LDPC coded OFDM over the atmospheric turbulence channel

Transcription

1 LDPC coded OFDM over the atmospherc turbulence channel Ivan B. Djordjevc, Bane Vasc, and Mark A. Nefeld Unversty of Arzona, Department of Electrcal and Computer Engneerng, Tucson, AZ 8571, USA Abstract: Low-densty party-check (LDPC) coded optcal orthogonal frequency dvson multplexng (OFDM) s shown to sgnfcantly outperform LDPC coded on-off keyng (OOK) over the atmospherc turbulence channel n terms of both codng gan and spectral effcency. In the regme of strong turbulence at a bt-error rate of 1-5, the codng gan mprovement of the LDPC coded sngle-sde band unclpped-ofdm system wth 64 sub-carrers s larger than the codng gan of the LDPC coded OOK system by.db for quadrature-phase-shft keyng (QPSK) and by 3.4dB for bnary-phase-shft keyng (BPSK). 7 Optcal Socety of Amerca OCIS codes: (6.451) Optcal communcatons; (1.133) Atmospherc turbulence; (6.48) Modulaton; (6.43) Multplexng; ( ) Orthogonal frequency dvson multplexng; ( ) Low-densty party-check (LDPC) codes References and Lnks 1. X. Zhu, and J. M. Kahn, Free-space optcal communcaton through atmospherc turbulence channels, IEEE Trans. Commun. 5, ().. M. A. Al-Habash, L. C. Andrews, and R. L. Phllps, Mathematcal model for the rradance probablty densty functon of a laser beam propagatng through turbulent meda, Opt. Eng. 4, (1). 3. X. Zhu, and J. M. Kahn, Markov chan model n maxmum-lkelhood sequence detecton for free-space optcal communcaton through atmospherc turbulence channels, J. Lghtwave Technol. 51, (3). 4. M.-C. Jeong, J.-S. Lee, S.-Y. Km, S.-W. Namgung, J.-H. Lee, M.-Y. Cho, S.-W. Huh, Y.-S. Ahn, J.-W. Cho, and J.-S. Lee, 8x1 Gb/s terrestral optcal free-space transmsson over 3.4 km usng an optcal repeater, IEEE Photon. Technol. Lett. 15, (3). 5. J. A. Anguta, I. B. Djordjevc, M. A. Nefeld, and B. V. Vasc, Shannon capactes and error-correcton codes for optcal atmospherc trubulent channels, J. Opt. Net. 4, (5). 6. R. Van Nee and R. Prasad, OFDM Wreless Multmeda Communcatons, (Artech House, Boston ). 7. Y. Wu and B. Caron, Dgtal televson terrestral broadcastng, IEEE Commun. Mag. 3, 46 5 (1994). 8. Q. Pan and R. J. Green, Bt-error-rate performance of lghtwave hybrd AM/OFDM systems wth comparson wth AM/QAM systems n the presence of clppng mpulse nose, IEEE Photon. Technol. Lett. 8, 78 8 (1996). 9. A. Km, Y. Hun Joo, and Y. Km, 6 GHz wreless communcaton systems wth rado-over-fber lnks for ndoor wreless LANs, IEEE Trans. Commun. Electron. 5, (4). 1. B. J. Dxon, R. D. Pollard, and S. Iezekel, Orthogonal frequency-dvson multplexng n wreless communcaton systems wth multmode fber feeds, IEEE Trans Mcrowave Theory Tech. 49, (1). 11. I. B. Djordjevc, O. Mlenkovc, and B. Vasc, Generalzed low-densty party-check codes for Optcal Communcaton Systems, J. Lghtwave Technol. 3, (5). 1. I. B. Djordjevc and B. Vasc, Nonbnary LDPC codes for optcal communcaton systems, IEEE Photon. Technol. Lett. 17, 4-6 (5). 13. O. Mlenkovc, I. B. Djordjevc, and B. Vasc, Block-crculant low-densty party-check codes for optcal communcaton systems, J. Sel. Top. Quantum Electron. 1, (4). 14. R. You and J. M. Kahn, Average power reducton technques for multple-subcarrer ntensty-modulated optcal sgnals, IEEE Trans. Commun. 49, (1). 15. H. X.-Yu, E. Eleftherou, D.-M. Arnold, and A. Dholaka, Effcent mplementatons of the sum-product algorthm for decodng of LDPC codes, n Proc. IEEE Globecom 1, E (1). 16. R. Hu, B. Zhu, R. Huang, C. T. Allen, K. R. Demarest, and D. Rchards, Subcarrer multplexng for hghspeed optcal transmsson, J. Lghtwave Technol., (). 17. J. G. Proaks, Dgtal Communcatons (McGraw Hll, Boston 1). (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 633

