BER Analysis of Optical Wireless Signals through Lognormal Fading Channels with Perfect CSI

Transcription

1 7th tratoal Cofrc o Tlcommucatos BER Aalyss of Optcal Wrlss Sgals through ogormal Fadg Chals wth rfct CS Hassa Morad, Maryam Falahpour, Hazm H. Rfa Elctrcal ad Computr Egrg Uvrsty of Olahoma Tulsa, OK, USA {hmorad, maryam, hazm}@ou.du tr. orst Elctrcal Egrg Uvrsty of Tulsa Tulsa, OK, USA pgl@ohm..utulsa.du Mohammd Atquzzama Computr Scc Uvrsty of Olahoma Norma, OK, USA atq@ou.du Abstract Du to cosstt atmosphrc codtos, scattrg ad sctllato of fr spac optcal (FSO sgal ca occur, thus gatvly flucg th rcvd sgal tsty. Th chal s usually modld as a ormalzd fadg coffct wth addtv aussa os. Optmal dtcto of th rcvd sgal s dsgd basd o a dcso rul,.g., Mamum lhood (M, assumg th rcvr ows th os statstcs ad fadg corrlato of th chal. Ths papr brfly dals wth aalyss o bt rror rat (BER of a wrlss optcal sgal passg through a logormally dstrbutd fadg chal, wh prfct owldg of chal stat formato (CS at th rcvr sd s avalabl. Two approachs wll b prstd to provd closd-form prssos for BER. O uss auss Hrmt quadratur appromato ad th othr o s basd o powr srs. Whl umrcal aalyss shows a vry small appromato rror wh th auss Hrmt approach s cosdrd, th powr srs approach dos ot uss ay appromato. Kywords: FSO, Chal stat formato (CS, BER, ogormal fadg, auss Hrmt quadratur, owr srs, Appromato rror.. NTRODUCTON Thr ar varous typs of udsrd oss wrlss optcal,.. FSO, chals whch advrsly affct th BER prformac of a commucato l. Bacgroud os, dar os ad thrmal os ar amog th most commo oss whch ar all modld as a addtv aussa os wth zro or v o-zro ma []. Th dsg ad practcal mplmtato of optmal dtctor, commoly basd o M fucto, s as smpl as wll b dscrbd hr latr. ths cas a smpl closd form prsso ca b drvd. Although th addtv os dgrads th prformac of th l trms of bt rror, t s ot th oly radom varabl that affcts th FSO chal. ay wrlss commucato l, th FSO chal facs fadg of th sgal du to propagato fr spac. Th fadg coms from th atmosphr ducd turbulc as sctllato whch s dffrt tha th vsblty cosdratos du to fog ad M scattrg whch dals wth sgal attuato, prvously vstgatd [] ad [3]. Th fadg radom varabl s usually modld as a logormal [4], amma [5] or amma-amma [6-7] dstrbuto modl ltratur. Th fadg ot oly dcrass sgfcatly th BER prformac of th commucato l, but t crass th complts of dsgg approprat rcvr basd o optmal dtcto [8]. ths papr, w apply a logormal fadg to th systm modl ad mathmatcally vstgat th chal bt rror rat th cas of prfct chal stat formato. t mas that th rcvr has th formato about th stataous valu of fadg coffct for ach symbol. Thus, th rcvr dos ot rqur th stmato of th chal to mtgat th ffct of th fadg o sgal. ths cas, th M basd dtcto mtrc ca b smply dsgd as smpl as a dtctor of aussa chal. But as mtod arlr, th BER prformac s stll affctd by th turbulc ducd fadg. W assum a o-off yg (OOK modulato format at th trasmttr ad drct (cohrt dtcto at th rcvr sd. No trsymbol trfrc cosdrd through th dscussos ths papr. Scto w df th systm modl. Scto provds th BER calculato, whr two approachs ar appld hr to drv closd-form prsso for BER a logormally fadd chal. O s th powr srs approach whch provds a act prsso whl th scod s basd o th hlp of auss Hrmt quadratur whch was prvously studd [4]. But hr w aalyz th appromato rror f th auss Hrmt s usd. Scto V w brfly prst th umrcal smulatos basd o drvd quatos. Fally, a brf cocluso wll b prstd Scto V.. CHANNE MODE A. Modulato ad Addtv Nos Th rcvd