NEW GAUSSIAN APPROXIMATION FOR PERFORMANCE EVALUATION OF OPTICAL RECEIVERS WITH ARBITRARY OPTICAL AND ELECTRICAL FILTERS

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1 NW GAUSSIAN APPROXIMATION FOR PRFORMANC VALUATION OF OPTICAL RCIVRS WIT ARBITRARY OPTICAL AND LCTRICAL FILTRS Jã L. Rba a Af V. T. Cata Abtact A w Gaua appmat (GA) whch ta t accut th fuc f abta ptca a ctca ft pp f th pfmac vauat f ptca pampf cv. Ug th w GA, th cv tvt vauat a tab pfmac ffc w bv f va Fab-Pt ptca ft. I. INTRODUCTION Th Gaua appmat (GA) ha b ut t vauat th pfmac f ptca pampf cv. It ht mpct, atv accuac, a ht cmputat tm hav b uffct a t u t. Sva GA hav b pp -6], hwv f thm ca fu accmmat th fuc f abta ptca a ctca ft. I ], ctagua ptca a ctca ft hav b aum, a w a ctat ga pw, a a mpf AG ha b v cg a appmat ga-pt vaac cacuat v th tta avag ga. I,] a ctagua ptca ft wth much ag bawth tha th bt at ha b c, a th GA pt aw t ta abta ctca ft t accut. I ], a GA f a ag bawth ptca ft a a ht-tm tgat a ctca ft ha b pt. I 5,6], a Ltza hap f th ptca Fab-Pt (FP) ft ha b aum, a th ma a vaac p gvau bta fm Kahu-Lèv ga a pa, a t pt pct th pc th ctca a ptca ft taf fuct. I th w, a w GA that accmmat abta ptca a ctca ft pt. Futhm, th GA a ta t accut th ffct f wavfm tt th ga a th ga-ptau m () bat. act p f th ma a vaac f th cut at th c ccut put a v, whch ca pt th pc th ptca a ctca ft taf fuct a ta t accut a pa pc abc. Ug th GA, th cv tvt vauat f fft f FP ptca ft a ctca ft. J. Rba a A. Cata a wth th Optca Cmmucat Gup, Dpt. ctca a Cmput gg, Ittut Tcmucaçõ, Ittut Sup Técc Pó I, Lba, Ptuga. Th w wa uppt b FCT a POSI wth pjct POSI/5576/CPS/000 - DWDM/ODC. J. Rba wu t tha FCT f upptg th w a u ctact SFR/BD/8/000. II. OPTICALLY PRAMPLIFID RCIVR MODL Th bc agam f th ptca cv hw Fg.. Th ptca cv cu a pampf wth fat ga G, whch ampf th cv ctca f ( t), a a () t t th ga. () t G Optca pampf () t A Pa A () t () t (f) Optca ft () t () t (t) PIN (f) ctca ft Fg. Bc agam f ptca cv. Sampg (t) Dc ccut Th ga a f a aum t hav abta paat. A abta paat ca b pa a a wght uppt f tw thga J vct 7]. c, th ga a f a gv, pctv, b, () () t () t () t =, () () t () t () t =, wh (t),, (t) a (t),, (t) a th cv ga a th cmpt v th thga ct f b th ut vct a, pctv. Uua, th ga f c wth paat tat -6], a th tuat, th tw thga paat m a f t a paa (t th ga) a thga m. Wth f gat, f w c th ga a pa ct, th a, th aumpt f ga paat tat accmph. Th aum a ptca atv wht Gaua (AWGN) wth pw pcta t ach thga ct gv b S = p (G-)hν wth p th ptau m fact a hν th pht g. Th pa tamt th ctca f cmpt th ct a bc th thga cmpt th ct. Th ffct f th pa ca b cb b A A () t G () t G p () t = ) ( () () t () t p () t =, ), ( () wh w t p= p=, pctv, th pc abc f a pa.

