these results s demonstrated n Fgs.,, 6-8 and, as seen n the sequel, the analytcal results follow the numercal results wth hgh accuracy. Then, to avod too much nformaton n each fgure, Fgs. 3-5, 9 report only the smulaton results. Note that n all fgures we have doublechecked the results wth the ones obtaned analytcally, and they match tghtly. The smulaton results are presented for homogenous setups. That s, dfferent RF-based hops follow the same long-term fadng parameters K,ω,, n -, and the FSO-based hops also experence the same long-term channel parameters,.e., λ,a and b n 4-5. Moreover, we set M = M j and R = R j,,j =,...,T total. In all fgures, we set P = N P cons such that the total consumed power at dfferent hops s the same. Then, usng 3, one can determne the output power of the RF-based antennas. Also, because the nose varances are set to, P n db, log P s referred to the SNR as well. In Fgs., and 5, we assume an deal PA. The effect of non-deal PAs s verfed n Fgs. 3, 4, 6-9. Wth non-deal PAs, we consder the state-of-the-art parameter settngs ϑ =.5,ǫ =.75,P max = 5 db,, [34] [37], unless otherwse stated.

19 9 Outage probablty of the RF hop Low SNR approxmaton of Lemma Ideal PA, M =, C =, R= N t =8 N t =8, 4, N t =4 Numercal result Approxmaton scheme of Lemma 3 Approxmaton scheme of Lemma 4 N t = SNR, db Fgure. On the tghtness of the results n Lemmas -4. Ideal PA, sngle RF-based hop, M =,R =,C =. The results are presented for N =,4, and 8,. The parameters of the Rcan RF PDF are set to ω =,K =.,, leadng to unt mean and varance of the channel gan dstrbuton f j g x,,j. Wth the exponental dstrbuton of the FSO-based hops, we x = λ consderf G e λx wthλ =,. Also, for the Gamma-Gamma dstrbuton we set x = a b a+b +b f G Γa Γb xa K a b a b x, a = ,b =.5636,, whch corresponds to Rytov varance of [4]. Fgures -8 consder mult-hop networks. The performance of mesh networks s studed n Fg. 9. Note that, as dscussed n Secton III, the results of cases wth long codewords and few number of retransmssons can be mapped to the cases wth short codewords, large number of retransmssons and a scaled code rate see 7-8. Fnally, t s worth notng that we have verfed the analytcal and the numercal results for a broad range of parameter settngs, whch, due to space lmts and because they lead to the same qualtatve conclusons as n the presented fgures, are not reported n the fgures. The smulaton results are presented n dfferent parts as follows. On the approxmaton approaches of Lemmas -5: Consderng an deal PA, M = as the worst-case scenaro, R = npcu, and C =,, Fg. verfes the tghtness of the approxmaton schemes of Lemmas -4. Partcularly, we plot the outage probablty of an RFbased hop for dfferent numbers of transmt antennas N,. Then, Fg. demonstrates the outage probablty of a dual-hop RF-FSO setup versus the SNR. Here, we set M =,R =, npcu, and C =, C = 3,N =,T =, T =, and the results are presented for cases wth deal PAs at the RF-based hops. As observed, the analytcal results of Lemmas -5 mmc the exact results wth very hgh accuracy Fgs. -. Also, Lemma properly approxmates the

20 Outage probablty Numercal result Approxmaton results of Lemmas & 5 Approxmaton results of Lemmas 3 & 5 Approxmaton results of Lemmas 4 & 5 Dual-hop network, M =, deal PA, C =, C = 3 R = R = SNR db Fgure. On the tghtness of the results n Lemmas -5. Ideal PA, dual-hop network, M =,R =,,C =, C = 3,T =, T =, and N =,. Outage probablty M =3 M = M = 3 db gan 7 db gan 3.5 db Non-deal PA, ε=.75, ϑ=.5, P max =5 db Ideal PA SNR db Fgure 3. Outage probablty of a dual-hop RF-FSO network for dfferent PA models and maxmum number of retransmssons, M,. Exponental dstrbuton of the FSO lnk, T =, T =,C =, C =,R = 3 npcu, and N = 6,. outage probablty at low and hgh SNRs, and the tghtness ncreases as the code rate decreases Fg.. Moreover, the tghtness of the approxmaton results of Lemmas 3-4 ncreases wth the number of RF-based transmt antennas Fg.. Ths s because the tghtness of the CLT-based approxmatons n Lemma ncreases wth N,. Fnally, although not demonstrated n Fgs. -, the tghtness of the CLT-based approxmaton schemes of Lemmas 3-5 ncreases wth the maxmum number of retransmssons M,. On the effect of HARQ and mperfect PAs: Shown n Fg. 3 s the outage probablty of a dualhop RF-FSO network for dfferent maxmum numbers of HARQ-based retransmsson rounds M,. Also, the fgure compares the system performance n cases wth deal and non-deal PAs. Here, the results are obtaned for the exponental dstrbuton of the FSO lnk, T =, T =

21 Outage probablty T = 4, T = 4 T =, T = T =, T = Non-deal PA, M =, R =, C =, C =, N = Exponental PDF of FSO hops Gamma-Gamma PDF of FSO hops SNR db Fgure 4. The outage probablty for dfferent numbers of RF- and FSO-based hops,.e., T and T. Non-deal PA, ϑ =.5,ǫ =.75,P max = 5 db, M =,N =,R = npcu, C =, and C =,. Outage probablty - -3 C = C =, C = C =, C = C = 3, C = C = 5, T =, T = T =, T = Number of transmt antennas n RF hops, N Fgure 5. Outage probablty for dfferent numbers of transmt antennas n the RF-based hops and channel coherence tmes. Exponental dstrbuton of the FSO-based hops, deal PA, R =.5 npcu, M =,T = T =,, and SNR = db.,c =, C =,R = 3 npcu, and N = 6,. As demonstrated, wth no HARQ, the effcency of the RF-based PAs affects the system performance consderably. For nstance, wth the parameter settngs of the fgure and outage probablty 4, the PAs neffcency ncreases the requred power by 3.5 db. On the other hand, the HARQ can effectvely compensate the effect of mperfect PAs, and the dfference between the outage probablty of the cases wth deal and non-deal PAs s neglgble for M >. Also, the effect of non-deal PA decreases at hgh

22 Ergodc achevable rate npcu Non-deal PA, Gamma-Gamma dstrbuton of FSO hops SNR= db SNR=5 db SNR= db SNR=5 db Numercal results Approxmaton results of Corollary 5 5 Number of transmt antennas n RF hop, N Fgure 6. Ergodc achevable rate for dfferent numbers of transmt antennas n the RF-based hops and SNRs. Gamma-Gamma dstrbuton of the FSO-based hops, non-deal PA ϑ =.5,ǫ =.75, and P max = 5 db,. SNRs whch s ntutvely because the effectve effcency of the PAs ǫ effectve = ǫ P P max ϑ,, s mproved as the SNR ncreases. Fnally, the mplementaton of HARQ mproves the energy effcency sgnfcantly. As an example, consder the outage probablty 4, an deal PA and the parameter settngs of Fg. 3. Then, compared to the open-loop communcaton,.e., M =, the mplementaton of HARQ wth a maxmum of and 3 retransmssons reduces the requred power by 3 and 7 db, respectvely. System performance wth dfferent numbers of hops: In Fg. 4, we demonstrate the outage probablty n cases wth dfferent numbers of RF- and FSO-based hops,.e., T, T. In harmony wth ntuton, the outage probablty ncreases wth the number of hops. However, the outage probablty ncrement s neglgble, partcularly at hgh SNRs, because the data s correctly decoded wth hgh probablty n dfferent hops as the SNR ncreases. Fnally, as a sde result, the fgure ndcates that the outage probablty of the RF-FSO based mult-hop network s not senstve to the dstrbuton of the FSO-based hops at low SNRs. Ths s ntutve because, at low SNRs and wth the parameter settngs of the fgure, the outage event mostly occurs n the RF-based hops. However, at hgh SNRs where the outage probablty of dfferent hops are comparable, the PDF of the FSO-based hops affects the network performance. On the effect of RF-based transmt antennas: Consderng an exponental dstrbuton of the FSO-based hops, deal PAs, R =.5 npcu, M =,T = T =,, and SNR = db, Fg. 5 demonstrates the effect of the number of RF transmt antennas on the network outage probablty. Also, the fgure compares the system performance n cases wth short and long

23 3 Ergodc achevable rate npcu Non-deal PA, Gamma-Gamma dstrbuton of FSO hops, N =8 Subplot: a FSO-lmted regon ε =.5 ε =.75 RF-lmted regon ε =.5 Numercal result Approxmaton result of Corollary SNR db Requred number of antennas, N Non-deal PA, Gamma-Gamma dstrbuton of FSO hops, Subplot: b ε =.5 ε =.5 ε =.5 Numercal result Approxmaton result of Corollary SNR db Fgure 7. On the tghtness of the analytcal results of Corollary. Subplot a: Ergodc achevable rate vs the SNR. Non-deal PA, Gamma-Gamma dstrbuton of the FSO hops, N = 8,. Subplot b: The mnmum number of transmt antennas n the RF-based hops to guarantee the same rate as n the FSO-based hops. Non-deal PA and Gamma-Gamma dstrbuton of the FSO hops. For the non-deal PAs, we have ϑ =.5,ǫ =.75, and P max = 5 db,. codewords,.e., n cases wth small and large values of C, C. As seen, wth short codewords, the outage probablty decreases wth the number of RF-based transmt antennas monotoncally. Ths s because, wth the parameter settngs of the fgure, the data s correctly decoded wth hgher probablty as the number of antennas ncreases. Wth long codewords, on the other hand, the outage probablty s almost nsenstve to the number of transmt antennas for N 3. Fnally, the outage probablty decreases wth C, C, because the HARQ explots tme dversty as more channel realzatons are experenced wthn each codeword transmsson.

24 4 Outage probablty Outage probablty Numercal result Approxmaton scheme of SNR db Numercal result and bounds of 36, M = Approxmaton scheme of 37, M = Numercal result, M = Upper bound of 36, M = RF-based hop -4 Lower bound of 36, M = -5 5 SNR db FSO-based hop M = Subplot: b M = Subplot: a Outage probablty -5 - RF-FSO dual-hop Exact results, upper and lower bounds of 37, M = Numercal result Boundng approach of Lemma 6 M = Subplot: c SNR db Fgure 8. Outage probablty n the cases wth short codewords. Non-deal PA, Gamma-Gamma dstrbuton of the FSO hops, R = npcu, M =,C =, C =,N = 6,. For the non-deal PAs, we haveϑ =.5,ǫ =.75, andp max = 5 db,. In subplots a-c, the outage probablty s presented for an FSO-based hop, an RF-based hop and a dual-hop RF-FSO network, respectvely. In Fg. 6, we plot the network ergodc achevable rates for cases wth Gamma-Gamma dstrbuton of the FSO-based hops, non-deal PAs and dfferent numbers of transmt antennas/snrs. Also, the fgure verfes the accuracy of the approxmaton schemes of Corollary. Note that, due to the homogenous network structure, the ergodc rate s ndependent of the number of RF- and FSO-based hops. As seen, at low/moderate SNRs, the network ergodc rate ncreases almost logarthmcally wth the number of RF antennas. At hgh SNRs, on the other hand, the ergodc rate becomes ndependent of the number of RF transmt antennas. Ths s because wth

25 5 Fgure 9. Mesh outage probablty non-deal PA, R = 3, M = 3, N = 6, C =, C =.5 db X =,T = T = 4 X =, T = T = 3 X =, T = T = X =, T = T = X =, T = T = X = 3, T = T = X = 4, T = T =. db SNR db Outage probablty of a mesh network for dfferent numbers of routes and hops. Non-deal PA, Gamma-Gamma dstrbuton of the FSO hops, R = 3 npcu, M = 3,C =, C =, and N = 6,. For the non-deal PAs, we have ϑ =.5,ǫ =.75, and P max = 5 db,. large number of RF-based antennas the achevable rate of the RF-based hops exceeds the one n FSO-based hops, and the network ergodc rate s gven by the achevable rate of the FSO-based hops. Fnally, the number of antennas above whch the ergodc rate s lmted by the achevable rate of the FSO-based hops ncreases wth the SNR. On the ergodc achevable rates: Along wth Fg. 6, we evaluate the accuracy of the results of Corollary n Fgs. 7a and 7b. Partcularly, Fg. 7a demonstrates the network ergodc rate for dfferent PA models and compares the smulaton results wth the ones derved n 3. Then, Fg. 7b verfes the accuracy of 3. Here, we show the mnmum number of requred RF transmt antennas versus the SNR whch determnes the ergodc rate of the FSO-based hops. As can be seen, the approxmaton results of Corollary are very tght for a broad range of parameter settngs. Thus, 3 and 3 can be effectvely used to derve the requred number of RF transmt antennas and the network ergodc rate, respectvely Fgs. 7a and 7b. The ergodc rate shows dfferent behavors n the, namely, FSO-lmted and RF-lmted regons. Wth the parameter settngs of the fgure, the ergodc rate s lmted by the achevable rates of the FSObased hops at low SNRs FSO-lmted regon n Fg. 7a. However, as the SNR ncreases, the achevable rates of the FSO-based hops exceed the ones n the RF-based hops and, consequently, the network ergodc rate s lmted by the rate of the RF-based hops RF-lmted regon n Fg. 7a. As a result, the effcency of the RF PAs affects the ergodc rate at hgh SNRs. Fnally, the fgure ndcates that at hgh SNRs the network ergodc rate ncreases almost lnearly wth the SNR.

