PREPRINT JANUARY 16 2023 1 Data-driven Enhancement of the Time-domain First-order Regular Perturbation Model

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PREPRINT, JANUARY 16, 2023 1

Data-driven Enhancement of the Time-domain

First-order Regular Perturbation Model

Astrid Barreiro

,Student Member, IEEE, Gabriele Liga

,Member, IEEE, and Alex Alvarado

,Senior

Member, IEEE

Abstract—A normalized batch gradient descent optimizer is

proposed to improve the ﬁrst-order regular perturbation coefﬁ-

cients of the Manakov equation, often referred to as kernels. The

optimization is based on the linear parameterization offered by

the ﬁrst-order regular perturbation and targets enhanced low-

complexity models for the ﬁber channel. We demonstrate that the

optimized model outperforms the analytical counterpart where

the kernels are numerically evaluated via their integral form.

The enhanced model provides the same accuracy with a reduced

number of kernels while operating over an extended power range

covering both the nonlinear and highly nonlinear regimes. A

6−7dB gain, depending on the metric used, is obtained with

respect to the conventional ﬁrst-order regular perturbation.

Index Terms—Channel modeling, perturbation methods, ﬁber

nonlinearities, gradient descent.

I. INTRODUCTION

CHANNEL models are the cornerstone in the design of

ﬁber-optic communication systems. Modeling provides

physical insights into the light propagation phenomena and

yields techniques to effectively compensate for nonlinear

interference (NLI), arguably the most signiﬁcant factor limiting

the capacity of long-haul coherent optical communication

systems [2, Sec. 9.1]. A channel model in the form of a

reasonably simple expression that, given the input to the chan-

nel, provides the corresponding output, is essential. Therefore,

research on modeling has been a central topic in ﬁber-optic

communications for many years [3]–[5]1.

The origin of most analytical models for coherent systems is

either the nonlinear Schrödinger (NLS) equation or the Man-

akov equation [2, Sec. 9.1]. These are the equations governing

the signal propagation in ﬁbers, and thus, ﬁnding their solution

is crucial for predicting the NLI. None of these equations

have closed-form solutions for arbitrary transmitted pulses.

However, approximated solutions exist in the framework of

perturbation theory [4], [5], [7]–[11]. Perturbation theory can

be used to develop fairly compact analytical expressions for

A. Barreiro, G. Liga, and A. Alvarado are with the Department of Electrical

Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The

Netherlands. e-mail: a.barreiro.berrio@tue.nl.

Parts of this work have been presented at the IEEE Photonics Conference

IPC Vancouver, Canada, Nov. 2022 [1].

The work of A. Barreiro and A. Alvarado has received funding from

the European Research Council (ERC) under the European Union’s Horizon

2020 research and innovation programme (grant agreement No 757791).

G. Liga gratefully acknowledges the EuroTechPostdoc programme under the

European Union’s Horizon 2020 research and innovation programme (Marie

Skłodowska-Curie grant agreement No. 754462).

1A comprehensive timeline of channel modeling efforts can be found in [6,

Sec. I-A].

computing the NLI under a ﬁrst-order approximation. One of

the most popular approaches uses ﬁrst-order perturbation on

the nonlinear ﬁber coefﬁcient, which, following [12], we refer

to in this paper as FRP.

FRP has been broadly employed in ﬁber-optic transmission

systems, both in the frequency and time domains. In this paper,

we focus on time-domain FRP since it is well suited to design

coded modulation systems tailored to the ﬁber channel. In

addition, time-domain FRP has proved great potential to be

used in the performance assessment of systems operating in

the pseudo linear regime2[4], [5], and for algorithm design in

the context of NLI compensation [13]–[16].

Although widely employed, time-domain FRP has two main

drawbacks. First, as discussed in [13], FRP requires the com-

putation of a generally large number of nonlinear perturbation

coefﬁcients, which are typically referred to as kernels. In

particular, to maintain a certain accuracy, the number of

kernels that need to be computed grows cubically as (2N+1)3,

with Nbeing the channel memory. The effective memory

of the channel increases with increments in bandwidth and

ﬁber length [17, Sec. II-B]. Thus, kernels’ evaluation becomes

computationally demanding as transmission bandwidth and

ﬁber length increase. This ﬁrst drawback can restrict the usage

of FRP to transmission scenarios limited in bandwidth and

distance.

