Maximum Likelihood Sequence Estimation Receivers for DWDM Lightwave Systems Hugo S. Carrer Diego E. Crivelli Mario R. Hueda Digital Communications Research Laboratory - National University of Cordoba - Argentina Av. V´elez Sarsfield 1611 - C´ordoba (X5016GCA) - Argentina - Email:
[email protected] Abstract— As the spacing between channels in dense wavelength division multiplexed (DWDM) links decreases and their number increases, nonlinear coupling effects such as four wave mixing (FWM) and cross-phase modulation (CPM) become more important. These impairments, combined with chromatic dispersion (CD), polarization mode dispersion (PMD), and amplified spontaneous emission (ASE) noise ultimately limit the maximum reach of the links. A receiver robust to these impairments would increase the maximum possible length of links without regeneration, reduce the number of required compensation elements such as dispersion compensation fibers (DCF) and optical amplifiers, and simplify link provisioning by making its fine tuning less critical and labor intensive. Maximum likelihood sequence estimation (MLSE) receivers have been previously proposed in [1], [2] to combat CD and PMD in the presence of ASE noise. In this paper we propose to extend the use of MLSE to DWDM links to combat the combined effect of nonlinear crosstalk, dispersion and noise. MLSE receivers can incorporate detailed knowledge of the statistical properties of noise and crosstalk into the decision process, therefore improving performance in the presence of these impairments. We derive a new analytical expression (necessary to implement the MLSE receiver) for the statistics of the received signal in the presence of FWM. Although in long haul DWDM links DCFs cannot be avoided even when electronic dispersion compensation is used, dispersion compensation with DCFs is not complete because even a compensated link exerts different dispersion effects on different wavelengths. Our results show that the performance of DWDM systems can be significantly improved by using an MLSE receiver, particularly in the presence of residual dispersion. For example, a 2000km 20-channel link can operate within the correction capabilities of a forward error correction (FEC) code (G.975 RS) in the presence of 1700ps/nm of residual dispersion even with an extremely high value of FWM crosstalk using the MLSE receiver proposed in this work. As a second important contribution of this paper, we present a theoretical performance analysis of MLSE in DWDM optical channels. This analysis is built upon the theory we developed in [3]. An excellent agreement is found between the prediction of the theory and simulation results.
I. I NTRODUCTION Nonlinear coupling effects such as four-wave mixing (FWM) and cross-phase modulation (CPM) are important performance limiting factors in dense wavelength division multiplexed (DWDM) links. The importance of these effects grows as the spacing between channels decreases and their number increases. These factors, combined with chromatic dispersion (CD), polarization mode dispersion (PMD), and amplified spontaneous emission (ASE) noise ultimately limit the maximum reach of the links. IEEE Communications Society Globecom 2004
Chromatic dispersion is a major impairment at data rates of 10Gb/s and higher. PMD arises because of birefringence in the optical fiber and also becomes important at high data rates [4]. Birefringence is caused by manufacturing defects, mechanical stress, and vibration. Its dependence on random changes in the polarization state of the laser, vibration, and temperature, makes PMD a nonstationary process. ASE noise is introduced by optical amplifiers deployed along the link. In intensity-modulation/direct-detection (IM-DD) channels, ASE noise becomes nongaussian and signal-dependent as a result of the nonlinear response of the photodetector. FWM is one of the main nonlinear effects observed in DWDM systems [5]. The efficiency1 of the FWM generation process increases with the total transmitted power [6], [7], and it also increases when the frequency separation between channels is reduced [5]. For this reason, the FWM effect sets a limit on the maximum admissible distance between optical amplifiers, and also limits the number of channels that can be allocated within the lowloss 1.55µm window [8]. Both limits directly reflect on the cost of the fiber optic link, and on the total link capacity. It is clear from this that FWM plays a major role in high-speed, high-density DWDM systems. In typical applications today, channel impairments are either small and left uncompensated, or they are compensated using optical techniques. These optical techniques require manual calibration and it is very difficult to make them adaptive. As a consequence, the capacity of high-speed DWDM networks is severely degraded by nonstationary effects such as PMD. The use of electronic signal processing in the receiver can mitigate this problem as well as adapt to changing channel conditions. Recent advances in technology have made elaborate signal processing functions at the receiver feasible, even at high data rates. Digital implementation of well-known receiver schemes such as maximum likelihood sequence estimation (MLSE) is now feasible even at data rates of 10Gb/s [3]. It has been shown [2] that MLSE receivers can efficiently cope with impairments such as nonlinear dispersion and nongaussian signal-dependent noise (which arises in intensity modulation/direct detection optical channels). In this paper we propose the use of MLSE receivers in high speed DWDM systems. MLSE receivers can incorporate detailed knowledge of the statistical properties of noise 1 Here efficiency represents the ratio at which power form the transmitted signals is transferred into the FWM-generated signals.
