Gaussian approach to the performance assessmentof optical multiplexer/demultiplexer concatenation intransparent optical networks

M. Leiria, J.L. Rebola and A.V.T. Cartaxo

Abstract: A Gaussian approach (GA) with two equivalent optical filters, one for the noise and theother for the signal, is proposed for analytically evaluating the transparent optical network (TON)performance for arbitrary TON topologies, optical multiplexers/demultiplexers and electrical filters.This GA helps in identifying, in a fast computational way, the different mechanisms that contribute toTON performance degradation. The accuracy of estimates obtained through the GA is assessed incomparison with an exact method. In the case of no laser misalignment, the estimates given by the GAare very accurate, with maximum discrepancies of 0.5 dB observed for infinite extinction ratio. Withextinction ratio reduction, the accuracy of GA estimates is further improved. With lasermisalignment, for an extinction ratio of 10, the GA estimates are reasonably accurate, and forinfinite extinction ratio, the accuracy is slightly reduced, but TON Q-factor behaviour is well assessedby the GA. Only for optical filters with large bandwidth and some laser misalignment, can GAestimates diverge, with discrepancies not exceeding 2.6 dB in the range of parameters investigated.

1 Introduction

Transparent optical networks (TONs) are composed ofseveral optical components that exhibit non-ideal charac-teristics and cause degradation of network performance.Cascading such components enhances this performancedegradation. The performance of TONs also degrades overtime, due to ageing and repair of system components,among other causes. So, ways of accurately assessing theperformance of TONs are required.

In earlier research, the cascadability of different types ofoptical multiplexers=demultiplexers ðMUX=DEMUXÞ hasbeen studied both theoretically and experimentally [1–11].However, these studies contain approximations concerningnetwork topology and performance [12]. Most of themcompute the eye-diagram opening at the output of the direct-detection receiver and disregard the noise impact whenassessing the network performance [1, 4, 8, 9].

A Gaussian approach (GA) has been proposed to evaluatethe performance of optically preamplified receivers witharbitrary optical and electrical filters [13]. Its inherentsimplicity, relative accuracy and short computation time arereasons to use it [13]. However, the influence of theconcatenation of MUX=DEMUX in transparent opticalnetworks cannot be rigorously described by the model usedin [13], because the signal and noise contributions of thevarious optical amplifiers of the TON are affected bydifferent equivalent optical filters.

Exact methods to obtain the error probability of opticalreceivers with arbitrary optical and electrical filters have

been proposed [12, 14, 15]. The methods in [12, 14] expandthe amplified spontaneous emission (ASE) noise inKarhunen–Loeve (KL) series and obtain the bit errorprobability from the moment generating function (MGF).The method proposed in [14] accommodates the parametricgain due to non-linear transmission over the optical fibre;however, intersymbol interference (ISI) and the dependenceof signal–ASE beat noise on signal waveform distortion areneglected. To obtain the MGF from the KL expansion, it isnecessary to solve a homogeneous Fredholm equation of thesecond kind in a numerical way [12]. However, itsnumerical computation can be difficult to implement. Thisnumerical problem is overcome by the method presented in[15]. However, there, the error probability is evaluated foran optically preamplified receiver only, and the influence ofthe concatenation of MUX=DEMUX in a TON is nottackled.

In this work, a GA that accommodates the influence of theconcatenation of arbitrary MUX=DEMUX in TONs isproposed. This GA also takes into account the effects ofwaveform distortion in the signal, the signal–ASE beatnoise and the different filtering that signal and noise sufferduring transmission along the chain of optical MUX=DEMUXs: Exact expressions for the mean and variance ofthe current at the decision circuit input are derived, whichclearly present the dependence on the optical and electricalfilters transfer functions. The accuracy of the estimatesobtained through the GA is assessed by comparison with ageneralisation of the exact method proposed in [15],improved to accommodate the influence of a concatenationof MUX=DEMUXs:

2 Theory

2.1 Network equivalent model

The network topology is depicted in Fig. 1. It consists of achain of N þ 1 equidistant transparent optical nodes,working as optical add–drop multiplexers (OADMs),separated by fibre spans of loss L. The optical signal is

q IEE, 2004

IEE Proceedings online no. 20040129

doi: 10.1049/ip-opt:20040129

The authors are with the Optical Communications Group, Instituto deTelecomunicacoes, Department of Electrical and Computers Engineering,Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Paper first received 6th January and in revised form 7th August 2003.Originally published online 18th May 2004

IEE Proc.-Optoelectron., Vol. 151, No. 3, June 2004 157

added at the first optical node, propagates through a chain ofN � 1 nodes and is dropped in the ðN þ 1Þth optical node.

