On the accuracy of the GN-model and on analytical correction terms to improve it

1

Abstract — The GN-model has been proposed to provide an

approximate but sufficiently accurate tool for predicting

uncompensated optical coherent transmission system

performance, in realistic scenarios. For this specific use, the

GN-model has enjoyed substantial validation, both simulative and

experimental. Recently, however, it has been pointed out that its

predictions, when used to obtain a detailed physical picture of

non-linear noise accumulation along a link, may be affected by

substantial error. In addition, it has been pointed out that part of

the non-linear interference (NLI) noise variance that it predicts

may be ascribed to long-correlation phase noise rather than

quasi-additive pseudo-white Gaussian noise.

In this paper we analyze in detail the GN-model errors and the

analytical correction terms that have been proposed to remove

them. We extend such analytical results to cover single-channel

non-linearity as well, and provide integral formulas for both

single and cross-channel effects. We also carry out a simulative

in-depth characterization of phase noise in realistic links.

Our findings show that, when used for its original purpose of

predicting system maximum reach and optimum launch power,

the GN-model provides rather reliable results, with 0.3 to 0.6 dB

error, always biased vs. being conservative. We also confirm prior

findings that the GN-model may substantially overestimate NLI

noise when used instead to characterize NLI noise accumulation

along a link, especially in the first spans of the link. We show that

previously proposed formulas for correcting the GN-model

tendency to overestimate NLI may actually substantially

underestimate it. We show why this happens and provide a new

overall self-consistent analytical model, that we call the enhanced

GN-model (EGN model), which completely corrects for these

problems. We extensively validate it simulatively, and discuss its

computational complexity. Finally, we show that long-correlation

phase noise is indeed present in actual links, but its impact on

performance is typically minimal for realistic system parameters.

Index Terms — optical transmission, coherent systems, GN

model

A. Carena, G. Bosco, V. Curri, P. Poggiolini, and Y. Jiang are with

Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Corso

Duca degli Abruzzi 24, 10129, Torino, Italy, e-mail: [email protected]; F. Forghieri is with CISCO Photonics, Via

Philips 12, 20052, Monza, Milano, Italy, [email protected]. This work was

supported by CISCO Systems within a sponsored research agreement (SRA)

contract.

I. INTRODUCTION

The GN-model of non-linear fiber propagation has been

recently proposed, based upon results from several prior

modeling efforts. For an extensive bibliography and a

comprehensive description of the underlying approximations

see [1].

Since the start, the GN-model main purpose has declaredly

been that of providing a simple but sufficiently accurate tool for

the prediction of the main system performance indicators in

uncompensated links that use coherent detection. Typical such

indicators are maximum reach and optimum launch power. For

this specific use, the GN-model has obtained substantial

validation, both simulative and experimental, by various

independent groups.

Recently, however, it has been pointed out that when the

GN-model is used to look at detailed span-by-span

characterization of non-linear interference (NLI) accumulation

along a link, its predictions may be affected by a substantial

error [2]-[5]. In particular in [2], the first peer-reviewed

published paper on the subject (simultaneously with [3]), we

presented for the first time a detailed picture of the predicted

and actual NLI noise variance accumulated along realistic links

based on PM-QPSK and PM-16QAM. We showed that the

GN-model overestimates the variance of NLI, most notably in

the first spans of the link, where this error may amount to

several dB's, depending on system parameters and modulation

format. We showed this error to be related to one of the

GN-model main approximations: the `signal Gaussianity'

assumption, which consists in assuming that the transmitted

signal, due to uncompensated dispersion, approximately

behaves as Gaussian noise. Especially in the first spans of the

link, this approximation is not accurate and generates

substantial error.

On the other hand, in [2] we also showed that such error abates

considerably along the link, so that the resulting inaccuracy in

the prediction of the maximum reach of a typical PM-QPSK or

PM-16QAM link is on the order of 0.3-0.6 dB for the coherent

GN-model and close to zero for the even simpler incoherent

On the Accuracy of the GN-Model and on

Analytical Correction Terms to Improve It

A. Carena, G. Bosco, V. Curri, P. Poggiolini, Y. Jiang, F. Forghieri

2

GN-model (for a discussion of these two versions of the model

see [1]).

Independently1 of [2], another paper [4] also recently focused

on the issue of the GN-model accuracy. Remarkably, [4]

succeeded in analytically removing the signal Gaussianity

assumption. A `correction term' to the conventional coherent

GN-model, limited to XPM2, was found. The results of [4]

constitute major progress, also because they showed that

removing the signal Gaussianity assumption does not lead to

unmanageably complex calculations, as we previously

believed3.