sgal d (t by OOK modulato ca b prssd by ( t h( t s( t ( t ( whr ( t {,} d s s th trasmttd sgal; h(t s th ormalzd chal fadg tsty du to atmosphrc turbulc ad cosdrd to b costat ovr a larg umbr of trasmttd bts; ad (t s total addtv os. For smplcty, vstgators glct trm tm (t th aalyss prstd hr. Although o-radom attuato du to propagato ad scattrg ca b also cludd th modl [], [3], t dos ot affct th rsults wh commucato s stochastcally aalyzd. Assumg th chal s ot ffctd by turbulc-ducd fadg, wll b th oly radom varabl usd th modl /9/$6. 9 EEE 493

2 Th avragd M-basd bt rror rat for such a aussa chal wth qually lly trasmttd bts s prssd trms of os ad sgal paramtrs, ad t : Rt, (,, t rfc ( whr ad ar th stadard dvatos of th os currts for symbols '' ad '', rspctvly, ad whr thy ar assumd to b dffrt [9]. ( R s th rcvr s rsposvty,.. optcal-to-lctrcal covrso coffct; t s th avrag of trasmttd powr; ad rfc(. s th complmtary rror fucto. For smplcty, w ca assum t R. Ths mmal rror probablty s provdd by th Mbasd dcso thrshold prssd by [9] D, (3 that ca b assumd costat for a gv obsrvato prod. (3 ( t R ad ar avrags of th gratd currts at th rcvr for symbols '' ad '', rspctvly. Addtoal dtals about os paramtrs ca b foud [9]. Not that for a FSO chal wth oly addtv aussa os, th avrag SNR ca b prssd by [] 4R t (4 ( Th BER ca b prssd trms of th avrag SNR, ( rfc (5 B. Dstrbuto of Fadg tsty A radom varabl B has a logormal dstrbuto f th radom varabl AlB has a ormal (.., aussa dstrbuto. So, f th ampltud of th radom path ga B s, th optcal tsty B s also logormally dstrbutd ths cas. Cosqutly, Th fadg chal coffct, whch modls th chal from th trasmt aprtur to th rcv aprtur, s gv by X h (6 m whch m s th sgal lght tsty, actually at th trasmttr, wthout turbulc; s th sgal lght tsty, actually at th rcvr, wth turbulc; ad log-ampltud X s th dtcally dstrbutd ormal radom varabl wth ma µ ad stadard dvato : ( X µ f ( X p (7 π Substtutg (6 (7, th dstrbuto of lght tsty fadg ducd by turbulc s a log-ormal dstrbuto, whch s prssd by [ ] l( h µ f ( h p (8 8π h 8 whr h from (6. Not that th chag of varabls troducs a addtoal /h trm outsd of th potal trm from th fact d (/ h dh. C. Ma ad Varac Sc varabl X s aussa, Eq. (6 dots that th pctd valu of th chal coffct h s qual to aussa momt-gratg fucto (MF of X at pot (9 Assumg th chal coffcts at dffrt tms ar dpdt, th varac of h ca b calculatd as E h E h ( [ ] [ ] 4 D. Normalzd Fadg sctllato fadg wth sctllato d S.., w ca grat avrag powr loss du to atmosphrc fadg uty, such that th fadg dos ot, o avrag, attuat or amplfy th optcal powr. W choos µ ( that lads us to µ. Thus, th varac wll b qual to 4 ( Ths paramtr s th so calld sctllato d, S.. t ca b s from ( that th paramtr s dffrt from th fadg stadard dvato. Authors rfr to as fadg tsty ths papr. Fally, th logormal dstrbuto (8 wll b gv by [ ] l( h f ( h p (3 8π h 8 whch s dfd as l( S.. (4 Not that for a logormal chal wth addtv aussa os, th stataous SNR from (4 wll b covrtd to 4h R t (5 ( Not that for a logormal chal wth addtv aussa os, th stat SNR from (4 wll b covrtd to E. Chal Stat formato (CS CS s th ralzato of th stataous fadg stat,.. fadg coffcts h, at ach symbol prod. Som vstgatos assum prfct avalablty of CS at th rcvr [5], []. Ths assumpto abls th dsg of th optmal dtctor ts smplst form so that th M-basd dcso thrshold s dfd smply as ( t Rh D, WthCS (6 Dtcto tas plac by th computato of (6 for ay dvdually rcvd symbol r, wh h s ow at th rcvr sd. A BER aalyss wll b prstd th followg scto. 494

3 That th rcvr has o owldg of th stataous fadg stat wll b dscussd subsqut sctos.. BER CACUATON A closd form prsso for th probablty of rror for a strog atmosphrc-ducd turbulc fadg chal wth amma dstrbuto ad prfct CS was prstd [5]. W td t for a amma-amma dstrbuto whch wll b (, α, β, ammaamma αβ α β α β,,,,,4 (7 5, 3, π Γ( α Γ( β α β Th approachs prstd ths scto ar tatd to fd prssos computatoally smplfd to achv ral-tm rror aalyss of BER logormal chals. th prsc of atmosphr turbulc,.. sctllato fadg wth sctllato d S.. as logormal dstrbuto, th rcvd powr wll b chagd to h t, thus BER wll b gv from Eqs. ( ad (6 by [4] X Rt t f f X rfc, (,,, ( ( (8 whr f (X s dfd by Eq. (7 ad rfc(. s th complmtary rror fucto dfd as rfc ( ( π p( r dr. By covrtg th varabl from X to h, th bt rror rat wll b prstd by (,,,, t [ l( h ] Rt h rfc dh (9 p h 3π ( 8 Calculato of tgrato (9, at tms appromatd, ca b do by mathmatcal ad umrcal mthods as follows: A. owr Srs Approach W may rwrt th prsso (9 by usg (6 th form of (,, [ X ] X rfc p ( 8π owr srs rprstato of rror fucto ca b dfd as ((3. [], ( rfc( ( π!( th X rfc( π Substtutg ths prsso, ( ( / ( /, t bcoms ( X!( (, (, 8π ( X ( / ( ( / ( X π!( Th aformtod tgrals ca b mapulatd as ( X, (, 8π 8π π π ( X ( t ca b show that ad ( ( / ( ( X ± ( X ± / ( /!( ( /!( m( X ( X ( X ( X π ± rfc ( X ( X (4 (3 (5 (6 π ( ± 4 ± rfc whr rfc( s th scald complmtary rror fucto gv by rfc ( rfc( ( π p( r dr (7 Howvr, w ow that rfc(rfc(-, ad th rfc(rfc(-. Fally th BER of a log-ormal fadg chal ca b summarzd as, (, ( / ( (4 (8 p ( / (! π Ul arlr prssos, ths s ot appromatd ad ot volvd wth zros ad wght factors rgardg Hrmt polyomal. Ths approach s vstgatd th followg scto. Howvr, a problm sts wth prsso (8, as t dos t covrg for small valus of. Evaluatg umrcally by smulato, Fg. shows th BER probablty by (8,,, vrsus avragd sgal-to-os rato,, for dffrt valus of fadg tsty,. B. auss-hrmt Quadratur Approach A gral soluto for umrcal calculato of th tgrals th form of f ( d s prstd [, ], whch s 495

4 f ( d w f ( R (9 whr s th umbr of sampl pots usd for appromato. Th ar th roots of th Hrmt polyomal H ( (,,...,; th assocatd wghts w ar gv by! π H ( ; ad R s th rmdr gv by ( ( f ( ξ! π (!. Not that f s th -th drvatv of f, ad ξ s som umbr btw - ad. Although, gral, th prcs valu of R s uow, ts aalytcal form usually s ow ad ca b usd to dtrm a uppr boud to R trms of []. Abramowtz t al. [, tabl 5.] offrs a tabl of abscssas ad wghts up to. Usg ths quato, th BER ca b appromatd by,, (, w rfc (3 π ca obsrv that th appromato rror has som local mmum pots for almost all of th thr valus of at dffrt valus of. Th appromato rror for th local mmum pots s zro. Howvr t dos ot show a zro valu Fg. 5 du to lmtato of computr smulatos. Abov aalyss was achvd wth th assumpto that ot oly th rcvr ows chal modl, fadg dstrbuto, fadg corrlato ad os statstcs, but also th stataous CS s avalabl. Howvr, th CS formato of th chal s ot asly avalabl at th rcvr. Th cas that th rcvr has o owldg of th stataous fadg stat wll b dscussd a futur wor by th authors. - - V. NUMERCA SMUATON ths scto w provd th umrcal smulato of a typcal fadg systm ad provd th aalyss of mathmatcal computatos th Scto. Evaluatg umrcally by smulato, Fg. shows th bt rror rat probablty by (3,,, vrsus sgal to os rato,, for dffrt valus of fadg tsty,. Th comparso to BER of a aussa chal s also show ths fgur. - BER SgmaX.4 SgmaX.3 SgmaX. SgmaX. Hrmt Appro Avragd SNR (db Fg : Th Hrmt-appromatd probablty of rror for polyomal ordr 5 ad dffrt valus of fadg tsty. BER SgmaX.4 SgmaX.3 SgmaX. SgmaX. aussa R Avragd SNR (db Fg. : Th bt rror probablty of logormal chal wth CS vrsus SNR for dffrt valu of fadg tsty. Th BER of a aussa chal (o fadg s also show to dmostrat prformac loss a fadg chal. Th BER of a auss Hrmt quadratur-basd prsso s umrcally plottd Fg.. Th rmdr R s llustratd Fgs. 3 ad 4, for dffrt valus of fadg tsty ad ordr. Th appromato rror (..,,,, /,, whch s th ormalzd rprstato of R, s also llustratd Fg. 5. crasg th SNR dos ot usually dcras th appromato rror. But wh th fadg tsty dcrass, a dcras of appromato rror tas plac. W Avragd SNR (db Fg 3: Th rmdr R from Hrmt-appromatd probablty of rror for dffrt appromato polyomal ordrs whl

5 SgmaX.4 SgmaX.3 SgmaX. SgmaX. th ormalzd appromato rror crass whl th SNR crass ad/or th fadg tsty crass. V. ACKNOWEDMENTS Ths wor s fudd by NSF grat umbr NSF-ECCS 758. R Avragd SNR (db Fg 4: Th rmdr R from Hrmt-appromatd probablty of rror for polyomal ordr 5 ad dffrt valus of fadg tsty. Appromato Error Avragd SNR (db Fg 5: Th appromato rror of Hrmt-appromatd probablty of rror for polyomal ordr 5 ad dffrt valus of fadg tsty. V. CONCUSONS Ths papr has vstgatd th BER aalyss of a logormally fadd chal wh prfct CS s avalabl at th rcvr. ths cas, although th complty of dsgg th rcvr wll b mmzd wh a optmal dtcto s rqurd, th BER s stll udr th ffct of th fadg. Howvr, th rcvr ows th act valu of chal coffcts at dffrt tm. A w closd-form prsso mployg powr srs was drvd ths papr. W also aalyzd th appromato rror wh a auss-hrmt Quadratur s usd as th scod approach for calculato of BER. W saw that v wth a valu of Hrmt polyomal ordr 3, th mamum of rror s 5 - for. 3. Also, crasg th SNR dos ot dcras th appromato rror whl dcrasg th fadg tsty dcrass th rror. Thus SgmaX.4 SgmaX.3 SgmaX. SgmaX. REFERENCES [] aglard, R. M., ad Karp, S., Optcal Commucatos, d dto, Joh Wly & Sos, c., 995. [] Km,.., ad Korvaar, E., Avalablty of Fr Spac Optcs (FSO ad Hybrd FSO/RF Systms, SE, Vol. 453, pp , Aug.. [3] Harrs, A., Sluss, J. J., Rfa, H. H., ad orst,.., Fr-spac optcal wavlgth dvrsty schm for fog mtgato a groudto-umadaral-vhcl commucatos l, Joural of Optcal Egrg, Vol. 45, p.p. -, 6. [4] Navdpour, S. M., Uysal, M., ad Kavhrad, M., BER rformac of Fr-Spac Optcal Trasmsso wth Spatal Dvrsty, EEE Tras. Wrlss Comm., Vol. 6, No. 8, pp , Aug. 7. [5] Tsftss, T. A., Sadalds, H.., Karagads,. K., ad Uysal, M., Optcal wrlss ls wth spatal dvrsty ovr strog atmosphrc turbulc chals, EEE Tras. Wrlss Comm., Vol. 8, ssu, pp , Fb. 9. [6] Baya, E., Schobr, R., ad Mall, R.K., rformac aalyss of MMO fr-spac optcal systms gamma-gamma fadg, EEE Tras. Comm., Vol. 57, ssu, pp , Nov. 9. [7] Navdpour, S. M., Uysal, M., ad, J., Aalyss of codd wrlss optcal commucatos udr corrlatd gamma-gamma chals, EEE VTC 4, p.p , 4. [8] Rdgr, M.. B, Schobr, R., ad amp,., Multpl-Symbol Dtcto for hoto-coutg MMO Fr-Spac Optcal Commucatos, EEE Tras. Wrlss Comm., Vol. 7, No., Dc. 8. [9] Morad, H., Rfa, H. H., orst,.., Atquzzama, M., A Estmato-Basd Optmum Rcvr for Fr Spac Optcs Usg lot-add Modulato, SE hotocs Wst,. [] tzps, N., Hollad,., ad Cowly, W. Th aussa fr spac optcal MMO chal wth Q-ary puls posto modulato, EEE Tras. Wrlss Comm., Vol. 7, ssu 5, art, pp , 8. [] Jffry, A., Hadboo of Mathmatcal Formulas ad tgrals, d dto, Acadmc rss,. [] Abramowtz, M. ad Stgu,. A., Hadboo of Mathmatcal Fuctos wth Formulas, raphs, ad Mathmatcal Tabls, US Dpartmt of Commrc,