An exhaustive study of transparent optical node archi-tectures and OADMs structures is beyond the scope of thispaper. The OADM structure is the same as considered in[12]. It consists of two optical amplifiers, an opticalMUX=DEMUX pair, an optical switch fabric for signaladd–drop, and a servo-controlled attenuator for powerequalisation, as shown in Fig. 1b of [12]. It is assumed thatthe TON is approximately linear. This means that the opticalnode components are modelled as linear devices and thetransmission along the fibre is modelled by its loss and delaydistortion due to chromatic dispersion. Hence, the distortioninduced by transmission along the fibre affects only the biterror probability through signal degradation. This model isvalid also for nonlinear fibre transmission in the presence ofalmost perfect dispersion compensation and reduced opticalpower, because the parametric gain effect is reduced in thiscase [14]. Furthermore, [16] states that it provides an upperbound for the cascadability of optical MUX=DEMUXs: It isalso assumed that signal attenuation due to the loss in eachfibre span is fully compensated by the optical amplifier atthe node input and that the output optical amplifiercompensates the intranode insertion losses when the opticalMUX and DEMUX are tuned to the optical carrierfrequency [12]. The servo-controlled attenuator compen-sates for the additional loss caused by filtering in order tokeep a constant average signal power at the input of theoptical amplifier at the node output. The additional loss pernode is considered to be within the operating range of theservo-controlled attenuators [12].

As described in [12], the above assumptions allow theTON topology to be simplified, as presented in the blockdiagram shown in Fig. 2. All intranode optical componentsare represented by equivalent models. Since the opticalamplifiers compensate for the fibre and the node insertionlosses, only the variable attenuations or gains are describedin the equivalent block diagram. The optical MUXs=DEMUXs are described by filters with transfer functionsnormalised to unit at the channel’s nominal frequency, �c;HMUXð�Þ and HDEMð�Þ respectively, where � is the opticalfrequency. The servo-controlled attenuators are representedby gain elements, gs;i; with gain equal to the excess losscaused by the optical MUX=DEMUX pair of the ith node

[12]. The noise introduced by the ith optical amplifier ismodelled by an independent Gaussian noise source withpower spectral density (PSD) in each orthogonal directiongiven by Sið�Þ: The receiver consists of a polariser(optional), a tuned optical filter with transfer functionH0ð�Þ; a p-i-n photodiode, an electronic low-pass filter withtransfer function Hrð f Þ; where f is the baseband frequency,a sampler and a decision device. The p-i-n photodetector ismodelled as a square-law device with responsivity Rs:

As described in [12], a linear TON can be represented bytwo equivalent optical filters, one for the signal Heq;sð�Þ; andother for the noise Heq;nð�Þ; as shown in Fig. 3. The reasonfor such a representation can be intuitively understood sincenoise is generated in a distributed fashion throughout theTON, so that the optical signal and noise are filtered by adifferent number of optical MUX=DEMUXs [16]. Thesignal equivalent optical filter takes into account thefiltering of the cascade. The TON gain gTON takes intoaccount the gain and losses of the cascade. From Fig. 2, thesignal transfer function between the transmitter output andthe p-i-n input is given by

H0eq;sð�Þ ¼ Hoð�Þ �YNi¼1

HMUX;ið�Þ · HDEM;iþ1ð�Þ ·ffiffiffiffiffiffigs;ip ð1Þ

To obtain the signal equivalent optical filter, we normaliseH0eq;sð�Þ by the TON gain gTON ¼ jH0eq;sð�sÞj2 so that thesignal equivalent optical filter gain at the carrier frequency�s is unity. Therefore, the signal equivalent optical filter is

