0 8th Intrnational Confrnc on Tlcommunications Cod Cross-Corrlation Effcts on th Prformanc of Optical CDMA Systms in th Prsnc of Rcivr Noiss S. M. Zabihi-Maddah M. Molavi Kakhki Faculty of Enginring, Frdowsi Univrsity of Mashhad, Faculty of Enginring, Frdowsi Univrsity of Mashhad, Mashhad, Iran Mashhad, Iran Email: zabihimaddah@stu-mail.um.ac.ir Abstract Incrasing th numbr of simultanous usrs is on of th challnging issus in incohrnt and asynchronous optical CDMA systms. On solution is to incras th valu of cross- corrlation, λ c. In [3] nglcting th ffct of nois, optimum valu of λ c was obtaind. In this work, with considring nois, th simultanous ffct of λ c and transmittd powr on th probability of bit rror (pb is invstigatd. Our invstigation is prformd by mploying OOC (λ c and prim cod (λ c and activ corrlator is chosn for its robustnss against th nois. Th prformanc is analyzd for thr common rcivr structurs, namly soft rcivr (without optical hard limitr, singl optical hard limitr (SHL and doubl optical hard limitrs (DHL. Our rsults show that undr th sam bandwidth fficincy, incrasing λ c in high powr rgims dcrass th pb but incrasing λ c in low powr rgims will incras pb. Kywords: Optical Cod division multipl accss (OCDMA, Hard limitr, Prim Cod, Cross-Corrlation. I. INTRODUCTION Optical CDMA (OCDMA schms hav rcntly rcivd substantial attntion in local ara ntworks. OCDMA offrs svral advantags in local ara ntworks. OCDMA allows simultanous usrs to snd thir data asynchronously through th assignmnt of uniqu signatur squnc. This squnc is usd to transmit bit "" whil nothing is snt for bit "0" (On- Off Kying. In th litratur som kind of cods has bn proposd for using as cod squnc, i.. optical orthogonal cods (OOCs and prim cods. In this papr, w study incohrnt and asynchronous OCDMA systms. Multipl accss intrfrnc (MAI is always a srious problm in th prformanc of such systms. In ordr to rduc th MAI, th signatur cod must hav small cross-corrlation. An ( L, w, λ a, λc OOC is a collction of binary L-tupls with Hamming wight w such that th maximum out of phas autocorrlation is boundd by λ a and th maximum cross-corrlation is boundd by λ c []. Th signatur cods with lowr corrlation will usually outprform th signatur cods with highr corrlation for th sam numbr of th simultanous usrs, (N. On of th problms associatd with systms using OOCs is th small numbr of usrs. It has bn shown that mor usrs can b providd with vn bttr prformanc for an ( L, w m,λ + m,λ compard with an (, w, λ, λ + OOC L OOC []. As shown in [3] for th sam numbr of usrs and th sam lngth (L, th prformanc of λ OOC may surpass th prformanc of c λc OOC. This is bcaus λ c OOC may hav a much largr valu of w, hnc compnsating th prformanc dgradation. Using soft rcivr (without HL in [4] th pb for λ c and λc OOCs was computd. It has bn shown in [5] that using optical hard limitr (HL could nhanc th bit rror rat (BER of OCDMA systms by xcluding som MAI pattrns. For λ c c OOCs, whr c is any arbitrary positiv intgr, th xact xprssion for th pb using hard limitd rcivr has bn givn in [6]. Latr, mploying a diffrnt approach and using gnralizd OOCs with λ c λ, Mashhadi and Salhi found anothr xprssion [3]. Thr ar two critria for prforming a comparison btwn two cod sts of optical CDMA systms, namly bit rror probability and bandwidth fficincy (BE. Prforming a fair