2 Th ga th ft b a ptca ft wth mpu p a taf fuct p, pctv, a h (t) a (f), a wth a -B bawth, B. W aum that th ptca ft act qua v th tw ct. Th utput ft ga () t () t, wth ( t) = A ( t) h ( t) (5) ( t) = ( t) h ( t) (6) A wh ta f th cvut pat. Th cmp vp f th, () t, a f th cv f, () t, a f, pctv b ] () t = R () t jπν { } t jπν { t } t () t = R () wh ν th ptca ca fquc. Th wpa quvat f th mpu p, h (t), a taf fuct, (f), f th ptca ft a f b h () t R h () t jπν { } t, (7) (8) = (9) ( f ) = ( f ν ) ( f ν ) (0) F th wpa-quvat m, fm p (), (), (5) a (6), w hav A () t = G () t G ( p ) ( t) () A () t =, () t ( p ), () t () () t = A () t h () t () () t = A () t h () t () wh () t a () t a th cmp vp f () t a, pctv. () t Th PIN phttct aum a a qua-aw tct wth pvt R, a th PIN utput cut (t) 8] () t R () t () t = (5) Th ubqut ctca ccut m b a ctca ft wth mpu p a taf fuct, pctv, h (t) a (f), a a -B bawth, B. It utput cut (t), whch gv b () t () t h () t = (6) amp a a c ccut c th tamtt mb. T tmat th cv pfmac thugh th GA, t ca t v p f th ma a vaac f (t). Fm ()-(6), (t) ca b p a h {, ] G ( τ), ( τ) ] G τ, τ ] G ( τ) ( τ) ]} () t = R G ( τ ) ( τ ) ( p ) ( ) ( ) ) h ) h ( t χ) τ τ χ, (7) III. GAUSSIAN APPROXIMATION Fm (7), th ma a vaac uct taghtfwa, but gth. Dta ca b fu App A. act p f th ma a vaac f (t) whch pct pt th pc th taf fuct f th abta ft a, pctv, µ () t = pr S ( 0) ( f ) f R G () τ h () τ ( p ) ( τ ) h ( τ ) h ( t ) τ (8) () t = R GS () τ h () τ ( p ) () τ h () τ pr S { ] h ( t )} h () τ { ] h ( t )} h () τ ( f ) ( f ) ( f ) f τ (9) Th ft tm f (8) th ma cut cmpt u t th ptau m a th c th ga cmpt. I (9), th ft tm th ga- bat a th c tm th - bat. Th cv pfmac tmat aatca thugh th pbabt. Ug a GA, th pbabt gv b 6] N F µ 0, N µ, F P = ( ) ( ) Q Q (0) N = 0 0, = 0, a = 0 a = wh N th tta gth f th ba quc; a th - th mb th 0 ; F th thh v; Q() f ]; a µ, a, a, pctv, th ma a taa vat f (t) ct b th mb a at th ampg tat, t =/Bt, wth =0,...,N-, wh t ca b ch t ptm th tm pfmac a B th bt at. Th ptma thh v, F pt, bta fm N = 0 0, ( = 0) a N = = 0, ( = ) a Fpt µ 0, p 0,, µ F p, pt = () whch ut fm ttg t z th vatv f (0) wth pct t F.

3 IV. COMPARISON WIT OTR AUTORS RSULTS Thugh apppat mpfcat, a cg that th ga pa th ct a p=, p (8) a (9) gat th ma a vaac p pt -5]. I th ct, th mpfcat a pa f th ca c Rf. ] a,]. A. Rctagua ptca ft I Rf.,], a ag bawth ctagua ptca ft aum wth wpa quvat taf fuct ( f ) ct( f B ), = () wh ct(f) ta f th ctagua fuct wth ut wth. It a aum that th ptca ft t affct th cv ga. Statg fm th at ( f ) ( f ) = B Λ( f B ), () wh Λ(f) th tagua fuct f b f B, f < B Λ ( f B ) = () 0, thw a ubttutg () a () (8) a (9), th ma a vaac p a gv, pctv, b () t = R S B ( ) R G P( χ ) h ( t χ ) µ 0 χ (5) () t = R S G P() t h () t R S ] R S B ( f ) f ( f ) f f (6) wh th ptca pw at th cv put gv b P(t)= (t). Sc B aum cab ag tha th bt at, th th tm f (6) ca b gct a, thu, th ma a vaac p cc wth th p pt,]. B. Rctagua ctca ft a ctat ga pw I at t th ctagua ptca ft, Rf. ] th ga c a havg ctat pw, (t)= P, a th ctca ft a ctagua ft wth bawth B a ut ga. It a aum that th ptca ft act v th a t affct th put ga. I th ca, p (5) a (6) mpf t µ () t = R S B R GP (7) () t = R S