1 On the Performance of Space-Tme MIMO Multplexng for Free Space Optcal Communcatons Mohammad Tagh Dabr, Hman Savojbolaghch and Seyed Mohammad Sajad Sadough Faculty of Electrcal Engneerng, Shahd Behesht Unversty G. C., 1983969411, Tehran, Iran Emal: {m dabr, h savojbolaghch, s sadough}@sbu.ac.r arxv:181.2167v1 [cs.it] 4 Oct 218 Abstract Due to the relatvely hgh cost of deployng optcal fbers, free space optcal (FSO) lnks have been developed as a cost-effectve alternatve technology for next generaton cellular networks. However, n order to have a practcal role n the physcal layer of future communcaton systems, data rate of FSO lnks must be mproved. To ths am, n ths paper we employ a multple-nput multple-output (MIMO) multplexng scheme wth two transcevers to ncrease the data rate of the consdered FSO system. Unlke MIMO dversty case, the performance of MIMO multplexng s sgnfcantly affected by nterference between parallel channels. To solve ths problem, we propose a novel space-tme scheme whch sgnfcantly reduces the nterference between parallel channels. We analyze the performance of the proposed scheme n terms of BER and outage probablty. Index Terms Free space optcal communcatons, atmospherc channels, BER, MIMO, multplexng, dversty. I. INTRODUCTION The ever ncreasng demand for moble data servces and the necessty of more effcent use of the rado spectrum are leadng network operators to ncrease the densty of base statons [1]. Ths densfcaton has become possble by smallcell deployment. Backhaul s needed to connect the small cells to the core network, nternet and other servces [1]. Optcal fber technologes offer hgh capacty whch are suffcent for next-generaton cellular networks. However, due to ts hgh cost, for some network operators, n some places, the use of fber s not always affordable. In such scenaros, free space optcal (FSO) communcaton systems have been developed as a cost-effectve alternatve technology for the backhaul of next-generaton cellular networks [2] [4]. Wth ts sgnfcant advantages such as large avalable bandwdth, low cost mplementaton, global lcense-free feature, low rsk of exposure and robustness to electromagnetc nterference, FSO communcaton has recently attracted a growng attenton for a wde range of applcatons [5], [6]. To ncrease the data rate of FSO lnks, recent efforts show that multple-nput multple-output (MIMO) spatal multplexng can sgnfcantly mprove the transmsson rate (see for nstance [7] [14]). In [7], the authors compared the spectral effcency of conventonal MIMO multplexng and spatalmode multplexng wth that provded by orbtal angular momentum (OAM) multplexng over turbulence channels. In [8], the authors nvestgated the nterest of spatal multplexng n MIMO FSO systems and compared ts performance to those acheved wth repetton codng orthogonal space tme block codes (OSTBC) and optcal spatal modulaton. In [1], the dversty-multplexng trade-off for log-normal channels were analyzed to optmze both dversty and multplexng gans when usng coherent modulatons and heterodyne recevers for FSO systems. In [11], the multplexng gan has been nvestgated for MIMO FSO systems when usng ntensty modulaton wth drect detecton (IM/DD). More precsely, n [11], the multple transmtted beams generate ndvdual ary patterns on the detector plane whch are separated due to the dfference n the angle of arrval (AOA). In [12], spatal-mode multplexng for practcal FSO systems usng DD s nvestgated. In [13] for two transcever pars and n [14] for M transcever pars, the authors have employed a spatal multplexng scheme to ncrease the data rate of an FSO system. More precsely, n [13], [14], assumng a Gaussan modulaton (whch s not a practcal assumpton), the performance of the proposed scheme s analyzed n terms of average achevable data rate. In order to complete recent results n [13], [14], n the frst part of ths paper we analyze the performance of MIMO multplexng scheme ntroduced n [13], [14] n terms of BER and outage probablty for pulse poston modulaton (PPM) sgnalng whch s a wdely used dgtal modulaton schemes n FSO systems. More precsely, we wll show how the tunable parameters such as beam wast at the recever can affect the performance of the consdered system. Unlke MIMO dversty case, the performance of MIMO multplexng s sgnfcantly degraded by the nterference between parallel channels. To solve ths ssue, n the second part of ths paper, we propose a novel space-tme scheme whch sgnfcantly reduces the nterference between parallel channels and mproves the performance of MIMO multplexng case, yet preservng the smplcty of the prevous scheme. II. SYSTEM MODEL We consder a MIMO-FSO communcaton system wth two transmtters (two laser sources) and two recever apertures, where each transmtter sends optcal sgnals toward the center of ts correspondng recever. At the transmtters, IM/DD wth PPM s exploted to modulate the optcal transmtted sgnals. A PPM scheme uses the poston of a pulse n two tme-slots to represent the value of an nformaton bt,.e., the presence