Heq;sð�Þ ¼ H0eq;sð�Þ=ffiffiffiffiffiffiffiffiffiffigTON

p ð2Þ

From Fig. 2, the noise PSD at the p-i-n input is the sum ofthe various contributions of the noise PSD introduced along

Fig. 1 TON topology

Fig. 2 Simplified block diagram of the TON

Fig. 3 TON block diagram with two different and equivalentfilters

IEE Proc.-Optoelectron., Vol. 151, No. 3, June 2004158

the chain of nodes. Each of those PSD is affected by thecascade of filter blocks, gain blocks, attenuators and fibrespan losses from the node where it is introduced, up to thep-i-n. From the simplified block diagram of Fig. 2, the noisePSD at the p-i-n input is given by

STð�Þ ¼�XN�1

i¼1

�S2i�1ð�Þ þ S2ið�Þ

�·YN

m¼iþ1

��HMUX;mð�Þ��2

·��HDEM;mð�Þ

��2 · gs;m þ S2Nð�Þ þ S2N�1ð�Þ�

� jHDEM;Nþ1ð�Þj2 · jHoð�Þj2

ð3ÞUsing the definition of filtered noise optical power([17], pp. 177, 178), the equivalent input noise en;eðtÞ isdefined as a fictitious white Gaussian noise source with PSDequal to the sum of the PSD of all optical amplifiers at thesignal carrier frequency. Therefore, the PSD of theequivalent input noise is given by [12]

Seq ¼X2 · N

i¼1

Sið�sÞ ð4Þ

Using expression (4) in (3), the noise PSD at the p-i-n inputcan be written as

STð�Þ ¼ Seq · jHTð�Þj2 ð5Þwhere HTð�Þ is an optical filter transfer function thatcharacterises the frequency dependence of the noise PSD atthe p-i-n input. HTð�Þ is obtained by replacing (3) and (4) in(5) except for the optical filter phase response, which isnot relevant. The noise PSD at the p-i-n input can berewritten as

STð�Þ ¼ Seq · jHeq;nð�Þj2 · F · gTON;n ð6Þ

where jHeq;nð�Þj2 ¼ jHTð�Þ=HTð�cÞj2 is the noise equival-ent optical filter normalised to unity at the channel’s

nominal frequency, gTON;n ¼ jHTð�sÞj2 is the TON noiseequivalent gain at the signal frequency, and F ¼ jHTð�cÞ=HTð�sÞj2 is the excess noise factor. The noise equivalentoptical filter describes the frequency dependence of thenoise PSD at the p-i-n input. The TON noise equivalent gainshows the existence of the gain or attenuation at the signalcarrier frequency that the noise experiences during trans-mission along the chain. For the simplified TON topologyshown in Fig. 2 with perfect loss compensation, the TONnoise equivalent gain is unity and is equal to the TON gain.However, for a different TON topology, even with perfectloss compensation, the TON noise equivalent gain may notto be unity. The excess noise factor describes the increase ofnoise power when the laser is misaligned. When the laser isaligned, the excess noise factor is unity. Expression (6)presents explicitly the influence of laser misalignment andof TON gain or attenuation on the accumulated noise PSD.This definition is advantageous over the one proposed in[12], where the effects described above are consideredmixed together by the excess noise factor definition.

Figure 3 summarises the equivalent model for signal andnoise resulting from considerations and parameters pre-sented above. This model is used to derive the GA in thefollowing. Signal and noise fields are assumed to havearbitrary polarisations. Two directions x and y areconsidered for the signal and noise description. Theassumptions used in [13] are considered for the signal andnoise representation. By considering the low-pass equival-ent model described by equations (9) and (10) of [13], andthe different optical filters for the signal and noise