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Tx. 1
Mux Tx. N Fiber Optical Amplifiers
Rx. 1 Demux Rx. N
Fig. 1: System model.
and crosstalk into the decision process, therefore improving performance in the presence of these impairments. We derive a new analytical expression for the statistics of the received signal in the presence of FWM. Previous work has been done on the statistics of the signal in the presence of ASE noise [9]–[11], and of FWM effects on DWDM transmissions [12]–[15]. However, a unified analysis that considers the joint effects of FWM induced crosstalk, CD, PMD, and ASE noise, has not been reported so far. This is necessary to allow the implementation of MLSE in DWDM receivers. Although in long haul links dispersion compensation fibers (DCF) cannot be avoided even when electronic dispersion compensation is used, in DWDM systems dispersion compensation with DCFs is not complete because even a compensated link exerts different dispersion effects on different wavelengths. Our results show that the performance of DWDM systems can be significantly improved by using an MLSE receiver, particularly in the presence of residual dispersion. For example, a 2000km 20-channel link can operate within the correction capabilities of the forward error correction code specified by the ITU-T G.975, (uncoded bit error rate (BER) threshold of 10−4 ) in the presence of 1700ps/nm of residual dispersion even with an extremely high value of FWM using the MLSE receiver proposed in this work. In this work we also present a novel theoretical performance analysis of MLSE receivers in DWDM optical channels. This analysis, built upon the theory we developed in [3], enables us to easily predict the effects of channel impairments on the performance of MLSE receivers in DWDM systems. An excellent agreement is found between the prediction of the theory and simulation results. The rest of the paper is organized as follows. The FWM generation process is revisited in Section II. In Section III, the new expression for the probability density function (pdf) of the signal considering the combined effects of dispersion, FWM, and ASE noise, is derived. In Section IV channel estimation methods are discussed. In Section V, a theory of the bit error probability of an MLSE-based receiver in DWDM lightwave systems is presented. In Section VI, computer simulation results are presented, while conclusions are drawn in Section VII. IEEE Communications Society Globecom 2004
II. F OUR - WAVE M IXING I NDUCED C ROSSTALK In Fig. 1 we show a model of the system under consideration. N intensity modulation (IM) transmitters, each one working at a different wavelength, are optically multiplexed into the fiber. The optical link spans several hundred kilometers. Optical amplifiers are placed at regular intervals along the link to compensate for the power loss of the previous fiber span. Dispersion compensating fibers are placed together with the optical amplifiers, to compensate the effects of chromatic dispersion. At the receiver end, the different channels are optically demultiplexed and detected by N different direct detection receivers. Optical dispersion compensation is assumed to be imperfect. Frequency spacing between the different channels is assumed to be constant. No optical preamplification is assumed. One of the dominant nonlinear effects in DWDM systems is FWM [5]. Owing to this effect, new wavelengths are generated through the nonlinear interaction of transmitted signals. In the case of equally frequency spaced systems, several of these new signals fall in the same wavelength of a given channel, originating crosstalk. Although this process is deterministic, we assume that the receiver of a specific channel has no knowledge of the information sent through any of the other channels, therefore crosstalk must be treated as noise. The amplitude of the electrical field component of the electromagnetic wave at the s-th channel demultiplexer output at instant n, taking into account only FWM interference and channel dispersion, can be written as,
