JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003 87

�-Factor Estimation and Impact ofSpontaneous-Spontaneous Beat Noise on the

Performance of Optically Preamplified Systems WithArbitrary Optical Filtering

João L. Rebola, Student Member, IEEE, and Adolfo V. T. Cartaxo, Member, IEEE

Abstract—Simple explicit expressions to estimate the -factorand the sensitivity of optically preamplified receivers with arbi-trary optical filtering, which only require the eye-diagram analysisand the knowledge of the optical and electrical filters transfer func-tions, are proposed. The physical insight and fast computation timeare its main advantages. The noise-equivalent bandwidths associ-ated with the nonuniformity of the amplified spontaneous emission(ASE) noise spectrum at the photodetector input are fully charac-terized and taken into account in the expressions derivation.

By using the noise-equivalent bandwidths, a simple way ofdesigning the optical receiver filters to bound the impact of theASE-ASE beat noise on the receiver performance is provided.Results show that this impact can be neglected for extinction ratiobelow 25 dB, as long as the optical-filter 3 dB bandwidth doesnot exceed 8 the bit rate.

Numerical results reveal that our -factor expression providesacceptable estimates. Only for systems where the impact of theASE-ASE beat noise is significant (high extinction ratio) or in caseof low (below 4) and high intersymbol interference (ISI), less ac-curate estimates have been found. In case of high ISI and low ,the accuracy is improved by taking the probability of occurrence ofthe nearest rails to the decision threshold in the eye-diagram intoaccount.

Index Terms—Eye-closure, intersymbol interference, optical fil-tering, power penalty, -factor, system sensitivity.

I. INTRODUCTION

THE -factor,whichiscloselyrelatedtotheerrorprobability,is becoming a widely used tool to estimate the performance

of optically preamplified systems [1]–[8]. Usually, the analyticalassessment of the receiver performance, through the -factor es-timation from numerical simulation, either relies on numericaltechniques [1]–[4] or on closed-form expressions based on sim-plified assumptions [5], [6]. Recently, a method of estimating the

-factor from experimental data, in the presence of waveformdistortions in optically amplified systems, has been proposed [7].This method is an improvement on the one proposed in [8] andrelies on the assumption that the error probability is determineddominantly by the eye-closure (nearest rails to the decision level).

Manuscript received March 15, 2002; revised August 2, 2002. This work wassupported by Fundação para a Ciência e a Tecnologia (FCT), Portugal, underGrant SFRH/BD/843/2000 and POSI within project POSI/CPS/35576/1999 –DWDM/ODC.

The authors are with the Optical Communications Group, Instituto de Tele-comunicações, Department of Electrical and Computer Engineering, InstitutoSuperior Técnico, 1049-001 Lisboa, Portugal (e-mail: joao.rebola@lx.it.pt;adolfo.cartaxo@lx.it.pt).

Digital Object Identifier 10.1109/JLT.2003.808619

It was shown that it is necessary to take into account the proba-bility of occurrence of those rail levels to obtain, through fittingtoexperimentaldata,accurateestimatesof the -factor[7]. Inthispaper, we propose to use that idea to determine more accuratelythe -factor through numerical simulation.

Recently, the analytical estimation of the receiver sensitivityof optically preamplified receivers has been also investigatedand several methods that allow us to consider arbitrary opticalfiltering at the optical receiver have been proposed [9]–[11]. Thesensitivity estimation through these methods relies on numer-ical techniques that could become time expensive, especially forthe exact method presented in [9], which is based on the mo-ment-generating function (MGF) of the current at the decisioncircuit input. The method considered in [10] and [11] is basedon the exhaustive Gaussian approximation (GA), which simpli-fies significantly the numerical calculations in comparison with[9] and gives sensitivity discrepancies inferior to 1.2 dB in com-parison with exact estimates [10]. However, it requires the con-sideration of the sum of contributions of all transmitted sym-bols to the bit-error probability and further optimization of thedecision threshold. This can demand a significant computationtime, especially for extensive optimization of the optical com-munication systems. Furthermore, the physical insight that canbe drawn using this method is limited.