2 18. W. E. Ryan, Concatenated convolutonal codes and teratve decodng, n Wley Encyclopeda n Telecommuncatons, J. G. Proaks, ed., (John Wley and Sons, 3). 19. C.-C. Ln, K.-L. Ln, H.-Ch. Chang, and C.-Y. Lee, A 3.33Gb/s (1,7) low-densty party check code decoder, n Proc. ESSCIRC 5, (5).. L. C. Andrews and R. L. Phlps, Laser beam propagaton through random meda, (SPIE Optcal Engneerng Press, 1998). 1. N. Levnson, The Wener RMS error crteron n flter desgn and predcton, J. Math. Phys. 5, (1947).. J. Durbn, Effcent estmaton of parameters n movng-average models, Bometrca 46, (1959). 3. A. T. A Wood and G. Chan, Smulaton of statonary Gaussan processes n [,1] d, J. Comp. Graph. Stat. 3, (1994). 1. Introducton Due to the hgh-complexty assocated wth coherent detecton, current free-space optcal (FSO) communcaton systems [1-5] employ ntensty modulaton wth drect detecton (IM/DD). Such systems use pont-to-pont communcaton between two optcal transcevers along a lne of sght. For example, a 8 1 Gb/s terrestral FSO transmsson over 3.4 km usng an optcal repeater was demonstrated n [4]. The IM/DD technque s also used n stateof-the-art fber-optc communcatons, and the avalablty of optcal components used n fber-based systems makes FSO communcaton a cost-effectve soluton for hgh-rate mage, voce and data transmsson [1-5]. However, an optcal wave propagatng through the ar experences fluctuatons n ampltude and phase due to atmospherc turbulence [1-5]. The atmospherc turbulence s caused by varatons n the refractve ndex of the transmsson medum due to nhomogentes n temperature and pressure caused by solar heatng and wnd. The atmospherc turbulence optcal channel has been ntensvely studed and varous models have been proposed to descrbe turbulence-nduced performance degradaton and ntensty fluctuatons [1-3]. The ntensty fluctuaton, also known as scntllaton, s one of the most mportant factors that degrade the performance of an FSO communcaton lnk. Due to constrants on the recever sze, t s not always possble to ensure that the recever aperture s sgnfcantly larger than the turbulent correlaton length. In such a case aperture averagng becomes neffectve, and alternatve technques to mtgate the ntensty fluctuatons are requred [1]. These technques can be placed nto two broad categores. Spatal-doman technques [1] nvolve dversty detecton usng multple recevers, and tme-doman technques [3] adaptvely optmze the decson threshold accordng to the maxmum lkelhood crteron. When the recever has knowledge of the jont temporal dstrbuton of ntensty fluctuatons, maxmum-lkelhood sequence detecton (MLSD) can be employed. MLSD has hgh computatonal complexty, and sub-optmal mplementatons of MLSD such as those based on sub-optmal per-survvor processng (PSP) [3] are more lkely to be mplemented n practce. At bt-error rate (BER) below 1-6 both MLSD and PSP requre the electrcal sgnal to nose rato larger than db even n the weak turbulence regme. Such sgnal powers are unacceptably hgh for many applcatons, and novel modulaton technques for IM/DD FSO systems are needed. In ths paper we show that orthogonal frequency dvson multplexng (OFDM) combned wth error control codng s a very good modulaton format for FSO IM/DD systems. OFDM [6-9] s a specal case of a multcarrer transmsson n whch a sngle nformatonbearng stream s transmtted over many lower rate subchannels. It has been used for dgtal audo broadcastng [6], hgh-defnton televson (HDTV) terrestral broadcastng [7], n dgtal subscrber lne (DSL) systems [6], n IEEE 8.11, n hgh-performance LAN type (HIPERLAN/) and multmeda moble access communcaton wreless LANs [6], and has been studed for use n lghtwave hybrd AM/OFDM cable systems [8], and n rado over fber-based networks [9,1]. It s nterestng to notce that modern dgtal TV broadcastng s based on OFDM [7], and that OFDM s ntensvely studed for wreless applcatons [6]. Because the FSO lnk s a cost-effectve soluton for transmsson of hgh-speed sgnals, the study of OFDM transmsson over the FSO lnk s becomng ncreasngly mportant. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6333