where
Ens = esn + ew n,
(1)
esn = f (Asn , Asn−1 , . . . , Asn−δ+1 ),
(2)
is the noise-free received optical signal for the selected channel, which is in general a nonlinear function of a group of δ consecutive transmitted symbols (Asn , Asn−1 , . . . , Asn−δ+1 ), and ew n is the FWM induced interference. In this work we model ew n as in [13], pqr ew Apn Aqn Arn Ppqr ejθn , (3) n = pqr
where Ain is the information symbol sent through channel i at time n with i = p, q, r. Ppqr and θnpqr are the peak power and phase of the signals generated by the four wave mixing process. These signals are the result of a combination of the signals transmitted through channels p, q and r that satisfy the condition fp + fq − fr = fs , with fi denoting the frequency of channel i.2 It is interesting to observe that some of the terms in the summation in (3) depend on the selected channel signal. This effect is relevant when the number of channels is small, but it can be ignored for a large number of channels. Therefore, the interference term in (1) can be considered as signal independent in the optical domain. 2 To simplify the analysis, in this paper the effects of CD or PMD in the FWM generation process are not considered.
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The interference caused by FWM is treated in the optical domain as a complex random variable (RV), where its real and imaginary parts are independent with zero mean [13]. This assumption is valid when the number of channels is sufficiently large [5], [13]. The variance of the real and imaginary component of ew n is given by [13] 2 2 2 2 = (Apn ) (Aqn ) (Arn ) Ppqr cos2 θnpqr−s σw I
+
2
2
2
2
2
2
(Apn ) (Aqn ) (Asn ) Ppqs cos2 θnpqs−s
II
+
(4)
(Apn ) (Apn ) (Arn ) Ppps cos2 θnppr−s ,
III
where the operator · means expected value, and θnpqr−s is the phase difference between the four wave mixing interference component and the selected channel. In Eq. (4) the summation of interference terms was rewritten as = + + = + + . (5) pqr
p=q=r=s p=q=r=s p=q=r
I
II
III
The values of Ppqr are obtained from [13] −αl 2 + 4e−αl sin2 (∆βpqr l/2) 2 3 −αl 1 − e Ppqr = d κP0 e 2 + α2 ∆βpqr ·
sin2 (Nf ∆βpqr l/2) , sin2 (∆βpqr l/2) 1024π 6 n4r λ2 c2 A2ef f
(3χ)2 ,
(7)
where P0 is the input power per channel, d = 1 or 2 for p = q or p = q respectively; α is the fiber loss coefficient, l is the fiber length between amplifiers, Nf is the number of fiber sections, Nf − 1 is the number of optical amplifiers, nr is the fiber refractive index, λ is the wavelength, c is the speed of light, χ is the third-order nonlinear coefficient, and Aef f is the effective mode field area. Parameter ∆βpqr represents the phase mismatch which, for a non-zero dispersion wavelength region, can be expressed as [12] 2πλ2 D , (8) ∆βpqr = (fp − fs )(fq − fs ) c where D is the fiber chromatic dispersion parameter. III. A NALYSIS OF THE S IGNAL S TATISTICS At the s-th receiver the photodetector generates a current proportional to the input optical power, yn = |Ens + ean |2 = In + nn
(9)
a where Ens = esn + ew n (see (1)) and en is a sample of the ASE noise. ASE noise is modeled as a Gaussian RV in the optical domain [9]. Since the total number of channels is considered large enough, it can be assumed that the FWM induced crosstalk bandwidth is much larger than the signal
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where N0 = (IASE +IF W M )/M is related to the noise power including both the FWM and the ASE noise components, M is the ratio of the optical to electrical bandwidth of the front-end. 2 are the mean currents generated IASE and IF W M = 2M σw by the sole effect of ASE and FWM noise, respectively. Finally Jm (·) is the m-th modified Bessel function of the first kind. Note that (10) is the pdf of the received signal in DWDM systems, considering the combined effect of CD, PMD, ASE, and FWM. This expression is used in this work to evaluate the metrics needed by the MLSE detector in DWDM receivers. IV. C HANNEL E STIMATION
