A Discrete-Time Polynomial Model of SingleChannel Long-Haul Fiber-Optic Communication

SystemsHoubing Song and Maıte Brandt-Pearce

Charles L. Brown Department of Electrical and Computer EngineeringUniversity of Virginia

Charlottesville, VA 22904Email: song@virginia.edu, mb-p@virginia.edu

Abstract—To mitigate various physical impairments of long-haul dense wavelength division multiplexing (DWDM) systemsand exploit their system capacity, there is a need to develop a two-dimensional (time and wavelength) discrete-time input-outputmodel which can become the foundation of signal processing foroptical communications. As the first step, this paper develops amodel for single channel multipulse multispan systems based onthe Volterra series transfer function (VSTF) method. This modelis suitable for high-bit-rate time-division multiplexed (TDM)transmission in the pseudo-linear regime and is easily extendableto the multichannel case. We overcome the well-known tripleintegral problem and reduce it to a simple integral. This modeltakes into account fiber losses, frequency chirp and photode-tection, which are ignored in the literature. Furthermore, withthis model we introduce coefficients quantifying the intersymbolinterference (ISI), self phase modulation (SPM), intrachannelcross phase modulation (IXPM) and intrachannel four wavemixing (IFWM), to characterize the impact of these effects onthe system performance. The model is in excellent agreementwith SSF (split-step Fourier) simulation. To illustrate how themodel might be applied, we develop a constrained coding schemethat uses the coefficients to suppress the impact of variousimpairments.

I. INTRODUCTION

High-capacity optical backbone networks are needed tosupport dramatically increasing data traffic demand. A solutionis dense wavelength division multiplexing (DWDM) whichcombines time-division multiplexing (TDM) with wavelength-division multiplexing (WDM). A total capacity of 69.1 Tb/swith 432 channels and 171 Gb/s per channel has been re-ported [1]. All-optical communications eliminate the bottle-neck of optical-electrical-optical (OEO) conversion over long-haul DWDM systems. However, this inevitably gives rise tosevere physical impairments which in turn adversely affectsystem performance [2]. The performance of DWDM systemsis fundamentally limited by dispersion, fiber nonlinearity, andnoise [3] [4] [5].

Sophisticated signal processing techniques are necessary tomitigate the physical impairments and fully exploit the systemcapacity [6]. These techniques cannot be developed without amathematical model which describes the input-output relation-ship of the long-haul DWDM systems, which are in practiceimplemented as multipulse multispan multichannel systems.

To better explore digital communications potential of thesesystems, this model should be a discrete-time model so thatvarious mature digital signal processing (DSP) techniques canbe applied and in two dimensions (2D: time and wavelength)so that both intrachannel and interchannel effects could besimultaneously mitigated. Such a model has the potential tobe applied in multichannel signal processing for intersymboland interchannel interference mitigation, constrained codingfor WDM systems, multiuser coding, multichannel detectionand path-diversity for all-optical networks.

This paper is limited to modeling single channel multi-pulse multispan systems for two reasons: (1) single channeldiscrete-time model is the first step in the development of themultichannel discrete-time 2D model; (2) high-bit-rate TDMsystems (100 Gb/s and above) could have advantages over theDWDM systems in many ways [7]. It is the use of fiber opticsin the pseudo-linear transmission regime that revives interestin high-bit-rate TDM systems [7].

The nonlinear Schrodinger (NLS) equation, which describesthe propagation of optical pulses over fibers, is a nonlinear par-tial differential equation (PDE) whose exact analytic solutionsgenerally are very difficult to obtain except for some specificcases. Its continuous time property also makes it difficult todirectly apply in our case. A large number of approximateanalytical and numerical methods have been developed tosolve the NLS equation. Linearization is a widely used ap-proximate analytical approach. Most linearization methods canbe classified into two categories: the Volterra series transferfunction (VSTF) method [8] and the regular perturbation (RP)method [9]. It has been shown that when the nonlinearity isdue to the Kerr effect alone, the n-order RP solution coincideswith the 2n+1-order VSTF solution [10]. Both the third-orderVSTF method and the first-order RP method reach the sametriple integral, which involves massive numerical evaluation ofiterated integration. Many researchers have tried to simplifythe triple integral problem. However, most attempts failed toreach a much simpler form with the exception of a singleintegral for a simplified single-span case in which both fiberloss and chirp parameter are ignored [7] [11].