3 OFDM uses the fast Fourer transform (FFT) algorthm for modulaton and demodulaton, and requres no equalzaton. At the same tme t provdes hgh spectral effcency. These features together wth ts mmunty to burst-errors due to ntensty fluctuatons make OFDM an nterestng canddate for FSO transmsson. Recently we have shown [5], [11-1] that sgnfcant performance mprovement can be obtaned by usng low-densty party-check (LDPC) codes and teratve decodng based on a sum-product algorthm that does not requre knowledge of the jont temporal probablty dstrbuton functons. LDPC codes have been shown to acheve mpressve codng gans for a varety channels [11-1]. They perform sgnfcantly better than turbo-product and Reed-Solomon (RS) codes n bursty-error channels such as the fber optcs communcaton channel at 4 Gb/s or above [11-1] and the FSO channel [5]; makng them an excellent error control codng scheme to combne wth OFDM. In ths paper we propose an LDPC coded free-space optcal OFDM (FSO-OFDM) system bult usng standard optcal and RF components. The key dea s to lower the symbol rate by usng OFDM, and, n combnaton wth nterleavng and LDPC codes, to obtan hgh tolerance to the deep fades that are nherent to a turbulent channel. On the other hand, OFDM s more senstve to phase nose, and has a relatvely large peak-to-average power rato [6]; so a careful desgn s needed n order to fully explot the advantages and mnmze the sde effects of OFDM. In partcular we note that the use of OFDM n the IM/DD system s not very power effcent f mplemented as suggested n [14]. To mprove the power effcency, we ntroduce sngle-sde band clpped- and unclpped-ofdm schemes. The paper s organzed as follows. The concept of FSO-OFDM transmsson s ntroduced n Secton, the smulaton model and error control codng scheme are descrbed n Sectons 3 and 4 respectvely; whle, the numercal results are presented n Secton 5. Secton 6 concludes the paper.. FSO-OFDM transmsson system FSO-OFDM systems support hgh data rates by splttng a hgh-rate data-stream nto a number of low-rate data-streams and transmttng these over a number of narrowband subcarrers. The narrowband subcarrer data-streams experence smaller dstortons than hgh-speed ones and requre no equalzaton. Moreover, most of the requred sgnal processng s performed n the RF doman. Ths s advantageous because mcrowave devces are much more mature than ther optcal counterparts and because the frequency selectvty of mcrowave flters and the frequency stablty of mcrowave oscllators are sgnfcantly better than that of correspondng optcal devces. Furthermore, the phase nose levels of mcrowave oscllators are sgnfcantly lower than that of dstrbuted feedback (DFB) laser dodes, whch means that RF coherent detecton s easer to mplement than optcal coherent detecton. Ths, n turn, allows a system archtect to drectly apply the most advanced coherent modulaton formats already developed for wreless communcaton. The basc FSO-OFDM transmtter and recever confguratons are shown n Fg. 1(a) and 1(b) respectvely. The correspondng FSO lnk s shown n Fg. 1(c). A 1-Gb/s nformatonbearng stream s demultplexed nto four.5-gb/s streams, each encoded by dentcal LDPC encoders (for an example of current avalable LDPC chps see [19]). The LDPC encoded outputs are further demultplexed, and parsed nto groups of B bts. The B bts n each group (frame) are subdvded nto K subgroups wth the th K subgroup contanng b bts, B = b. = 1 The b bts from the th subgroup are mapped nto a complex-valued sgnal from a b -pont sgnal constellaton such as quadrature-ampltude modulaton (QAM), whch s consdered n ths paper. The complex-valued sgnal ponts from K subchannels are consdered to be the values of the dscrete Fourer transform (DFT) of a multcarrer OFDM sgnal (for more detals see Ref. [6]). Therefore, the symbol length (the tme between two consecutve OFDM symbols) n an OFDM system s T=KT s, where T s s the symbol-nterval length n an equvalent sngle-carrer system. By selectng K, the number of subchannels, suffcently large, the OFDM symbol nterval can be made sgnfcantly larger than the dspersed pulsewdth n a sngle-carrer system, resultng n an arbtrarly small ntersymbol nterference. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6334

4 Followng the descrpton gven n Ref. [6], the complex envelope of a transmtted OFDM sgnal can be wrtten as st () = sofdm () t+ b, (1) where N ( ) FFT / 1 j π t kt TFFT jπ frf t s t = Re w t kt X e e s defned for () ( ) OFDM k, k= = N / FFT kt TG / Twn t kt + TFFT + TG /+ T. wn In the above expresson X,k denotes the k-th OFDM symbol n the -th subcarrer, w(t) s the wndow functon, and f RF s the RF carrer frequency. The duraton of the OFDM symbol s denoted by T, whle T FFT s the FFT sequence duraton, T G s the guard nterval duraton (the duraton of cyclc extenson), and T wn s the length of the wndowng nterval. The detals of the resultng OFDM symbol are shown n Fgs. 1(d)-1(e). The symbols are constructed as follows. N QAM (=K) consecutve nput QAM symbols are zero-padded to obtan N FFT (= m,m>1) nput samples for nverse fast Fourer transform (IFFT), then N G samples are nserted to create the guard nterval T G and fnally the OFDM symbol s multpled by the wndow functon (rased cosne functon s used n Ref. [6], but the Kaser, Blackman-Harrs and other wndow functons are also applcable). The purpose of the cyclc extenson s to preserve the orthogonalty among subcarrers when the neghborng OFDM symbols partally overlap, and the purpose of the wndowng s to reduce the out-of band spectrum. The cyclc extenson, llustrated n Fg. 1(d), s performed by repeatng the last N G / samples of the FFT frame (of duraton T FFT wth N FFT samples) as the prefx, and repeatng the frst N G / samples (out of N FFT ) as the suffx. (Notce that wndowng s more effectve for smaller numbers of subcarrers.) After a D/A converson and RF up-converson, we convert the RF sgnal to the optcal doman usng one of two optons: () for symbol rates up to 1 Gsymbols/s the OFDM sgnal drectly modulates the DFB laser, and () for symbol rates above 1 Gsymbols/s the OFDM sgnal drves the Mach-Zehnder modulator (MZM). The DC component [b n Eq. (1)] s nserted to enable noncoherent recover of the QAM symbols. In the remnder of ths Secton three dfferent OFDM schemes are presented..1 Based-OFDM sngle-sde band scheme The frst scheme s based on ntensty modulaton, and shall be referred to as the based- OFDM (B-OFDM) scheme. Because bpolar sgnals cannot be transmtted over an IM/DD lnk, t s assumed that the bas component b s suffcently large so that when added to s OFDM (t) the resultng sum s non-negatve. For llustratve purposes the DFB laser drvng sgnal (whch s dentcal to the MZM RF nput sgnal of schemes B and C) s shown n Fg. (a). The man dsadvantage of the B-OFDM scheme s the poor power effcency.. Clpped-OFDM sngle-sde band scheme To mprove the power effcency we propose two alternatve schemes. The frst of these, whch we shall refer to as the clpped-ofdm (C-OFDM) scheme, s based on sngle-sde band (SSB) transmsson, wth clppng of the negatve porton of the OFDM sgnal after bas addton. The bas s vared to fnd the optmum one for fxed optcal launched power. It was found that the optmum case s one n whch ~5% of the total electrcal sgnal energy before clppng s allocated for transmsson of a carrer. The MZM RF nput sgnal for C-OFDM scheme s shown n Fg. (b). (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6335