The second drawback of time-domain FRP is the loss of

precision at high powers. Since the nonlinear contribution in

the Manakov/NLSE equation can no longer be considered

a perturbation at high powers, higher-order terms in the

perturbative expansion become signiﬁcant. Such behavior sets

a power threshold up to which FRP can accurately predict the

channel output. This drawback can be avoided by considering

higher order terms in the regular perturbation expansion [8],

[18]. However, such an approach makes the solution analyti-

cally more complex, which renders the FRP solution preferable

in practice despite its reduced accuracy at high powers.

In the context of the design of nonlinearity compensa-

tion/mitigation algorithms, multiple researchers have targeted a

reduction of the FRP computational complexity. For instance,

the pulse shape was designed to simplify the kernels’ compu-

tation in [19], [20]. Other approaches, e.g., [13], [16], reduce

the number of kernels by considering the temporal phase-

matching symmetry. Such symmetry enables the pruning of

coefﬁcients having zero contribution due to the isotropic phase

distribution of the transmitted symbols [21, Sec. VIII]. Other

2A comprehensive summary can be found in [2, Chap. 9.4].

arXiv:2210.05340v2 [eess.SP] 13 Jan 2023

2 PREPRINT, JANUARY 16, 2023

FRP approximation in (3)

MOD

h(t)N

Fiber channel (2)

Ideal

CDC

h(t)

L0, T, 2T,...

ra Q(t, 0) Q(t, L)N

√Es

Fig. 1. System model under consideration in this work. The transmitter uses linear modulation (MOD) to generate the transmit signal using the normalized

pulse shape h(t). The receiver applies ideal chromatic dispersion compensation (CDC), matched ﬁltering (MF), and sampling. The coefﬁcient √Esis used

to tune the launch power. The optical channel considers a ﬁber of length L, and propagation of the transmit signal through the ﬁber is given by (2).

works impose a quantization of the kernels [22]–[24]. The

quantization procedure equates sets of perturbation coefﬁcients

to a single value, leading to a signiﬁcant reduction in the

number of kernels to be computed for the FRP approxi-

mation. More recently, a signiﬁcant number of data-driven

approaches empowered by machine learning algorithms have

been reported to perform the optimization of equivalent FRP

coefﬁcients [25]–[28] to be used in nonlinearity compensation

algorithms.

In this work, we propose a data-driven optimization of FRP-

like kernels to generate an equivalent numerical FRP model

addressing the two main drawbacks of the analytical FRP

model. Our enhanced model provides the same accuracy with

a reduced number of kernels while operating over an extended

power range. The main contributions of our work compared to

previous works are i) the kernels’ optimization in this paper

speciﬁcally targets improved low-complexity models for the

ﬁber channel; ii) the optimization makes use of the linear pa-

rameterization offered by the FRP formalism. Optimizing the

kernels in the FRP formalism reduces the modeling complexity

and provides useful insights into some channel properties such

as its effective memory length. The numerical results show

that optimized kernels yield an effective FRP model that for

the system under consideration extends the validity region of

FRP 6-7 dB above the pseudo-linear threshold, signiﬁcantly

improves the model matching in phase and magnitude to the

true value, and generates a memory reduction at a ﬁxed model

precision that translates into a complexity reduction of the

model computation.

The paper is organized as follows. In Sec. II we introduce

the transmission system model and brieﬂy review the essentials

of the time-domain FRP, also reported in [29]. In Sec. III

we assess the FRP performance that is the baseline for the

subsequent analysis. In Sec. IV we present the essentials of

our optimizer, where some examples are given to illustrate the

vectorization of gradient descent. In Sec. Vwe discuss the per-

formance of the optimized model and discuss the implications

over the validity region of FRP and its complexity. Section VI

is devoted to conclusions.

II. TRANSMISSION SYSTEM MODEL

Throughout this paper, a dual-polarization single-span un-

repeated ﬁber-optic transmission system is considered. The

block diagram in Fig. 1illustrates the system model under

study. As the purpose of this model is to mainly study NLI,

ampliﬁed spontaneous emission (ASE) noise is not taken

into account. Furthermore, we only consider single-channel

transmission for simplicity of illustration of the proposed

enhanced model.

First, a sequence3of two-dimensional complex symbols

a=. . . , an−1,an,an+1, . . . is used to linearly modulate the

energy-normalized real pulse shape h(t), i.e., R∞

−∞ h2(t)dt=

1. The transmit signal Q(t, 0) at location z= 0 is given by

Q(t, 0) = pEs

∞

n=−∞

anh(t−nT ),(1)

where Tis the symbol duration, and Esis the average energy

per transmitted symbol. Assuming the symbols anare taken

form a normalized constellation, the parameter Es=P/2Rs

in (1) deﬁnes the total transmitted optical power, where P

and Rs= 1/T are the launched power and the symbol rate,

respectively.