transmission, the current at the decision circuit input isgiven by

idðtÞ ¼�

Rs ·ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF · gTON;n

p·�ne;xðtÞ · xþ ðp� 1Þ · ne;yðtÞ · y

�� �� heq;n;lðtÞþ

ffiffiffiffiffiffiffiffiffiffigTON

p·�se;xðtÞ · xþ ðp� 1Þ · se;yðtÞ · y

�� heq;s;lðtÞj2

�� hrðtÞ ð7Þ

where � stands for convolution, ne;xðtÞ and ne;yðtÞ are thecomplex envelopes of the noise en;eðtÞ at the noiseequivalent optical filter input, respectively, in the x and ydirections; heq;n;lðtÞ and heq;s;lðtÞ are the low-pass equivalentimpulse responses of the noise and signal equivalent opticalfilters; se; xðtÞ and se;yðtÞ are the complex envelopes of thetransmitted signal, es;eðtÞ at the signal equivalent opticalfilter input, respectively, in the x and y directions; hrðtÞ isthe impulse response of the electrical filter; and we set p ¼ 1or p ¼ 2; in the presence or absence of a polariser. Weassume that the polariser, if present, selects the x direction.

Expressions (1) and (3) show that laser source misalign-ment may cause different effects in the low-pass equivalentfilters of the signal and noise. It leads to a frequencytranslation of the signal equivalent filter transfer functionand to a change of the shape of the transfer function of thenoise low-pass equivalent filter when the noise sources arenon-white.

2.2 Gaussian approach

From (7), the deduction of the mean and variance isstraightforward, but lengthy. The deduction is similar tothe one found in the Appendix of [13]. The major novelty isthat different optical filters for signal and noise areconsidered. Exact expressions for the mean and varianceof the current at the decision circuit input are given,respectively, by

mðtÞ ¼ pRs ·F ·gTON;n ·Seq ·Hrð0ÞZ þ1�1jHeq;n;lð f Þj2df

þRs ·gTON

Z þ1�1

�jðp�1Þ · se;yðwÞ �heq;s;lðwÞj2

þjse;xðwÞ �heq;s;lðwÞj2�

·hrðt�wÞdw ð8Þ

�2ðtÞ ¼ 2R2s ·F ·gTON ·gTON;n ·Seq

Z þ1�1

���ðse;xðwÞ

�heq;s;lðwÞÞ · hrðt�wÞ��heq;n;lðwÞ

��2dw

þ2R2s ·F ·gTON ·gTON;n ·Seq ·ðp�1Þ2

Z þ1�1

���ðse;yðwÞ

�heq;s;lðwÞÞ ·hrðt��Þ��heq;n;lðwÞ

��2dw

þpR2s ·F2 ·g2

TON;n ·S2eq

Z þ1�1jHrð f Þj2

�jHeq;n;lð f Þj2

� jHeq;n;lð�f Þj2�df ð9Þ

where Heq;n;lð f Þ is the low-pass equivalent transfer functionof the noise equivalent optical filter.

In (8) and (9), the dependence of mean and variance ofthe current on the transfer function of the equivalent filters,electrical filter, gain of TON, equivalent noise PSD, TONnoise equivalent gain and excess noise factor is explicitlypresented. In (8), the first component is the mean current dueto the accumulated noise and the second one is the signalcomponent. In (9), the first two terms are the signal–noisebeat noise variance �2

s�nðtÞ and the third term is the noise–noise beat noise variance, �2

n�n:

IEE Proc.-Optoelectron., Vol. 151, No. 3, June 2004 159

Using an exhaustive GA, the TON bit error probability isgiven by [13]

Pe ¼1

Nb

·XNb�1

k¼0ðak¼‘0’Þ

QIopt � m0;k

�0;k

� �þXNb�1

k¼0ðak¼‘1’Þ

Qm1;k � Iopt

�1;k

� �264

375

ð10Þ

where Nb is the total length of the binary sequence; ak is thekth symbol, either ‘0’ or ‘1’; Iopt is the optimised thresholdlevel [13]; Q(x) is defined in ([17], Table T.6); mi;k and �i;k

are, respectively, the mean and standard deviation of idðtÞconditioned to the symbol i and at the sampling instant,tk ¼ k=Bþ to; with k ¼ 0; . . . ;Nb � 1; where to can bechosen in order to optimise the system performance, and Bis the bit rate.