In this paper we adopt a similar approach to that indicated in [4]

and derive for the first time the GN-model ‘correction terms’

for single-channel non linearity (that is, self-channel

interference or SCI), which was not addressed in [4]. We also

provide explicitly dual-polarization integral expressions for the

overall power spectral density (PSD) of NLI noise due to both

SCI and XCI (cross-channel interference). We also discuss the

impact of MCI (multi-channel interference) and show it to be

non-negligible in certain important scenarios, namely with

low-dispersion fibers, such as TrueWave RS or LS. We provide

the formulas needed to account for the MCI contributions as

well. Overall, we supply a complete set of equations that fully

correct the GN-model for the effect of signal non-Gaussianity.

We call this overall analytical set the enhanced GN model

(EGN model).

We study the EGN model in realistic system scenarios and

compare it with simulation results and with those of [4]. We

find that the formulas proposed in [4] may substantially

underestimate the XCI due to the channels directly adjacent to

the one of interest. They also completely neglect MCI which

may be significant, especially with low-dispersion fibers.

Another aspect addressed in [4] and, by the same group, in [5],

is the approximation used by the GN-model, as well as by most

prior models, according to which NLI noise is Gaussian and

additive, so that its system impact can be assessed simply by

1 Even though [4] has a later publication date than [2], in [4] the authors

claim to be the first to point out the inaccuracy problems of the GN-model. This

is because the authors of [4] were unaware of [2], as it has been later found out

through private communications between the two groups. 2 In [4] the traditional taxonomy of non-linear effects is used. Here we prefer

to use a taxonomy that more naturally relates to the GN-model (see [6]), which

divides NLI into SCI, XCI and MCI. The definition of these NLI contributions is recalled in Sect. II. The relation between XPM in [4] and our taxonomy is

also dealt there. While XPM roughly coincides with XCI, it neglects

contributions that may be substantial. This aspect is studied in Sect. II. 3 A survey of the procedure used for the GN model derivation showed us

that accounting for signal non-Gaussianity would generate final model

formulas containing triple and quadruple integrals over the WDM frequency range, whereas in the standard coherent GN model only a double integral is

present. We reckoned that such level of complexity would make both the

analytical and the numerical evaluation of the model too complex. Therefore, we refrained from undertaking this path. In this respect, [4] has marked

substantial progress, as it showed that this approach is doable.

summing its variance to that of ASE noise. The claim in [4] is

that a very substantial part of the XCI contribution to NLI is in

fact phase noise and hence non-additive. In addition, such

phase noise would appear to show a very long correlation time,

on the order of tens or even hundreds of symbols.

The presence of a non-linear noise component with very long

correlation time had first been pointed out in [7], there too

attributed to `cross-phase modulation'. The correlation results

in [4] actually agree well with those found earlier in [7]. Both

papers, however, concentrate on a single-polarization, lossless

fiber scenario to assess the strength of the long-correlated

phase-noise component of NLI. In that idealized context, the

phase noise component indeed turns out to be very large. In this

paper, we provide a detailed analysis of the phase-noise

component of NLI in realistic system scenarios, both due to

XCI and, studied for the first time, due to SCI. We show them

to be substantial in the first link spans, but we find their impact

to decrease along the link, so that their effect on maximum

reach estimation is small or negligible, depending on link and

fiber parameters.

Finally, we devote a section of this paper to the discussion of

our results and those of [4] in relation to the estimation of

realistic system maximum reach and optimum launch power,

also in comparison with the standard coherent and incoherent

GN model. Our bottom-line findings are that, when used for its

original purpose of predicting PM-QAM systems maximum

performance, the GN-model error is always conservative and

its error is typically quite contained. Its simpler `incoherent'

version possibly represents the best current compromise

between accuracy and complexity, providing rather accurate

maximum reach predictions in most practical scenarios with

minimal computational effort. However, if very accurate

system performance prediction is critical or when a

span-by-span detailed picture of NLI is of interest, then the GN

model must be supplemented by correction terms, i.e., the EGN

model must be used.

To summarize, this paper aims at taking a comprehensive look

at the recent findings about the accuracy of the GN-model. It

provides further substantial contributions, both in terms of

theoretical and analytical advancement, as well as in terms of

validating and putting in realistic context the available results.