(6)
with κ=
bandwidth [12] and ew n can be considered as white Gaussian noise [5]. In = fa (an ) is the noise-free received signal, which is in general a nonlinear function of a group of δ consecutive transmitted bits an = (an , an−1 , ..., an−δ+1 ). Finally nn is a sample of the total noise which is nongaussian and signal-dependent. Using (9), it is possible to verify [9], [13] that the probability density function of zn conditioned to In is a noncentral chi-square distribution with 2M degrees of freedom: M2−1 yn 1 yn + In exp − fy|I (yn |In ) = N0 In N0 (10) √ yn In · JM −1 2 , N0
In most practical cases the receiver does not have a priori knowledge of the pdf parameters (such as IF W M , M , or σw ) and the dispersion function of the channel. Therefore they must be estimated from the received signal itself. The channel estimation method we consider in this paper is decision directed, in other words, it assumes that the receiver is operating normally and making decisions with a sufficiently low error rate. At the beginning of the operation, the channel estimator can be initialized with relatively crude approximations to the expected value of the signal and the pdf of the noise (for example the former could ignore the intersymbol interference and the latter could be initialized with a Gaussian function). Although this will result in a high initial error rate, the estimation algorithm will typically converge and the error rate will be gradually reduced as the channel estimation improves, until convergence is completed. Channel estimation methods may be parametric or nonparametric. Parametric methods assume that the functional form for the pdf of the signal is known but its parameters are not, whereas nonparametric methods do not assume any knowledge of the pdf. In this paper we consider a nonparametric method, the histogram method. In this method, 2δ histograms (one for each combination of values of the receiver estimates of the δ ˆn−1 , . . . , a ˆn−δ+1 ) are created. most recent received bits a ˆn , a The signal is assumed to be quantized to R bits, therefore each histogram consists of at most 2R bins, where R is a design parameter. Notice that each histogram can be uniquely associated with a branch in the trellis diagram of the receiver. Assuming that the number of signal samples collected is large, the histogram (normalized so that the sum of all its bins is
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unity) is an estimate of fy|I (yn |In ). The histogram is updated iteratively, based on the observed data. The main difficulty with the histogram method is that a large number of samples is needed to obtain accurate estimates. This is particularly problematic in the tail regions of the pdf, where it may take an inordinate amount of time to obtain enough samples. Also, samples in the tail regions could be corrupted by decision errors, even at low BER, although this problem can be virtually eliminated using forward error correction. In these situations, the pdf obtained by evaluating (10) using typical values could be a reasonable initial approximation to the true pdf, but actual information collected from the samples of the received signal would significantly improve the accuracy of the channel estimation. To apply this method, the bins of the histogram are initialized using the best approximation to the pdf available before signal samples are collected, which we call the prior pdf. The prior could be obtained using any available source of prior knowledge about the channel. Then the prior is improved using collected signal samples.