2 of a pulse n the frst tme-slot s characterzed by a 1 and n the second tme-slot s characterzed by a. The receved sgnal at the th recever s denoted r for {1,2} (2 s the number of transcever pars) s expressed at any dscrete symbol tme as [ ] r (1) r = r (2) Rg h P t T s s (1) + 2 j=1 j = Rg h P t T s s (2) + 2 j=1 j Rg j h j P t T s s (1) j +n (1) Rg j h j P t T s s (2) j +n (2). where s (1) {,1} and s (2) = 1 s (1) are respectvely the transmtted sgnals n the frst and second tme-slot correspondng to the BPPM symbol; r (1) and r (2) are the receved electrcal sgnals n the frst and second tme-slot. We consder two spatal sgnalng method: ) MIMO multplexng where each transmtter sends ndependent optcal sgnals and ) MIMO dversty or repetton codng where all transmtters send same optcal sgnals,.e., s (1) 1 = s (1) 2. In the case of MIMO dversty, (1) can be smplfed as r = [ r (1) r (2) ] = [ 2 j=1 Rg jh j P t T s s (1) j +n (1) 2 j=1 Rg jh j P t T s (1 s (1) j )+n (2) where R s the photo detector s responsvty, P t s the transmtted sgnal power, n (1) and n (2) are the sgnal-ndependent addtve whte Gaussan nose (AWGN) wth zero mean and varance σn=n 2 T s /2, T s denotes the tme-slot duraton and h j s the atmospherc turbulence coeffcent between jth transmtter and th recever whch s assumed perfectly known at the recever. Notce that ths s a practcal assumpton due to the slow fadng property of FSO lnks [15] [17]. Moreover, for h j = 1, the fracton of the collected power at th recever due to transmtted sgnal by jth transmtter can be wrtten as ( ) g j = A exp 2 (x p +d) 2 +(y p ) 2 wz 2, (3) eq where [d,] are dstance between two recevers n the [x,y] plane, A = (erf(ν)) 2 denotes the maxmal fracton of the collected power, ν = πr a πerf(ν) 2wz, wz 2 eq = wz 2 2νexp( ν 2 s the ) equvalent beamwdth and erf(z) = 2 z π e x2 dx s the error functon. At the recever aperture plane, we can express the radal dsplacement vector as r p = [x p,y p ], where x p and y p, denote respectvely the dsplacements located along the horzontal and elevaton axes at the recever whch can be modeled as zero mean Gaussan dstrbuted random varables (RV),.e., x p N(,σxp) 2 and y p N(,σyp). 2 Lastly, we consder the well-known gamma-gamma dstrbuton for modelng the atmospherc turbulence. Ths way, the PDF of the normalzed channel coeffcent h s gven by [18] (1) ], (2) 2(αβ)α+β 2 f h (h) = hα+β 2 1 k α β (2 αβh), (4) Γ(α)Γ(β) where Γ(.) s the gamma functon, k m (.) s the modfed Bessel functon of second knd of order m, 1/β and 1/α are the varances of the small and large scale eddes, respectvely. A. BER Analyss III. PERFORMANCE ANALYSIS 1) MIMO Multplexng: Accordng to (1) and after some mathematcal calculatons, the average BER of MIMO multplexng can be obtaned as where and P e,m = P e,m p = P e,m p f xp (x p )f yp (y p )dx p dy p, (5) P e,m p,h f h11 (h 11 )f h21 (h 21 )dh 11 dh 21, P e,m p,h = 1 ( ) 2 Q RPt Ts (g 11 h 11 +g 21 h 21 ) (6) + 1 ( ) 2 Q RPt Ts (g 11 h 11 g 21 h 21 ). (7) 2) MIMO Dversty: Accordng to (2), the BER of MIMO dversty condtoned on h and r p can be obtaned as ( 2 2 RPt Ts =1 j=1 P e,d p,h = Q g ) jh j. (8) 2 Fnally, substtutngp e,d p,h n (6) and (5) nstead ofp e,m p,h, the average BER of MIMO dversty s obtaned. B. Outage Probablty Analyss 1) MIMO Multplexng: For a gven target BER P e,t, the outage probablty s defned as the probablty that the communcaton system can not support P e,t [19]. Accordng to (7) and for low values of outage probablty (for outage probablty lower than 1 2 ), the outage probablty of MIMO multplexng case can be closely expressed as P M out = Prob { P e,m p,h > P e,t } Prob{g 11 h 11 g 21 h 21 < A th1 }, (9) where A th1 = N RP t Ts Q 1 (2P e,t ). 2) MIMO Dversty: Accordng to (8), the outage probablty of MIMO dversty case can be obtaned as Pout D = Prob{ } P e,d p,h > P e,t 2 2 = Prob g j h j < A th2, (1) =1 j=1 where A th2 = 2N RP t Ts Q 1 (P e,t ).