2.3 Exact method of TON error probabilitycomputation

The exact method used to estimate the TON errorprobability is based on the analytical formulation presentedin [15]. The main difference arises from the TON modelpresented in Fig. 3 with two equivalent filters. This leads todifferences from equation (A.3) of [15], where the signal isfiltered by heq;s;lðtÞ; and from equation (A.6) of [15], withthe noise being filtered by heq;n;lðtÞ: However, furtherdevelopment of the analytical formulation and derivation ofthe MGF follow the analysis presented in [15] [Note 1].

The advantage of the formulation in [15] over [12] is thatits analytical approach overcomes the difficulty of comput-ing the solution of the homogeneous Fredholm equation ofthe second kind as described in [12], by considering aperiodic signal and by using a Fourier expansion for theGaussian noise. The remaining problem is the assessment ofnormalised eigenvectors and eigenvalues of the matrix(A.20) of [15], what is much simpler than solving theFredholm equation, which depends on the noise autocorre-lation function.

In this work, the error probability is obtained from theMGF through the saddlepoint approximation, which givesvery accurate results in comparison with the exact method[13, 15].

3 System considerations

For simplicity, in the analysis of the GA accuracy, a single-channel system operating at lc ¼ 1550 nm is considered.In the future, our model will be generalised to wavelength-division multiplexed (WDM) systems where the impact ofcrosstalk will be taken into account. The optical amplifiersare considered to be erbium doped fibre amplifiers (EDFAs).The noise introduced by each EDFA is considered as anindependent white Gaussian noise source with low-passequivalent PSD in each orthogonal direction given bySið f Þ ¼ nsp · ðgi � 1Þ · �c · h; where gi is the ith EDFA gain,nsp is the spontaneous emission noise factor and h is thePlanck constant. It is assumed that all fibre spans have equallength with 22 dB loss. The loss of the servo-controlledattenuator at the normal operation point, plus the insertionlosses of the MUX, DEMUX and switch fabric is 22 dB.

With this assumption, the EDFA gains are equal in allnodes. Therefore, the PSD of the low-pass equivalent inputGaussian noise, obtained from (4), is simplified to Seq ¼2 · N · nsp · ðg� 1Þ · �c · h; with g equal to 22 dB. IdenticalMUX and DEMUX frequency responses modelled by asingle-stage Fabry–Perot [18] with free spectral range of3000 GHz are considered. The receiver optical filter is atuned 6th-order Butterworth filter with a bandwidth of100 GHz.

The signal at the output of the transmitter is a 10Gbit=snon return-to-zero signal with rectangular pulse shape andan extinction ratio of r. Due to the servo-controlledattenuators, the average signal power is assumed to be thesame at the output of each servo-controlled attenuator.

The following parameters are considered: Rs ¼ 1 A=W;nsp ¼ 2; absence of polariser ð p ¼ 2Þ; a deBruijn sequenceof 26 bits and the electrical receiver filter is a 2nd-orderButterworth filter with a �3 dB bandwidth of 6.5 GHz. Thesampling instant is the time instant corresponding to themaximum eye opening.

4 Accuracy of the Gaussian approach

In this Section, the accuracy of the GA is investigated indifferent practical situations. We compare the resultsobtained using the GA with the results obtained using theexact method a described in 2.3.

Recently, the Q-factor has been utilised for evaluating theperformance of optically amplified systems [19–23]. TheTON performance is assessed through the Q-factor, which isestimated, for the GA and the exact method, from theTON bit error probability, by numerically solving theequation [21]

Q ¼ffiffiffi2p

· erfc�1ð2PeÞ ð11Þ

where erfcð · Þ is the complementary error function [17].The difference, in dB, between the estimates of the two

methods is characterised by

DQ ¼ 20 log10ðQgÞ � 20 log10ðQeÞ ð12Þ

where Qg and Qe are, respectively, the Q-factor assessed bythe GA and the exact method.

4.1 Dependence of the GA accuracy on theoptical filters bandwidth

Figure 4 shows DQ as a function of the �3 dB bandwidth ofthe optical filters normalised to the bit rate and of theaverage power after the optical amplifier at the node outputfor 10 OADMs. Figures 4a and 4b correspond, respectively,to r ¼ 10 and r ¼ 1: Figure 4a shows that the discrepan-cies in estimates between the GA and the exact method arevery small, not exceeding 0.03 dB, showing a very goodagreement between the methods for this practical situation.Figure 4b shows that, for r ¼ 1; the estimates differencebetween the GA and the exact method are again small, notexceeding 0.5 dB. Higher differences occur for high opticalfilter bandwidths. Theory and experimental work show thatpure signal–ASE beat noise is nearly Gaussian [24]. Withr ¼ 10; the signal–ASE beat noise is dominant over theASE–ASE beat noise and the probability density functions(PDFs) of the decision current corresponding to both logicallevels approach Gaussian shape. For r ¼ 1; the influence ofthe ASE–ASE beat noise is enhanced significantly. In thecase of negligible ISI, and when this impact becomessignificant, the PDF of the current at the decision circuit

Note 1: In [15], equation (A.12) should be corrected. The matrix Q shouldhave its elements given by qij ¼ Hrð j� iÞ=To: In this case, it is expectedthat the GA would be even more accurate than as presented in [13].

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input corresponding to the zero logical level, deviates fromGaussian distribution [24]. Therefore, the estimate differ-ences between the GA and the exact method increase whenthe impact of ASE–ASE beat noise is strengthened.

Figures 5a and 5b show DQ as a function of the �3 dBbandwidth of the optical filters normalised to the bit rate andof the average power after the optical amplifier at the nodeoutput, for two OADMs, respectively, for r ¼ 10 andr ¼ 1: Differences in estimates similar to the situation of10 OADMs are obtained. These results reveal that theaccuracy of the GA approach is weakly dependent on thenumber of optical nodes in the case of tuned MUX=DEMUX: Instead, the results show that the accuracy of theGA has some dependence on the extinction ratio. To getmore insight into this dependence, Figs. 6a and 6b show DQas a function of the �3 dB bandwidth of the optical filtersnormalised to the bit rate and of the extinction ratio, for anaverage signal power after the optical amplifier at the nodeoutput of 5 dBm, respectively, for 10 and 2 OADMs. Thediscrepancies in estimates between the GA and the exactmethod increase slightly as the extinction ratio increase, inagreement with the results of Figs. 4 and 5. This behaviouris due to the increase of the impact of the ASE–ASE beatnoise on system performance. However, these differences

are almost negligible, for the usually experimental extinc-tion ratios (less than 30 dB).

4.2 Dependence of the GA accuracy on thelaser carrier frequency misalignment from thechannel’s nominal frequency

All results presented in this Section are obtained for 5 dBmof average signal power after the optical amplifier at thenode output.

Figures 7a and 7b show the Q-factor, assessed by theGA and the exact method, as a function of laser carrierfrequency misalignment from the channel’s nominalfrequency for 2, 5 and 10 OADMs, Bo ¼ 40GHz;respectively, for r ¼ 10 and r ¼ 1: Due to the symmetryof the optical signal spectrum and optical filter frequencyresponse, the Q-factor is the same for laser frequencymisalignments of the same magnitude but with oppositesign. Therefore, only positive laser frequency misalign-ments are displayed. Figure 7a shows that the estimatesgiven by the GA are reasonably accurate. Discrepancy in theestimates increases when laser frequency misalignmentincreases, but the discrepancies do not exceed 1.5 dB.However, the largest discrepancy observed occurs for

Fig. 4 DQ as a function of the average signal power after optical amplifier at the node output and of the MUX=DEMUX filterbandwidths normalised to the bit rate, for 10 OADMs

Thick lines correspond to the Q-factor of 17 dB obtained using the exact methoda r ¼ 10b r ¼ 1

IEE Proc.-Optoelectron., Vol. 151, No. 3, June 2004 161

detuning greater than the bit rate. Figure 7b shows that, forr ¼ 1; the estimates predicted by the GA give maximumdiscrepancies of 0.8 dB. However, the behaviour of the TONQ-factor is very well assessed by the GA. Figure 7 showsalso that the accuracy of estimates increases with increase inthe number of nodes. This happens because the estimatesgiven by GA become more accurate with increase in ISI[13]. The increase in ISI enhances the impact of the signal–ASE beat noise on the system performance over the ASE–ASE beat noise impact. Hence, as shown in [23], the GAaccuracy is increased with this higher signal–ASE beatnoise power, especially for ‘0’ bits.

For 10 OADMs, the Q-factor increases when the detuningincreases, reaches a maximum value and, then, decreases forgreater laser misalignment. This behaviour occurs onlywhen the equivalent bandwidth of the OADM cascadebecomes comparable to bit rate and is due to the vestigialsideband filtering [12], which increases the eye-opening ofthe decision current. This can be seen in Fig. 8, which showsthe noiseless eye-diagrams of the decision current for 10OADM, r ¼ 10; and Bo ¼ 40GHz: Figures 8a and 8b,correspond, respectively, to an aligned laser and a laser

misalignment of 5.3 GHz (which leads to the maximumQ-factor presented in Fig. 7a).

Figures 9a and 9b show, respectively, the mean andvariance of the decision current at the sampling instants forboth cases of misalignment. The mean is almost the same inthe two cases; however, when the laser frequency is tuned,the mean variation is higher than when the laser frequency ismisaligned. When the laser is misaligned, the variance ishigher than in the tuned situation. Due to laser misalign-ment, the attenuation of the servo-controlled attenuator isreduced and the excess noise factor is increased. Due to thisincrease, the noise power increases, resulting in a highervariance. From (10), the bit error probability is essentiallyimposed by the bits that lead to the lower argument of theQ-function. When the laser frequency is aligned, despite thesmaller variance, the variance and mean variations lead tohigher and lower arguments of the Q-function in compari-son with the misaligned case. Hence, the bits that lead tolower arguments of the Q-function are responsible for theworst performance when the laser frequency is tuned. Forlaser frequency misalignments higher than the optimumvalue, the eye-opening at the decision circuit input

Fig. 5 DQ as a function of the average signal power after optical amplifier at the node output and of the MUX=DEMUX filterbandwidths normalised to the bit rate, for 2 OADMs

Thick lines correspond to Q-factors obtained using the exact methoda r ¼ 10b r ¼ 1

IEE Proc.-Optoelectron., Vol. 151, No. 3, June 2004162

decreases continuously, hence, degrading the TONperformance.

Figures 10a and 10b show the Q-factor, assessed by theGA and by the exact method, as a function of the lasercarrier frequency misalignment from the channel’s nominalfrequency for 2, 5 and 10 OADMs and Bo ¼ 80GHz;

respectively, for r ¼ 10 and r ¼ 1: Figure 10a shows that,with increase in laser misalignment, the GA estimates apartfrom the estimates given by the exact method, havemaximum discrepancies of 2.6 dB. However, the discre-pancies in estimates for laser misalignment smaller than thebit rate do not exceed 0.25 dB. These discrepancies areattributed to the non-Gaussian behaviour of the PDFs of thecurrent at the decision circuit input corresponding to bothlogical levels, which is enhanced for some laser misalign-ments and TON characteristics (misalignment of about30 GHz and 5 OADMs). In Fig. 10, the TON Q-factor doesnot show any improvement with laser misalignmentincrease, because the equivalent bandwidth of the opticalOADM cascade is much higher than the bit rate. Figure 10bshows larger differences in estimates than with a bandwidthof 40 GHz, with maximum discrepancies of 2.6 dB.However, the discrepancies in estimates for laser misalign-ment smaller than the bit rate do not exceeding 0.32 dB.Even so, similar dependence on laser misalignment of thebehaviour of Q-factor is provided by the GA and the exactmethod.