transmitter sent sequence a(j) . P a(i) |a(j) can be computed based on (10). To obtain the probability of error of the MLSE receiver, we shall apply a lower bound developed in [3]. To better understand this lower bound we will start by analyzing the case of no intersymbol interference and a two-symbol transmitted constellation. The typical procedure to obtain the probability of selecting the wrong symbol given the transmitted symbol can be divided into two steps. First determine the error region by finding the crossing point of the two pdfs conditioned to each constellation symbol. And second perform the integration of the transmitted symbol pdf over this error region. In [3] an extension of this method is made for the case of N dimensional signals and nongaussian noise. First, the crossing surface L(u, v) of the function F (y, u) with the function F (y, v) is found. L(u, v) is the locus of all points in RN such that F (y, u) = F (y, v),
(13)
where V. P ERFORMANCE A NALYSIS OF MLSE R ECEIVERS IN DWDM S YSTEMS
F (y, u) =
In the following, we analyze the application of an MLSE receiver to combat the channel-induced intersymbol interference. Let N be the total number of symbols transmitted. The MLSE receiver chooses among the 2N possible sequences, the one that minimizes the metric mr =
N k=1
(r) − log fy|I (yk |Ik ) ,
(r) Ik
(11)
i=j (j)
(j)
(j)
where a(j) = (a1 , a2 , . . . , aN ) represents the transmit(i) (i) (i) ted sequence, a(i) = (a1 , a2 , . . . , aN ) is an erroneous sequence, N is the total number of symbols transmitted, P (a(i) |a(j) ) is the probability of the error event a(j) → a(i) , (the Viterbi decoder chooses sequence a(i) instead of a(j) ), (i) (j) and WH (a(i) , a(j) ) is the Hamming weight of a ∧a (∧ is the exclusive OR operator). P a(j) is the probability that the IEEE Communications Society Globecom 2004
b (yk , uk )
(14)
b (yk , vk )
(15)
k=n0
F (y, v) =
n1 +δ−1 k=n0
b (yk , uk ) = − log fy|I (yk |uk ) ,
(16)
and un = fa (an , an−1 , . . . , an−δ+1 ) vn = fa (ˆ an , a ˆn−1 , . . . , a ˆn−δ+1 )
(r) (r) (r) fa (ak , ak−1 , ..., ak−δ+1 ), N N
in this equation = and (r) (r) (r) (a1 , a2 , . . . , aN ) with r = 1, . . . , 2 are the 2 candidate sequences. The minimization of (11) can be efficiently implemented using the Viterbi algorithm. For a channel with δ symbols of memory as in (2), the trellis has 2δ−1 states. It is interesting to notice that, unlike in the Gaussian channel where the branch metrics are simple Euclidean distances, in the optical channel computation of the branch metrics in general requires the evaluation of different functions for each branch. This is the result of the fact that the noise in the electrical domain is signal dependent. The probability of error of the Viterbi decoder [16] is upper bounded by WH a(i) , a(j) P a(i) |a(j) P a(j) , (12) Pe ≤
n1 +δ−1
u,v, y ∈ RN , ˆ differs from the assuming that the erroneous sequence a transmitted sequence a only for n0 ≤ n ≤ n1 . Then a vector y in L(u, v) is found that minimizes F (y, u). Once y and F (y, u) are known, the probability of the error event can be computed by integrating f (y, u) over the entire half-space containing v, where RN is treated as divided into two half-spaces by L(u, v). Given that the integral is hard to compute, a second order Taylor expansion of F (y, u) around the minimum y called FQ (y, u) is used. Since F (y, u) is a sum of terms where each term depends on only one component of y, the expansion is: n1 +δ−1
FQ (y) = F (y, u) +
pi (yi − y i ) + qi (yi − y i )2 ,
i=n0
(17) 2 (y,u) 1 ∂ F (y,u) where pi = ∂F∂y , and q = . It is 2 i 2 ∂yi i y=y y=y convenient to define the new variables: √ (18) zi = qi (yi − y i ) , then
1008
FQ (z) = F (y, u) +
n1 +δ−1
γi zi + zi2 ,
(19)
i=n0
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√ where γi = pi / qi . Let
0 (20) (21)
−2
N. Ch. = 44
(22)
−4
N. Ch. = 24
−6
N. Ch. = 20
It is clear that k is a unit vector normal to L(u, v) at y. It is always possible to define an orthogonal transformation V such that V k = e1 = (1, 0, . . . , 0)T . Also define s = V T z, α = V T γ . Then we can rewrite (19) in terms of variables si as: FQ (s) = F (y, u) +
n1 +δ−1
αi si + s2i .