In this paper, we present simple expressions to estimate the-factor and the receiver sensitivity, which take into account

arbitrary optical and electrical filters, intersymbol interference(ISI), and extinction ratio. These expressions only require theknowledge of the optical and electrical filters transfer functionsand analysis of the eye-diagram at the decision circuit input toevaluate the system performance. Its main advantages are itssimplicity, short computation time, and physical insight. Theaccuracy of the estimates obtained through these expressions isassessed by comparison with the exact method and the GA. Theimpact of the amplified spontaneous emission (ASE)-ASE beatnoise on the optical receiver performance is also investigated andan easy way to assess this impact is proposed. A bandwidth ratio,which depends on the noise-equivalent bandwidths associatedwith the ASE-ASE and signal-ASE beat noises, is defined. Byusing this bandwidth ratio, a simplified formula depending on theoptical and electrical filters transfer functions and system param-eters (extinction ratio, eye-opening) is provided to describe theimpact of the ASE-ASE beat noise on the system performance.

0733-8724/03$17.00 © 2003 IEEE

88 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003

II. THEORY

The following analysis accompanies closely the derivationpresented in [12], but the effect of the ASE-ASE beat noise isadded to the analysis. The receiver model is the same as thatconsidered in [9]–[12].

Exhaustive GA that takes into account arbitrary optical andelectrical filters has been proposed in [10] and [11] to evaluatethe performance of an optically preamplified receiver and exactexpressions for the mean and variance of the current at the deci-sion circuit input were presented. These expressions present ex-plicitly the dependence on the transfer functions of the opticaland electrical filters. For ASE noise dominance and by assumingthat the signal at the optical receiver input has constant power,the mean and variance of the current at the decision circuit inputpresented in [10] and [11] can be written, respectively, as

(1)

(2)

where is the preamplifier gain; is the p-i-n responsivity;is the signal power at the optical receiver input; is

the transfer function of the electrical filter that models the elec-tronic circuitry of the receiver including the frequency responseof the p-i-n photodetector; is the optical loss due to the opticalfilter, which is given by , where is thelowpass equivalent of the transfer function of the optical filter;

is the power spectral density of the ASE noise, given by, with as the spontaneous emission

noise factor and the photon energy; is the mean currentdue to the ASE noise; describes the effect of a polarizer be-tween the preamplifier and the optical filter, and we set or

, respectively, in its presence or absence. In (2),is the noise-equivalent bandwidth of the electrical filter, gener-alized for nonwhite signal-ASE beat noise, defined by [12]

(3)

and presents explicitly the influence of the optical filtering ofthe ASE noise on the signal-ASE beat noise. In (2),is the noise-equivalent bandwidth of the ASE-ASE beat noiseas defined in [13], given by

(4)

and is the noise-equivalent bandwidth of theelectrical filter, generalized for nonwhite ASE-ASE beat noise,given by

(5)

The definitions of generalized noise-equivalent bandwidthsfor nonwhite noise, as given by (3) and (5), result from a gener-alization of the noise-equivalent bandwidth definition for whitenoise given in [14]. In [15], an equivalent rectangular opticalbandwidth associated with the ASE-ASE beat noise has beenproposed; however, we believe that the physical insight givenby (4) and (5) is more powerful and useful than the one pro-vided by the definition proposed in [15].

From (4), when the optical-filter 3 dB bandwidth broadens,increases also. From (5), for optical-filter band-

widths much larger than the electrical filter bandwidth, suchthat the ASE-ASE beat noise spectrum can be consideredwhite over the electrical filter bandwidth, is onlyimposed by the electrical filter. So, with the optical-filter band-width enlargement, the ASE-ASE beat noise variance given in(2) increases with the increase of the filtered ASE noise power.