Most numerical methods can be classified into two cate-

978-1-61284-233-2/11/$26.00 ©2011 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

gories: finite-difference methods and pseudospectral methods[12]. One extensively used pseudospectral method is the split-step Fourier (SSF) method, which runs much faster than mostfinite-difference methods due in part to the use of the fast-Fourier-transform (FFT) algorithm. The SSF method is usuallytaken as the standard of accuracy for validating other methodsin the absence of experimental data due to its well-establishedability to accurately simulate the pulse propagation in fibers.

The purpose of this paper is to develop a discrete timemodel for single channel multipulse multispan systems withperiodic amplification and dispersion management. In thispaper, we extend the VSTF method to the multispan case andreduce the triple integral to a simple integral for Gaussianpulses, gaining computation efficiency advantage over theSSF method with comparable accuracy. The resulting modelis a polynomial model which takes into account the fiberlosses, the dispersion, the pulse chirp, the number of spansand various nonlinearities, and facilitates the suppression ofvarious physical impairments.

The paper is organized as follows. In Section II, we in-troduce the NLS equation, various intrachannel effects andvarious analytical methods and numerical methods. In SectionIII, we develop our model for single channel multipulse multi-span systems. In Section IV, we describe a constrained codingscheme based on the proposed model. Section V concludesthe paper and proposes future avenues for research.

II. NONLINEAR SCHRODINGER EQUATION

The nonlinear Schrodinger (NLS) equation describes thepropagation of optical pulses inside single-mode fibers (SMF).For pulse widths T0 > 5 ps, the NLS equation is of the form[12]

∂A

∂z= −α

2A − iβ2

2∂2A

∂t′2+ iγ|A|2A (1)

where A = A(t′, z) is the slowly varying complex envelope

of the propagating field; t′

is measured in a frame of referencemoving with the pulses at the group velocity vg (t

′= t−z/vg);

z is the propagation distance measured along the fiber; α isthe attenuation constant, a measure of total power loss from allsources during transmission of optical signals inside the fiber;β2 is the group-velocity dispersion (GVD) parameter, a mea-sure of chromatic dispersion which induces pulse broadening;γ is the nonlinear parameter. Hereinafter we drop the primeover t for notational simplicity. The three terms on the righthand side describe, respectively, the effects of fiber losses,dispersion, and nonlinearity on the pulses propagating throughoptical fibers.

A. Fiber Nonlinearities

The field of a single channel can be represented as a sumof the fields of individual pulses, A =

∑K−1k=0 Ak, where Ak

is the field representing the kth of K pulses centered at kT ,where T is the bit period. By substituting this sum into (1)we obtain

K−1∑k=0

(∂Ak

∂z+

α

2Ak + i

β2

2∂2Ak

∂t2) = iγ

K−1∑l,m,n=0

AlA∗mAn (2)

The nonlinear terms on the right hand side of (2) can beidentified as follows: when l = m = n we have self phasemodulation (SPM); when l = m �= n or l �= m = n itis intrachannel cross phase modulation (IXPM), and whenl �= m �= n or l = n �= m it is intrachannel four wave mixing(IFWM). In the pseudo-linear regime, the location of thenonlinear interaction is given approximately by (l−m+n)T .This relation is analogous to the phase-matching conditionused to determine the frequency location of a wave generatedby four-wave mixing (FWM).

SPM leads to spectral broadening of optical pulses. IXPMresults in timing jitter of the pulses. IFWM is responsible fortwo effects that degrade the system performance: amplitudejitter and ghost pulse generation. These three nonlinear effectsare caused by the Kerr effect which is due to the intensitydependence of the refractive index which, in turn, gives riseto an intensity-dependent phase shift of the optical field.