Each symbol anin the sequence is an= (ax,n, ay,n)T,

where ax,n and ay,n represent the complex symbols mapped

onto two arbitrary orthogonal polarization states xand y.

The noiseless propagation of the two-dimensional complex

envelope Q(z, t)through the ﬁber link is governed by the

Manakov equation [30, eq. (57)]

∂Q

∂z =−β2

∂2Q

∂t +γ 8

9e−αz|Q|2Q,(2)

where for simplicity (z, t)is omitted. In (2), αis the attenu-

ation coefﬁcient, β2the group-velocity dispersion coefﬁcient,

and γthe nonlinear coefﬁcient. The ﬁeld Qis considered to be

attenuation-normalized [31, eq. (3.1.3)]. The ﬁrst contribution

on the RHS of (2) is associated with linear propagation, while

the second is accounting for nonlinear propagation.

At the receiver side in Fig. 1, ideal chromatic dispersion

compensation is performed on the propagated ﬁeld Q(t, L).

The resulting ﬁeld is then matched-ﬁltered and sampled at

the symbol rate. Throughout this work, we assume that the

matched ﬁlter is matched to the transmitted pulse h(t). The

sequence of received symbols r=. . . , rn−1,rn,rn+1, . . . is

3Notation convention: We use boldface letters to denote column vectors,

e.g., u. Underlined bold letters represent inﬁnite sequences of vectors, e.g., u.

|·|denotes absolute value. When |·|is applied to a set, it denotes cardinality.

For any pair of vectors uand v, we use to denote the element-wise division,

i.e., w=uvimplies wi=ui/vi. The operations (·)Tand (·)†are the

transpose and the Hermitian transpose respectively, and calligraphic letters

are used to denote sets. Zdenotes the set of integers, while Cis the set of

complex numbers. Throughout this paper, we often use triple indexation (e.g.,

Gklm), which we sometimes write using separating commas (e.g., Gk,l,m ).

BARREIRO et al.: ON A DATA-DRIVEN ENHANCEMENT FOR THE FIRST-ORDER REGULAR PERTURBATION MODEL 3

obtained by scaling the samples by 1/√Es. Each output sym-

bol rnin the sequence contains two orthogonal polarization

states, namely rn= [rx,n, ry,n]T. Average phase rotations on

the received constellations are in practice compensated by DSP

algorithms. In this paper, we do not consider any additional

DSP step beyond what is included in Fig. (1) (CDC and MF) to

set as a benchmark a system that is energy preserving. Based

on this choice, we are interested in benchmarking the accuracy

of FRP (and enhanced versions thereof) with respect to the

SSFM.

For small enough values of the nonlinear coefﬁcient γ,

FRP approximates the exact solution to the Manakov equation

yielding the following input-output relation in discrete-time

[29, eq. (3)]:

rn≈an+8

9γEsX

(k,l,m)∈Z3a†

n+kan+lan+mSklm.(3)

In (3), Sklm are complex perturbation coefﬁcients that model

self-phase modulation (SPM). They are deﬁned as [29, eq. (4)]

Sklm ,ZL

e−αz Z∞

−∞

h∗(z, t)h∗(z, t −kT )

h(z, t −lT )h(z, t −mT ) dtdz,

(4)

where with a slight abuse of notation, we used h(z, t)to denote

the solution of (2) when γ= 0 and Q(t, 0) = h(t).

In general, (4) could be strictly nonzero for all (k, l, m)∈

Z3(time-unlimited pulses) which would require the compu-

tation of an inﬁnite number of kernels to evaluate (3). How-

ever, in practice, due to the exponential decay of the kernel

magnitude as a function of the 3D index squared magnitude

k2+l2+m2[13, Fig. 5], the sums in (3) can be truncated

with limited loss in accuracy, yielding a ﬁnite memory channel

model. In this work, we consider the following truncation

rn≈an+ ∆an,(5)

where

∆an,8

9γEsX

(k,l,m)∈S a†

n+kan+lan+mSklm,(6)

and

S,{(k, l, m)∈Z3:−M≤k, l, m ≤M}.(7)

Since the model in (5) resembles the heuristic ﬁnite-memory

channel model introduced in [17], we call it ﬁnite-memory

FRP. Accordingly, Mcan be interpreted as the model’s mem-

ory size, and 2M+1 deﬁnes the size of the interfering window.

The symbols within this window are needed to compute the

ﬁnite-memory model’s output (5). Henceforth, we refer to

ﬁnite-memory FRP simply as FRP.