From (4) and (5), we see also that the detuning of theoptical filter, with respect to the optical carrier, will not affect

and, consequently, the ASE-ASE beat noise vari-ance as given in (2). The ASE noise spectrum is much broaderthan the optical-filter bandwidth. Hence, the power of the ASEnoise filtered by a tuned or detuned optical filter is practicallythe same in both situations. The ASE-ASE beat noise resultsfrom the squared detection of the filtered ASE noise performedby the photodetector. If the optical filter is detuned by ,the filtered ASE noise at the photodetector input is centered at

, where is the optical-carrier frequency. By squaringthis “detuned” noise, noise components centered at zero and

appear at the photodetector output. The lowpassfiltering eliminates the noise component centered at highfrequencies and passes the other noise component, which hasthe same noise power as the one obtained for a tuned opticalfilter. Thus, the detuning of the optical filter does not affect theASE-ASE beat noise variance at the decision circuit input.

Our analysis is based on the observation of a noiseless eye-diagram. Hence, we define , , and as the means andstandard deviations of the current at the decision circuit inputat the sampling instant, respectively, for the symbols “1” and“0” [12]. and can be written in terms of equivalent powerlevels and at the optical receiver input, respectively, forthe symbols ’1’ and ’0’ [12]. By using (1), and are given,respectively, by

(6)

(7)

From (2), the variance of the symbols “1” and “0” can bewritten in terms of and , respectively, as

(8)

(9)

REBOLA AND CARTAXO: -FACTOR ESTIMATION AND IMPACT OF SPONTANEOUS-SPONTANEOUS BEAT NOISE 89

We normalize the maximum eye-opening to unit and, there-fore, for a nonreturn to zero signal, and are given, respec-tively, by

(10)

(11)

where is the average signal power at the optical receiverinput; is the extinction ratio defined accordingly with Inter-national Telecommunications Union Telecommunications Stan-dardization Sector (ITU-T) [16] as the ratio between the averageoptical power levels obtained for a logical “1” and “0” andand define the normalized eye-closure [12].

The -factor is given by [1]–[6], [8]

(12)

By substituting (6)–(9) in (12) and by using (10) and (11),after straightforward algebraic manipulation, the -factor canbe written in a closed-form expression for the worst-case of ISI,as in (13), shown at the bottom of the page, where

and .Expression (13) provides an easy and simple way of esti-

mating the -factor for arbitrary optical filtering. The compar-ison with the -factor expression presented in [6] reveals thatthe terms , , and can be identified, respectively, as

, , andand thus, the link budgeting between distortion and noise as pro-posed in [6] can be assessed for arbitrary optical filtering. By as-suming that the ASE-ASE beat noise can be neglected in (13),we are in a signal-dependent noise-dominant situation, and the

-factor presented in [12] is obtained.The average power at the optical receiver input can be ob-

tained as a function of the -factor from (13), which after al-gebraic manipulation, is given, in a closed-form expression, asin (14), shown at the bottom of the page, where is the nor-malized eye-opening given by . Expression(14) is a useful, simple, and fast tool to help the design of an

optical communication system. Expression (14) provides also away of assessing separately the ISI, signal-ASE, and ASE-ASEbeat noises and extinction ratio impacts on the system perfor-mance. If we neglect the ASE-ASE beat noise, the second terminside the square root of (14) disappears, and (14) degeneratesin the average power expression given in [12].

The power penalty due to the ASE-ASE beat noise is givenby

(15)

where is the average signal power for the referencesituation corresponding to null ASE-ASE beat noise power( ). It is convenient to define the bandwidthratio as

(16)

which represents the ratio between the noise-equivalent band-widths associated with the ASE-ASE and the signal-ASE beatnoises. For a large optical-filter bandwidth, is approximatedby

(17)

where is the noise-equivalent bandwidth of the electricalfilter.