B. VSTF Description

The VSTF method expresses the NLS equation as a poly-nomial expansion in the frequency domain. Retaining only thefirst-order and the third-order Volterra kernels, the frequency-domain output of the fiber at length L is given as [8]

A(ω,L)≈H1(ω,L)A(ω, 0) +∫ ∫

H3(ω1, ω2, ω−ω1+ω2, L)

A(ω1, 0)A∗(ω2, 0)A(ω − ω1 + ω2, 0)dω1dω2 (3)

whereH1(ω,L) = exp(−α

2L + i

β2

2ω2L), (4)

H3(ω1, ω2, ω − ω1 + ω2, L)=iγ

4π2H1(ω,L)

∫ L

0

exp[−αz +

iβ2z(ω1 − ω)(ω1 − ω2)]dz (5)

In (3), A(ω, z) is the Fourier transform of A(t, z), i.e., A(ω, 0)and A(ω,L) are the fiber input and the fiber output in thefrequency-domain, respectively. H1(ω,L) and H3(ω1, ω2, ω−ω1 + ω2, L) are the first-order and the third-order Volterrakernels which are the linear and nonlinear transfer functionsof an optical fiber of length L, respectively. Substituting (4)and (5) into (3) yields the well-known triple integral. Returningto the time domain can result in yet another integral.

III. ANALYTICAL MODEL

This section describes the development of our analyticalmodel of single channel multipulse multispan systems. Fig.1 shows a schematic of a typical single-channel fiber-opticcommunication system with periodic dispersion compensationand amplification. At the optical transmitter, an encoder andan external modulator modify the incoming binary sequence{bk} of duration Tb into a new M -ary symbol sequence {ak}of duration T = Tblog2M ; a laser diode converts {ak} intothe corresponding transmitted optical signal s(t) and thenlaunches it into the optical fiber serving as the communicationchannel. With proper carrier and modulation selection, s(t)can be an amplitude-shift keyed (ASK), frequency-shift keyed

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Fig. 1. Schematic of a typical fiber-optic communication system

(FSK), or phase-shift keyed (PSK) signal. On-off keying(OOK) and differential binary phase-shift keying (DBPSK) aretwo commonly used modulation formats for optical commu-nication systems. s(t) is transmitted through the fiber spanby span and becomes the receiver input signal r(t). Thesignal r(t) is then passed through a square-law photodetectorto convert back into the electrical form y(t) and a decisiondevice. The amplified spontaneous emission (ASE) noise fromthe optical amplifiers and the postdetection electrical filer arenot considered in this paper.

Suppose the system under consideration consists of Nspans with each span of length L. The output of each spanafter dispersion compensation and amplification becomes theinput of the next span. In the frequency domain, disper-sion compensation is implemented by exp(−iβ2

2 ω2L) andamplification is implemented by exp(α

2 L). So the combinedeffects of dispersion compensation and amplification could berepresented by

H−11 (ω,L) = exp(

α

2L − i

β2

2ω2L). (6)

Applying (3) and (6) span by span yields the multispan VSTFexpression.

We define S(ω) and R(ω) as the input and output of thefiber as the communication channel in the frequency domain,which correspond to s(t) and r(t) in the time domain, asshown in Fig. 1. In the case of K successive independentGaussian input pulses of RMS width T0, the input field of thefiber is given as

S(ω) =K−1∑k=0

akP120

√2πT0 exp

[−ω2T 2

0

2− iωkT + iΦk

](7)

where ak is the modulated symbol, which could be complex-valued in the case of DPSK (M ≥ 4); P

120 is the peak

amplitude of the Gaussian pulses, where P0 is the launchedpeak power; T 2

0 = T 20

1+iC , where C is the chirp parameterwhich governs the frequency chirp imposed on the pulse; Tis the symbol period; and Φk is the phase of the kth pulse.

Substituting (7) into the multispan VSTF expression andsolving the resulting triple integral, we obtain the total output

field in the frequency-domain

R(ω) =√

2π

K−1∑k=0

akP120 T0 exp

(−ω2T 2

0

2− iωkT + iΦk

)

+ iNγ√

2πT 20 exp

(−3ω2T 2

0

2

)K−1∑l=0

K−1∑m=0

K−1∑n=0

ala∗man

P320 exp[−iω(l − m + n)T + i(Φl − Φm + Φn)]

∫ L

0

exp{−αz + [2ωT 2

0 +i(l−m)T ][2ωT 20 +i(n−m)T ]

3T 20 +iβ2z

}√

3T 20 + β2

2z2

T 20

− i2β2z

exp

⎡⎣− (l − n)2T 2

3T 20 + β2

2z2

T 20

− i2β2z

⎤⎦ dz (8)