5 .3 Unclpped-OFDM sngle-sde band scheme The second power-effcent scheme, whch we shall refer to as the unclpped-ofdm (U- OFDM) scheme, s based on SSB transmsson employng LNbO 3 MZM n a fashon smlar to that used n duobnary optcal transmsson. To avod dstorton due to clppng, the nformaton bearng sgnal s transmtted by modulatng the electrcal feld (nstead of ntensty modulaton employed n the B-OFDM and C-OFDM schemes) so that the negatve part of the OFDM sgnal s transmtted to the photodetector. Dstorton ntroduced by the photodetector, caused by squarng, s successfully elmnated by proper flterng, and the recovered sgnal dstorton s nsgnfcant. Notce that U-OFDM s less power effcent than C-OFDM because the negatve porton of the OFDM sgnal s transmtted and then dscarded [see Fg. (c)]. For U-OFDM the detector nonlnearty s compensated by post-detecton flters that reject (potentally useful) sgnal energy and compromse power effcency. Despte ths drawback we fnd that U-OFDM s stll sgnfcantly more power effcent than B-OFDM. Note that the DC bas shfts the average of the C-OFDM sgnal towards postve values, whle n the case of B-OFDM a much larger bas s needed to completely elmnate the negatve porton of the sgnal. The MZM RF nput sgnal for U-OFDM s shown n Fg. (c), and the recovered constellaton dagram for 16-QAM SSB s shown n Fg. (g). The transmtted sgnal s recovered wth neglgble dstorton. 1-Gb/s data stream 1x4 DEMUX.5-Gb/s data streams LDPCE LDPCE DEMUX Constellaton Mapper IFFT P/S converter D/A converter RF upconverter DC bas addton + Clppng DFB MZM to FSO lnk from FSO lnk LDPC encoders (a) PD Carrer suppresson RF and downconverter A/D converter FFT (b) Expandng FSO-OFDM telescope transmtter Fber Input data optcal amp. Collmatng lens N G / samples N G / samples Orgnal N FFT samples LDPCD Bt relablty DEMUX calculaton LDPCD LDPC decoders FSO-OFDM Detector recever Output data Compressng Lght beam through telescope turbulent channel (c) T G / T FFT T G / 1-Gb/s data stream 4:1 MUX Preffx Suffx Effectve part T wn OFDM symbol after cyclc extenson, N FFT + N G samples T=T FFT +T G +T wn OFDM symbol duraton (d) (e) Fg. 1. FSO-OFDM system: (a) transmtter confguraton, (b) recever confguraton, (c) FSO lnk, (d) OFDM symbol cyclc extenson, (e) OFDM symbol after wndowng. LDPCE-LDPC encoder, LPDCD-LDPC decoder, DFB-dstrbuted-feedback laser, MZM-Mach-Zehnder modulator, S/P-seral-to-parallel, P/S-parllel-to-seral. kt (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6336

6 The pont-to-pont FSO system consdered here, shown n Fg. 1(c), conssts of an FSO- OFDM transmtter, propagaton medum and an FSO-OFDM recever. The modulated beam s projected toward the recever usng the expandng telescope. At the recever, an optcal system collects the lght, and focuses t onto a detector, whch delvers an electrcal sgnal proportonal to the ncomng optcal power. Notce that no aperture averagng s appled. The recever commonly employs the transmpedance desgn, whch s a good compromse between nose and bandwdth. A PIN photodode plus preamplfer or an avalanche photodode are typcally used as optcal detectors. Durng propagaton through the ar, the optcal beam experences ampltude and phase varatons caused by scatterng, refracton caused by atmospherc turbulence, absorpton, and buldng sway. The photodode output current can be wrtten as { } () t a() t s * OFDM () t a() t b = R + = R a() t sofdm () t + a() t b + Re a() t sofdm () t a () t b,() where a(t) denotes the ntensty fluctuaton due to atmospherc turbulence, and R denotes the photodode responsvty. The sgnal after RF down-converson and approprate flterng, can be wrtten as () = () ( ω ) ( τ) + ( ) r t t krf cos RFt he n t, () where h e (t) s the mpulse response of the low-pass flter (havng the transfer functon H e (jω)), n(t) s electronc nose n the recever, commonly modeled as a Gaussan process, k RF denotes the RF downconverson factor, and the s the convoluton operator. Fnally, after the A/D converson and cyclc extenson removal, the transmtted sgnal s demodulated by the FFT algorthm. The soft outputs of the FFT demodulator are used to estmate the bt relabltes that are fed to four dentcal LDPC teratve decoders based on the sum-product algorthm [15]. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6337

7 MZM RF nput, v RF [V] Power spectral densty, PSD [dbm/hz] Based-OFDM 1 3 Tme, t [ps] (a) MZM output, SSB OFDM Frquency relatve to optcal carrer, f [GHz] (d) Power spectral densty, PSD [dbm/hz] SSB, SCM MZM RF nput, v [V] Power spectral densty, PSD [dbm/hz] PD output PSD Frequency, f [GHz] (g) Tme, t [ps] (b) Clpped-OFDM Frequency, f [GHz] (e) Power spectral densty, PSD [dbm/hz] MZM RF nput, v [V] Imagnary axs MZM output, DSB OFDM Frquency relatve to optcal carrer, f [GHz] (h) 3 1 Unclpped-OFDM 1 3 Tme, t [ps] (c) Real axs (f) Fg.. Waveforms and power spectral denstes of the SSB OFDM sgnal wth 64 sub-carrers at dfferent ponts durng transmsson of an OFDM sgnal n a back-to-back confguraton: (a) MZM RF nput for B-OFDM, (b) MZM RF nput for C-OFDM, (c) MZM RF nput for U- OFDM, (d) PSD after MZM (U-OFDM), (e) photodectector (PD) output PSD (U-OFDM), (f) recever constellaton dagram for 16-QAM (U-OFDM), (g) PSD of SCM sgnal wth four OFDM channels (U-OFDM). The PSD of double-sde band OFDM sgnal after MZM s shown n Fg. (h) (U-OFDM). The parameters of the overall OFDM-FSO system must be carefully chosen, as explaned later n ths secton, so that the reconstructed sequence constellaton dagram suffers mnmal dstorton n a back-to-back confguraton (when the nfluence of the channel s neglected). For the sake of llustraton, consder the sgnal waveforms and power-spectral denstes (PSDs) at varous ponts n the OFDM system gven n Fg.. These examples were generated usng SSB transmsson n a back-to-back confguraton. The bandwdth of the OFDM sgnal s set to.5 GHz, and the RF carrer to 7.5 GHz. The number of OFDM sub-channels s set to 64. The OFDM sequence s zero-padded and the FFT s calculated usng 18 ponts. The guard nterval s obtaned by a cyclc extenson of x16 samples as explaned above. The wndowng (x16 samples) s based on the Blackman-Harrs wndowng functon. The average transmtted launched power s set to dbm. The RF drver amplfer and MZM operate n lnear regme [see Fgs. (a)-(c)]. The PSD for an SSB OFDM MZM output sgnal s shown n Fg. (d), and the photodetector output sgnal (for SSB OFDM transmsson) s shown n Fg. (e). The OFDM term after beatng n the PD [the thrd term n Eq. ()], the low-pass term, and the squared OFDM term [the frst term n Eq. ()] can easly be dentfed. If a 16-QAM OFDM system employng 64 sub-carrers s used n combnaton wth 39 Mb/s sub-channels, the OFDM system proposed here allows transmttng a 1 Gb/s sgnal (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6338

8 over a.5 GHz bandwdth, thereby ncreasng the spectral effcency of OOK. To mprove the spectral effcency further, OFDM may be combned wth sub-carrer multplexng (SCM) n a smlar fashon as proposed for fber-optc communcaton [16]. In ths case [PSD shown n Fg. (g)] the spectral effcency of 4 1Gb/s/11.5GHz =3.55bts/s/Hz s acheved, whch s sgnfcantly better than that for OOK transmsson over an FSO lnk. (For llustratve purposes, the PSD of DSB OFDM sgnal s also provded [see Fg. (h)]. Ths hgh-level descrpton of the system allows us now to ntroduce a statstcal model of atmospherc turbulence based on gamma-gamma dstrbuton (Secton 3), and an effcent LDPC error correcton scheme based on block-crculant LDPC codes sutable for combnng wth OFDM (Secton 4). 3. An atmospherc turbulence model One turbulence model that s commonly used n the lterature assumes that the varatons n medum temperature and pressure due to solar heatng and wnd can be understood as ndvdual cells of ar or eddes of dfferent dameters and refractve ndces [1-5]. These eddes may be consdered as lenses randomly refractng the optcal wave front, and producng a dstorted ntensty profle at the recever sde of an FSO communcaton lnk. The most wdely accepted theory of turbulence s due to Kolmogorov [5], and t assumes that knetc energy from large turbulent eddes, characterzed by the outer scale L, s transferred wthout loss to eddes of decreasng sze down to szes of a few mllmeters characterzed by the nner scale l. The nner scale represents the cell sze at whch energy s dsspated by vscosty. The refractve ndex vares randomly across the dfferent turbulent eddes and causes phase and ampltude perturbatons to a propagatng optcal wave front. Turbulence can also cause random drfts of optcal beams a phenomenon usually referred to as wanderng and can nduce beam focusng. In our study t s assumed that the outer scale s nfnte, and that the nner scale s zero; however, t straghtforward to extend ths analyss to the case of non-zero nner scale [, 5]. To quantfy the strength of the turbulence we use the untless Rytov varance, gven by [5] 7/6 11/ 6 σ R = 1.3C n k L, (4) where k = π/λ s the optcal wave number, λ s the wavelength, L s the propagaton dstance, and C n s the refractve ndex structure parameter, whch we assume to be constant for horzontal paths. Although the Rytov varance represents the scntllaton ndex of an unbounded plane wave n the weak turbulence regme, t can also been used as an ntutve measure of turbulence strength that brngs together all relevant physcal parameters. Throughout the paper we often refer to σ R smply as the turbulence strength. The refractve ndex structure parameter C n vares from about 1-17 m -/3 for very weak turbulence to about 1-13 m -/3 for strong turbulence [1, 5]. In order to characterze the FSO channel from a communcaton theory perspectve the rradance s factored nto the product of two ndependent random processes representng the large-scale and small-scale rradance fluctuatons, both wth Gamma dstrbutons []. The resultng probablty densty functon (PDF), known as the gamma-gamma dstrbuton, can be wrtten as []: ( ) ( α+ β αβ )/ ( ) ( )/ 1 (5) α+ β f I = I Kα β ( αβ I), I >, Γ( α) Γ( β) where I s the sgnal ntensty, Γ( ) s the gamma functon, and K α β ( ) s the modfed Bessel functon of the second knd and order α β. α and β are parameters of the PDF descrbng the scntllaton experenced by plane waves, and n the case of zero-nner scale (l = ) are gven by [] σ.51 exp R σ, (6) α = 1, β exp R 1 1/5 7/6 = 1/5 5/6 ( σr ) ( σr ) (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6339