III. ACCURACY AND LIMITATIONS OF FRP

In order to study the limitations of the FRP, we ﬁrst investi-

gate a representative transmission scheme outlined within the

400ZR implementation agreement. In this section, we address

the FRP drawbacks from a nonlinearity modeling perspective

as opposed to focusing on the compensation approach, as

it has been done in for example [32]. To the best of our

TABLE I

FIBER AND PULSE SHAPE PARAMETERS

Nonlinear parameter γ1.2W−1km−1

Fiber attenuation α0.2dB/km

Group velocity dispersion β2−21.7ps2/km

Pulse shape h(t)Root-raised-cosine (RRC)

RRC roll-off factor 0.01

knowledge, a thoughtful performance assessment has not been

previously addressed from a modeling perspective. Hence we

present a study of the loss of accuracy of FRP at high powers

via a precise quantiﬁcation of the discrepancies with reliable

simulations of ﬁber propagation.

The study case considers a L= 120 km standard single-

mode ﬁber span, for a dual-polarization transmission in a

single 60 Gbd channel using 16-QAM. Table Isummarizes

the considered ﬁber parameters. The single-span transmission

system constitutes the building block for the ﬁrst stage of a

multi-span EDFA system. When a multi-span transmission is

considered, we do not expect the results to be qualitatively

different from the ones observed in a single-span scenario.

In a multi-span scenario, the model’s mathematical formalism

remains the same and that second-order effects are mainly

affected by the transmitted power rather than transmission

distance.

Throughout this paper, the standard split-step Fourier

method (SSFM) is used for the simulation of ﬁber propaga-

tion according to (2). The simulations are performed with a

sampling rate equal to four times the transmission bandwidth

and a uniform step size equal to 10 m. Although ASE noise

is neglected in Fig. 1and in (2), in some of the results we

consider an erbium-doped ﬁber ampliﬁer (EDFA). In such

cases, a 5dB noise ﬁgure is assumed. That system is used

to set a baseline to study the performance of FRP around

and beyond the optimum launch power. In addition, a root-

raised-cosine (RRC) is chosen as pulse shape to numerically

calculate the nonlinear coefﬁcients in (4). Table Isummarizes

the considered pulse and parameters.

In what follows, we introduce and discuss the results of

four metrics used in this work to assess the performance of

FRP. The ﬁrst three metrics are average metrics, while the

last one is a point-wise metric. Our general intention in this

section is to characterize the well-known FRP drawbacks to set

a baseline for the analysis of the results following the model

optimization.

A. Signal-to-noise ratio (SNR)

Let Ax/yand Rx/ybe complex random variables corre-

sponding to the x/ycomponent of the transmitted and re-

ceived symbols in Fig. 1, respectively. Throughout this paper

we assume that the transmitted symbols are drawn from a

polarization-multiplexed format, and thus, the 4D complex

symbols are the Cartesian product of a constituent 2D complex

constellation by itself. The support of the random variable Ax/y

is the constellation A ⊂ C, given by |A| constellation points

A,{s1, s2, . . . , s|A|}.

4 PREPRINT, JANUARY 16, 2023

0 5 10 15 20

P∗= 7 dBm 3dB

Input power [dBm]

SNR [dB]

FRP M= 0

FRP M= 3

FRP M= 9

FRP M= 15

SSFM

SSFM+ASE

FRP+ASE M= 15

9 10 11

0.39 dB

Fig. 2. The SNR in (10) as a function of launch power Pwith (“+ASE”)

and without ASE noise. Increments in memory size close the gap between

SSFM and FRP in the absence of ASE noise for powers up to 10 dBm. The

optimum launch power for the “+ASE” case P∗= 7 dBm is shown. The

gray area delimits the region covered for M∈[0,15]. The FRP prediction

(with and without ASE) starts to fail at about 3dBm above P∗.

Let µ∈Cand σ2∈Rbe two functions of the random vari-

able Ax/yrepresenting the conditional mean and conditional

variance of the general constellation point Ax/y, respectively.

These quantities are deﬁned as

µ(Ax/y),E{Rx/y|Ax/y},(8)

σ2(Ax/y),E{|Rx/y−µ(Ax/y)|2|Ax/y}.(9)

We assume that Ax/yis zero mean, and unit energy

(E{|Ax/y|2}= 1). Additionally, Axand Ayare assumed

independent. For reasons that will become clear in Sec. III-B,

it is assumed that Adoes not include 0 + 0and that it is a

constellation with more than one symbol per ring.