Taking the inverse Fourier transform of (8) yields the totaloutput field in the time domain

r(t) =K−1∑k=0

akP120 exp

[− (t − kT )2

2T 20

+ iΦk

]+ iNγ

K−1∑l=0

K−1∑m=0

K−1∑n=0

ala∗manP

320 exp[i(Φl − Φm + Φn)]

exp

(− t2NL

6T 20

)∫ L

0

exp[−αz−K2(z, l, n)]

exp

⎧⎨⎩−3{ 2tNL3 +(l−m)T}{ 2tNL

3 +(n−m)T}T 2

0

[1+

i3β2z

T20

]⎫⎬⎭

K1(z)dz

(9)

wheretNL = t − (l − m + n)T (10)

K1(z) =

√1 +

i2β2z

T 20

+3β2

2z2

T 40

(11)

K2(z, l, n) =(l − n)2T 2

T 20

[1 + i2β2z

T 20

+ 3β22z2

T 40

] (12)

If both the frequency chirp and the attenuation are ignored, anda single span is considered, (9) reduces to the form obtainedby the perturbation approach [7] [11].

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

The resonance location of nonlinear effects for any indextriplet [lmn] is (l − m + n)T . For a 3-pulse case, there are33 = 27 index triplets [lmn] and 7 possible locations because−2 ≤ (l−m+n) ≤ 4. For example, the output field at t = Tis affected by 7 triplets of [lmn]. The triplet [111] contributesto the SPM effect. The IXPM effects are caused by [001],[221], [100], and [122], whereas the triplets for IFWM effectsare [012], and [210]. The type and location of the nonlineareffects generated by a pulse triple located at 0, T and 2T aresummarized in Table I.

TABLE IINDEX TRIPLETS [LMN] FOR A TRIPLE PULSE CASE

Nonlinearity Location q-2 -1 0 1 2 3 4

SPM 000 111 222IXPM 011 001 002

022 221 112110 100 200220 122 211

IFWM 020 010 121 012 101 102 202021 210 201120 212

Rather than the total output field in (9), we are more inter-ested in the sampled output field at discrete times tq = qT ,where q takes on integer values in the range [min(l−m+n),max(l − m + n)]. To characterize the individual impact ofvarious impairments on the fiber output, we introduce thefollowing definitions:

ISI coefficient (for q �= k):

ρISIq,k = exp

[− (q − k)2T 2

2T 20

](13)

SPM coefficient (for l = m = n):

ρSPM = iγ

∫ L

0

exp(−αz)K1(z)

dz (14)

IXPM coefficient (for l = m �= n or l �= m = n):

ρIXPMl,m,n = iγ

∫ L

0

exp [−αz − K2(z, l, n)]K1(z)

dz (15)

IFWM coefficient (for l �= m �= n or l = n �= m):

ρIFWMl,m,n = iγ

∫ L

0

exp [−αz − K2(z, l, n)]K1(z)

exp

⎧⎪⎨⎪⎩−3(l − m)(n − m)T 2

T 20

[1 + i3β2z

T 20

]⎫⎪⎬⎪⎭ dz (16)

Table II gives the SPM, IXPM and IFWM coefficientscorresponding to the triplets in Table I if α = 0.2 dB/km,β2 = −22 ps2/km, γ = 2 W−1km−1, L = 100 km, T0 = 6.25ps, T = 25 ps and C = 0.

With the above four definitions, the output field at time tqis obtained as follows:

r(tq) = aqP120 exp(iΦq) + P

120

K−1∑k=0;k �=q

akρISIq,k exp(iΦk)

+ N |aq|2aqP320 ρSPM exp(iΦq)

+ NP320

∑l=m�=n;l �=m=n

ala∗manρIXPM

l,m,n

exp[i(Φl − Φm + Φn)]

+ NP320

∑l �=m�=n;l=n�=m

ala∗manρIFWM

l,m,n

exp[i(Φl − Φm + Φn)] (17)

This is our proposed discrete-time model of single channelmultipulse multispan systems. It is a polynomial model withwhich we can easily investigate not only the impact of variousintrachannel effects on the fiber output, but also the effectsof frequency chirp, pulse width, data rate, attenuation andchromatic dispersion.