We first retrace the breakthrough analytical findings of [4], [5]

and provide a further correction for XCI which improves on the

accuracy of [4]. We derive here for the first time the relevant

correction for SCI. We show that MCI may be significant too

and provide the corresponding analytical formulas, thus

completing a new comprehensive model that we call the EGN

model. We validate these formulas vs. simulations and discuss

their computational complexity. We address the issue of

long-correlation phase noise through a detailed simulative

study. We then discuss the different model versions in a

realistic system performance prediction scenario, providing a

comprehensive set of guidelines and recommendations.

3

II. THE EGN MODEL

When removing the signal Gaussianity assumption, the power

spectral density (PSD) of NLI turns out to be expressed by

various contributions, which are then summed together. One is

the standard coherent GN-model, the others can be thought of

as correction terms which take the effect of signal

non-Gaussianity into account. In the following, we call the

corrected model as the `enhanced GN model’, or EGN-model.

In this section we orderly present the EGN model formulas

according to the type of NLI that they address, namely SCI,

XCI and MCI. The reason for resorting to this subdivision is the

fact that the different contributions have rather different

features. Before proceeding, we recall the definition of the three

NLI types, for the readers’ convenience.

The NLI impinging on the channel-under-test (CUT) of a

WDM comb is the sum of three types of NLI contributions:

Self-channel interference (SCI): it is NLI caused by

the CUT on itself.

Cross-channel interference (XCI): it is NLI affecting

the CUT caused by each single interfering channel

(INT)

Multiple-channel interference (MCI): it is NLI

affecting the CUT caused by the non-linear beating of

any two or any three INT channels

A more precise definition and more details can be found4 in [6].

In the following, we assume a multi-span link, with lumped

amplification and all identical spans. We also assume that

channels have rectangular spectra with same bandwidth, equal

to the symbol rate sR . These limiting assumptions could be

removed but are kept here to avoid excessive complexity in the

resulting formulas. We assume dual polarization throughout.

The main symbols used here are, with units that make the

subsequent formulas self-consistent:

z : the longitudinal spatial coordinate, along the link

[km]

: optical field fiber loss [1/km], such that the optical

field attenuates as ze

; note that the optical power

attenuates as 2 ze

2 : dispersion coefficient in [ps2/km]

: fiber non-linearity coefficient [1/(W km)]

sL : span length [km]

2

eff 1 2sLL e

: span effective length [km]

4 In [6] a different but equivalent definition of the types of NLI can be found,

based on where the actual signal frequency components are located, whose

non-linear beat affects the CUT.

sN : total number of spans in a link

sR : symbol rate of an individual channel [TBaud]

sT : duration of a symbol, equal to 1

sR [ps]

CUT

s t , INT

s t the pulses used by either the CUT or

the INT channels, respectively

CUT

s f , INT

s f Fourier transforms of the above,

assumed rectangular with bandwidth sR and flat-top

value equal to 1/ sR , centered at f =0 for the CUT

and at f = cf for an INT channel

cf the center frequency of the INT channel, such that

the CUT and INT channel do not overlap [THz]

xa ,ya , xb ,

yb : random variables (RVs), representing

the generic symbols transmitted in either the CUT (

‘ a ’ RVs), or the INT channels (‘ b ’ RVs),

respectively, on either polarization (subscripts x and

y )

According to the above definitions, the CUT overall

transmitted signal can be written as:

CUCUT T, ,

ˆ ˆ( ) x n y n s

n

S t a x a y s t nT

and similarly for the INT channel, with ‘ b ’ RV’s in the

formula. The average transmitted power in the CUT and INT

channels are then given by:

2 22 2

CUT INTE , Ex y x yP a a P b b

A. Self-Channel Interference (SCI)

The NLI produced by a CUT on itself is SCI. Its contribution

can be rather substantial. In a densely packed, full C-band

system, operating at 32 GBaud, it may approximately range

between 25% to 35% of the total NLI power perturbing the

CUT.

In [4] SCI was not dealt with and all calculations/simulations

assumed that SCI was removed. In theory, removing SCI may

be possible using electronic non-linear-compensation (NLC).

While NLC is a fervid field of investigation, so far it is unclear

whether NLC can be effectively implemented in DSP. At

present, there are no commercial products incorporating it. It

therefore seems quite necessary to consider SCI as well, in

dealing with a GN-model upgrade.

