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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 33, NO. 18, SEPTEMBER 15, 2015 3763

Modified Nonlinear Decision Feedback Equalizerfor Long-Haul Fiber-Optic Communications

Deepyaman Maiti and Maıte Brandt-Pearce

Abstract—In a long-haul optical fiber communication system,fiber attenuation, dispersion, and nonlinearity combined with non-deterministic noise from optical amplifiers used for periodic regen-eration cause adverse effects on system performance. Several opti-cal and electrical signal processing techniques have been proposed,and implemented to extract the transmitted data; some provide bet-ter performance than others, but at a cost of higher computationalcomplexity. We present a modified nonlinear decision feedbackequalizer designed for use in a legacy optical communication sys-tem with periodic dispersion compensation. The effects of noiseand nonlinearity on the equalizer coefficients are investigated, anda suboptimal convergence algorithm to reduce such effects is pro-posed and verified. Our equalizer provides performance compa-rable to that obtained using digital backpropagation, while beingcomputationally simpler, compensating linear and nonlinear phys-ical impairment effects effectively even at high power levels wherefiber nonlinearity is significant. Performance prediction of the de-signed DFE is also discussed, using a numerical method, with andwithout error propagation.

Index Terms—Decision feedback equalizer, long-haul fiberoptic communication, nonlinear equalization, nonlinear fiberimpairments.

I. INTRODUCTION

O PTICAL fibers offer potential for high speed communi-cation over long distances. In the dispersive and nonlin-

ear optical channel, the random amplified spontaneous emission(ASE) noise from amplifiers interacts nonlinearly with the deter-ministic signal, while the signal interacts with itself, as does thenoise, to generate crosstalk at the receiver. Intersymbol and in-terchannel interference, self, cross and interchannel cross phasemodulation, four wave mixing and amplifier noise all affect theperformance of the communication system, especially the pe-riodically dispersion compensated legacy systems consideredhere [1].

Fiber nonlinearity is a major limiting factor for long-haulhigh-speed wavelength division multiplexed (WDM) opticalcommunication. For such communication systems, coherent de-tection is often preferred since it supports spectrally efficientmodulation formats, and offers potential for superior signalsensitivity and the possibility of electronic compensation algo-rithms [2]. These spectrally efficient modulation formats require

Manuscript received December 31, 2014; revised March 9, 2015; acceptedMarch 12, 2015. Date of publication June 9, 2015; date of current version August9, 2015. This work was supported in part by the NSF Grant CCF-1422871.

The authors are with the Charles L. Brown Department of Electrical andComputer Engineering University of Virginia, Charlottesville, VA 22904 USA(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JLT.2015.2444273

higher signal-to-noise power ratio. However, nonlinear impair-ment effects are also greater at higher power levels.

An equalizer extracts the transmitted information by methodssuch as removing impairment effects using filters or searchingfor likely transmitted sequences using probability measures [3].Optical and electrical techniques have been developed to re-duce the effects of physical impairments and amplifier noiseand extract the transmitted signal with a desired degree of accu-racy. Electrical solutions can be integrated into electronic chipsand mass-produced at a reasonable cost, and can be applied toexisting optical networks.

Equalizers can be divided into two categories: linear and non-linear. Linear equalizers may be designed using a linear(ized) oraffine model for the communication channel. The simplest formof equalization uses a linear filter that applies the inverse of thechannel transfer function on the received symbols to estimatethe transmitted data. Equalizers using finite impulse response(FIR) type filters have been studied in works such as [4]. Disper-sion compensation techniques [5] also fall under this category.Linear equalizers can only compensate the linear impairments.

On the other hand, a nonlinear equalizer can successfully mit-igate nonlinear impairment effects as well and are thus muchmore suitable for long-haul optical communication system ap-plications. Nonlinear equalizers utilizing methods such as max-imum likelihood sequence estimation (MLSE) or maximum aposteriori (MAP) sequence detection have been well studied.MLSE equalizers require a Viterbi decoder and soft decisionfor decoding [6]. The computational complexity is considerabledue to the large number of states.

The complexity of MLSE and MAP-based methods growsexponentially with the equalizer memory size and the numberof WDM channels since it needs a look-up table of possiblesequences of transmitted symbols. MAP-based equalizers formulti-channel transmission or WDM coherent systems havinggood performance but high signal processing complexity havebeen proposed in [7] and [8], respectively.

Frequency domain equalization methods [9] have also beenstudied, using a look-up table or a constant-modulus based algo-rithm to adjust tap-weights (in polarization multiplexed trans-mission systems), or by estimating the channel transfer functionin optical orthogonal frequency division multiplexed (OFDM)transmission [10].

Some authors have designed nonlinear equalizers based onanalytical closed form approximations from appropriate orderVolterra kernels to mitigate nonlinear effects. Using third orderinverse Volterra theory on a mathematical description of anoptical system that includes all the impairments effects, a model-centric nonlinear equalizer for single channel systems is found

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3764 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 33, NO. 18, SEPTEMBER 15, 2015

in [11]. The disadvantage of a Volterra-model-based design is itscomplexity and truncated models are often used [12]. A Wiener–Hammerstein model based equalizer (for OFDM transmission)is presented as a simpler alternative [13].

Digital backpropagation [14] jointly compensates linear andnonlinear impairments. The basic principle is to solve an inversenonlinear Schrodinger equation through the fiber to estimate theinput signal. In the absence of stochastic noise, backpropaga-tion can reconstruct the originally transmitted sequence withperfect accuracy. The main disadvantage of backpropagation isits excessive computational complexity. It is also practically dif-ficult to implement with hardware or at high data-rates due toits oversampling rate requirement.

We propose to apply a nonlinear decision feedback equalizer(DFE). In general, for communication systems with additivenoise and pre and post-cursor interference effects (such as ours),DFEs offer a good compromise between complexity and perfor-mance [15], and can be applied to any modulation format. Lineardecision feedback approaches have been used before in relatedfields such as wireless optical communication [16] or channelestimation for OFDM [17]. A DFE based structure with a non-linear Volterra filter feedback for recording systems is foundin [18], and with nonlinear Volterra filters in feedforward andfeedback paths for reading information stored in optical discs in[19]. The structure and design of a digital DFE is conceptuallysimple, and it can be implemented using hardware [20]. Com-pared to a single feedforward equalizer, a nonlinear DFE (witha nonlinear feedforward and a nonlinear feedback filter (FBF))can mitigate pre and post-cursor impairment effects simultane-ously without enhancing noise while requiring a lower numberof filter coefficients.

Only a few applications of decision feedback in optical fibercommunication systems are found in literature. In [21], the au-thors study the impact of the channel bandwidth on performanceand the maximization of spectral efficiency in a multi-channelsystem using a constant modulus algorithm feed forward equal-izer with a feedback loop (to further compensate for ISI). Theequalizer filter coefficients are calculated from the impulse re-sponses of the feed forward and feedback paths. In [22], a DFEarchitecture integrating error detection code capability and con-sisting of one or more linear or nonlinear Volterra filters isinvestigated for a single channel system. This work is concep-tually similar to the Volterra filter based equalizers discussedearlier, with the addition of a decision feedback loop, as done in[19]. The authors conclude that integration of the error detectioncapability with a Volterra filter in the forward path provide goodperformance over a range of input power. The transmission dis-tance is 480 km, and the equalizer performance is not comparedagainst that of standard techniques such as backpropagation.

The novelty of the current work is in identifying a standalonenonlinear DFE suitable for a long-haul coherent optical fibersystem, understanding that system noise and uncompensatednonlinearity affect tuning of the coefficients, and presenting amodified DFE training that uses multiple iteration least meansquares (LMS) algorithm [23] to mitigate it. We report resultsfrom long-haul legacy systems having total lengths varying be-tween 2800 and 4000 km and find the equalizer performance

Fig. 1. Block diagram of the optical fiber communication system.

is comparable to that from digital backpropagation while beingcomputationally simpler.

The paper is organized as follows. In Section II, we decideon the DFE structure from an input-output model of the opti-cal system and describe the equalizer in training and decisionmodes. In Section III, we study the effects of noise and uncom-pensated impairments on the filter coefficients. Finding that thecoefficients do not achieve steady-state values with the conven-tional tuning method (in general), we implement and verify asuboptimal convergence training approach. In Section IV, weevaluate the performance of the DFE, and compare the resultswith other approaches including backpropagation. In Sections Vand VI, we investigate the computational complexity and theperformance analysis of the DFE. Section VII concludes thepaper, also presenting future research directions.

II. DFE STRUCTURE AND COEFFICIENT TUNING

The basic DFE structure contains two filters [15], a feed for-ward filter (FFF), a FBF and also includes a nonlinear decisiondevice. The FBF feeds back a function of the decisions on thepast symbols. The filters may both be linear, or at least one ofthem can be nonlinear. If the past decisions were correct, weare in fact feeding back a certain function of the actual pasttransmitted symbols. Thus the DFE can suppress or cancel thepost-cursor interference effects for a number of past symbols.

A. Fiber-Optic Communication System Model

Our long-haul single-channel optical communication system(see Fig. 1) is a legacy network. At the transmitter, an M-aryelectrical symbol sequence aq is converted into the correspond-ing optical signal s(t) after pulse shaping and modulation. Wechoose a square-root raised cosine pulse shaping filter to mini-mize interference effects. The optical signal s(t) is then trans-mitted through a finite number of spans of standard single modefibers (SSMF) and is affected by the linear and nonlinear effects(and amplifier noise) to evolve into r(t), the input to the receiver.Our channel model has n spans of SSMF fibers with each spanof length L, with dispersion compensating fiber (DCF) sectionsand optical amplifiers (EDFA- erbium doped fiber amplifiers)

MAITI AND BRANDT-PEARCE: MODIFIED NONLINEAR DECISION FEEDBACK EQUALIZER FOR LONG-HAUL FIBER-OPTIC 3765

at the end of each span (after the DCF). We assume full (100%)and ideal dispersion compensation, and any loss or nonlinearityin the DCF is neglected.

The coherent receiver downconverts the optical signal intoelectrical signal pulses and it consists of an optical filter to limitthe ASE noise, a photo-detection device, and an electrical filterto reduce (electrical) noise [1]. Our receiver-end electrical filteralso has a square-root raised cosine pulse shape, and is matchedto the transmitter filter. The narrow bandpass nature of the elec-trical filter removes all second order nonlinearity [24] from thephotodetector output. Sampling once per symbol period T at op-timal discrete times tq = qT (ignoring synchronization issues)then generates yq = y(tq ) which is the input to the equalizermodule. Note that, since the pulses are broadened in frequencydue to the impairment effects, sampling once per symbol timeis suboptimal and does not capture all the nonlinear effects.

When ASE noise is not considered, using the result in [25],we determine that yq is a (mainly third order) nonlinear functionof the modulating data. The general form of this relation wouldbe

yq =∞∑

i=−∞ρiaq+i +

∞∑

i,j,k=−∞ρijkaq+ia

∗q+j aq+k

+ H.O.T. (1)

The coefficients ρi’s and ρijk ’s are related to the fiber impair-ments, and the abbreviation H.O.T indicates fifth and higherorder terms.

For the equalization problem, we need to estimate the trans-mitted symbols aq ’s from the yq ’s. Assume that from (1), byapplication of inverse Volterra theory [11], we can obtain

aq =N −q−1∑

i=−q

Ciyq+i +N −q−1∑

i,j,k=−q

Cijk yq+iy∗q+j yq+k

+ H.O.T., (2)

where N is the total size of transmitted data, still ignoring theeffect of ASE noise in the system.

B. Design Decisions

Analytical methods are sometimes used in comparable sce-narios to determine the DFE filter coefficients, but lacking anaccurate description of the system noise to include in (1), wetune these numerically with training data. From (2), we deter-mine that a DFE with third order filters in both the forward andfeedback paths would compensate the impairments, the signifi-cant nonlinearity being of third order. Since the noise affects thecalculation of the filter coefficients (tuning), we retain only thelargest of the third-order nonlinear terms (from the few adjacentsamples), and not the smaller third order and all higher orderones, the latter having a greater chance of being inaccurate andadversely affecting the equalization.

Thus, our nonlinear DFE has the structure

pq = decision[f1(yq) + f2(pq)

](3)

Fig. 2. Decision feedback based equalizer training and decision modes.

where the FFF output is

f1(yq) =n1∑

i=−n1

ciyq+i +n2∑

i,j,k=−n2

cijk yq+iy∗q+j yq+k , (4)

and the output of the FBF is

f2(pq) =−1∑

i=−n1

dipq+i +−1∑

i,j,k=−n2

dijkpq+ip∗q+j pq+k . (5)

The vector yq holds values of past, current and future receivedsymbols; that is, yq = [yq−n1 , yq−n1 +1 , . . . , yq+n1 −1 , yq+n1 ].Likewise, pq is the vector of past decisions; pq =[pq−n1 , pq−n1 +1 , . . . , pq−1 ]. The linear and nonlinear coeffi-cients for the FFF are ci and cijk , and those for the FBF aredi and dijk , respectively. The length of the linear parts of thetwo filters is denoted by n1 , and that of the nonlinear parts by n2 .

The FFF and the FBF blocks contain the necessary memoryand circuits to form the functions f1(yq) and f2(pq) from yq

and pq , respectively, for each q. In the absence of any decisionerrors, pq = aq for each q.

C. Training and Decision Modes

The coefficients of the two filters (FFF and FBF) are tunedduring a training period. A known sequence of aq is transmit-ted, and the same is fed back to the FBF (instead of the previousdecisions from the decision device that is disconnected fromthe FBF in this mode). The error signal, which is the differencebetween the training signal and the decision device input, is min-imized by adapting the FFF and FBF coefficients, as presentedin Fig. 2.

Let wq be a combined coefficient vector composed ofthe filter coefficients (as yet untuned) ci(i = −n1 , . . . , n1),cijk (i, j, k = −n2 , . . . , n2), di(i = −n1 , . . . ,−1) anddijk (i, j, k = −n2 , . . . ,−1). xq is the combined vector ofthe appropriate first and third order terms from the receivedsymbols and the past decisions and their conjugates such thatwq × xq = f1(yq) + f2(pq).

Recursive least squares (RLS) and LMS are two popular al-gorithms used in adaptive filtering. RLS-based algorithms havegood performance when applied in non-stationary channel mod-els [26] but at the costs of increased computational complexity


Algorithm 1 Tuning of FFF and FBF coefficientsfor q = 1 to Nt do

for i = 1 to imax doe(i)

q ← pq − w(i)q xq

w(i+1)q ← w(i)

q + 2μe(i)q xq

end forw

(1)q+1 ← w(i)

q

end for

and possible instability [27]. Since the optical fiber channelchanges only very slowly, we choose the LMS algorithm. Ithas been successfully used before to adapt the tap weights fornonlinear equalizers [18] [19]. We use multiple iterations forevery symbol to increase the convergence rate of the algorithm,as previously done in [23]. The tuning process is stopped aftera certain number of iterations imax (Algorithm 1). Nt denotesthe size of the training dataset, and e(i)

q denotes the (generallycomplex) error signal after iteration i. The convergence param-eter μ must be carefully selected for proper convergence. μ canbe a scalar, or if separate parameters for the different first andthird order coefficients are desired, a vector. The coefficients aretuned once for each q.

After the training is completed, it can be expected that theequalizer coefficients are properly tuned. The coefficients canthen be obtained from w

(im ax )Nt

. Let the decision device inputbe denoted by zq . That is,

zq =n1∑

i=−n1

ciyq+i +n2∑

i,j,k=−n2

cijk yq+iy∗q+j yq+k

+−1∑

i=−n1

dipq+i +−1∑

i,j,k=−n2

dijkpq+ip∗q+j pq+k . (6)

A decision is made on zq according to standard demodulation.We do not update the filter coefficients in the decision mode.This would add to the complexity and overhead. More impor-tantly, if a particular decision is wrong, the resulting adjustmentsto the filter coefficients would be incorrect and may adverselyaffect future performance by increasing the probability of sub-sequent decision errors. The decision device output pq = aq isthe decision on the received symbol yq , and the FBF forms itsinput from the decision device output.

III. NATURE AND CONVERGENCE OF COEFFICIENTS

During training, the equalizer must decide on a set of co-efficients ci , cijk , di and dijk that would—with help from thedecision device—correctly map the received symbols yq ’s to thetransmitted symbols aq ’s, in the presence of all the linear andnonlinear impairments, noise and crosstalk.

A. Conventional Training of Coefficients

When the equalizer is trained conventionally as described inSection II, the coefficients never converge to particular fixedvalues except under special circumstances, such as low fibernonlinearity coupled with absence of ASE noise. (If we did have

Fig. 3. Coefficient c1 versus training data length for conventional training.

Fig. 4. Coefficient c000 versus training data length for conventional training.

a model for the additive noise to use in (1), this observation mightnot have been true.) During the training phase, an optimizationproblem is solved, for each q, so that zq equals the correspondingtraining symbol pq = aq . For any particular symbol pair, theoptimization is performed very efficiently indeed, giving rise tovery low values of individual e(im ax )

q . Also, the effects of theresidual third order nonlinearity, the higher order nonlinearityand the ASE noise unaccounted for in (3) all enter into thecoefficient tuning process in an unpredictable manner for eachq, and are deposited into the coefficients of the other terms.

In Figs. 3 and 4 we illustrate the behavior of the coefficientsc1 and c000 from a (single) simulation experiment run on QPSKmodulated data transmitted at 21.4 Gb/s over 50 SSMF spansat 2 dBm input power. The inline amplifiers have a noise figure(NF) of 5 dB. We found n1 = n2 = 3 to be a good compromisebetween performance and computational complexity. Other rel-evant system parameters are given in Table I. Although noisein the system is relatively low, nonlinearity and nonlinear phasenoise (NLPN) are not. The coefficients fail to converge to steadyvalues even as the training size increases. (Increasing the filterlengths does not seem to affect this.)

B. Proposed Training of Coefficients

To reduce the oscillation in the coefficients, at least to somedegree, we modify the way the coefficients are tuned by forcingsuboptimal convergence of zq to pq for each q. This schemeconsists of a two stage training mode. The stages are


TABLE ISIMULATION PARAMETERS

Description Symbol Value

pseudo-random binary sequence length 21 5

transmission wavelength 1550 nmdata-rate B 21.4 to 50 Gb/sbits per symbol k

symbol rate Fs B /koversampling rate (at transmitter) ns 16sampling frequency F ns Fs

attenuation constant α 0.2 dB/kmgroup-velocity dispersion parameter β2 −20 ps2 /kmnonlinear parameter γ 2/(W·km)span length L 80 kmnumber of spans n 35 to 50transmitter filter 3 dB bandwidth 1/Fs

transmitter filter rolloff factor 0.5amplifier NF (practical) 5 dBamplifier NF (ideal) 3 dBoptical filter bandwidth 2B /Freceiver filter 3 dB bandwidth 1/Fs

receiver filter rolloff factor 0.5tolerance ε 0.5

Algorithm 2 Modified coefficient trainingfor q = 1 to Nt do

for i = 1 to imax doe(i)

q ← pq − w(i)q xq

if |e(i)q | ≥ ε then

w(i+1)q ← w(i)

q + 2μe(i)q xq

elsew(im ax +1)

q ← w(i)q

i ← imaxend if

end forw

(1)q+1 ← w(i)

q

end for

sequential—the second stage starts after the first ends. The train-ing set is divided into two distinct subsets (which need not beequal in size), one for each stage. In the first stage, we train theequalizer conventionally to get the coefficients in the ballpark.In the second, instead of tuning the coefficients until zq equalspq , we set a condition so that the coefficient updating stops assoon as zq is sufficiently close (to be defined) to pq , and thetraining moves on to the next (q + 1)th symbol.

Recalling that in the training mode as described in Section II,for each q, the coefficients are updated such that after eachiteration in the LMS algorithm, zq moves little by little towards

pq . Also, w(1)q+1 = w(im ax )

q . That is, for the first iteration forthe (q + 1)th training (yq+1 , pq+1) pair, the coefficients are the

same as that for the last iteration for the qth pair. Then w(i)q+1 ,

i = 1, 2, . . ., imax , may move away from w(im ax )q as zq+1 =

w(i)q+1 × xq+1 gets closer to pq+1 with every iteration.

We would like w(im ax )q+1 to be close to w(im ax )

q to identify thefilter coefficients to be used in the decision mode. We definea region around pq so that if zq is in that region, we consider

Fig. 5. Coefficient c1 versus training data length for suboptimal convergence.

Fig. 6. Coefficient c000 versus training data length for suboptimal conver-gence.

it close enough (a decision device would map the region to adesired point). For QPSK modulated data; for the normalizedconstellation point pq = ±1 ±

√−1, we consider the optimiza-

tion sufficiently good if |pq − zq | < ε; ε being a non-negativereal number to be tuned empirically, after considering the de-sired margin against possible noise that could hurt performance.The modified second training stage is illustrated in the pseu-docode given in Algorithm 2. The result of this modificationis not the same as setting imax to be low: a low imax may notalways guarantee the desired amount of convergence, and theadditional criterion minimizes the computational effort. Sincethe non-steadiness of the coefficients cannot always be perfectlyeliminated, we use the arithmetic means of the filter coefficientsover each q in the second training phase to find the correspond-ing fixed coefficients for use in decision mode.

Besides ensuring the coefficients do not change much withq, the suboptimal convergence modification allows us to runthe LMS algorithm, for each q in the second training stage, asmaller number of iterations, since we neither seek not wantperfect convergence of zq to pq . Simulation runtimes indicatethat this cuts down on computations by 30% or more.

In Figs. 5 and 6 we have the behavior of the same coefficientsc1 and c000 from a simulation experiment run at 2 dBm inputpower when suboptimal convergence is applied. The secondtraining stage starts at q = 2000. (As long as the second train-ing set is of a reasonable length, the actual starting point is not


TABLE IICOEFFICIENT OF VARIATION (CV) OF c1 AND c000

Conventional Training Suboptimal Convergence

|CV(c1 )| 0.5605 0.2301|CV(c0 0 0 )| 1.9560 0.3805

Fig. 7. BER performance (21.4 Gb/s QPSK, 50 spans, 5 dB amplifier NF).

important, since all the coefficients are affected at the same timeby the training procedure for proper equalization.) These arethe counterparts to Figs. 3 and 4, respectively, and are generatedunder identical conditions. In Table II, we present the absolutevalues of the coefficient of variations (ratio of standard deviationto mean) of c1 and c000 for conventional and suboptimal con-vergence training, and conclude that suboptimal convergence issuccessful in keeping the coefficients reasonably constant evenas q changes.

IV. RESULTS

In this section, we apply our DFE to data obtained fromsimulation experiments (using the split-step Fourier method)conducted under different system conditions and modulationformats (OOK or QPSK) to verify that the performance is sat-isfactory. We primarily compare the performance of the DFEagainst that of digital backpropagation, which is considered abenchmark in the literature [14]. The relevant simulation pa-rameters are indicated in Table I.

A. Performance on QPSK Modulated Data

In Fig. 7, we have plotted the bit error rate (BER) curves forthe unequalized, linearly equalized (phase rotation filter), con-ventional nonlinear DFE equalized, modified nonlinear DFEequalized (with n1 = n2 = 3), backpropagated (with full non-linearity compensation) and the no-nonlinearity limit cases. TheBER for the unequalized case is quite high, mostly due to thenonlinear phase rotation [28]. The first order filter performswell at lower levels of nonlinearity. At the lower end of theinput power range, the performances of all four techniques arecomparable and close to the no-nonlinearity limit. At higher

Fig. 8. BER performance (50 Gb/s QPSK, 40 spans, 5 dB amplifier NF).

TABLE IIIMINIMUM BER FOR QPSK MODULATED DATA (×10−3 )

Conditions Unequalized Nonlinear DFE Backpropagation

A 4.3838 0.133 0.133B 10.4412 0.343 0.343C 11.0841 0.457 0.429

levels of nonlinearity, backpropagation and the nonlinear DFEoutperform the other techniques.

The slight differences in performance between the DFE andthe backpropagation method can be attributed to the fact thatthe modified DFE equalizer makes an adjustment for the un-compensated terms so that the equalized constellations are in aregion around the respective symbol points. Backpropagation,on the other hand, works methodically span by span undoingthe effects of the linear and nonlinear impairments. Noise inthe system causes inaccuracy in the backpropagation solution,regardless of the backpropagation step size.

Next, in Fig. 8, we have the BER curves from a simulationexperiment run on a higher rate system. This plot is of a similarnature as Fig. 7. At this higher data-rate, the nonlinear impair-ment effects are greater [25], as is the effect from the interactionof the ASE noise with the nonlinearity. The nonlinear DFE andbackpropagation again both perform better than the linear filteror the conventionally trained DFE.

Additional results on the performance of the nonlinear DFEare presented in Table III. The conditions for the simulationexperiments are:

A: 21.4 Gb/s data-rate, 50 SSMF spans, 3 dB amplifier NF,B: 50 Gb/s data-rate, 30 SSMF spans, 5 dB amplifier NF,C: 50 Gb/s data-rate, 40 SSMF spans, 3 dB amplifier NF.From Figs. 7 and 8 and Table III, we conclude that the pro-

posed nonlinear DFE works satisfactorily for QPSK modulateddata transmitted at different data-rates over different lengths offibers, with different noise characteristics, and its performancematches that of backpropagation.


Fig. 9. BER performance (25 Gb/s OOK, 40 spans, 5 dB amplifier NF).

TABLE IVMINIMUM BER FOR OOK MODULATED DATA (×10−3 )

Conditions Unequalized Nonlinear DFE Backpropagation

D 9.658 0.667 0.667E 28.459 4.345 3.886F 61.261 5.031 5.029

B. Performance on OOK Modulated Data

In Fig. 9 we have results from a simulation experiment run onOOK modulated data. Comparing with Fig. 7, we first note thatthe unequalized BER is higher for QPSK modulated data. Thisis due the effect of NLPN on PSK modulation schemes. Next,we note that the minimum BER after equalization is lower forQPSK. QPSK is a more power efficient modulation technique,and modulated data has constant amplitude.

Additional results on the performance of the nonlinear DFEare presented in Table IV. The conditions for the simulationexperiments are:

D: 25 Gb/s data-rate, 50 SSMF spans, 3 dB amplifier NF,E: 25 Gb/s data-rate, 50 SSMF spans, 5 dB amplifier NF,F: 50 Gb/s data-rate, 40 SSMF spans, 5 dB amplifier NF.We can draw similar conclusions as from the corresponding

QPSK performance results, that is, lower amount of noise pro-duces a lower BER in the equalized data, as do a lower numberof spans. A higher data-rate hurts the minimum equalized BER.From Fig. 9 and Table IV, we conclude that the DFE is justas effective in suppressing linear and nonlinear physical im-pairments for OOK modulated data transmitted under differentconditions.

V. COMPLEXITY ANALYSIS

In this section, we study if the modified DFE algorithm iscomputationally attractive. We would like to note that the onlyoverhead in our proposed equalization algorithm arises from theneed to train the filter coefficients. Once the tuning process iscomplete, there is no further introduction of overhead since the

TABLE VNOTATIONS

Description Symbol

data block size N

size of data for first training stage N1

size of data for second training stage N2

total training data size Nt

number of filter coefficients nc

iterations per symbol for first training stage im a x

mean iterations per symbol for second training stage iav g

optical transmission link changes only very slowly compared tothe data-rate. The notations used are explained in Table V.

For the modified DFE, the number of calculations duringfirst training stage is (N1nc + N1imaxnc + N1imax). Since nc

and imax are much smaller than the size of the training datafor either stage, the complexity of this stage is O(N1). Thenumber of calculations during the second training stage is(N2nc + N2iavgnc + N2iavg), and the complexity of this stageis O(N2). Since N1 ≈ N2 , the complexities are of the sameorder. The second training stage involves a comparison to checkif |pq − zq | < ε; however iavg is usually less than half of imaxto compensate for this.

For the conventional DFE, the number of calculations dur-ing the single training stage is (Ntnc + Ntimaxnc + Ntimax),and thus the complexity is O(Nt), with Nt � N due to thevery slowly changing channel. For either DFE configuration,the decision mode (actual operation) is identical. The numberof calculations required for a decision on each symbol is of theorder of nc .

For backpropagation, an oversampling rate of 3 or 4 is suffi-cient [14]. The computational complexity of back- propagationalgorithm is O(N logN) [29]. The complexity of digital back-propagation is also a function of the number of the segments,which can be huge. For reduced complexity backpropagationmethods [30], the complexity remains of the same order. Thisshows that the DFE or the modified DFE is computationally lesstaxing than digital backpropagation for a transmission link withperiodic dispersion compensation. For an uncompensated link,the channel will have a larger memory necessitating a largernumber of equalizer taps and a corresponding increase in the al-gorithm complexity. The complexity of digital backpropagationwill remain unaffected.

VI. PERFORMANCE ANALYSIS

The error probability evaluation of a DFE can be studiedin two subcategories—with and without error propagation. Awrong decision on a particular symbol increases the probabilityof subsequent errors by worsening the post-cursor impairmenteffects. Thus the error is propagated. In the absence of errorpropagation, we assume that as far as post-cursor compensationis concerned, all previous decisions are correct.

Due to residual and higher order uncompensated nonlineari-ties, unmodeled ASE noise and NLPN in the system, and pos-sibility of decision errors in the equalizer, zq in (6) is differentfrom aq in (2). Let us call this deviation δq = zq − aq . For no


error propagation, using (4)–(6),

δq = zq − aq = f1(yq) + f2(pq) − aq

= f1(f3(aq )) + f2(aq ) − aq , (7)

where

f3(aq ) =N −q−1∑

i=−q

ρiaq+i +N −q−1∑

i,j,k=−q

ρijkaq+ia∗q+j aq+k (8)

from (1) when the higher order terms are ignored. If error prop-agation is considered,

zq = f1(yq) + f2(pq) = f1(f3(aq )) + f2(decision[zq ]), (9)

setting up a recursive equation to be solved.There will be an error in decision for a particular q if δq crosses

a certain threshold. For normalized QPSK modulated data, theconditions for error in the in-phase and quadrature componentsare, respectively, |Re[δq ]| > 1 and |Im[δq ]| > 1.

Theoretically, in the absence of noise, we can obtain δq an-alytically to find the error probability. However, if there is nonoise in the system, then the FFF and FBF lengths can be cho-sen large enough such that the error probability is zero at allbut the highest input power levels (the effect of the higher ordernonlinearities become more significant with increasing powerlevel). When noise is present, having large filter lengths can beuseless or hurt the design, as discussed earlier.

Related work in existing literature, such as [31], [33] makeuse of limitations and assumptions about the channel, signalsand the noise in the system which are not satisfied by our non-linear optical communication system that has interacting andco-propagating amplifier noise. The problem is further compli-cated since we use training to tune the filter coefficients. Thuswe can not directly use such results.

A. DFE Performance Analysis for No Error Propagation

The condition that a particular decision error does not influ-ence future error probability is enforced by setting the previouspq ’s to the corresponding aq ’s while a decision is being madefor the current yq . We measure the deviations δq = zq − aq

and analyze the distributions of the in-phase and the quadraturecomponents of δq . A numerical estimate of the DFE error prob-ability (BER) can be obtained by observing the percentage ofeach component of δq having magnitudes greater than unity.

We provide an illustration in Fig. 10. The optical amplifiersdo not introduce any noise in the system, only for this example.The nonlinearity is quite high due to the large number of spansand the high power level, but the effect of the residual third orderand unaccounted higher order terms are only just beginning toaffect the performance. The distribution of indicates that thetails are stretching a little over the −1 and +1 limits. The BERis found to be 0.0011.

B. Performance Analysis Considering Error Propagation

We illustrate how previous decisions can affect the equalizerperformance. We consider a situation where all inline amplifiershave a NF of 5 dB. The numerical best case error probability

Fig. 10. Distributions of δq (21.4 Gb/s QPSK, 60 spans, 3 dBm power).

Fig. 11. Effect of error propagation on DFE performance (21.4 Gb/s QPSK,50 spans, 5 dB NF).

estimates (assuming all previous decisions to be correct) andactual BERs obtained via a simulation experiment are plottedacross a range of input power levels in Fig. 11. In addition, weplot the BER that would have been obtained if half the previousdecisions from the DFE were incorrect.

The equalizer performance is close to the no error propagationBER curve. Even in the presence of a decision error, the DFEcan provide a large enough margin to provide correct decisionsand recover quickly. We conclude that not considering enoughhigher order terms and the contribution from all signals withinthe range of influence are the more important factors affectingthe BER performance. The difference between the measuredBER and the no-error-propagation error estimate curves is alittle larger at higher power levels since the post-cursor nonlinearimpairment enhancement in case of a decision error is magnifiedwith more nonlinearity.

We can model the error propagation mechanism as a Markovchain [32] using stochastic states, transition probabilities (to bedetermined numerically), and use some simplifications since thenumber of FBF taps is not unreasonably large. Also, most opticalcommunication systems employ modulation formats for which


the number of constellation points (and distinct magnitudes ofpossible errors) is not very large either.

We consider separately the decisions on the in-phase andquadrature components. For each component, each previousdecision that enters into the current decision calculation caneither be correct or wrong. Thus the total possible number ofcombinations of previous decision errors is 22n2 , including thecase where all previous 2n2 decisions are correct. A state isassigned to each error combination.

We denote a correct decision with 0, and an erroneous onewith 1. As an example, assume that we have n2 = 3, and that theDFE is making a decision on the qth symbol. Also assume thecurrent state of the system is Vq = [000000], where the first twobits indicate the decision on the (q − 1)th symbol, the middletwo that on the (q − 2)th one, and the last two on the (q − 3)thone, with the first, third and fifth bits in a state reserved forin-phase symbol bits. The state Vq = [000000] indicates thedecisions on all three past symbols were correct. If the decisionon the in-phase bit is incorrect, the next state will be Vq+1 =[100000], and so on. This allows us to create a state transitiontable, or equivalently, a state transition diagram, for the errorpropagation mechanism. With the knowledge of current state,corresponding distributions of the deviation δq are generated toyield the transition probabilities.

The error propagation mechanism system can reside in ex-actly one state at any sampling instant q, and the number oftotal possible states is finite, as is each of the states. State tran-sition probabilities are determined only by the current state andare independent of all previous states. Also, these probabilitiesare constant over time, as long as the optical communicationsystem parameters are unchanged. Thus the error propagationmechanism system is a Markov chain process.

The transition matrix of this Markov chain contains the statetransition probabilities. Also, by the nature of definition of thestates, none of them are absorbing. The limiting distributionis numerically obtained by raising the transition matrix to asufficiently high power and observing any row.

The above methodology allows us to predict the probabilitythat the error propagation mechanism system is in a certainstate, the probabilities being constant in the limiting case. Theappropriate state probabilities are then added to compute therequired probability of error of the in-phase or the quadraturebit, or the symbol error probability.

Assume that we wish to numerically estimate of the bit errorprobability when the input power level is 2 dBm in the system inthe previous example. With n1 = n2 = 3, the number of statesis 22×3 = 64. With the filter coefficients fixed from training, weapply the DFE to a known dataset. The current state Vq containsthe previous decisions and we check the probability of a wrongdecision for all possible Vq ’s. The resulting percentage of errorsin the in-phase and the quadrature bits generate the transitionprobabilities. Sections of the state transition table are given inTable VI.

The transition probabilities in the state transition table formthe elements of the transition matrix (with 64 rows and 64columns). By raising the transition matrix to a sufficiently highpower, each row of the limiting distribution of the transition

TABLE VISTATE TRANSITION TABLE FOR DFE ERROR PROPAGATION SYSTEM

Present State Next State Transition Probability

S0 = [000000] S0 = [000000] 994.23 × 10−3

S0 = [000000] S1 = [010000] 2.57 × 10−3

S0 = [000000] S2 = [100000] 3.20 × 10−3

S0 = [000000] S3 = [110000] 0S1 = [010000] S4 = [000100] 933.11 × 10−3

S1 = [010000] S5 = [010100] 61.69 × 10−3

S1 = [010000] S6 = [100100] 5.20 × 10−3

S1 = [010000] S7 = [110100] 0...

.

.

....

S6 2 = [101111] S5 6 = [001011] 926.25 × 10−3

S6 2 = [101111] S5 7 = [011011] 4.57 × 10−3

S6 2 = [101111] S5 8 = [101011] 69.18 × 10−3

S6 2 = [101111] S5 9 = [111011] 0S6 3 = [111111] S6 0 = [001111] 839.70 × 10−3

S6 3 = [111111] S6 1 = [011111] 76.09 × 10−3

S6 3 = [111111] S6 2 = [101111] 84.21 × 10−3

S6 3 = [111111] S6 3 = [111111] 0

TABLE VIIBER AS A FUNCTION OF CURRENT STATES

Vq BER

[001000] 3.69 × 10−3

[000100] 3.43 × 10−3

[000010] 3.26 × 10−3

[000001] 3.20 × 10−3

[001100] 3.49 × 10−3

[001010] 3.20 × 10−3

[001001] 3.52 × 10−3

[000110] 3.57 × 10−3

[000101] 3.23 × 10−3

[000011] 3.12 × 10−3

matrix is numerically determined to be[982.31 2.54 3.17 0 · · · 0 0 0 0

]× 10−3 .

Finally, from the probabilities of the system being in the relevantstates, we determine the BER to be 3.09 × 10−3 , which is a goodestimate for the simulated BER of 3.83 × 10−3 . We also checkthe BER corresponding to particular current state Vq ’s, and thisis illustrated in Table VII.

VII. CONCLUSION AND FUTURE RESEARCH

This paper discusses the application of a modified nonlinearDFE in a long-haul coherent optical fiber communication sys-tem. The channel is lossy, dispersive, nonlinear and containsrandom amplifier noise along with NLPN. We design the equal-izer structure from a mathematical description of the system,and study the effects of noise, NLPN and residual or uncom-pensated nonlinearity on the filter coefficients when the DFE istrained conventionally, and also using our modified convergencemethod.

We note the possible benefits of the modified training schemeand apply the designed DFE to a fiber-optic system under var-ious conditions and modulation formats, comparing with re-sults obtained from backpropagation. We conclude that the


performance is satisfactory, and that the computational com-plexity is modest. Finally we study the problem of performanceprediction of the DFE when applied to an optical communica-tion system, and use a numerical method for the same both inabsence and in presence of error propagation.

In the future, we would like to apply a DFE to a dispersionunmanaged system where the effects of dispersion, nonlinear-ity and their interaction are stronger. A rigorous mathematicalanalysis on the effects of noise and NLPN on filter coefficientsis also desirable. This would enable us to design more effectivetraining schemes to tune the coefficients. Finally, when errorpropagation is taken into account, analytical performance pre-diction of a DFE when applied to fiber-optic communicationsystems is a topic that requires further investigation.

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Deepyaman Maiti received the B.S. degree in electronics and telecommunica-tion engineering from Jadavpur Univerity, Kolkata, India, in 2009, and the M.S.degree in electrical engineering from the University of Virginia, Charlottesville,VA, USA, in 2011, where he is currently working toward the Ph.D. degree atthe Charles L. Brown Department of Electrical and Computer Engineering. Hiscurrent research interests include fiber optics communication and signal pro-cessing applications.

Maıte Brandt-Pearce received the Ph.D. degree in electrical engineering fromRice University, Houston, TX, USA, in 1993. She is currently a Professorat the Charles L. Brown Department of Electrical and Computer Engineering,University of Virginia, Charlottesville, VA, USA. She coedited the book entitled“Cross-Layer Design in Optical Networks” Springer Optical Networks Series(New York, NY, USA: Springer, 2013). Her research interests include nonlineareffects in fiber optics, free-space optical communications, cross-layer design ofoptical networks subject to physical layer degradations, body area networks, andradar signal processing. She was the General Chair of the Asilomar Conferenceon Signals, Systems, and Computers in 2009, and the Chair of the OpticalNetworks and Systems Symposium at the IEEE GLOBECOM in 2015. Shereceived an NSF CAREER Award and an NSF RIA. She coreceived the BestPaper Awards at ICC 2006 and GLOBECOM 2012.

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