comparison, on can xamin th systm prformanc by on critria whil th othr on is rmaind fixd. In othr words, w ar intrstd in choosing th cod st which givs lowr valu of bit rror probability, P, for a givn pair of (L,N or choosing th cod st which givs highr BE, which is dfind as N BE [7], for a minimum rror probability L ( P min. Sinc OCDMA systms using OOC with λ c suffr from low numbr of simultanous usrs, an incrasing intrst has bn focusd on cods with highr cross-corrlation valus. For this rason, a prformanc of OCDMA systm was analyzd mploying AND logic gat rcivr structur and gnralizd optical orthogonal cod [3]. In th mntiond papr, th minimum bit rror probability was obtaind by firstly finding optimum valu of λ c ( λ opt, rgardlss of cod wight and thn using this valu to dtrmin th optimum valu of cod wight w. opt 978--4577-004-8//$6.00 0 IEEE 978--4577-004-8//$6.00 0 IEEE 544
In all mntiond works th ffct of rcivr noiss was not includd. An analysis was prformd by Kwon [8] to valuat th pb of OCDMA systms in prsnc of rcivr noiss mploying OOC with λ c, lctrical ncoding and two rcivr structurs, namly soft rcivr and using singl hard limitr (SHL. It was shown that whn th numbr of simultanous usrs is much lss than th pulsd lasr's modulation xtinction ratio and for both of mntiond rcivr structur, OCDMA systms utilizing optical ncoding prform in a similar way as thir lctrical ncoding countrparts [9]. It should b notd that, OCDMA systms using doubl hard limitrs (DHL rcivrs and activ corrlator outprform thir countrparts which ar using logic gat rcivrs or chip lvl dtction [0]. In this papr, in ordr to choos th optimum cod, w study th ffct of varying λ c on cod wight and w calculat bit rror probability in th prsnc of rcivr noiss as wll as intrfrnc. Our analysis ar prformd using activ corrlator and mploying thr common rcivr structurs, namly soft rcivr, SHL and DHL for incohrnt and asynchronous OCDMA systms using prim cod ( λ c. Furthrmor, in all of th rcivrs APD is chosn as photo dtctor and its output is modld by a Gaussian function and optical dcodr is mployd. Our rsults show that for a givn (L, N, cods with lowr valus of λ c could prform bttr than ons with gratr valus. It should also b notd that this bttr valu of pb, for low cross-corrlation cods, is achivd in lowr cod wights in comparison with thir high cross-corrlation countrparts. II. SYSTEM MODEL In our systm modl, data bits driv a pulsd lasr with a rat of Tb. Thn, th output of th pulsd lasr is convyd to an array of tappd dlay lins (optical ncodr. Aftr that, th ncodd usrs data pass through a passiv star couplr (PSC and finally rach th output ports of PSC which th rcivrs ar placd. In this work, w nglct th powr loss in th output ports of PSC. Figur shows th modl of soft rcivr. Th rcivd signal is multiplid by a stord rplica of dsird cod. Th dcodd signal is fd into an APD. Th rcivr intgrats th APD output ovr a bit tim T b FTc. Thn, by passing th signal through a samplr with frquncy Tb, th dcision variabl Z is achivd. Lastly, by comparing Z with a thrshold lvl Th, th final dcision on th snt bit is mad. If an optical HL is placd in front of th optical corrlator, th rcivr with SHL is obtaind. Furthrmor, by placing an optical HL bfor and on aftr th optical corrlator, th rcivr with DHL is obtaind. It was shown that OCDMA systms using DHL outprform th systms using SHL and th latr outprform OCDMA systms using soft rcivr considring th ffct of nois in high powr rgims [, 9]. Th probability that a spcifid numbr of photons ar absorbd from an incidnt optical fild by an APD dtctor ovr a chip tim with duration T c is givn by Poisson distribution with rat λ s, which is [8]: η P ( λs hυ whr P th rcivd powr in front of powr splittr during a mark chip tim whn th dsird usr is transmits bit "", η is th APD quantum fficincy with which th APD convrts incidnt photons to photolctrons, h is Plank s constant and υ is th optical frquncy. Primary photolctron-hol pairs in th APD dtctor undrgo an avalanch multiplication procss that rsults in th output of n lctrons from th APD in th rspons to th absorption of λ s T c primary photons on th avrag. By Rprsnting λ as th total photon absorption rat du to signal and APD bulk lakag currnt, w hav: λs + Ib / λ λs / M + Ib / for bit for bit whr is an lctron charg, I b / rprsnts th contribution of th APD bulk lakag currnt to th APD output, and M is th modulation xtinction ratio of th lasr. It is shown that random variabl (r.v. Z could b approximatd with a Gaussian r.v. with man and varianc of [8]: μ Z IbTb G c + s + "" "0" IsT IbTb I stb σ var( Z G F cs + + + σ whr cs th man valu of absorbd photon in T b, G is th man valu of APD random gain and I s is th surfac lakag currnt which is inhrntly modld in APD. F is th xcss nois factor dfind as F kff G + ( kff whr kff is G th ffctiv ionization ratio and its valu dpnds on th wavlngth and matrial usd in fabrication of photo dtctor. Also, σ th is thrmal nois varianc of APD and its valu is rlatd to systm paramtrs as k BTrTc σ th whr kbis th RL Boltzmann constant, T r is th rcivr nois tmpratur and R L is th rcivr load rsistor (s Tabl I [8]. r(t Stord Rplica of Cod APD n (t T b 0 Figur -Modl of soft rcivr. t T b Z b Th < th ( (3 (4 545
TABLE I. TYPICAL PARAMETER VALUES OF OCDMA SYSTEMS USING APD Nam Symbol Valu Lasr frquncy υ.44 0 4 Hz APD quantum fficincy η 0.6 APD avrag gain G 00 Effctiv ionization ratio of APD κ ff 0.0 Bulk lakag currnt Ib 0. na Surfac lakag currnt Is 0 na Modulation xtinction ratio M 00 Rcivr nois tmpratur Tr 00 K Rcivr load rsistanc RL 030 Ω Figur - Bit rror probability vrsus cod wight for two cosscorrlation for L89, N7. III. EFFECT OF VARRYING λc ON CODE WEIGHT AND BIT ERROR PROBABILITY An uppr bound on pb for OCDMA systms mploying SHL and nglcting th ffct of rcivr noiss using gnralizd OOC is givn in [3]: w kw w PE + ( k k λ L ( w λ( w λ...( w λ k + (...( w w w k + N Considr two cod st of OOC λ c, λ c 3 in a fixd BE. W ar going to mak a fair comparison btwn thm. St valus of cod lngth and th numbr of simultanous usrs rspctivly as L89 and N7. Basd on [3, (45], on can gt λ opt 3 and thn w opt 35. Dcrasing λ c as λ c will rsult in w opt 7. Th givn uppr bound of two abov cod sts using (5 has bn plottd in Figur. As can bn sn from this figur, comparison of abov two cass rvals that lowr cross-corrlation cod st, i.. λ c, rsult in bttr prformanc in th lowr valu of cod wights. Th abov rsult will b mor intrsting whn w ar taking transmittd powr into account. This powr is usd for dsird usr whn bit "" has transmittd. This is du to th fact that for a fixd transmittd powr - dcrasing w - incrass th signal to nois ratio during th dsird (mark chip tims which dirctly lads to lowr bit rror probabilitis. Furthrmor, according to this figur, incrasing cod wight in low transmittd powrs incrass bit rror probability whn th ffcts of rcivr noiss ar includd. Among all cod sts which thir cardinality ar gratr than or qual to N, th cod st with lowr λ c is prfrabl. Thus, λ opt in th abov xampl is λ opt and using Johnson bound th corrsponding cod wight is w opt w max 7. (5 A prim cod can b shown ( p, p, p,. In th abov xampl lt p 7 and thn constructs a prim cod st. It is apparnt that th paramtrs of this prim cod st ar clos to an OOC with λ c. In othr words, considring th constraint on cod wight causs prim cod (89, 7, 7, with lowr cross-corrlation to b bttr than OOC with λ c 3. It also worth noting that, considring λ a p, it's possibl to choos valus of cod wight abov p, but prim cod is limitd to hav w p. Th prformanc of OCDMA systms using prim cod is bttr than thir countrparts using optimum OOC with λ opt 3. Scondly, for low transmittd powrs, lss cod wights rsults in bttr prformanc. Sinc thr is not a closd form formula for th pb of OOC with λ c 3, a comparison is mad btwn OCDMA systms mploying OOC with λ c, as a cod with highr cross-corrlation, and OCDMA systms mploying OOC with λ c, as a cod with lowr cross-corrlation. Sinc w ar also intrstd in making this comparison undr fixd BE, prim cod is chosn as an accptabl OOC with λ c. To mak th abov comparison w hav usd th drivd bit rror probabilitis of OCDMA systm using OOC with λ which ar givn in [8] and [9]. c IV. PERFORMANCE ANALYSIS OF OCDMA SYSTEMS USING PRIME CODE Lt l and l b rspctivly th numbr of intrfring usrs which hit xactly on and two marks of dsird cod in OCDMA systms using prim cods. Probability distribution function (pdf of intrfrnc pattrn has multinomial distribution with paramtrs N, q and q : 546
Pr( l, l ( N! l! l!( N l l! l l N l l q ( q q q whr ( p + ( p q, p + p + 7 p p q and q 0 p 6 p p []. Morovr, it should b notd that th total numbr of intrfring usrs is l l + l and th total numbr of intrfring pulss (marks is l + l. A. Prformanc Analysis for OCDMA Systm with Soft Rcivr In this part w study th pb of OCDMA systms using prim cods with soft rcivr. W dfin th st A as: { l l { 0,,,..., N } l + l } A N (7, As mntiond arlir, r.v. Z is Gaussian and has th conditional pdf as: P Z ( z I i,snt bit b ( z μ ( i xp σ ( i 0,, furthrmor w hav: whr { } Pr( rror snt bit b 0, Th Pr πσ ( i Th μ 0( l + l Q Pr( l, l 0( l l σ + l, l A ( rror snt bit b, Th Th μ ( l + l Q Pr( l, l ( l l σ + l, l A y x whr Q( x dy μ π c I s + LTc and w hav: ( i GT ( w + i λ + wn ( w + i s λs M Ib + L λ ( ( s Ib σ i G FTc w + i λs + ( wn w + i + L M I s + LTc + σ th Finally, th bit rror probability is obtaind as: (6 (8 (9 (0 ( ( Pr( rror Pr Min{ Pr( rror snt bit b 0, Th Th ( rror snt bit b, Th } + (3 B. Prformanc Analysis for OCDMA Systm with SHL In this part w study th pb of OCDMA systms using prim cods with SHL rcivr in prsnc of rcivr noiss. Pr i m i whr, Achiving this goal, it is ssntial to hav ( i is th intrfrnc pattrn vctor and i is its wight or numbr of nonzro lmnts of i [6]. W obtain: Pr ( rror snt bit b 0, Th ( w, l + l min 0 Q l, 0 0 0( l A m σ m m ( q + q + q Pr( l, l 0 Th μ ( w N m m w w Th μ ( snt bit b, Th Q σ (4 Pr rror (5 whr in ths two quations w hav: Ib I s μ 0 ( GTc λs + L + LTc (6 Ib Is σ 0 ( G FTc λs + L + LTc + σ th (7 Ib Is μ GTc wλs + L + LTc (8 Ib Is σ G FT c wλs + L + LTc + σ th (9 C. Prformanc Analysis for OCDMA Systm with DHL In this part w study th pb of OCDMA systm using prim cod with DHL rcivr. In this cas, th bit rror is occurrd if intrfring usrs hit all th mark positions of dsird usrs cod, i.. i w. Pr( rror snt bit b 0, Th w Th μ pass Q ( ( σ l, l A m 0 ( pass w m (0 547
Pr N m w m q0 q q Pr( l, l m + + w w min( w, l + l Th μ( block + Q l, l A 0 m 0 σ ( block m w ( m N m m q0 + q + q Pr( l, l w w Th μ σ ( pass ( ( pass rror snt bit b, Th Q ( a: (N5, p7 and c: ( N5, p9 Prim cod with nois P tr -0dBW using (3-9; b:(n5, p7 and d:(n5, p9 Prim cod idal link [6, (6]; :(N5, L89, w4 and g:(n5, L36, w5 OOC with nois P tr -0dBW [9, (6, (35, (40]; f:(n5, L89, w4 and h:(n5, L89, w5 OOC idal link [5, (34]. Figur 3- Bit rror probability vrsus normalizd thrshold using prim cod and OOC mploying SHL rcivr. whr in ths two quations w hav [9]: Ib Is μ ( block GLTc + LTc Ib Is σ ( block G FLTc + LTc + σ th Ib μ ( GTc wλs + L pass + Is LTc Ib Is σ ( pass G FT c wλs + L + LTc + σ th ( (3 (4 (5 V. NUMERICAL RESULTS In this sction, our numrical rsults for OCDMA systms mploying prim cod and OOC for thr diffrnt rcivr structurs, namly soft rcivr, SHL and DHL ar prsntd. In ths calculations paramtr valus givn in Tabl I ar usd. Throughout this sction, by "Normalizd Thrshold" w man th thrshold lvl which is fittd appropriatly to covr all variations of our plots. It is notd that by -P tr - w man that th transmittd lasr powr of dsird usr during T whn th b transmittd bit is "". Th bit rror probability for OCDMA systms utilizing SHL rcivr and using prim cods and OOCs vrsus normalizd thrshold ar plottd in Figur 3. Equations (4, (5, (3 with paramtr st of ( N 5, p 7 and ( N 5, p 9 hav bn usd to plot this figur. All of curvs in this figur hav bn plottd for P tr -0dBW. Our numrical rsults ar also compard with th idal link [6, (6]. Two optical orthogonal cod sts of ( L 89, w 4, N 5, ( L 36, w 5, N 5 which ar quivalnt to th prvious cod sts from th BE point of viw hav also bn prsntd in Figur 4. Bit rror probability of OCDMA systms using SHL in thir rcivr structur is calculatd using [9, (6, (35, (40]. For plotting th idal link w hav also usd [5, (34]. A comparison of OCDMA systms utilizing DHL rcivr using prim cod and OOC vrsus transmittd powr is mad in figur 4. Two fair comparisons ar mad btwn prim cod and OOC in a fixd bandwidth fficincy. Choosing N9 and stting p9, w hav th prim cod family of ( 36,9,9, and it's OOC countrpart is th family of ( 36,4,,. As can b sn from this figur, th prformanc of systms using prim cod is bttr than ons using OOC whn th transmittd lasr powr is gratr than -5.5dBW. Th prformanc supriority of systms using OOC rgards to systms using prim cod is bcaus of having lss cod wight and thrfor rciving lss nois powr. Figurs 5 and 6 show bit rror probability vrsus numbr of simultanous usrs for both OOCs and prim cods and mploying DHL rcivr whn th rcivd lasr powr is qual to -50dBW and -55dBW, rspctivly. For xampl, for making a comparison btwn prim cod and OOC in a fixd BE in figur 5, lt p7, hnc, th maximum numbr of simultanous usrs N max 7. OOC with th paramtr st of ( L 89, w 4, λ a, λ c is assumd. As can b sn from this figur, OCDMA systms using prim cod outprform ons using OOC considrably. But for lowr transmittd powr (.g. -55dBW OCDMA systms using OOC with λ c prform bttr than ons using prim cod (s figur 6. 548
VI. CONCLUSION In this papr, w studid asynchronous fibr optic CDMA systms mploying cods with λ c in th prsnc of rcivr noiss. W drivd a closd form formula for ach rcivr structur, namly soft rcivr, SHL and DHL. Th bit rror probability of systms using prim cod was compard with thir countrparts using OOC. W concludd that incras th valu of λ c from λ c to λ c by itslf could not mak th prformanc bttr or wors but it dpnds on transmittd lasr powr. In othr words, for high transmittd lasr powrs, incrasing th valu of λ c which is ld to incrasing cod wight (for achiving minimum bit rror probability will nhanc th prformanc. But, for low transmittd powrs, dcrasing th valu of λ c which is ld to dcrasing cod wight will nhanc th prformanc. Figur 4-Bit rror probability vrsus transmittd lasr powr using DHL rcivr for prim cod and OOC. Figur 5- Bit rror probability vrsus numbr of simultanous usrs at fixd bandwidth fficincy. Figur 6- Bit rror probability vrsus numbr of simultanous usrs at fixd bandwidth fficincy. ACKNOWLEDGMENT Authors apprciat Iran Tlcommunication Rsarch Cntr (ITRC for financial supporting this rsarch. Th first author also thanks Abbasali Ghorban Sabbagh for his hlpful commnts. REFERENCES [] J. A. Salhi, "Cod Division Multipl-Accss Tchniqus in Optical Fibr Ntworks-Part I: Fundamntal Principals," IEEE Trans. Commun., vol. 37, no. 8, pp. 84-833, Aug. 989. [] G.-C. Yang and T. Fua, "Optical orthogonal cods with unqual auto and cross-corrlation constraints," IEEE Trans. Inform. Thory, vol. 4,pp. 96 06, Jan. 995. [3] S. Mashhadi and J. A. Salhi, "Cod-Division Multipl-Accss Tchniqus in Optical Fibr Ntworks Part III: Optical AND Logic Gat Rcivr Structur With Gnralizd Optical Orthogonal Cods," IEEE Trans. Commun., vol. 54, no. 8, pp. 457-468, Aug. 006. [4] M. Y. Azizoglu, J. A. Salhi, and Y. Li, "Optical CDMA via Tmporal Cods," IEEE Trans. Commun., vol. 40, no. 7, pp. 6-69, July 99. [5] J. A. Salhi, and C. A. Bracktt, "Cod Division Multipl-Accss Tchniqus in Optical Fibr Ntworks-Part II: Systms Prformanc Analysis," IEEE Trans. Commun., vol. 37, no. 8, pp. 834-84, Aug. 989. [6] J. J. Chn, and G. C. Yang, "CDMA Fibr-Optic Systms with Optical Hard Limitrs," J. Lightwav Tchnol., vol. 9, no. 7, pp. 950-958, Jul. 00. [7] P. Kamath, J. D. Touch and J. A. Bannistr, "Th Nd for Mdia Accss Control in Optical CDMA Ntworks," IEEE Computr and Communications Socitis, vol. 4, pp. 08-9, Mar. 004. [8] M. Kwon, "Optical Orthogonal Cod-Division Multipl - Accss Systm Part I: APD Nois and Thrmal Nois," IEEE Trans. Commun., vol. 4, no. 7, pp. 470-479, July 994. [9] A. Ghorban Sabbagh, M. Molavi K. and M. Mirsalhi, "Prformanc Analysis of Optical CDMA Systms Using APD for two typs of rcivr structurs in th Prsnc of Intrfrnc and Rcivr Noiss, IST008, Iran Tlcommunication Rsarch Cntr (ITRC, indxd by IEEE, Aug. 008. [0] S. Zahdi, and J. A. Salhi, " Analytical Comparison of Various Fibr- Optic CDMA Rcivr Structurs," J. Lightwav Tchnol., vol. 8, no., pp. 78-77, Dc. 000. [] T. Ohtsuki, "Prformanc Analysis of Dirct-Dtction Optical Asynchronous CDMA Systms with Doubl Optical Hard-Limitrs," J. Lightwav Tchnol., vol. 5, no. 3, pp. 45-457, Mar. 997. [] G. C. Yang, Prim Cods with Applications to CDMA Optical and Wirlss Ntorks, Boston London: Artch Hous, 00. 549