Using (8) and (9), we deﬁne the signal-to-noise ratio (SNR)

as the average SNRs across the two polarizations, i.e.,

SNR ,1

2E{|µ(Ax)|2}

E{σ2(Ax)}+E{|µ(Ay)|2}

E{σ2(Ay)}.(10)

Fig. 2shows the SNR in (10) obtained using SSFM and FRP

for different input powers Pand three model-memory sizes.

The gray area depicts the region covered by FRP with memory

size M∈[0,15]. Fig. 2also includes results with ASE noise,

for which P∗= 7 dBm is found to be the optimum launch

power.

Fig. 2shows that already for M > 1, the SNR prediction of

FRP matches SSFM within 0.5 dB (0.39 dB) in the linear and

pseudo-linear regimes. This good ﬁt is lost when P > 10 dBm,

a region where the nonlinear distortions are large. For powers

above 10 dBm, the SNR curves of FRP and SSFM begin to

diverge. Therefore FRP’s accuracy extends up to 3dB above

the optimum launch power. The divergence observed above

10 dBm implies that FRP is (a) underestimating the NLI, (b)

it is making an inaccurate prediction of the conditional means,

or (c) is doing both, (a) and (b), simultaneously. In the highly

nonlinear regime (i.e., P > 15 dBm), the FRP model shows a

saturation trend that is as more visible as Mincreases. This

saturating behavior will be discussed in Sec. III-B.

To understand the divergent curves in Fig. 2, a more

qualitative comparison between FRP and SSFM is displayed

in Fig. 3. Three scenarios are considered: 10 dBm for M= 5

(a), 13 dBm for M= 5 and M= 15, (b) and (c) respectively.

In Fig. 3(a), a mismatch between the FRP (purple) and SSFM

(red) is already noticeable even though the memory size con-

sidered (M= 5) is relatively large. The constellation clouds,

in this case, appear to have still similar average variance, but

in the SSFM case show an extra phase rotation not accounted

for by FRP. In Fig. 3(b) we show the comparison again

for M= 5 but at P= 13 dBm (6dB above P∗). The

mismatch between the constellations in Fig. 3(b) worsens.

We note that the FRP clouds here are not simply rotated

compared to SSFM, but also scaled (up). In Fig. 3(b) we

illustrate the rings where the conditional means of SSFM and

FRP fall, and it is visible they do not overlap. Notice that the

rings do not match and that the FRP ring has a radius larger

than SSFM. Therefore, we conclude that FRP is making an

inaccurate prediction of conditional means. Lastly, Fig. 3(c)

shows results at P= 13 dBm when the model’s memory

size is increased from M= 5 to M= 15. Despite the

increment in memory size, the mismatch persists and the radii

discrepancy between SSFM and FRP enlarges (a characteristic

further explored in Sec. III-B). This shows that the model’s

proximity to SSFM quickly saturates as a function of memory

size. We further investigate the constellation mismatch based

on the three metrics we introduce.

B. Radii and Phase difference

To better assess the observed mismatch between the received

constellations in Fig. 3and understand the reason for the SNR

prediction mismatch shown in Fig. 2, we consider two metrics

that quantify how different the conditional means (µ(s)in

(8)) of SSFM and FRP are with respect to the transmitted

constellation points (A). The ﬁrst metric we introduce is the

normalized radii difference, deﬁned as

∆r,1

2E|µ(Ax)|−|Ax|

|Ax|+E|µ(Ay)|−|Ay|

|Ay|.

(11)

The normalized radii difference is such that −1≤∆r < ∞.

Three cases are of interest. When ∆r= 0, the condi-

tional means perfectly match the transmitted symbols. When

∆r < 0, the magnitude the of conditional means of the

received constellation are on average smaller with respect

to the transmitted symbols. In other words, the constellation

is “compressed”. Conversely, when ∆r≥0, the received

constellation experiences an expansion.

Secondly, we compare the average phase rotation expe-

rienced by the conditional means of SSFM and FRP with

respect to transmitted constellation points. The average phase

difference is deﬁned as

∆φ,1

2E∠µ(Ax)−∠Ax

ϕ(Ax)+E∠µ(Ay)−∠Ay

ϕ(Ay),

(12)

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PREPRINT,JANUARY16,20231Data-drivenEnhancementoftheTime-domainFirst-orderRegularPerturbationModelAstridBarreiro,StudentMember,IEEE,GabrieleLiga,Member,IEEE,andAlexAlvarado,SeniorMember,IEEEAbstractAnormalizedbatchgradientdescentoptimizerisproposedtoimprovetherst-orderregularperturbationcoef-cient...

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