Optical Signal-to-Noise Ratio and the Q-Factor in Fiber-Optic Communication Systems

Transcription

1 Applcaton ote: FA-9.0. Re.; 04/08 Optcal Sgnal-to-ose Rato and the Q-Factor n Fber-Optc Communcaton Systems Functonal Dagrams Pn Confguratons appear at end of data sheet. Functonal Dagrams contnued at end of data sheet. UCSP s a trademark of Maxm Integrated Products, Inc. AVAIAB

2 Optcal Sgnal-to-ose Rato and the Q-Factor n Fber-Optc Communcaton Systems Introducton The rato of sgnal power to nose power at the receer of a fber-optc communcaton system has a drect mpact on the system performance. Many electrcal engneers are famlar wth sgnal-to-nose rato (SR) concepts when referrng to electrcal sgnal and nose powers, but hae less famlarty wth the equalent optcal sgnal and nose powers. The purpose of ths applcaton note s to show the relatonshp between the electrcal and optcal sgnal-to-nose rato (SR), and then ntroduce the Q-factor. Whle the prncples outlned n ths applcaton note may be appled to many types of systems, the scope of the dscusson s lmted to bnary dgtal communcatons oer optcal fber. Wthn ths scope, there are only two possble symbols that can be transmtted, where these symbols represent a bnary one or a bnary zero. Thus, the symbol rate and the bt rate are equalent. Sgnal Power The power n an arbtrary electrcal waeform can be defned as the oltage multpled by the current, whch s wrtten mathematcally as: P ( ( ( () Usng ohm s law, we can substtute ( (R, or alternately ( (/R, nto equaton () to get: P ( ( / R ( R () Applcaton ote FA-9.0. (Re.; 04/08) where R oltage/current s the resstance n ohms. In bnary dgtal communcatons, the sgnal s lmted to two dscrete leels. Based on ths, we can represent the electrcal sgnal power at any gen tme by ether: S R or S R (3) R R where S represents electrcal sgnal power and the subscrpts and represent the low or hgh power, oltage, or current leels assocated wth a bnary zero or one respectely. ow we wll repeat the aboe deratons for the case of optcal sgnals usng electromagnetc ector notaton. Usng ths notaton, the power n an optcal sgnal can be defned as the magntude of the ector cross product of the electrc and magnetc felds, whch can be wrtten and smplfed as follows: P O ( ( ( ( ( ( (4) η η where the notaton X represents the magntude of µ X, and η s the optcal mpedance of the ε fber (µ permeablty and ε permttty). Recognzng that there are only two dscrete power leels leads to the optcal equalent of equaton (3),.e., SO and SO (5) η η where S O represents optcal sgnal power and the subscrpts and represent the low and hgh power or low and hgh electrc feld strengths assocated wth a bnary zero or one respectely. 3 ose Power ose can be defned as any unwanted or nterferng sgnal other than the one that s ntended or expected. The arous types of nose and ther sources are beyond the scope of ths applcaton Maxm Integrated Page of 7

3 note. For purposes of llustraton we wll model the nose power as random, normally dstrbuted, zero mean, and addte (the most common type of nose). The random nature of the nose means that the nstantaneous alue of the nose ampltude s unpredctable. Thus, nstead of classfyng the nose n terms of ts actual alue at any gen tme, we use statstcal aerages and probabltes. We wll classfy the nose ampltude n terms of ts rootmean-square (rms) aerage, whch s commonly gen the symbol. The nose power s smlarly expressed n terms of ts mean-square aerage (equalent to the statstcal arance), whch s gen the symbol. In general, the noses assocated wth the hgh and low sgnal leels n bnary optcal dgtal communcatons each hae a dfferent alue. The mean-square aerage electrcal and optcal nose powers can be computed mathematcally usng the followng equatons: T T ) 0 ( t R dt R Applcaton ote FA-9.0. (Re.; 04/08) T ( dt T (6) 0 R R T o ( ) T t dt o 0 η η (7) where s the nose power, T s the ntegraton perod, s the mean-square aerage power, and the subscrpt sgnfes that the assocated current, oltage or electrc feld s classfed as nose. 4 Sgnal Plus ose Addton of the sgnal and nose ampltudes ersus addton of the sgnal and nose powers can sometmes cause mathematcal confuson. For example, f the combned sgnal and nose ampltude s wrtten as ( ), then the power would seemngly be ( ) ( R ( ) R, whch, when multpled out, s equal to ( R. But, addton of the results of equatons (3) and (6) ges S ( ) R. So, why s there a dfference n the two results? The answer les n the fact that when we add the sgnal (a constan and the nose (an aerage alue), we compute the result as an aerage. (We don t need to know the alue of the sgnal plus nose at eery nstant of tme we only care about the aerage alue.) In the aerage, the cross term s equal to zero. The reason for ths s that the probablty densty functon (pdf) of the nose was defned as zero mean and normally dstrbuted. Snce ths pdf s symmetrc about the mean, multplcaton by a constant wll not change the mean, whch wll reman zero,.e., n the aerage, the result wll always be zero. 5 Sgnal-to-ose Rato (SR) Knowledge of the rato of the sgnal power to the nose power (sgnal-to-nose rato or SR) s mportant because t s drectly related to the bt error rato (BR) n dgtal communcaton systems, and the BR s a major ndcator of the qualty of the oerall system. Drawng from the results of the precedng sectons, we can mathematcally express the electrcal SR as SR S R R Smlarly, the optcal SR s SR o R (8) R So η (9) o o o η In practce, optcal powers are rarely measured drectly. Instead, the optcal power s conerted to a proportonal electrc current usng a dece such as a PI photodode, and then the current s measured. The rato between the output current and the ncdent optcal power s called the responsty (mathematcally represented usng the symbol R ), whch has the unts of Amperes per Watt (A/W). It s mportant to note that the conerson between optcal power (unts of Watts) and electrcal current Maxm Integrated Page 3 of 7