By using (14) and (15), the minimum , which leads to apower penalty equal to or greater than , can be written as

(18)

where is a reference bandwidth ratio and corresponds tothe lowest value of . is obtained from bysetting infinite extinction ratio and null eye-closure and is givenby

(19)

(13)

(14)

90 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003

where 10 . In (18), the factor depends onlyon the eye-closure, extinction ratio, and power penalty due toASE-ASE beat noise and is given by

(20)

which gives for infinite extinction ratio and null eye-closure.

III. INFLUENCE OF THE OPTICAL FILTERING OF THE ASE-ASEBEAT NOISE ON THE RECEIVER PERFORMANCE

In this section, the influence of the optical filtering of theASE-ASE beat noise on the receiver performance is assessedthrough the study of the bandwidth ratio and by estimatingthe sensitivity using (14).

Different optical filters are considered throughout this paper:single-cavity Fabry–Perot (FP) with a free spectral range FSR

GHz [17] and a uniform Bragg grating (UBG) with(which ensures a 10-dB sidelobes suppression) [18].

The following parameters are considered throughout thispaper: , dB, A W, and the bit rateis Gb s. We assume a sequence length of 2 b1

and nonreturn to zero rectangular pulse shapes at the opticalreceiver input. The electrical filter is modeled by a second-orderButterworth filter with a 3 dB bandwidth of 26 GHz.

Fig. 1 shows the variation of with the power penaltyfor different and . Fig. 1 shows that the absence of

a polarizer demands lower bandwidth ratio for a spe-cific power penalty than with its presence . This hap-pens because the absence of a polarizer doubles the ASE-ASEbeat noise power at the decision circuit input in comparison with

. For lower error probability (higher ), higher signal av-erage power at the optical receiver input is required. With thisincrease of average power, the weight of the signal-ASE beatnoise on the receiver performance is higher and for the samepower penalty due to ASE-ASE beat noise, the bandwidth ratioshould increase (as shown in Fig. 1). For power penalties above1 dB, should be above 2.5, and for very large optical-filterbandwidths, by considering (17), should be at least2.5 higher than .

Henceforth, we consider and . Fig. 2 showsas a function of the extinction ratio for a power penalty of

1 dB, eye-openings of , , and , andseveral eye-closure asymmetries. Fig. 2 shows that when the ex-tinction ratio increases, the ratio decreases and approaches aconstant value. For , the ratio reaches its minimum

and therefore, tends to . With the extinctionratio reduction, the signal-ASE beat noise increases and higher

is required for the same power penalty due to ASE-ASE beat

1This sequence length was found sufficient to obtain accurate estimates in allresults presented in the paper.

Fig. 1. � as a function of the power penalty due to ASE-ASE beat noise� . � � �, � � �; � � �, � � �; � � �, � � �; � � �, � � �.

Fig. 2. � as a function of the extinction ratio for a power penalty� � � dBand for � � �, � � ���, and � � ���. Several eye-closure asymmetriesare shown.

noise. For extinction ratio lower than 25 dB, higher than 2.5should be observed to have 1 dB of power penalty. By using(17) and obtained from (19), shouldexceed 8 . So in a rough approximation, the optical-filter

3 dB bandwidth should exceed 8 the electrical filter 3 dBbandwidth. This value of optical-filter 3 dB bandwidth is un-feasible in optical communication systems, where optical-filterbandwidths should be comparable with the bit rate. So in thiscase, the impact of the ASE-ASE beat noise is practically neg-ligible for extinction ratios below 25 dB.

Fig. 2 also reveals that is very dependent on the eye-clo-sure asymmetry, particularly for extinction ratio above 10 dB.When the eye-closure for the “0” symbol is more significantthan for the “1” symbol, remarkably higher is required for1 dB of power penalty due to ASE-ASE beat noise. This hap-pens because the signal-ASE beat noise power is higher in thiscase. When the eye-closure decreases for the “0” symbol, theratio reduces. When the eye-closure occurs only on the “1”symbol, the discrepancies between the several correspondingto different eye-openings can be approximated by achievedfor . In fact, the corresponding to is the lowerbound of .