log 10(BER)
G(z) =F (z, u) − F (z, v) h =∇G(z)|z=0 h . k= ||h||
−8 −10 −12
Exact Our theory Simulation 5
(23)
FQ (s) = F (y, u) +
n1 +δ−1 i=n0
si +
αi 2 αi2 − . 2 4
(24)
VI. S IMULATION R ESULTS AND D ISCUSSION In this section we present computer simulation results that confirm the above theory . We show results for a typical singlemode fiber as specified by the ITU G.652 Recommendation [17] used in the third telecommunications window (1550nm), which leads to a dispersion parameter D = 17ps/km/nm. The extinction ratio (r10 ) defined below was set to 10dB. Channel frequency separation was set to 50GHz, with the central channel located at 1550nm. Transmitted power per channel was set at a very high value of P0 = 15mW in order to enhance non-linear effects. IEEE Communications Society Globecom 2004
25
30
35
We define:
H
where Q(x) = 12 erfc √x2 . To compute an approximation to the receiver bit error rate, equation (12) is used. As is common practice, the sum over error events in (12) is replaced by its largest term, whose value is approximated using (25). However, before (25) can be computed, the pairs of sequences that correspond to minimum distance error events must be identified. This can be done by exhaustive search over a set of error events of limited length.
20
Fig. 2: Bit-error rate as a function of OSNR, on a perfectly dispersion compensated DWDM channel for different numbers of transmitted channels. Solid lines represent the numerical integration of (10). Dashed lines represent the theory of Section V. Crosses represent simulation results. M = 3, Bit-rate=10Gb/s.
ˆ can be approximThe probability of the error event a → a ated as e−FQ (y) dy (25) P (ˆ a|a) ≈ where H is the error region for the pair (u, v). Substituting variables, approximating the error region as the region where s1 ≥ 0, and using (24), (25) can be expressed as:
γ 2 − F (y, u) P (ˆ a|a) ≈ e−FQ (s) ds = exp 4 s1 ≥0 T (26) n1 +δ−1 1/2 γ k π , Q √ qi 2 i=n0
15
OSNR [dB]
i=n0
By completing squares, (23) can be rewritten as:
10
N. Ch. = 10
I1 , Optical Signal to Noise Ratio 2IASE I1 = , Extinction Ratio, I0
OSNR =
(27)
r10
(28)
where I s (s = 0, 1) is the current generated by each constellation symbol in a nondispersive channel. The receiver used was an eight-state Viterbi decoder where exact knowledge of the channel response was assumed. Expression (10) was used for the branch metric computations. Other parameters used in this section were: Nf = 20, l = 100km (including DCF sections), α = 0.2dB/km, χ = 6×10−15 cm3 /erg, M = 3, and Aef f = 50µm2 . The values of nr used are the typical values for fused silica and are dependent on the channel wavelength [7]. The channel under consideration is always the central one giving a worst case situation from the standpoint of FWM effects. In Fig. 2, the exact performance (obtained by numerical integrations based on (10)) is compared with the theory of Section V and with Montecarlo system simulations, for a channel with perfect chromatic dispersion compensation by means of DCF. Close agreement between simulation, exact, and our theory can be observed. In Fig. 3, simulation results of the MLSE-based decoder are compared against the theory presented in Section V for a total uncompensated link dispersion of 1700ps/nm. Again, note the excellent agreement between simulation and the theory presented in Section V. From Fig. 3 we can observe that for a total uncompensated link dispersion of 1700ps/nm, MLSE can provide an uncoded bit error rate of less than 10−4 for a 20-channel system with a power per channel of P0 = 15mW (this high value of peak power per channel would allow for an increase in the
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DWDM systems has been clearly demonstrated by comparisons with simulation results. Our study shows that MLSE based receivers are excellent candidates to (i) compensate the different impairments in DWDM optical channels, and (ii) reduce the cost of the link and the outage probability owing to nonstationary nature of the channel response.
log 10(BER)
0 −2
N. Ch. = 44
−4
N. Ch. = 24 N. Ch. = 20
−6
R EFERENCES
N. Ch. = 10
[1] W. Sauer-Greff, A. Dittrich, R. Urbansky, and H. Haunstein, “Maximumlikelihood sequence estimation in nonlinear optical transmission systems,” Lasers and Electro-Optics Society (LEOS 2003). The 16th Annual Meeting of the IEEE, vol. 1, pp. 167–168, Oct. 2003.