26 6 As shown n Fg. 7b, the requred number of RF-based transmt antennas to guarantee the same rate as n the FSO-based hops ncreases consderably wth the SNR and, consequently, the ergodc rate of the FSO-based hops. Moreover, the PAs effcency affects the requred number of antennas sgnfcantly. As an example, consder the parameter settngs of Fg. 7b and SNR = 3 db. Then, the requred number of RF-based antennas s gven by 33, 49 and 98 for the cases wth PA effcency 75%, 5% and 5%, respectvely. Thus, hardware mparments such as the PA neffcency affect the system performance remarkably and should be carefully consdered n the network desgn. However, selectng the proper number of antennas and PA propertes s not easy because the decson depends on several parameters such as complexty, nfrastructure sze and cost. Performance analyss wth short codewords: In Fgs. 8a, 8b and 8c, we study the outage probablty of an FSO-based hop, an RF-based hop and a dual-hop RF-FSO network, respectvely. Partcularly, consderng non-deal PAs and Gamma-Gamma dstrbuton of the FSO-based hops, the results are obtaned for C = C =, M =,, and we evaluate the accuracy of dfferent bounds/approxmatons n Lemma 6 and 37. As demonstrated, the bound of 33 matches the exact values derved va smulaton analyss of φ exactly n cases wth M =. Also, the boundng/approxmaton methods of 33, 36 and 37 mmc the numercal results wth hgh accuracy n cases wth a maxmum of M =,, retransmssons. Thus, the results of Secton III.B can be effcently used to analyze the RF-FSO systems n cases wth small values of M,C, C,. On the performance of mesh networks: In Fg. 9, we study the outage probablty of mesh networks for dfferent numbers of routes. Here, we consder non-deal PAs, exponental PDF of the FSO hops, M = 3, N = 6, C =, C =, and R = 3 npcu. The results are presented for cases wth one RF- and one FSO-based hop n each route. Also, we compare the outage probablty of the mesh network wth that of a sngle route setup consstng of dfferent numbers of RF- and FSO-based hops. Note that there are the same total number of RF- and FSO-based hops n each case wth X = n,t = T =, =,...,X, and X =,T = T = n. As demonstrated n Fg. 9, n contrast to the sngle-route setup where the outage probablty ncreases wth the number of hops, the outage probablty of the mesh network decreases consderably by addng more parallel routes nto the network. For nstance, consder the parameter settngs of the fgure and outage probablty of 6. Then, compared to cases wth a sngle route, the requred SNR at each hop decreases by almost. db f the data s transferred through

27 7 two routes. Ths s ntutve because the probablty that the data s correctly receved by the destnaton ncreases wth the number of routes. However, the relatve effect of addng more routes decreases wth the number of routes, and, for the example parameters consdered, there s about.5 db energy effcency mprovement f the number of routes ncreases from X = to X = 3. Fnally, whle we dd not consder t n Secton III.C, the performance of the mesh network s further mproved f the sgnals from dfferent routes are combned at the destnaton. V. CONCLUSION We studed the performance of RF-FSO based mult-hop and mesh networks n cases wth short and long codewords. Consderng dfferent channel condtons, we derved closed-form expressons for the networks outage probablty, the ergodc rates as well as the requred number of RF transmt antennas to guarantee dfferent achevable rate qualty-of-servce requrements. The results are presented for cases wth and wthout HARQ. As demonstrated, dependng on the codeword length, there are dfferent methods for analytcal performance evaluaton of the mult-hop/mesh networks. Moreover, there are mappngs between the performance of RF-FSO based mult-hop networks and the ones usng only the RF- or the FSO-based communcaton. Also, the HARQ can effectvely mprove the energy effcency and compensate for the effect of hardware mparments. Fnally, the outage probablty of mult-hop networks s not senstve to the large number of RF-based transmt antennas whle the ergodc rate s sgnfcantly affected by the number of antennas. ACKNOWLEDGEMENT The research leadng to these results receved fundng from the European Commsson H programme under grant agreementn G PPP mmmagic project, and from the Swedsh Governmental Agency for Innovaton Systems VINNOVA wthn the VINN Excellence Center Chase. REFERENCES [] A. Vavoulas, H. G. Sandalds, and D. Varoutas, Weather effects on FSO network connectvty, IEEE J. Opt. Commun. Netw., vol. 4, no., pp , Oct.. [] L. Yang, X. Gao, and M.-S. Aloun, Performance analyss of relay-asssted all-optcal FSO networks over strong atmospherc turbulence channels wth pontng errors, J. Lghtw. Technol., vol. 3, no. 3, pp. 4 48, Dec. 4.

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