To validate the model’s accuracy, consider the transmissionof a 40-Gb/s signal over a 16-span standard SMF. We usethe normalized squared deviation (NSD) as a measure of thedifference between the output fields calculated by our modeland the SSF method. The NSD between the output fields afterN spans is defined as

NSD(N) =

∫KT

0|rModel(t) − rSSF (t)|2dt∫KT

0|rSSF (t)|2dt

(18)

The NSD between the output fields obtained by the proposedmodel and the SSF method for two modulation formats andthree power levels are plotted in Fig. 2. The output calculatedby the model is in excellent agreement with that from the SSFmethod, even for high launched power levels.

2 4 6 8 10 12 14 16

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

x 10−4

Number of Spans

Nor

mal

ized

Squ

ared

Dev

iatio

n

P0=1 mW; OOKP0=3 mW; OOKP0=10 mW; OOKP0=1 mW; DBPSKP0=3 mW; DBPSKP0=10 mW; DBPSK

Fig. 2. Normalized squared deviation of the output fields between ourproposed model and the SSF simulation

To convert the received signal r(t) back into electrical formand recover the data transmitted through the system, r(t) is

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

TABLE IISPM, IXPM AND IFWM COEFFICIENTS

Coefficients Location q-2 -1 0 1 2 3 4

ρSPM -0.901+6.316i -0.901+6.316i -0.901+6.316iρIXPM

l,m,n 0.062+2.263i 0.062+2.263i 0.035+1.272i0.035+1.272i 0.062+2.263i 0.062+2.263i0.062+2.263i 0.062+2.263i 0.035+1.272i0.035+1.272i 0.062+2.263i 0.062+2.263i

ρIFWMl,m,n -0.042-0.088i 0.640-0.233i 0.640-0.233i -0.651-0.197i 0.640-0.233i 0.154-0.251i -0.042-0.088i

0.154-0.251i -0.651-0.197i 0.154-0.251i0.154-0.251i 0.640-0.233i

passed through a square-law photodetector and a sampler.The resulting sampled output varies for different modula-tion schemes: for OOK, it is defined as y(tq) = |r(tq)|2;

for DBPSK, it is defined as y(tq) =∣∣∣ r(tq)+r(tq−1)

2

∣∣∣2 −∣∣∣ r(tq)−r(tq−1)2

∣∣∣2 if the photodetector is balanced1.The resulting sampled output for OOK is

y(tq) = |aq|2P0 + P0

∣∣∣∣∣∣K−1∑

k=0;k �=q

akρISIq,k exp(iΦk)

∣∣∣∣∣∣2

+ 2P0Re

⎧⎨⎩aq

K−1∑k=0;k �=q

ak(ρISIq,k )∗ exp[i(Φq − Φk)]

⎫⎬⎭+ 2NP 2

0 Re{aq(ρSPM )∗

}+ N2|aq|6P 3

0

∣∣ρSPM∣∣2

+ 2NP 20 Re{aq

∑ala

∗man(ρIXPM

l,m,n )∗K3(l,m, n)}

+ N2P 60

∣∣∣∑ ala∗manρIXPM

l,m,n

∣∣∣2+ 2NP 2

0 Re{aq

∑ala

∗man(ρIFWM

l,m,n )∗K3(l,m, n)}

+ N2P 60

∣∣∣∑ ala∗manρIFWM

l,m,n

∣∣∣2 (19)

where

K3(l,m, n) = exp

{− [q − (l − m + n)]2T 2

6(T 20 )∗

}exp{i[Φq − (Φl − Φm + Φn)]} (20)

Due to space limitation, the resulting sampled output forDBPSK is not presented. From (19), we can see that thesampled output consists of nine terms: the first term representsthe contribution of the qth transmitted bit; the eight other termsrepresent various intrachannel interferences. Their summationis defined as intrachannel interference (ICI) exerted by thesequence of bits {bk} on the qth transmitted bit, denoted asICI{bk},q, which varies with different input bit patterns. In thisway, we establish a mapping from the binary input vector tothe sampled output vector. Given a binary input vector, we canuse the model to calculate the corresponding sampled outputwithout the need of time-consuming SSF simulation. TableIII shows a mapping calculated from our model and obtained

1Without loss of generality, the responsivity is taken to be unity.

from the SSF simulation for a 3-bit case when the launchedpower is 3 mW.