9 where σ R s the Rytov varance as gven n (4). The PDF parameters α and β represent the effectve number of large-scale and small-scale cells, respectvely []. Therefore, the PDF of the ntensty fluctuatons at the recever can be predcted from the physcal turbulence condtons. The predcted dstrbuton matches very well the dstrbutons obtaned from numercal propagaton models [], rangng from weak turbulence (when t resembles the lognormal dstrbuton) to the strong turbulence regme. (a) (b) (c) (d) Fg. 3. Receved constellaton dagrams of QPSK (a)-(c) and 16-QAM (d) SSB FSO-OFDM systems wth electrcal SNR per bt of 18 db under the weak turbulence (σ R =.6) for: (a),(d) U- OFDM scheme, (b) C-OFDM scheme, and (c) B-OFDM scheme. The nfluence of both the atmospherc turbulence and electronc nose on QPSK and 16- QAM SSB FSO-OFDM systems s llustrated n Fg. 3. Results for an SSB OFDM system wth 64 sub-carrers are shown. The average launched power s set to dbm, the electrcal sgnal-to-nose rato at the PD s set to 18 db, and the receved sgnal constellaton dagrams are obtaned assumng weak atmospherc turbulence (σ R =.6). The atmospherc turbulence changes the symmetry of clusters from crcular for AWGN channel to ellptc (see Fg. 3). Both C-OFDM and U-OFDM schemes are more mmune to the atmospherc turbulence than the B-OFDM scheme. The U-OFDM system s only slghtly more mmune to the atmospherc turbulence than C-OFDM scheme. It appears that the better power effcency of C-OFDM compensates the dstorton ntroduced by clppng. The reason s smple. The average launched power s fxed for all three OFDM schemes meanng that more energy per bt s allocated n the C-OFDM scheme (because the power n DC bas s lower), and as a consequence the scheme s more mmune to electrcal nose. Hgher mmunty to electrcal nose may result n slghtly better BER performance of the C-OFDM scheme when compared to the U-OFDM scheme (see Secton 5). (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 634

10 4. Block-crculant LDPC codes and teratve decodng In ths secton we gve a bref descrpton of the codng scheme used n ths paper, as well as an algorthm for calculatng the requred bt lkelhoods n the teratve decoder. LDPC codes have been shown to sgnfcantly outperform turbo-product codes n burstyerror prone channels such as the fber-optcs channel n the presence of ntrachannel nonlnear effects [11-1]. The block-crculant LDPC codes smlar to those proposed n Ref. [13] are sutable for hgh-speed mplementaton because ther party-check matrces have smple cyclc or quas-cyclc structure, and the encoders can be mplemented based on shft regsters and modulo adders. As we have shown n Refs. [1-13] the party-check matrx, H, of a regular block-crculant LDPC code can be wrtten as 1 3 P P P... Pl Pl P P... Pl 1 H =, P = , l r l r 3 l r 4 P + P + P +... Pl r+ 1 the exponents n H are selected as elements from the followng set L = { : p 1, θ + θ GF ( p ) } p s a prme, and θ s the prmtve element of the fnte feld GF(p ). The structure of the party-check matrx of a block-crculant code facltates a low-complexty decoder mplementaton because t s hghly regular and only the dmenson of the permutaton matrx P and the exponents are to be stored. The decoder s based on an effcent realzaton of the sum-product algorthm gven n Ref. [15]. Bt relabltes fed to the teratve decoder are calculated as explaned below. In FSO communcatons the recever electroncs nose s commonly modeled as a Gaussan nose (see e.g., [1], [3, 5]). If r I s the n-phase demodulator sample, and r Q s the quadrature demodulator sample, then the symbol log-lkelhood rato (LLR) s calculated as ( ( ) ( ri si ) ( rq sq) (7) λ s= si, sq ) =, σ σ where s I and s Q are the coordnates of a transmtted sgnal constellaton pont and AWGN varance (σ ) s determned from the requred electrcal sgnal-to-nose rato (SNR) per bt E b /N o E E{ k, } s b P. (8) = o No log M σ P o s the normalzed receved power [6], and s,k denotes the QAM symbol n the k-th subcarrer channel of the -th OFDM frame. (Wth M we denote the number of ponts n the correspondng constellaton dagram.) Notce that the defnton of electrcal SNR per bt, common n dgtal communcatons [Eq. (8)] (see Refs. [6] and [17]), s dfferent from that used n Refs. [1], [3], and [5]. The symbol energy-to-nose densty rato E s /N s equal to the bt energy-to-nose densty rato E b /N multpled by the number of bts per symbol whch s typcally large. For example, for an OFDM system wth 56 sub-carrers usng BPSK, E s /N s about 4 db larger than E b /N. One must be careful when comparng OFDM system BER performance wth OOK, not to confuse the bt energy-to-nose densty rato (E b /N ) wth symbol energy-to-nose densty rato (E s /N ). For more detals on OFDM prncples the nterested reader s referred to Ref. [6], and for more detals on comparson of dfferent modulaton schemes to Ref. [17]. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6341