Fig. 6 DQ as a function of extinction ratio and of the MUX=DEMUX filter bandwidths normalised to the bit rate, for 5 dBm ofaverage signal power after optical amplifier at the node output

Thick lines correspond to Q-factors obtained using the exact methoda 10 OADMsb 2 OADMs

Fig. 7 Q-factor, in dB, as a function of the laser carrierfrequency misalignment from the channel’s nominal frequency,for a TON with 2, 5 and 10 OADMs and Bo ¼ 40 GHz

The dotted line corresponds to the GA estimates and the solid line to theexact method estimatesa r ¼ 10b r ¼ 1

Fig. 8 Noiseless eye diagram of the decision current for 10OADM, r ¼ 10 and Bo ¼ 40 GHz

a Aligned laser frequencyb Laser frequency misalignment of 5.3 GHz

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Similar studies have been performed for other types ofoptical filter, namely Gaussian and Butterworth. Nosignificant differences from the behaviour presented in thisSection have been observed.

5 Conclusions

A GA with two equivalent optical filters, one for the noiseand the other for the signal, has been proposed as a fast andsimple tool for analysing linear TON performance forarbitrary TON topologies, transparent optical nodes struc-tures, optical MUX=DEMUXs and electrical filters. ThisGA is also valid for nonlinear fibre transmission in thepresence of almost perfect dispersion compensation andreduced optical power.

This GA helps in providing rapid identification of theinfluence of optical filtering on the signal and on the noiseaccumulation, and of the different mechanisms that mainlycontribute to degradation in TON performance.

Without laser misalignment, for an extinction ratio of 10,the estimates given by the GA are very accurate incomparison with the exact method, with Q-factor estimatedifferences below 0.03 dB. For an infinite extinction ratio,the GA estimates are also accurate in comparison with theexact method, with maximum discrepancies of 0.5 dB. It hasalso been found that the accuracy of GA estimates increaseswith extinction ratio reduction.

Regarding laser misalignment, for an extinction ratio of10, the GA estimates are reasonably accurate for most of thelaser misalignments considered. For an infinite extinctionratio, the accuracy of GA estimates is slightly reduced, butthe Q-factor estimates provided by the GA follow the samebehaviour as the exact estimates. Only for optical filters withlarger bandwidths and for some laser misalignment do theGA estimates diverge from the estimates of the exactmethod, with discrepancies not exceeding 2.6 dB within therange of parameters investigated.

The results show that the estimates given by the GA canbe either pessimistic or optimistic, depending clearly on thesystem parameters. Several references [24–26] show thatthe GA gives pessimistic results. Those references usuallyconsider infinite extinction ratio. They also estimate theerror probability for Q-factors below 7 (16.9 dB). In thiscase and in agreement with those references, our resultsshow that the GA usually provides pessimistic results. Forsmaller extinction ratios and higher Q-factors, it is shownthat this is not true. Furthermore, our results show that theGA usually provides optimistic estimates of the Q-factor inthe case of high laser misalignment.

All numerical results indicate that the GA is an importanttool to rapidly design TON with tight optical filtering,typical of high-density WDM systems. Generalisation of theGA to WDM networks, where crosstalk between channelscan have a significant impact, is in progress.

6 Acknowledgments

This work was supported by Fundacao para a Ciencia e aTecnologia (FCT) from Portugal and POSI within projectPOSI/CPS/35576/1999 - DWDM/ODC. J. Rebola wouldlike to thank FCT for supporting his work also undercontract SFRH/BD/843/2000.

7 References

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Fig. 10 Q-factor, in dB, as a function of the laser carrierfrequency misalignment from the channel’s nominal frequency fora TON with 2, 5 and 10 OADMs, and Bo ¼ 80 GHz

The dotted line corresponds to the GA estimates and the solid line to theexact method estimatesa r ¼ 10b r ¼ 1

Fig. 9 Mean and variance of the decision current at the samplinginstants for 10 OADMs, r ¼ 10 and Bo ¼ 40 GHz

The solid line corresponds to a tuned laser carrier frequency and the dottedline to a detuning of 5.3 GHz. The first half of the plot corresponds totransmitted ‘1’ bits and the second half to transmitted ‘0’ bits

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