TABLE IIIMAPPING FROM BINARY INPUT TO SAMPLED PHOTODETECTOR OUTPUT

Input (bit) Model Output (mW) SSF Output (mW)0 0 0 0 0 0 0 0 00 0 1 0.0001 0.0001 2.8395 0.0001 0.0001 2.83850 1 0 0.0001 2.8395 0.0001 0.0001 2.8385 0.00010 1 1 0.0019 2.8364 2.8364 0.0019 2.8364 2.83641 0 0 2.8395 0.0001 0.0001 2.8385 0.0001 0.00011 0 1 2.8504 0.0001 2.8504 2.8487 0.0001 2.84871 1 0 2.8364 2.8364 0.0019 2.8364 2.8364 0.00191 1 1 2.9443 2.6185 2.9443 2.9435 2.6193 2.9435

IV. EXAMPLE APPLICATION: CONSTRAINED CODING

In this section, we present an example of how the proposedmodel can be used to improve the performance of fiber-opticcommunication systems.

The presence of various interference terms introduces unde-sirable error in the decision device at the receiver output. Theseinterferences degrade the system performance, even in theabsence of noise, and must be suppressed. Some constrainedcoding schemes have been developed to suppress ghost pulsesthrough disallowing certain bit patterns to minimize the prob-ability of forming ghost pulses and resulted in significant Qfactor improvement: up to 9.75 dB for rate 4/5 constrainedcode and up to 6.73 dB for rate 2/3 constrained code [13][14].

We develop a new constrained coding scheme based on theproposed model. In the model, an ICI value is specified by boththe benchmark bit and the bit pattern. To take into accountall the intrachannel interference exerted on every input bitover all possible bit patterns, we propose to use the valueof ICI{bk} =

∑K−1q=0 |ICI{bk},q| as a metric measuring the

severity of ICI to determine whether a K-bit vector is to beavoided through constrained coding. We rank the bit patternsin order of increasing ICI and choose only the vectors weexpect to give the lowest metric, i.e., the best eye-opening.For example, for a set of 8-bit binary sequences, if we choosea rate 7/8 code, every 7-bit input sequence with bad ICI metricis mapped to one of the 8-bit sequences with good ICI metric,where good is defined as the ones that provide the low ICImetric and bad is defined as the ones that provide the highICI metric.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

A straightforward criterion to measure the effectiveness ofthe constrained coding scheme is the bit error rate (BER),but BER is very difficult to evaluate in practice. There is aparametric relationship: BER= 1

2erfc( Q√2), where erfc is the

complementary error function [15]. In our example, first weuse our model to calculate the average sampled outputs for1 and 0 bits (μ1 and μ0) and the corresponding standarddeviations (σ1 and σ0):

μi = E[y(tq)|aq = i], i = 0, 1 (21)

σ2i = E{[y(tq) − μi]2|aq = i}, i = 0, 1 (22)

Then we use the approximation: Q = μ1−μ0σ1+σ0

to obtain the Qfactor of the system before the constrained coding is applied.Next we rank the bit patterns by the bit pattern-specific ICImetric ICI{bk} and choose a fraction of the bit patterns tobe constrained. Then we obtain the Q factor of the systemafter the constrained coding is applied. The performance of theconstrained code scheme depends on the number of encodedsequences, the modulation format and the power levels. Therelationship between the Q factor and the proportion of thesequences constrained is shown in Fig. 3. Unlike [13] [14],the data rate here is kept fixed at 40 Gb/s, i.e., the codedsymbol rate is larger. For example, when one half of thesequences are constrained, this corresponds to a rate 7/8 codeand the corresponding symbol rate is 40/(7/8) = 45.71 Gb/s.The constrained coding scheme is more effective for higherpower levels because higher ICI always accompanies higherpower levels. Also, better performance is obtained in the caseof DBPSK than that in the case of OOK, as expected.