11 The ntal bt lkelhoods, provded to the teratve decoder, are calculated from the symbol LLRs, λ(s), as s: s 1exp[ ( )] ( ) j = λ s (9) L sj = log. s: s exp[ λ ( ) j = s ] The Gaussan assumpton n Eq. (7) may lead to BER performance degradaton because the jont dstrbuton s actually a convoluton of the Gaussan and gamma-gamma PDFs. In order to reduce complexty, we use the Gaussan approxmaton n the calculaton of symbol relabltes. Nevertheless, dramatc performance mprovement of an LDPC coded FSO- OFDM system over an LDPC coded FSO OOK system s obtaned, as shown n Secton 5. In calculatng bt relabltes from symbol ones [Eq. (9)] the followng max-star operator, defned as max*(x,y)= log(e x +e y ), s appled recursvely [18] max * (x,y)=max(x,y)+log(1+e - x-y ). x y Notce that the correcton factor log ( 1+ e ) n max * -operator for hgh-speed applcatons can be ether tabulated or even omtted, wthout sgnfcant degradaton n performance [18]. 5. Smulaton Results Smulaton results of an LDPC coded SSB U-OFDM system for two dfferent turbulence strengths are gven n Fg. 4. The nfluence of the atmospherc turbulence channel s ncluded by perturbng the ntenstes of the OFDM samples (before photodode) accordng to the PDF n Eq. (5). Therefore, the worst case scenaro s observed. No aperture averagng s appled. For BPSK and QPSK, the codng gan mprovement of an LDPC encoded FSO-OFDM system over an LDPC encoded OOK FSO system ncreases as the turbulence strength ncreases. However, the 16-QAM FSO-OFDM system s not able to operate n the regme of strong turbulence. For weak turbulence (σ R =.6) [see Fg. 4(a)] the codng gan mprovement of LDPC coded FSO-OFDM system wth 64 sub-carrers over the LDPC encoded FSO OOK system s 8.47 db for QPSK and 9.66 db for BPSK, at the BER of 1-5. For strong turbulence (σ R =3.) [see Fg. 4(b)] the codng gan mprovement of the LDPC coded FSO-OFDM system over the LDPC coded FSO OOK system s.4 db for QPSK and 3.38 db for BPSK. In both cases the block-crculant [13] LDPC code (43,34) of rate.75 s employed. The comparson of dfferent LDPC coded SSB OFDM schemes, under the weak turbulence (σ R =.6), s gven n Fg. 5. The C-OFDM scheme slghtly outperforms the U- OFDM scheme. Both C-OFDM and U-OFDM schemes outperform the B-OFDM scheme by approxmately 1.5dB at BER of 1-5. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 634

12 Bt-error rate, BER σ R =.6 BPSK SSB OFDM: QPSK SSB OFDM: 16-QAM SSB OFDM: OOK: Bt-error rate, BER Electrcal SNR, E b /N [db] (a) σ R =3. BPSK SSB OFDM: QPSK SSB OFDM: 16-QAM SSB OFDM: OOK: Electrcal SNR, E b /N [db] (b) Fg. 4. BER performance of LDPC-coded SSB U-OFDM system wth 64-subcarrers under: (a) the weak turbulence (σ R =.6), and (b) strong turbulence (σ R =3.). Bt-error rate, BER σ R =.6 QPSK SSB Unclpped-OFDM: QPSK SSB Clpped-OFDM: QPSK SSB Based-OFDM: OOK: Electrcal SNR, E b /N [db] Fg. 5. Comparson of dfferent LDPC coded SSB FSO-OFDM systems wth 64-subcarrers under the weak turbulence (σ R =.6). (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6343

13 For the results shown n Fgs. 4-5, the receved ntensty samples are consdered to be ndependent and uncorrelated, smlarly as n Refs. [], [5] and [14]. In realty, especally at hgh bt rates, the channel has temporal correlaton, and consecutve bts propagate experence smlar channel condtons. Because of the lack of lterature on the temporal statstcs n the FSO channel and the complexty of multdmensonal jont dstrbutons, we restrcted our study (see Fgs. 4-5) to the ndependent and uncorrelated case. In many OFDM systems ths approach s reasonable for the followng reasons: () when the channel condtons do not vary, a smple channel estmaton technques based on plot sgnals (see Ref. [6] for more detals) can be used to overcome the temporal correlaton, and () the mmunty to temporal correlaton can further be mproved by usng nterleavng. The nterleavng can be vsualzed as the formng an LxN (N s the codeword length) array of L LDPC codewords (the parameter L s known as nterleavng degree) wrtten row by row, and transmttng the array entres column by column. If the orgnal code can correct a sngle error burst of length l or less, then the nterleaved code can correct a sngle error burst of length ll. Therefore, nterleaved OFDM can successfully elmnate temporal correlaton ntroduced by the FSO channel. To llustrate the applcablty of LDPC-coded OFDM n the presence of temporal correlaton we performed smulatons by employng the jont temporal correlatve dstrbuton model from [1], whch descrbes the fadng n an FSO channel at a sngle pont of space at multple nstances of tme. Ths method s based on the Rytov method to derve the normalzed log-ampltude covarance functon for two postons n a recevng plane perpendcular to the drecton of propagaton [1], []: BX ( P, Pj ) bx ( dj) =, (1) BX ( P, P ) where d j s the dstance between ponts P and P j. B X denotes the log-ampltude covarance functon: BX ( P, Pj ) = E X ( P ) X ( Pj ) E X ( P ) E X ( Pj ), (11) and X s the log-ampltude fluctuaton. The jont temporal dstrbuton of n ntensty samples (I 1,I,,I n ) s gven by Ref. [1]: I1 I n f ( I1, I,..., In ) = exp - ln...ln, (1) I n n / 1/ ( ) 8 I I π I π CX = 1 where C X s the covarance matrx of ntensty samples: T ( n 1) T σx σxbx d... σxbx d τ τ T ( n ) T σxbx d σx... σxbx d CX = τ τ ( n 1) T ( n ) T σxbx d σxbx d... σ X τ τ σ X denotes the varance of the log-normally dstrbuted ampltude, whch for plane wave can be approxmated as Ref. [] L 7/6 σ.56 k C x L x dx, (13) X n ( )( ) 5/6 where the wave number k, propagaton length L, and the refractve ndex structure parameter C n were ntroduced earler. T s the tme nterval between observatons, whch corresponds to (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6344

14 the OFDM symbol perod; whle τ s the coherence tme. Notce that expressons (1)-(13) are vald n the weak turbulence regme. In the same regme the covarance functon (11) s found to be exponental for both plane and sphercal waves [] 5/3 τ b X ( τ ) = exp. (14) τ The typcal values of coherence tme τ are n the range from 1μs to 1ms. The results of smulatons usng the model descrbed by Eq. (1) Eq. (14) are shown n Fg. 6. The standard devaton σ X s set to.6 (notce that σ X s dfferent from Rytov standard devaton σ R used earler, and for horzontal paths σ X.498σ R ). It s clear from Fg. 6 that LDPC-coded OFDM wth or wthout nterleaver provdes excellent performance mprovement even n the presence of temporal correlaton. The BER performance can further be mproved by usng the nterleaver wth larger nterleavng degree than that used n Fg. 6 (the star curve), at the expense of ncreasng encoder/decoder complexty. Notce the on-off keyng (OOK) modulaton scheme enters BER floor for ths value of standard devaton (σ X =.6), and even advanced FEC s not able to help too much. However, LDPC-coded OOK s able to operate properly at lower standard devatons σ X. To generate temporally correlated samples we used two dfferent methods, the frst one s based on the Levnson-Durbn algorthm [1], [], and the second one s based on an algorthm due to Wood and Chan [3]. Bt-error rate, BER τ = 1 μs, σ X =.6: Uncoded, QPSK-OFDM, QPSK-OFDM Uncoded, 16-QAM-OFDM, 16-QAM-OFDM 16-QAM-OFDM,, L=3 Uncoded, OOK τ = 1 μs, σ X =.1: Uncoded, OOK, OOK Electrcal SNR, E b /N [db] 6. Concluson Fg. 6. BER performance of LDPC-coded OFDM n the presence of temporal correlaton We have descrbed an LDPC-coded IM/DD-OFDM system, and a novel modulaton/codng scheme for FSO systems over atmospherc turbulence channels that provdes a number of advantages: () excellent codng gans (defned at BER of 1-5 ) rangng from 8.47 db n the regme of weak turbulence (for QPSK) to 3.38 db n the regme of strong turbulence (for BPSK) compared to LDPC-coded OOK, () sgnfcant spectral effcency mprovement, () no channel equalzaton s requred, and (v) a smple FFT s used for modulatng and demodulatng. To further mprove spectral effcency, the FSO-OFDM SSB transmsson scheme may be combned wth sub-carrer multplexng. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6345

15 Acknowledgments Ths work s funded n part by the NSF under Grant ITR The authors would lke to thank J. A. Anguta for hs nvolvement n an earler work on LDPC codes for FSO communcaton and for useful dscussons. (C) 7 OSA 14 May 7 / Vol. 15, No. 1 / OPTICS EXPRESS 6346