GPB R S ( B B B ) (8) whch cp t th ma a vaac p pt ]. V. NUMRICAL RSULTS I th ct, th cv tvt tmat b th w GA cmpa wth th bta wth th GA pt,] a th fatu a cu. Th, th w GA u t tmat th cv pfmac f va ptca a ctca ft. Wth f gat, th ga pa th ct. Th paamt c f th cv a G=0B, p =, B=0Gbt/, R =A/W, N=6, p=, ctagua pu hap a aum f th ba quc. Th tvt cacuat f th pbabt f 0 -. I Fg., th tvt bta wth th w GA a th GA,] a pct. A g-cavt FP ft 9] wth f pcta ag f FSR=00Gz, a a Buttwth ctca ft wth B =0.65B a c. I t cmput th cv tvt ug th GA pt,], a ctagua ptca ft wth th am quvat bawth (wth th FSR) a th FP ft aum, a th ptca ga ft b th FP ptca ft. Stvt Bm] Nw GA GA ] a ] B Gz] Fg. Stvt v ptca ft bawth. Fm Fg., cpac btw th ut tmat b th tw GA a bvu. Stvt ffc f 5B a fu f w B. F ag B, th ma a th vaac achv b th GA pt,] t t (8) a (9), a hw ct IV.A. Th w GA ca a pa tmat a ptma bawth f th ptca ft, wha th GA pt,] t gv th fmat ca. I Fg. a, th w GA u t tmat th cv pfmac wth th fft ptca ft: th gcavt FP ft, th ub-cavt FP ft 9] a th thm FP ft 0]. I Fg. a Fg., th ctca ft a a a 6 th Buttwth wth B =0.65B, pctv. Fg. a hw that th th-m FP ft a t w tvt bhavu. Th tvt bhavu f th g- a ub-cavt FP ft v a, but th ub-cavt FP ft pfmac ght w tha th g-cavt FP ft pfmac.

4 Stvt Bm] g-cavt FP ub-cavt FP th-m FP B Gz] Fg. Stvt v ptca ft bawth f th Buttwth ft. Stvt Bm] g-cavt FP ub-cavt FP th-m FP B Gz] Fg. Stvt v ptca ft bawth f th 6 th Buttwth ft. F w ptca bawth, th tmb tfc (ISI) ffct mat th tvt gaat. S ft wth hap cut-ff a hgh a p th paba g (a th th-m FP) a t w cv pfmac, bcau th tuc hgh ISI. Wth th ca f th ptca ft bawth, th fuc f th ca a th ISI tuc b th ptca ft ca. Th a pt wh th tw mpamt a t th mamum tvt. That mamum cp t th ptma ptca ft bawth. F th th-m FP ft th ptma bawth cp t 50Gz a 5Gz, wth th a 6 th Buttwth ctca ft, pctv. F th g- a ub-cavt FP ft, th bawth abut 55Gz a 7 Gz, wth th a 6 th Buttwth ctca ft, pctv. B th ptma bawth th tvt gaat ma u t th ga- bat. Th chc f th ctca ft a mptat. Fm Fg. a, th cv pfmac wt f th ctca ft wth hap cut-ff a hgh a p (6 th Buttwth ft). Th pfmac ffc ca b abv B f th ptma a ag ptca bawth. Th tvt bhavu bv Fg. t a mth a th bv Fg.. Th u t th ctca ft bawth chc. I Fg. 5, th ptmat f th ctca ft -B bawth tmat f a g-cavt FP ft wth B =0Gz a f th a 6 th Buttwth ft. Fg. 5 hw that th ptma ctca ft bawth abut 0Gz a Gz, f th a th 6 th Buttwth ft, pctv. Th bawth f th ctca ft u Fg. a wa B =Gz (0.65B). F th ctca ft bawth, a hw Fg. 5, th 6 th Buttwth ft a t a tvt gaat abv B, wha th Buttwth a t tha 0.5B gaat, cmpa wth th ptmum ctca ft bawth bv Fg. 5. Th pa th wt pfmac f th 6 th Buttwth ft bv Fg.. Stvt Bm] Butt Butt B Gz] Fg. 5 Stvt v ctca ft bawth f th gcavt FP ft. Stvt Bm] Gz Gz B Gz] Fg. 6 Stvt v ptca ft bawth f th 6 th Buttwth ft wth fft -B bawth. Fg. 6 hw that wth th ptm ctca ft bawth, th cv tvt tu ut t b much btt tha wth B =Gz f th 6 th Buttwth ft, f th ptma ptca ft bawth a abv. VI. CONCLUSIONS Th w GA a fat a mp mth t aa th cv pfmac f abta ptca a ctca ft, a th abta a gat avatag abv th GA -7]. It aw th tmat f th ptma ptca a ctca ft bawth, a hc t a mptat t th g f ptca pampf cv wth tght ptca ftg, uch a th tpca f hgh -wavgth v mutpg tm.