REBOLA AND CARTAXO: -FACTOR ESTIMATION AND IMPACT OF SPONTANEOUS-SPONTANEOUS BEAT NOISE 91

(a)

(b)

Fig. 3. Sensitivity (with and without ASE-ASE beat noise) and ratio � ��as a function of the optical-filter�3 dB bandwidth� normalized to the bit ratefor an extinction ratio of � ��. (a) FP optical filter and (b) UBG optical filter.The ratio � �� is also shown.

Fig. 3 shows the sensitivity as a function of the optical-filter3 dB bandwidth normalized to the bit rate for .

Fig. 3(a) and (b) correspond, respectively, to FP and UBG op-tical filters. The sensitivity is computed from (14) and plottedfor two situations: one without ASE-ASE beat noise (by setting

) and the other with it. The ratiosand are also presented. By comparison of the twosensitivity curves, we conclude that, for a power penalty largerthan 1 dB, should exceed and , respec-tively, for the FP and UBG optical filters. This means that, for

dB, the receiver exhibits a very small eye-closure. Ifthe ratio presented in Fig. 2 for is considered, an ac-ceptable approximation of for dB is obtained.So to bound the power penalty due to ASE-ASE beat noise to1 dB for any extinction ratio, the electrical and optical filtersshould be designed to have lower than obtained for

.

Fig. 3 shows that approaches with the increaseof the optical-filter bandwidth. This means that is anacceptable approximation of , for large optical-filter band-widths. For filters with sharper cutoff provides a more

accurate approximation of , as can be seen in Fig. 3(b). Fig. 3reveals also that the ASE-ASE beat noise power increases withthe optical-filter bandwidth increase, and consequently, the sen-sitivity diminishes with this increase. For narrow optical-filterbandwidths, the ISI and the signal-ASE beat noise are the majorfactors affecting the receiver sensitivity. With the increase ofthe optical-filter bandwidth, the maximum sensitivity is reachedfor the optimal optical-filter bandwidth due to a balance be-tween signal-ASE beat noise and ISI, and less significantly tothe ASE-ASE beat noise. With the further increase of the op-tical-filter bandwidth, just a slight reduction of the receiver sen-sitivity due to the signal-ASE beat noise can be noticed, but theASE-ASE beat noise power impact on the receiver performancestill increases.

The previous analysis provides an easy way of designing theoptical and electrical filters in optically preamplified receiversto bound the power penalty due to ASE-ASE beat noise, for ex-ample, by bounding the power penalty to 1 dB, considering

and and using (19), . It has been shown thatASE-ASE beat noise has its higher impact on the power penaltyfor infinite extinction ratio. So to get an upper bound of powerpenalty, infinite extinction ratio and , i.e., zero ISI, areconsidered. These conditions lead to . Then,by setting or (what is also acceptable in a roughapproximation) to , the optical and electrical fil-ters can be settled using the noise-equivalent bandwidths givenby (3)–(5). This means that in a rough approximation, the op-tical-filter 3 dB bandwidth should not exceed 3.3 the elec-trical filter bandwidth to ensure that the power penalty due toASE-ASE beat noise is lower than 1 dB.

IV. - FACTOR ESTIMATION AND COMPARISON

WITH EXACT METHOD

In this section, we assess the accuracy of the -factor esti-mated through (13) by comparison with the -factor obtainedfrom the error probability computed through the exact methodpresented in [9]. By using [1]

(21)

where is the complementary error function and is theerror probability, the -factor can be obtained using a numericalmethod. The error probability is obtained from the MGF throughthe saddlepoint approximation (SA) [9].

To characterize the difference, in dB, between two -factorestimations, we define

(22)

where is the -factor estimated through (13).The accuracy of the -factor is investigated in two different

situations: in the optical-filter bandwidth dependence and prop-agation of the signal in an optical fiber with an increasing dis-persion.

92 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003

(a)

(b)

Fig. 4. �� as a function of the average power and of the optical-filter�3 dBbandwidth (normalized to the bit rate) for an extinction ratio of � ��. (a) FPoptical filter and (b) UBG optical filter. The thick line corresponds to � � �

(obtained through the SA).

A. Optical-Filter Bandwidth Dependence

Fig. 4 shows the difference as a function of the optical-filter 3-dB bandwidth (normalized to the bit rate) and of theaverage power at the optical-receiver input for . Fig. 4(a)and (b) correspond, respectively, to FP and UBG optical filters.For average powers between 32 and 20 dBm, the -factorvaries from about 5.5 to 22. Fig. 4 shows that the maximum dis-crepancy between and the -factor estimated using the SA is1.4 dB. This discrepancy happens for larger optical bandwidths(above 2 ), i.e., where the ASE-ASE beat noise becomes im-portant and, consequently, the noise statistics differ more signif-icantly from the Gaussian distribution. For optical-filter band-widths below 2 , our expression provides estimates with rea-sonable accuracy. However, for very narrow bandwidths and es-pecially for the UBG filter, due to its higher introduced ISI, the

difference achieves 1 dB, i.e., the estimate is conser-vative.

Fig. 5 shows the difference as a function of the FP op-tical-filter 3 dB bandwidth (normalized to the bit rate) and ofthe average power at the optical receiver input for . In thiscase, the -factor varies from about 4 to 17. The maximum dis-

Fig. 5. �� as a function of the average power and of the FP optical-filter�3 dB bandwidth (normalized to the bit rate) for an extinction ratio of � � ��.The thick line corresponds to � � � (obtained through the SA).

crepancy achieved in this case is 1 dB and happens for low op-tical-filter bandwidths, i.e., high ISI and very low (of about 4).For larger bandwidths, the differences are practically negligible.In comparison with , the main difference is on the weightof the ASE-ASE beat noise on the receiver performance. Asseen in Section III, for optical-filter bandwidths between 1.5and 4 , and , the ASE-ASE beat noise impact on the re-ceiver performance is practically negligible, while for ,this is not true. So one can conclude that the -factor estimationthrough (13) is less accurate for systems where the ASE-ASEbeat noise plays an important role or the ISI is high. The rig-orous ASE-ASE beat noise distribution is presented in [9] and[19] and is clearly a non-Gaussian distribution. The -factor es-timates are less accurate for high ISI, because the ISI is an effectthat does not follow a Gaussian statistic.

Comparisons between and the -factor estimated throughthe GA have been also performed and only for optical-filter

3 dB bandwidths below , where the ISI is relevant, signifi-cant differences (not exceeding 1 dB) have been found. This isdue to the increase of accuracy of the exhaustive GA with theincrease of ISI [10].

B. Propagation of the Signal in an Optical Fiber

In this section, the linear propagation of the optical signalin an optical fiber with an increasing dispersion is considered.The optical-filter 3 dB bandwidth is kept constant at 1.5 . Weconsider a chirpless nonreturn to zero signal with rectangularpulse shape at the optical-fiber input.

Fig. 6 shows the difference as a function of the averagepower at the optical receiver input and of the total fiber dis-persion for the FP optical filter. Fig. 6(a) and (b) refer,respectively, to and . For and

, the -factor varies, respectively, from 4 to 22 and 3 to 17.Fig. 6 shows that the estimates start losing accuracy withthe increase of waveform distortion and differences exceeding

1.7 dB are observed for a of about 100 ps/nm (). In this case, the -factor is very low (between 3 and 4).

This variation with the ISI increase can be also observedin Figs. 4 and 5 for low optical-filter bandwidths. For

REBOLA AND CARTAXO: -FACTOR ESTIMATION AND IMPACT OF SPONTANEOUS-SPONTANEOUS BEAT NOISE 93

(a)

(b)

Fig. 6. �� as a function of the average power and of the total dispersion�� [ps/nm] for the FP optical filter with a�3 dB bandwidth of 1.5�, with (a)� �� and (b) � � ��. The thick line corresponds to� � � (obtained throughthe SA).

and in the presence of low ISI (low ), gives estimates withgood accuracy. For , the ASE-ASE beat noise impact onthe receiver performance is higher than for , and for low

, the estimates give a difference of about 1 dB in compar-ison with the -factor obtained through the SA. This behavioris also observed for other optical filters.

Comparisons between and the -factor estimated throughthe GA have been also performed. For lower ISI, the GAestimates are very alike with the estimates obtained from (13).For very high ISI ( ps/nm), differences exceeding

1.7 dB similar to the ones shown in Fig. 6(b) have beenfound.

V. DEPENDENCE OF -FACTOR ON THE OCCURRENCE OF THE

RAIL LEVELS

The -factor estimation by the formula proposed in [7]assumes that the error probability is mainly determined bythe rail levels responsible for the eye-closure and that theyhave the probability of occurrence and , respectively, forthe symbols “1” and “0”. In this case, the -factor is given

(a)

(b)

Fig. 7. �� as a function of the average power and of the total dispersion��[ps/nm] for the FP optical filter with a�3 dB bandwidth of 1.5�with (a) � ��and (b) � � ��. The �-factor is estimated using (23).

by [7]

(23)

with , , , and given by (6)–(9). If we consider that thecurrent levels that define the eye-closure have probabilities

and , we obtain the -factor as given by (13). Afterextensive computation, we found that the rail levels that differfrom the rails corresponding to the worst-case of ISI by 1% ofthe eye opening of an eye with , provide probabilities

and that allow to estimate the -factor with very goodaccuracy.

Fig. 7 shows the as a function of the average powerand of the total fiber dispersion for the FP optical filter.Fig. 7(a) and (b) refer, respectively, to and .is the -factor estimated using (23) and is compared with the

-factor estimated using the SA. We observe that the -factor

94 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 1, JANUARY 2003

estimates obtained through (23) differ from the ones obtainedthrough the rigorous method by less than 1 dB. These differ-ences happen for below 40 ps/nm (low ISI) and for ,where the ASE-ASE beat noise impact on the system perfor-mance is higher. For systems where this impact is less signifi-cant, and with above 40 ps/nm, the discrep-ancies between the estimates obtained through (23) and the SAare very accurate (below 0.4 dB).

Moreover, in case of low and high ISI, the -factorgiven by (23) is more accurate than the -factor given by(13) [Fig. 6(a) and (b)]. For low ( above 80 ps/nm andaverage power below 30 dBm), discrepancies between thosetwo estimates that exceed 1 dB can be observed. For above80 ps/nm, with the increase of average power, the -factorpredictions using (13) become more accurate and similar to theestimates obtained using (23) and the SA.

For systems where the impact of ASE-ASE beat noiseplays an important role, (13) and (23) predict almost the same

-factor. This means that the consideration of the probabilityof occurrence of the rail levels nearest to the decision thresholdonly leads to more accurate predictions of the -factor forsystems where the ISI is significant and low is achieved.

VI. CONCLUSION

In this work, simple explicit expressions of estimating the-factor and the receiver sensitivity of optically preamplified

receivers with arbitrary optical filtering from the eye-diagramhave been proposed. These expressions depend only on the eye-closure and optical and electrical filters shape and take into ac-count the impact of the optical filtering of the ASE noise on thesignal-ASE and ASE-ASE beat noises, ISI, and extinction ratio.The noise-equivalent bandwidths associated with the nonunifor-mity of the ASE noise spectrum at the photodetector input havebeen fully characterized, presented, and taken into account inthe derivation of these expressions.

By using these noise-equivalent bandwidths and the de-rived sensitivity expression, a simple way of designing theoptical and electrical filters to bound the power penalty dueto ASE-ASE beat noise has been also proposed. A bandwidthratio, depending on the noise-equivalent bandwidths associatedwith the ASE-ASE and signal-ASE beat noises, has beendefined. A minimum bandwidth ratio, which only dependson the eye-closure, extinction ratio, presence or absence ofa polarizer, and -factor has been found for a given powerpenalty. For bandwidth ratios higher than this minimum,higher power penalties are observed. Hence, by employing theminimum bandwidth ratio and the bandwidth ratio definitions,the optical and electrical filters can be designed for a specificpower penalty due to ASE-ASE beat noise, thus controlling theimpact of ASE-ASE beat noise on the receiver performance.Results have shown that this impact can be almost neglectedfor extinction ratio below 25 dB as long as the optical-filter

3 dB bandwidth does not exceed 8 the bit rate.The accuracy of the -factor obtained using these expres-

sions has been also assessed. The -factor estimate is less ac-curate for systems where the impact of the ASE-ASE beat noiseon the system performance is significant, i.e., high extinction

ratio. In comparison with rigorous estimates, maximum over-estimate differences of 1.4 dB have been found. The -factorestimates also become less accurate in case of low (below 4)and high ISI. Maximum underestimates differences of 1.7 dBhave been found. With the increase of the -factor, these differ-ences diminish and become practically negligible. The consid-eration of the probability of occurrence of the rail levels nearestto the decision threshold provide more accurate estimates of the

-factor in case of low and significant ISI, with almost negli-gible differences from the exact method. However, for systemswhere the impact of the ASE-ASE beat noise on the system per-formance is significant, the -factor estimates obtained takingand not taking into account those probabilities give similar dis-crepancies with respect to the rigorous method (not exceeding1.4 dB).

ACKNOWLEDGMENT

The authors wish to thank M. Leiria, Instituto de Telecomu-nicações, Lisboa, Portugal, for his helpful suggestions and com-ments during the preparation of the manuscript.

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João L. Rebola (S’01) was born in Lisbon, Portugal, on March 29, 1976. Hereceived the “Licenciatura” degree in electrical engineering in telecommunica-tions and computers from the Instituto Superior Técnico (IST), Lisbon TechnicalUniversity, Lisbon, Portugal, in 1999. He is currently working toward the Ph.D.degree in electrical engineering at IST.

Currently, his main research interests include optical filtering and per-formance evaluation in dense-wavelength-division-multiplexing (DWDM)systems.

Adolfo V. T. Cartaxo (S’89–A’89–M’98) was born in Montemor-o-Novo, Por-tugal, on January 10, 1962. He received the “Licenciatura” degree in electricalengineering, the M.S. degree in telecommunications and computers, and thePh.D. degree in electrical engineering from the Instituto Superior Técnico (IST),Lisbon Technical University, Lisbon, Portugal, in 1985, 1989, and 1992, respec-tively.

From 1987 to 1990, he was an Assistant Lecturer with IST, and from 1990to 1992, he conducted postgraduate research work at IST in the area of clockrecovery circuit optimization in direct-detection optical communications. From1992 to 2002, he was an Assistant Professor at IST, until he was promoted toAssociate Professor. He is a Senior Researcher of the Instituto de Telecomuni-cações, conducting research on optical communications and networks. He hascoordinated the IST’s participation in several projects in the context of the Eu-ropean Union and Portuguese programs on research and development in thetelecommunications area. In the past few years, he has acted as a Technical Au-ditor and Evaluator for projects included in European Union research and de-velopment programs. He has authored more than 20 journal publications and 50international conference papers. His main research interests include fiber-opticcommunication systems and networks.

He has served as a Reviewer for the Institute of Electrical and ElectronicsEngineers (IEEE) and the Institution of Electrical Engineers (IEE) magazinesin the field of optical communications.