To derive the SCI formulas we used an approach similar to [4],

suitably taking into account the effect of the non-Gaussianity of

the signal. Derivations are shown in Appendix A. The NLI

power spectral density (PSD) emerging at a generic frequency

f within the CUT, due to the interference of a single other

INT channel, in dual polarization, is given by:

4

SCISCI

3

1 2 3( ) ( ) ( ) ( )a aG f P f f f

Eq. 1 where:

4

222a

a

a

6 4

2 23 29 12

a

a a

a a

CUT CUT CUT

/2 /2

2 3

1 1 2

/2 /2

2 2 2

1 2 1 2

2 2

1 2 1 2

16( )

27

( ) ( ) ( )

, , , ,

s s

s s

R R

s

R R

f R df df

s f s f s f f f

f f f f f f

CUT CUT CUT CUT

CUT

/2 /2 /2

2 2

2 1 2 3

/2 /2 /2

2

1 2 3 1 2

1 3 1 2 1 3

1 2 1 3

1 2 1 3

/2 /2

2 2

1 2

/2 /2

80( )

81

( ) ( ) ( ) ( )

( ) , , , ,

, , , ,

, , , ,

16

81

s s s

s s s

s s

s s

R R R

s

R R R

R R

s

R R R

f R df df df

s f s f s f s f f f

s f f f f f f f f f

f f f f f f

f f f f f f

R df df

CUT

CUT CUT CUT CUT

/2

3

/2

2

1 2

1 2 1 2 3 3

1 2 1 2 3 3

1 2 1 2 3 3

1 2 1 2 3 3

( )

( ) ( ) ( ) ( )

, , , ,

, , , ,

, , , ,

s

s

R

df

s f f f

s f s f s f f f s f

f f f f f f f f

f f f f f f f f

f f f f f f f f

CUT CUT CUT CUT CUT

CUT

/2 /2 /2 /2

2

3 1 2 3 4

/2 /2 /2 /2

1 2 1 2 3 4

3 4 1 2 3 4

1 2 3 4 1 2 3 4

16( )

81

( ) ( ) ( ) ( ) ( )

( ) , , , ,

, , , , , , , ,

s s s s

s s s s

R R R R

s

R R R R

f R df df df df

s f s f s f f f s f s f

s f f f f f f f f f

f f f f f f f f f f f f

Assuming lumped amplification, the factors , and are

defined as:

2

2 1 22 4 ( )( )

1 2 2

2 1 2

1, ,

2 4 ( )( )

s sL j f f f f L

e ef f f

j f f f f

2

2 1 2

1 2 2

2 1 2

sin 2 ( )( ), ,

sin 2 ( )( )

s s

s

f f f f N Lf f f

f f f f L

2

2 1 22 ( )( )( 1)

1 2, ,

s sj f f f f N Lf f f e

The term related to 1( )f is exactly the standard (coherent)

GN model. The other two terms are corrections that take signal

non-Gaussianity into account. Note the need to include both a

4th

and a 6th

-order moment of the transmitted symbol sequence,

through the coefficient a . Note also the quite substantial

complexity of the overall formula vs. the standard GN-model

term 1( )f alone.

In Fig. 1 and Fig. 2 we show the result of the SCI calculation vs.

simulation, for the system described in the previous section,

with SMF and NZDSF, respectively. We looked at the NLI

normalized average power SCI

defined as follows:

SCI CUT SCI

/2

3

/2

s

s

R

R

P G f

Eq. 2

This parameter collects the total SCI noise spectrally located

over the CUT, normalized through CUT

3P so that

SCI itself does

not depend on launch power. The system data are as follows:

single channel PM-QPSK at sR =32 GBaud

roll-off 0.02

SMF fiber with D =16.7 [ps/(nm km)], =1.3 [1/(W

km)], dB =0.22 dB/km

NZDSF fiber with D =3.8 [ps/(nm km)], =1.5

[1/(W km)], dB =0.22 dB/km

span length sL =100 [km]

Note that we chose not to use ideally rectangular spectra, to

avoid possible numerical problems due to the truncation of

excessively long, slowly decaying signal pulses. On the other

hand, the very small value of roll-off employed has a negligible

effect on non-linearity generation.

The plot shows that Eq. 1 has excellent accuracy, as soon as

there is some accumulated dispersion. The apparent gap

between analytical and simulative results in the first few spans

is currently being investigated. Outside of the first few spans,

the agreement is excellent.

Note also that the difference between simulation (or the EGN

model) and the standard coherent GN-model tends to close up

5

for large number of spans, with only about 2.1 dB residual gap

at 50 spans for NZDSF and only 1.1 dB for SMF.

Fig. 1 : Plot of normalized Self-Channel Interference (SCI),

SCI , vs.

number of spans in the link, assuming a single PM-QPSK channel over

SMF, with span length 100 km. Red dashed line: simulation. Blue solid

line: standard coherent GN model. Green solid line: EGN model Eq. 1).