The strength of this scheme is its flexibility in both thefraction of the sequences constrained and the type of inter-ference to be suppressed. For example, if we are interestedin suppressing ghost pulses only, we just need to modify thedefinition of ICI{bk},q to IFWM{bk},q based on (19).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.58

10

12

14

16

18

20

22

Proportion of Sequences Constrained

Q fa

ctor

(dB

)

OOK; 1 mWOOK; 3 mWOOK; 10 mWDBPSK; 1 mWDBPSK; 3 mWDBPSK; 10 mWOOK; 1 mWOOK; 3 mWOOK; 10 mWDBPSK; 1 mWDBPSK; 3 mWDBPSK; 10 mW

Encoded

Uncoded

Fig. 3. Q factor achieved by our constrained coding scheme

V. CONCLUSION

In this paper we develop a discrete-time input-output modelfor single channel multipulse multispan fiber-optic commu-nications systems based on the VSTF method. This modelhas been shown to be in excellent agreement with the SSFmethod and its effectiveness has been demonstrated by a newconstrained coding scheme to suppress various intrachannelinterferences. We are currently extending this model to thegeneral case, which includes ASE noise, postdetection elec-trical filtering, and multichannel transmission. The ultimatemodel will be a two-dimensional model (time dimension andfrequency dimension) and have the potential to be applied ina variety of applications of signal processing for optical com-munications, such as multichannel equalization for intersym-bol and interchannel interference mitigation, development ofconstrained coding schemes for WDM systems, and multiusercoding and path-diversity for all-optical networks.

ACKNOWLEDGMENT

This work was supported in part by the U.S. NationalScience Foundation (NSF) under grant CCF-0916880.

REFERENCES

[1] A. Sano, H. Masuda, and T. Kobayashi, “69.1-Tb/s (432*171-Gb/s)C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” in Optical FiberCommunication Conference (OFC), Mar 21-25 2010.

[2] P. Bayvel and R. Killey, “Nonlinear optical effects in WDM transmis-sion,” in Optical Fiber Telecommunications IV-B: Systems and Impair-ments, I. P. Kaminow and T. Li, Eds. Academic Press, 2002.

[3] E. Basch, R. Egorov, S. Gringeri, and S. Elby, “Architectural tradeoffsfor reconfigurable dense wavelength-division multiplexing systems,”IEEE J. Sel. Topics Quantum Electron., vol. 12, no. 4, pp. 615–626,Jul-Aug 2006.

[4] A. Gnauck, R. Tkach, A. Chraplyvy, and T. Li, “High-capacity opticaltransmission systems,” J. Lightw. Technol., vol. 26, no. 9, pp. 1032–1045,May 2008.

[5] M. Wu and W. Way, “Fiber nonlinearity limitations in ultra-dense WDMsystems,” J. Lightw. Technol., vol. 22, no. 6, pp. 1483–1498, Jun 2004.

[6] M. Taghavi, G. Papen, and P. Siegel, “On the multiuser capacity ofWDM in a non-linear optical fiber: Coherent communication,” IEEETrans. Inf. Theory, vol. 52, no. 11, pp. 5008–5022, Nov 2006.

[7] R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear trans-mission of high-speed TDM signals: 40 and l60 Gb/s,” in Optical FiberTelecommunications IV-B: Systems and Impairments, I. P. Kaminow andT. Li, Eds. Academic Press, 2002.

[8] K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transferfunction of single-mode fibers,” J. Lightw. Technol., vol. 15, no. 12, pp.2232–2241, Dec 1997.

[9] M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in commu-nication systems with strong dispersion management,” Opt. Lett., vol. 23,no. 21, pp. 1668–1670, Nov 1998.

[10] A. Vannucci, P. Serena, and A. Bononi, “The RP method: a new tool forthe iterative solution of the nonlinear Schrodinger equation,” J. Lightw.Technol., vol. 20, no. 7, pp. 1102–1112, Jul 2002.

[11] G. P. Agrawal, Lightwave Technology: Telecommunication Systems.Wiley-Interscience, 2005.

[12] ——, Nonlinear Fiber Optics, 4th ed. Academic Press, 2006.[13] N. Kashyap, P. Siegel, and A. Vardy, “Coding for the optical channel:

the ghost-pulse constraint,” IEEE Trans. Inf. Theory, vol. 52, no. 1, pp.64–77, Jan 2006.

[14] I. B. Djordjevic and B. Vasic, “Constrained coding techniques forthe suppression of intrachannel nonlinear effects in high-speed opticaltransmission,” J. Lightw. Technol., vol. 24, no. 1, pp. 411–419, Jan 2006.

[15] G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. Wiley-Interscience, 2002.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings