Performance analysis of RF-FSO multi-hop networks

Transcription

1 Performance analyss of RF-FSO mult-hop networks Behrooz Makk, Tommy Svensson, Mate Brt-Pearce Mohamed-Slm Aloun Chalmers Unversty of Technology, Gothenburg, Sweden, {behrooz.makk, Unversty of Vrgna, Charlottesvlle, VA, USA, Kng Abdullah Unversty of Scence Technology KAUST, Thuwal, Saud Araba, arxv:73.766v [cs.it] Mar 7 Abstract We study the performance of mult-hop networks composed of mllmeter wave MMW-based rado frequency RF free-space optcal FSO lnks. The results are obtaned n the cases wth wthout hybrd automatc repeat request HARQ. Takng the MMW characterstcs of the RF lnks nto account, we derve closed-form expressons for the network outage probablty. 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 the FSO lnks on the system performance. Fnally, we present mappngs between the performance of RF-FSO mult-hop networks the ones usng only the RF- or the FSO-based communcaton, n the sense that wth approprate parameter settngs the same outage probablty s acheved n these setups. The results show the effcency of the RF-FSO setups n dfferent condtons. Moreover, the HARQ can effectvely mprove the outage probablty/energy effcency, compensate the effect of hardware mparments n RF-FSO networks. For common parameter settngs of the RF-FSO dualhop networks, outage probablty 4 code rate 3 nats-perchannel-use, the mplementaton of HARQ wth a maxmum of 3 retransmssons reduces the requred power, compared to the cases wth no HARQ, by 3 7 db, respectvely. I. INTRODUCTION To address the dems on the next generaton of wreless networks, a combnaton of dfferent technques are consdered among whch free-space optcal FSO communcaton s very promsng []. FSO systems provde fber-lke data rates through the atmosphere usng lasers or lght emttng dodes LEDs. Thus, FSO can be used for a wde range of applcatons such as last-mle access, back-haulng multhop networks. In the rado frequency RF doman, on the other h, t has been recently concentrated on mllmeter wave MMW communcaton as a key enabler to obtan suffcently large bwdths so that t s possble to acheve data rates comparable to those n the FSO lnks. In ths way, the combnaton of FSO MMW-based RF lnks s consdered as a powerful cdate for hgh-rate relable communcaton n, e.g., vehcle- nfrastructure-to-nfrastructure networks. Ths s partcularly nterestng because, as we show n the followng, wth proper parameter settngs the outage probablty of the RF-FSO networks can be mapped to the ones n the cases wth only the RF- or the FSO-based communcaton. The RF-FSO related works can be dvded nto two categores. The frst group are papers on sngle-hop setups where the lnk relablty s mproved va the jont mplementaton of RF FSO systems. Here, ether the RF the FSO /4/$3. c 4 IEEE lnks are consdered as separate lnks 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 [4] [6]. Also, the mplementaton of hybrd automatc repeat request HARQ n RF-FSO lnks has been consdered n [6] [8]. The second group are the papers studyng the performance of mult-hop RF-FSO networks. For nstance, [9] analyzes decode--forward technques n multuser relay networks usng RF-FSO. Then, [], [] study RF-FSO based relayng wth an RF source-relay lnk an FSO or RF-FSO relaydestnaton lnk. Also, consderng Raylegh fadng condtons for the RF lnk amplfy--forward relayng technque, [], [3] derve the end-to-end error probablty of the RF- FSO based setups compare the system performance wth RF-based relay networks, respectvely. Fnally, the mpact of pontng errors on the performance of dual-hop RF-FSO systems s studed n [4], [5]. In ths paper, we analyze the performance of mult-hop RF- FSO systems from an nformaton theoretc pont of vew. Consderng the MMW characterstcs of the RF lnks heterodyne detecton technque n the FSO lnks, we derve closed-form expressons for the system outage probablty Lemmas -3. Our results are obtaned for the decode-forward relayng approach n dfferent cases wth wthout HARQ. Specfcally, we show the HARQ as an effectve technque to compensate for the mperfect propertes of the RF-FSO system to mprove the network relablty/energy effcency. Fnally, we present mappngs between the performance of RF- FSO-based hops Corollary, analyze the effect of varous parameters such as the power amplfers PAs effcency, dfferent coherence tmes of the lnks number of antennas on the network outage probablty. As opposed to [] [8], we consder mult-hop systems. Also, our paper s dfferent from [] [5] because our analytcal/numercal results on the outage probablty as well as our dscussons on the effect of mperfect PAs HARQ have not been presented before. The dfferences n the problem formulaton the channel model makes our analytcal/numercal results as well as our conclusons completely dfferent from the ones n the lterature. Our results show that there are mappngs between the performance of RF-FSO mult-hop networks the ones usng only the RF- or the FSO-based communcaton, n the sense that wth proper scalng of the channel parameters the

2 same outage probablty s acheved n these setups Corollary. Whle the outage probablty s senstve to the number of RF-based transmt antennas for short codewords, the outage probablty reducton due to ncreasng the number of antennas s neglgble for the cases wth long codewords. The PAs effcency affects the network outage probablty consderably. However, the HARQ protocols can effectvely compensate 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 4 code rate 3 nats-per-channel-use npcu. Then, compared to the cases wth open-loop communcaton, the mplementaton of HARQ wth a maxmum of 3 retransmssons reduces the requred power by 3 7 db, respectvely. II. SYSTEM MODEL Consder a T total -hop RF-FSO system, wth T RF-based hops T = T total T FSO-based hops. As seen n the followng, the outage probablty s ndependent of the order of the hops. Thus, we do not need to specfy the order of the RF- FSO-based hops. The -th, =,...,T, RFbased hop uses a multple-nput-sngle-output MISO setup wth N transmt antennas. 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 ts correspondng receve antenna. Here, we present the analytcal results for the Rcan channel model of the RF-based hops, whch s an approprate model for near lne-of-sght condtons has been well establshed for dfferent mllmeter wave-based applcatons, e.g., [6] [8]. Let us denote the probablty densty functon PDF the cumulatve dstrbuton functon CDF of a rom varable X by f X F X, respectvely. Wth a Rcan model, the channel gan g j,,j, follows the PDF x = K +e K f g j e K +x Ω I Ω K K +x Ω,,j, where K Ω denote the fadng parameters n the -th hop I n s the n-th order modfed Bessel functon of the frst knd. Also, the sum channel gan G = N follows f G x = K +e KN K +x Ω K N Ω e K +x Ω I N j = gj N K K +N x Ω,. 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 [9, eq. 3], [, eq. ] P P cons = ǫ P P max ϑ P = ϑ ǫ P cons P max ϑ,. 3 Here, P,P max P cons,, are the output, the maxmum output the consumed power n each antenna of the - th hop, respectvely, ǫ [,] denotes the maxmum power effcency acheved at P = P max ϑ [,] s a parameter dependng on the PA classes. The FSO lnks, on the other h, are assumed to have sngle transmt/receve termnals. Revewng the lterature dependng on the channel condton, the FSO lnk may follow dfferent dstrbutons. Here, we present the results for the cases wth exponental Gamma-Gamma dstrbutons of the FSO lnks. For the exponental dstrbuton of the -th FSO hop, the channel gan G follows x = λ f G e λx,, 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 Γx = u x e u du s the Gamma functon. Also, a b, =,..., T, are the dstrbuton shapng parameters whch can be expressed as functons of Rytov varance, e.g., [8]. A. Data Transmsson Model We consder the decode--forward technque where at each hop the receved message s decoded re-encoded, f t s correctly decoded. Thus, 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 lowest outage probablty [], 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 codeword of length M L channel uses. Then, the codeword s dvded nto M sub-codewords of length L channel uses whch are sent n the successve retransmssons. Thus, the equvalent data rate at the end of round m s q ml = R m npcu where R = q L denotes the ntal code rate n the -th hop. In each round, the data s decoded based on all sub-codewords receved up to the end of that round. The retransmsson contnues untl the message s correctly decoded or the maxmum permtted transmsson round s reached. Fnally, note that settng M =,, represents the cases wthout HARQ,.e., open-loop communcaton. III. ANALYTICAL RESULTS Consderng the decode--forward approach because ndependent channel realzatons are experenced n dfferent hops, the system outage probablty s gven by T T PrOutage = φ φ. 6 = = Here, φ φ denote the outage probablty n the -th RF FSO-based hops, respectvely. Thus, to analyze the outage

3 probablty, we need to fnd φ φ,. Followng the same procedure as n, e.g., [] usng the propertes of the mperfect PAs 3, the outage probablty of the -th RF- FSO-based hops are obtaned as φ = Pr M C M mc m=c=m C + log φ = Pr M C M m C m=c=m C + + ϑ ǫ P cons P max ϑ G c R M, 7 log + P G c R, 8 M respectvely. Here, P represents the transmsson power n the -th, =,..., T, FSO-based hop we have consdered heterodyne detecton technque n 8. Also, C, =,...,T, C, =,..., T, represent the number of channel realzatons experenced n each HARQ-based transmsson round of the -th RF- FSO-based hops, respectvely for smplcty, we present the results for the cases wth normalzed symbol rates. Usng the same approach as n [5], t s straghtforward to represent the results wth dfferent symbol rates of the lnks. In the sequel, we present closed-form expressons for 7-8,, consequently, 6. Here, we concentrate on the cases wth long codewords where multple channel realzatons are experenced durng data transmsson n each hop,.e., C C,, are assumed to be large. For performance comparson n the cases wth short long codewords, see Fg. 4. Indeed, to fnd the expressons, we need to mplement approxmaton technques. However, as seen n Secton IV, the analytcal results mmc the numercal results wth hgh accuracy. To approxmate 7, we frst represent an approxmaton 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 rom varable Z NN ζ,n ν wth ζ = S, ν = S 4 S S n =. n Ω K + Γ + n L n K. Here, L n x = e x d n n! dx e x x n denotes the Laguerre polynomal of the n-th n order K,Ω are the fadng parameters as defned n. Proof. Usng central lmt Theorem CLT for moderate/large number of antennas, the rom varable G = N j = gj s approxmated by the Gaussan rom varable Z NN ζ,n ν. Here, from, ζ ν are found as ζ = xf j g xdx = K +e K Ω xe K +x K Ω K +x I dx, 9 Ω ν = ρ ζ, ρ = x f j g xdx = K +e K Ω x e K +x Ω I K K +x dx, Ω respectvely. Then, usng the varable transform t = x, some manpulatons the propertes of +c the Bessel functon b x n+ e x b I cx b dx = b n n Γ + n L c b, c,b,n, the mean the varance ζ,ν are determned as stated n the lemma. Lemma : The outage probablty 7 s approxmated by 5 wth ˆµ ˆσ gven n -3, respectvely. Proof. Replacng the rom varable M mc M C m= c=m C log ǫp + + ϑ cons G P max ϑ c by ts equvalent CLT-based Gaussan rom varable U N ˆµ, M C ˆσ, the probablty 7 s approxmated by φ Pr U R,U N ˆµ, M M C ˆσ, where ˆµ = a = Q Q log ϑ + ϑ Y xf Z xdx ǫ P cons ϑ P max ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ x f G xdx,,n ζ,n ν,s,,n ζ,n ν, +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 a3 x erf a 4 a4 π a e a 3 x a 4, ˆσ = ˆγ ˆµ, = ˆγ = b = T log + ϑ Y xf Z xdx ϑ ǫ P cons P max ϑ ǫ P cons P max ϑ x,,n ζ,n ν,s f G xdx

4 T ϑ ǫ P cons P max ϑ,,n ζ,n ν, +T r,θ r d,n ζ,n ν, T r,θ r d,n ζ,n ν,s, erf T a,a,a 3,a 4,x =. x +a 3 x a3 π e a 4 a4 a 4 a e a 3 x a 4 a a 3 +x+a + πe x +a 3 a 4 a a 3 +a 4+a a a 3 +a. 3 Here, a b n 3 come from approxmatng f G x by f Z x defned n Lemma the approxmaton ǫp log + ϑ cons x Y x where Y x = s = r = ϑ { P max ϑ ϑ ǫp cons x, x [,s P max ϑ ] θ +r x d, x > s, θ ϑ ǫp cons P max ϑ ǫ P cons P max ϑ e θ e θ,d = ϑ ǫp cons P max ϑ e θ ϑ ǫp cons P max ϑ,. 4 Then, usng the CDF of Gaussan rom varables the error functon erfx = x π dt, s determned as e t φ M C R M +erf ˆµ, θ >. 5 ˆσ Note that, n 5, θ > s an arbtrary parameter, based on our smulaton, accurate approxmatons are obtaned for a broad range of θ >. Fnally, the followng lemma represents the outage probablty of the FSO-based hops. Lemma 3: The outage probablty of the FSO-based hop,.e., 8, s approxmately gven by φ M C R M +erf µ. 6 σ Here, µ σ are gven by 7-8 [8, eq ] for the exponental the Gamma-Gamma dstrbutons of the FSO lnks, respectvely. Proof. Usng CLT, the rom varable M mc M C m= log + P c=m C G c s + replaced by ts equvalent Gaussan rom varable R N µ, M C σ, where for the exponental dstrbuton of the FSO lnk we have µ = xlog+ P f G xdx c F = G x λ P + P x dx = e P E λ, 7 P σ = ρ µ, ρ = E{log+ P G } = d = P Here, Ex = e t dt x t represents the exponental ntegral functon. Moreover, c d are obtaned by partal ntegraton. Then, denotng the Euler constant by E, e comes from, varable transform + P x = t, some manpulatons as well as the defnton of Gamma ncomplete functon Γs,x = x ts e t dt the generalzed hypergeometrc functon n F n. For the Gamma-Gamma dstrbuton, on the other h, the PDF n 7-8 s replaced by 5 f G the mean varance are calculated by [8, eq. 43] [8, eq. 44], respectvely. In ths way, followng the same arguments as n Lemma, the outage probablty of the FSO-based hops s approxmated by 6. f G xlog + P xdx e λx + P x log+ P xdx e = H H, H x = e λ λ P x 3 F 3,,;,,; λ x P P + logx λ log x +E P Γ, λ x +logx. 8 P Lemmas -3 lead to the followng corollary statement about the performance of mult-hop RF-FSO systems. Corollary : Wth long codewords, there are mappngs between the performance of FSO- RF-based hops, n the sense that, wth proper parameter settngs, the same outage probablty s acheved n these hops. Proof. The proof comes from Lemmas -3 where for dfferent hops the outage probablty s gven by the CDF of Gaussan rom varables. Thus, wth parameter settngs of ˆµ,ˆσ µ, σ n Lemmas -3, the same outage probablty s acheved n these hops. In ths way, the performance of RF- FSO based mult-hop networks can be mapped to the ones usng only the RF- or the FSO-based communcaton. IV. SIMULATION RESULTS Throughout the paper, we presented dfferent approxmaton technques. The verfcaton of these results s demonstrated n Fg., as seen n the sequel, the analytcal results follow the smulatons wth hgh accuracy. Then, to avod too much nformaton n each fgure, Fgs. -4 report only the smulaton results. Note that n all smulatons we have checked the results wth the ones obtaned analytcally 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 -, the FSO-based hops also experence the same long-term channel parameters,.e., λ,a b n 4-5. Also, we consder M = M j R = R j,,j =,...,T. In all fgures, we set P = N P cons such that the total consumed power at dfferent hops are the

5 Outage probablty Dual-hop network, M =, deal PA, C =, C = SNR db 5-4 Exact result Approxmaton results of Lemmas -3 R = R = Fgure. On the tghtness of the results n Lemmas -3. Ideal PA, dual-hop network, M =,R =,,C =, C = 3,T =, T =,N =. Outage probablty M =3 M = Non-deal PA, ε=.75, ϑ=.5, P max =5 db Ideal PA 3 db gan 7 db gan M = SNR db Fgure. Outage probablty of dual-hop RF-FSO network for dfferent PA models maxmum number of retransmssons, M,. Exponental dstrbuton of the FSO lnk, T =, T =,C =, C =,R = 3 npcu, N = 6,. same. Then, usng 3, one can determne the output of the RFbased antennas. Also, because the nose varances are set to, P n db, log P s referred to the SNR as well. In Fgs. 4, we assume an deal PA. The effect of non-deal PAs s verfed n Fgs. 3. Wth non-deal PAs, we consder the state-of-the-art parameter settngs ϑ =.5,ǫ =.75,P max = 5 db,, unless otherwse stated [9], []. The parameters of the Rcan RF PDF are set to ω =,K =.,, leadng to unt mean varance of the channel gan dstrbuton f j g x,,j. Wth the exponental dstrbuton of the FSO-based hops, we consder f G x = λ e λx wth λ =,. Also, for the Gamma-Gamma dstrbuton we set x = +b f G aba +b Γa Γb xa K a b a b x, a = ,b =.5636,, whch corresponds to Rytov varance [8]. The smulaton results are presented n dfferent parts as follows. On the approxmaton approaches of Lemmas -3: Consderng an deal PA M =,R = npcu, C =,, Fg. verfes the tghtness of the approxmaton schemes of Lemmas -3. Partcularly, we plot the outage probablty of a dual-hop RF-FSO setup versus the SNR. Here, we set M =,R =, npcu, C =, C = 3,N =,T =, T =, the results are presented for the cases wth deal PAs at the RF-based hops. As t s observed, the analytcal results of Lemmas -3 mmc the exact results wth very hgh accuracy. As a result, 5 6 can be effectvely used to analyze the data transmsson effcency of the RF-FSO mult-hop networks, as well as the mult-hop networks wth only the RF- or the FSO-based communcaton. On the effect of HARQ mperfect PAs: Fgure shows the outage probablty of a dual-hop RF-FSO network for dfferent maxmum numbers of HARQ-based retransmsson rounds M,. Also, the fgure compares the system performance n the cases wth deal non-deal PAs. Here, the results are presented for the exponental dstrbuton of the FSO lnk, T =, T =,C =, C =,R = 3 npcu, 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 outage probablty 4, the PAs neffcency ncreases the requred power by 3.5 db. On the other h, the HARQ can effectvely compensate the effect of mperfect PAs, the dfference between the outage probablty of the cases wth deal non-deal PAs s neglgble for M >. Fnally, the HARQ mproves the energy effcency sgnfcantly. As an example, consder the outage probablty 4, an deal PA the parameter settngs of Fg.. Then, compared to the open-loop communcaton,.e., M =, the mplementaton of HARQ wth a maxmum of 3 retransmssons reduces the requred power by 3 7 db, respectvely. System performance wth dfferent numbers of hops: In Fg. 3, we show the outage probablty n the cases wth dfferent numbers of RF- FSO-based hops,.e., T, T. As expected, 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, 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. Intutvely, ths s because at low SNRs 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: Fgure 4 demonstrates the effect of the number of RF transmt antennas on the network outage probablty. Also, the fgure compares the system performance n the cases wth short long codewords,.e., n the cases wth small large values of C, C. Here, we consder Gamma-Gamma dstrbuton of the FSO-based hops, deal PAs, R =.5 npcu, M =,T = T =,, SNR = db. As seen, wth short codewords, the outage probablty decreases wth the number of RFbased transmt antennas monotoncally. Wth long codewords, however, the outage probablty s almost nsenstve to the number of antennas for N 3. Ths s because, wth the parameter settngs of the fgure, the data s correctly decoded

6 Outage probablty T = 4, T = 4 T =, T = T =, T = Gamma-Gamma PDF of FSO hops Non-deal PA, M =, R =, C =, C =, N = Exponental PDF of FSO hops SNR db Fgure 3. The outage probablty for dfferent numbers of RF- FSO-based hops,.e., T T. Non-deal PA, ϑ =.5,ǫ =.75,P max = 5 db, M =,N =,R = npcu, C =, C =,. Outage probablty C = C =, C = C =, C = C = 3, T =, T = T =, T = Number of transmt antennas n RF hops, N Fgure 4. Outage probablty for dfferent number of transmt antennas n the RF-based hops. Gamma-Gamma dstrbuton of the FSO-based hops, deal PA, R =.5 npcu, M =,T = T =,, SNR = db. wth hgh probablty as the number of antennas ncreases. Fnally, the outage probablty decreases wth C, C, because the HARQ explots tme dversty as more channel realzatons are experenced wthn each codeword transmsson. V. CONCLUSION We studed the performance of RF-FSO mult-hop networks usng long codewords. Consderng dfferent channel condtons, we derved closed-form expressons for the network outage probablty n the cases wth wthout HARQ. As demonstrated, there are mappngs between the performance of RF-FSO based mult-hop networks the ones usng only the RF- or the FSO-based communcaton. Moreover, the HARQ can effectvely mprove the energy effcency compensate the effect of hardware mparments. Fnally, whle the outage probablty decreases wth the number of RF transmt antennas, the network outage probablty s not senstve to the number of RF antennas for long codewords. Extenson of our results can be found n [] where we analyze the network ergodc rate the requred number of antennas n dfferent condtons. ACKNOWLEDGEMENT The research leadng to these results receved fundng from the European Commsson H programme under grant agreement n G PPP mmmagic project, from the Swedsh Governmental Agency for Innovaton Systems VINNOVA wthn the VINN Excellence Center Chase. REFERENCES [] M. Usman, H. C. Yang, M.-S. 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