5 Futu w hu vtgat th accuac f th tvt tmat thugh th w GA b cmpa wth tmat bta ug m gu mth. APPNDIX A Th aumpt f th a AWGN a t th ppt () t ] = 0 A() t A() t ] = S ( t t) () t () t ] 0 A (9) δ (0) A = A () wh A (t) ta th f ct. Oth mptat ppt a: th mmt f gaua pc wth z avag a u, a thga cmpt a ucat. Ug (7), (9), (0) a (), th ma gv b h, () t ] = µ () t = R G ( τ ) ( τ ) G( p ) ( τ ) ( τ ) ps δ ( τ τ )] h ( χ τ ) ( χ τ ) h ( t χ ) τ τ χ () wh th at (p-) =(p-) f p= a u. Fm w, th tmat f (8) fm () mp a vv th tga mpfcat. Th autcat tm gv b () t () t ] = R G, ( τ ), ( τ ), ( τ), ( τ) G ( p ), ( τ ), ( τ ), ( τ) ( τ) G ( p ) ( τ), ( τ ), ( τ) ( τ) G ( p ), ( τ ), ( τ ), ( τ), ( τ ) pgs, ( τ ), ( τ) δ( τ ) G( p ) S, ( τ ), ( τ ) δ( τ ) GS, ( τ ), ( τ) δ( τ ) G( p ) S, ( τ ), ( τ ) δ( τ ) GS, ( τ ), ( τ) ( τ ) G( p ) S, ( τ ), ( τ) δ( τ ) pgs, ( τ ), ( τ ) δ( τ ) ( p ) GS, ( τ ), ( τ) ( τ ) psδ ( τ ) δ( τ ) psδ ( τ ) ( τ )] h, ( χ ) h, ( χ ) h, ( χ ) h, ( χ ) ( t χ) h ( t χ) ττ τ τ χ χ δ δ δ h () wh th at (p-) =(p-) a (p-) =(p-) f p= a a u. Fm (7), w bta G () t ] () t G ] = R G ( τ ) ( τ ) ( τ ) ( τ ) G ( p ), ( τ ), ( τ ), ( τ ) ( τ ) ( p ), ( τ ), ( τ ), ( τ ) ( τ ) ( p ) ( τ ) ( τ ) ( τ ) ( τ ) pgs ( τ ),,, δ δ h ( τ ) δ ( τ ) G( p ) S ( τ ) ( τ ) ( τ ) pgs ( τ ) ( τ ) δ ( τ ) ( p ) ( τ ) ( τ ) δ ( τ ) p S δ ( τ ) ( τ )] h ( χ ) h ( χ ) h ( χ ), ( χ ) h ( t χ ) h ( t χ ) τ τ τ τ χχ GS () B ubtactg () b (), th vaac ca b wtt a ( t) = R GS ( τ ) ( τ ) ( p ) ( τ ) h ( τ)] δ( τ ) GS ( τ) ( τ) ( p ) ( τ) ( τ)] δ( τ ) psδ ( τ ) δ( τ )] h, ( χ ) ) h ) h ) h ( t χ ) h ( t χ ) τ τ τ τ χ χ (5) Fm p (5), th vaac pt (9) ca b bta b pfmg th tgat v τ, τ, τ a τ, a th, b mpfg th utg tga. VII. RFRNCS ] N. O, Lghtwav tm wth ptca ampf, J. Lghtwav Tch., pp , Ju 989. ] L. Rb J. a Rcha, J. Pt Pfmac vauat f DFA pampf cv tag t accut tmb tfc, J. Lghtwav Tch., pp. 5-, Fb ] S. Da t a., Dta tattc f a ptca pampf ct tct cv, J. Lghtwav Tch., pp , Ma 995. ] P. umbt, M. Azz ~ gu, O th bt at f ghtwav tm wth ptca ampf, J. Lghtwav Tch., pp , Nv ] I. M, G. a, Bt vauat f ptca pampf ct tct cv wth Fab-Pt ptca ft, J. Lghtwav Tch., pp , Aug ] C. Lawtz a J. Catg, Pfmac f ptca pampf cv wth Fab-Pt ptca ft, J. Lghtwav Tch., pp. 67-7, Nv ] B. Sah, M. Tch, Fuamta f phtc, Jh W & S, Ic., Ch. 6, 99. 8] L. Kazv, S. Btt A. W, Optca fb cmmucat tm, Atch u, Ch., ] P. umbt, W. am, Cta aa a ft ptmzat f g- a ub-cavt Fab-Pt ft, J. S. Aa Cmm., pp , Aug ] J. St, L. Stuz, A. Sah, Th-m fb Fab- Pt ft f ptma g, ct. Ltt., pp , Ju 990.