3 C. Smulaton Results In ths part, the performance of MIMO multplexng and MIMO dversty communcaton systems are numercally studed and the behavor of the consdered system s studed versus dfferent tunable parameters such as beam wast at the recever and dstance between recevers. The values of the parameters used for our numercal analyss are set as follows: aperture radus of receverr a = 1 cm, optcal wavelengthλ = 1.5 µm, lnk length 1 km, slot duraton T s = 1 ns, α = 11.7 and β = 1.2. The average BER of MIMO multplexng and MIMO dversty cases versus SNR s depcted n Fg. 1 for dfferent values of d = 1.2, 1,.8,.6,.4 m. As expected, MIMO dversty case has better performance compared to the MIMO multplexng case. However, note that the transmtted rate of multplexng case s twce as large as dversty case. As we observe from Fg. 1, by ncreasng d, the performance of MIMO multplexng case mproves, however, the performance of MIMO dversty s a decreasng functon of d. The reason for ths s that the nterference between two parallel channels ncreases by decreasng d. Another tunable parameter whch sgnfcantly affects the performance of the consdered MIMO system s the optcal beam wast at the recever w z. To show the effect of w z on the performance of the consdered system, the average BER versus SNR s plotted n Fg. 2 for dfferent values of w s =.6,.8,.9,1,1.2 m. As we observe from Fg. 2, the performance of both multplexng and dversty system sgnfcantly depends on w z. For nstance, n the cases of MIMO multplexng and dversty, the best performance s acheved for w z = 1.2. At frst t mght be thought that, the performance of both dversty and multplexng cases are mproved by ncreasng w z. To clarfy ths pont, n Fgs. 3 and 4, the BER of consdered system are depcted versus w z and d, respectvely. Fgure 3 shows that, the optmum value of w z for both cases depend on d. For nstance, for MIMO multplexng case, the optmum values for w z are.95,.85 and.65 m for d=1,.75 and.5 m, respectvely. For MIMO dversty, the optmum values for w z are 1.15, 1.1 and 1 m for d=1,.75 and.5 m, respectvely. Moreover, results presented n Fg. 4 confrm that the performance of MIMO multplexng s an ncreasng functon of d whle, MIMO dversty s a decreasng functon of d. IV. SPACE-TIME SIGNALING FOR MIMO MULTIPLEXING It s well known that the nterference between two parallel channels degrades the performance of MIMO multplexng case. To reduce the nterference between parallel channels, n the sequel we propose a new space-tme codng scheme for MIMO multplexng case when the optcal sgnals are modulated by PPM. The dea behnd ths scheme s that the start tme of two parallel channels to transfer optcal sgnal are dfferent. We consder the start tme of frst transmtter s zero and second transmtter s startng to send optcal sgnal after a delay equal to T s /2. Fgure 5 s provded for two ndependent sgnal sequence where the frst sequence s transmtted by 5 1 15 2 25 Fg. 1. Average BER of MIMO multplexng and dversty cases versus SNR for dfferent values of d. 5 1 15 2 25 Fg. 2. Average BER of MIMO multplexng and dversty cases versus SNR for dfferent values of w z. the frst transmtter at start tme equal to zero and the second sequence s transmtted by the second transmtter at a start tme equal to T s /2. In Fg. 6, we have also depcted the receved sgnal equvalent to the transmtted sequence n Fg. 5 wthout consderng the effect of recever noses and when h j = 1 and g j = 1 for,j {1,2}. Ths sgnalng at the transmtter causes approxmately same nterference n two tme-slot of PPM. For nstance, when the second transmtter sends bt 1, at the frst recever, the nterference n frst and second slot of PPM are Rg 21 h 21 P t T s /2. When the second transmtter sends bt, at the frst recever, the nterference depends on the
4 Frst: second: Fg. 5. Start tme of frst and second transmtters are zero and T s/2, respectvely..5 1 1.5 Fg. 3. Average BER of MIMO multplexng and dversty cases versus w z for dfferent values of d. Frst s propagaton delay Second Fg. 6. Receved sgnal equvalent to the transmtted sequences n Fg. 5 wthout consderng the effect of recever noses and when h j = 1 and g j = 1 for,j {1,2}..5 1 1.5 Fg. 4. Average BER of MIMO multplexng and dversty cases versus d for dfferent values of w z. next transmtted bt of second transmtter. For transmtted bts, and, 1, the nterference at the frst slot of frst recever are Rg 21 h 21 P t T s /2 and Rg 21 h 21 P t T s, respectvely, and the nterference at the second slot are Rg 21 h 21 P t T s /2 and Rg 21 h 21 P t T s /2, respectvely. The BER condtoned on h and r p of the consdered space-tme scheme s obtaned as P e,st p,h = 1 ( ) 2 Q Rg11 h 11 P t Ts + 1 ( ) 4 Q Rg11 h 11 P t Ts + 1 ( ) 8 Q RPt Ts (g 11 h 11 +g 21 h 21 /2) + 1 ( ) = 3 ( ) 4 Q Rg11 h 11 P t Ts + 1 ( ) 8 Q RPt Ts (g 11 h 11 +g 21 h 21 /2) + 1 ( ). (11) Fnally, substtutng P e,st p,h n (6) and (5) nstead of P e,m p,h, the average BER of consdered space-tme scheme s obtaned. For low values of d n whch the nterference between channels s large, (11) can be closely approxmated as P e,st p,h 1 ( ). (12) Accordng to (12) and for low values of outage probablty (for outage probablty lower than 1 2 ), outage probablty
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