4 (unts of Amperes) essentally results n a square root operaton. In other words, as we recall from equaton (), electrcal power s related to the square of the oltage or current, and, as we recall from equaton (5) optcal power s related to the square of the magntude of the electrc feld. The result s that the conerson between optcal sgnal or nose power, (S o or o both related to ) and electrcal current results n what s essentally a square root relatonshp,.e., R and nose o R (0) sgnal S o Also, the optcal SR, when conerted to an electrcal SR, s equal to the square root of the equalent electrcal SR. Ths s llustrated mathematcally by combnng equatons (8) and (0) as follows: S sgnal SoR SR or 6 The Q-Factor Applcaton ote FA-9.0. (Re.; 04/08) SR o () As dscussed preously, there are only two possble sgnal leels n bnary dgtal communcaton systems and each of these sgnal leels may hae a dfferent aerage nose assocated wth t. Ths means that there are essentally two dscrete sgnalto-nose ratos, whch are assocated wth the two possble sgnal leels. In order to calculate the oerall probablty of bt error, we must account for both of the sgnal-to-nose ratos. In ths secton we wll show that the two SRs can be combned nto a sngle quantty prodng a conenent measure of oerall system qualty called the Q-factor. In the followng dscusson, we wll assume the sgnals are electrcal oltages, but, as demonstrated n the preous sectons, the concepts can easly be extended to electrcal current sgnals or optcal sgnals. To begn ths dscusson, we consder the decson crcut n a fber-optc receer, whch smply compares the sampled oltage, (, to a reference alue,, called the decson threshold. If ( s greater than, t ndcates that a bnary one was sent, whereas f ( s less than, t ndcates that a bnary zero was sent. Assumng perfect synchronzaton between the bt stream and the bt clock, the major obstacle to makng the correct decson s nose added to the receed data. V CC If we assume that addte whte Gaussan nose (AWG) s the domnant cause of erroneous decsons, then we can calculate the statstcal probablty of makng such a decson. The probablty densty functon for ( wth AWG can be wrtten mathematcally usng the Gaussan probablty densty functon (pdf) as follows: ( ) t s [ ( ), ] x PROB t x e () π x where S s the oltage sent by the transmtter (the mean alue of the densty functon), ( s the sampled oltage alue n the receer at tme t, and s the standard deaton of the nose. quaton () s llustrated n Fgure. PROB [(] Fgure. Block-dagram of a fber-optc receer Fgure. AWG probablty densty functon S Decson Crcut If we assume that S can take on one of two oltage leels, whch we wll call and, then the probablty of makng an erroneous decson n the receer s: Maxm Integrated Page 4 of 7 D Q P DATA CK (

5 P[ε] P[( > S ] P[ S ] P[( < S ] P[ S ] (3) where P[ε] s the probablty of error and P[x y] represents the condtonal probablty of x gen y. If we further assume an equal probablty of sendng ersus (50% mark densty), then P[ S ] P[ S ] 0.5. Usng ths assumpton, equaton (3) can be reduced to: P[ε] P[( > S ] 0.5 P[( < S ] 0.5 PROB t dt [ ( ), ] PROB [ ( t ), ] dt (4) where PROB[(, x ] s defned n equaton (). Ths result s llustrated n Fgure 3. ( Sgnal ose P[( S ] ( From Fgure 3 and equatons (3) and (4) we can conclude that the probablty of error s equal to the area under the tals of the densty functons that extend beyond the threshold,. Ths area, and thus the bt error rato (BR), s determned by two factors: () the standard deatons of the nose ( and ) and () the oltage dfference between and. P[( S ] PROB[(] P[( > S ] P[( < S ] Fgure 3. Probablty of error for bnary sgnalng It s mportant to note that for the specal case when, the threshold s halfway between the low and hgh leels (.e., ( )/). But, for the more general case when, the optmum threshold for mnmum BR wll be hgher or lower than ( )/. In order to sole equaton (4) we need a practcal way to compute the result of the ntegrated Gaussan pdf ( PROB[(, x ] ) that s defned n equaton (). Snce there s no known closed form soluton to ths ntegral, t must be ealuated numercally. To mantan compatblty wth exstng numercal solutons, equaton () can be re-wrtten n ts equalent standardzed (zero mean and standard deaton of one) form. In order to conert to the standardzed form, we use the well known z (x µ)/ substtuton, where x ( and µ S n equaton (). For example, we start wth PROB[ x, ] dx xµ e π (from equatons () and (4) ) µ and then, substtutng z x (so that x z µ and dx dz ) results n: dx Applcaton ote FA-9.0. (Re.; 04/08) Maxm Integrated Page 5 of 7

6 π z e x z µ dz, whch s defned as the error functon z r z dz ( ) (5) π z e (ote that there are a number of aratons of ths functon publshed n the lterature.) The error functon ges the area under the tal of the Gaussan pdf (mean S and standard deaton x ) between ( and nfnty. Ths form of the error functon s useful because numercal solutons are aalable n both tabulated form and as bult-n functons wthn many software utltes (e.g., r(x) - ORMSDIST(x) n Mcrosoft xcel). In terms of r(z), equaton (4) can be rewrtten as : P[ ε ] r r (6) It s nterestng to note that the arguments of the error functons n equaton (6) represent the square root of the sgnal power dded by the square root of the nose power, whch, we recall from equaton (), s equalent to the optcal sgnal-to-nose rato. Thus, equaton (6) can be rewrtten as follows: P[ ε ] r [ SRO ] r [ SR O ] (7) where SR O and SR O are the optcal SRs for the hgh and low leels. The optmum threshold leel, opt, s defned as the threshold leel that results n the lowest probablty of bt error. Further, settng the optmum threshold leel also results n the same probablty of bt error when a hgh sgnal s transmtted as when a low sgnal s transmtted. Ths means that for the specal condton of opt, SR O SR O, whch leads to the followng defnton of the Q-factor 3 : opt opt Q (8) Applcaton ote FA-9.0. (Re.; 04/08) By substtutng the defnton of Q from equaton (8) nto equaton (6) we fnd that, when opt P [ ε ] r [ Q] r [ Q] r[ Q] (9) ext, we sole equaton (8) for opt to get opt (0) and then substtute ths expresson for opt back nto equaton (8) to get Q () It should be noted that multplyng the nddual terms n equaton () by resstance, mpedance, or responsty wll conert the expresson for Q to equalent terms of current or optcal power,.e., Q P O O () Fnally, we can substtute equaton () nto the result from equaton (9) to get P[ ε ] r[ Q] r (3) 7 Conclusons The Q-factor defned n equatons (8) and () represents the optcal sgnal-to-nose rato for a bnary optcal communcaton system. It combnes the separate SRs assocated wth the hgh and low leels nto oerall system SR. The form of the Q- factor gen n equaton () smplfes both the measurement of SR and the calculaton of the theoretcal BR due to addte random nose. For example, measurement of the Q-factor can be performed wth the ertcal hstogram functon on many communcatons osclloscopes. Ths can be done by dsplayng a porton of the data pattern and alternately applyng the ertcal hstogram to the hgh (one) leel and the low (zero) leel. The osclloscope hstogram functon wll estmate the P Maxm Integrated Page 6 of 7

7 mean ( or ) and the standard deaton ( or ), whch can then be used drectly to compute the Q-factor. The Q-factor s also useful as an ntute fgure of mert that s drectly ted to the BR. For example, the BR can be mproed by ether () ncreasng the dfference between the hgh and low leels n the numerator of the Q-factor, or () decreasng the nose terms n the denomnator of the Q-factor. Fnally, the Q-factor allows smplfed analyss of system performance. The most drect measure of system performance s the BR, but calculaton of the BR requres ealuaton of the cumulate normal dstrbuton ntegral. Snce ths ntegral has no closed form soluton, ealuaton requres numercal ntegraton or the use of tabulated alues. A much smpler method of analyzng system performance s to optmze the Q-factor, knowng that ths wll result n optmzed BR. B. Sklar, Dgtal Communcatons: Fundamentals and Applcatons, nglewood Clffs, ew Jersey: Prentce all, pp G. Agrawal, Fber-Optc Communcaton Systems, ew York,.Y.: John Wley & Sons, pp. 7, S. Bergano, F.W. Kerfoot, and C.R. Dadson, "Margn Measurements n Optcal Amplfer Systems, " n I Photoncs Technology etters, ol.5, no. 3, pp , Mar Applcaton ote FA-9.0. (Re.; 04/08) Maxm Integrated Page 7 of 7