−8 N. Ch. = 6
−10 −12
Our theory Simulation 5
10
15
20
25
OSNR [dB] Fig. 3: Bit-error rate as a function of OSNR, on a DWDM channel with residual dispersion 1700ps/nm a for different numbers of transmitted channels. Dashed lines represent the theory of Section V. Crosses represent simulation results. M = 3, Bit-rate=10Gb/s.
amplifier span). Note that rates of BER≈ 10−4 , which are within the correction capabilities of the FEC recommended by G.975, can be achieved by the MLSE receiver without the need for extremely accurate optical dispersion compensation schemes (e.g., residual dispersion as high as 1700ps/nm can be allowed). These facts not only would decrease the cost of the link but also would reduce the outage probability due to changes in the channel response (i.e., PMD) [6]. These important topics will be addressed in detail in a future work. VII. C ONCLUSIONS In this paper we have proposed the use of MLSE based detectors in DWDM links to combat the combined effect of nonlinear crosstalk, dispersion, and noise. MLSE receivers can incorporate detailed knowledge of the statistical properties of noise and crosstalk into the decision process, therefore improving performance in the presence of these impairments. We have presented a unified expression for the statistics of the received signal in an IM/DD DWDM system affected by dispersion, ASE, and FWM noise. Building on this expression, we have evaluated the performance of MLSE receivers using computer simulation and an analytical study based on the novel theory we developed in [3]. The excellent accuracy of this theory in evaluating MLSE receiver performance in
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[2] H. F. Haunstein, K. Sticht, A. Dittrich, W. Sauer-Greff, and R. Urbansky, “Design of near optimum electrical equalizers for optical transmission in the presence of PMD,” in Proc. of the Optical Fiber Communication Conference and Exhibit (OFC), vol. 3, pp. 558–560, 2001. [3] O. E. Agazzi, D. E. Crivelli, and H. S. Carrer, “Maximum likelihood sequence estimation in the presence of chromatic and polarization mode dispersion in intensity modulation/direct detection optical channels,” in IEEE Proc. of the International Conference on Communications (ICC), vol. 5, pp. 2787–2793. [4] J. P. Gordon and H. Kogelnick, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci., vol. 97, pp. 4541– 4550, Apr. 2000. [5] I. P. Kaminow and T. L. Koch, Optical Fiber Telecommunications IIIA. Academic Press, 1997. [6] G. P. Agrawal, Fiber-Optic Communication Systems. Wiley-Interscience, 1997. [7] G. P. Agrawal, Nonlinear Fiber Optics. Academic Press, 2001. [8] R. Ramaswami and K. Sivarajan, Optical Networks: a Practical Perspective. Morgan Kaufmann, 2002. [9] D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol., vol. 8, pp. 1816–1823, Dec. 1990. [10] D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol., vol. 9, pp. 505–513, Apr. 1991. [11] P. A. Humblet and M. Azizo˜glu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol., vol. 9, pp. 1576– 1582, Nov. 1991. [12] M. Maeda, W. Sessa, and W. Way, “The effect of four-wave mixing in fibers on optical frequency-division multiplexed systems,” J. Lightwave Technol., vol. 8, pp. 1402–1408, Sept. 1990. [13] K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol., vol. 12, pp. 1423–1439, Aug. 1994. [14] D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol., vol. 9, pp. 121–128, Jan. 1991. [15] K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol., vol. 10, pp. 1553–1561, Nov. 1992. [16] G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Commun., vol. 18, pp. 363–378, May 1972. [17] International Telecommunications Union ITU-T Recommendation G.652, Characteristics of Single-Mode Optical Fibre and Cable, Mar. 2003.
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