Fig. 2 : Plot of normalized Self-Channel Interference (SCI),

SCI , vs.

number of spans in the link, assuming a single PM-QPSK channel over

NZDSF, with span length 100 km. Red dashed line: simulation. Blue solid

line: standard coherent GN model. Green solid line: EGN model Eq. 1).

B. Cross-Channel Interference (XCI)

A key aspect of XCI is that the contributions of each single

INT channel in the WDM comb simply add up. As a result, one

can concentrate on analytically finding the XCI due to a single

INT channel, then the total XCI will be the sum of formally

identical contributions.

1) The XPM approximation provided in [4]

We started out from the formula provided in [4] in summation

form, which the authors define as ‘XPM’. We re-wrote it in

integral dual-polarization form and in such a way as to make it

represent the NLI power spectral density (PSD) emerging at a

generic frequency f within the CUT. It is:

CUT INTXPM

2

11 12( ) ( ) ( )bG f P P f f

Eq. 3

where:

4

222b

b

b

CUT INT INT

/2 /2

2 3

11 1 2

/2 /2

2 2 2

1 2 1 2

2 2

1 2 1 2

32( )

27

( ) ( ) ( )

, , , ,

s c s

s c s

R f R

s

R f R

f R df df

s f s f s f f f

f f f f f f

CUT INT INT

INT INT

/2 /2 /2

2 2

12 1 2 3

/2 /2 /2

2

1 2 3

1 2 1 3

1 2 1 3

1 2 1 3

1 2 1 3

80( )

81

( ) ( ) ( )

( ) ( )

, , , ,

, , , ,

, , , ,

s c s c s

s c s c s

R f R f R

s

R f R f R

f R df df df

s f s f s f

s f f f s f f f

f f f f f f

f f f f f f

f f f f f f

As argued in [4], the 11( )f corresponds to a GN-model-like

contribution, whereas 12( )f represents a correction that takes

into account the non-Gaussianity of the transmitted signal. As

said, these formulas account for a single INT channel.

Considering a WDM system, the same calculations shown

above must be repeated for each INT channel and the results

summed together.

Note that in [4] XPM is not proposed as a partial contribution to

NLI, but as an overall NLI estimator, accurate enough to

represent the whole non-linearity affecting the CUT (excluding

SCI). In other words, it is proposed as a more accurate overall

estimator of NLI than the conventional coherent GN-model

(again, excluding SCI). In the next subsection we will discuss

this claim.

2) The overall XCI

Eq. 3, derived from [4], neglects various XCI contributions

arising when the INT channel is directly adjacent to the CUT.

To provide a graphical intuitive description of what was left out

in [4], in Fig. 3 we show a plot of the 1 2,f f domain where

integration occurs for the 11( )f contribution. As pointed out

in [6], each point of this 1 2,f f plane represents a triple of

frequencies, namely 1 2 3, ,f f f , with 3 1 2f f f , that

6

produce a FWM beat at frequency f , contributing to NLI

there. The example shown in Fig. 3 refers to NLI forming at

0f , that is at the center of the CUT. It considers XCI due to

a single INT channel adjacent to the CUT, placed at higher

frequency than the CUT.

The formulas reported in [4] and hence Eq. 3 take into account

the two D1 domains only. They neglect D2, D3 and D4. For

each one of these further regions there exist a GN-model-like

contribution and also one or more related correction terms that

take signal non-Gaussianity into account.

Fig. 3: Integration regions to obtain the power spectrum of XCI ,

XCI

( )G f ,

at 0f (i.e., at the center of CUT), due to a single adjacent INT channel,

whose center frequency is slightly higher than the symbol rate. The XPM

approximation of Eq. 3 considers the D1 regions only. The full XCI

formula of Eq. 4 accounts for all D1-D4 regions.

The complete resulting XCI formula is:

CUT INT

CUT INT

CUT INT

INT

XCI

2

11 12

2

21 22

2

31 32

3

41 42 43

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

b

a

a

b b

G f P P f f

P P f f

P P f f

P f f f

Eq. 4

where:

4 4

2 22 22 , 2

a b

a b

a b

and

6 4

2 23 29 12b

b b

b b

The index m refers to the domain number according to Fig. 3.

Index n is 1 for the GN-model-like contribution and 2 for the

non-Gaussianity correction. The functions mn f are

reported in Appendix B, whereas in Appendix C the complete

derivation is shown. As in the SCI formula, when these further

contributions are addressed, both 4th

and 6th

order moments of

the transmitted symbol sequences must be considered (seeb ).

Note that the XCI domains D2-D4 are non-empty as long as the

INT channel adjacent to the CUT is not too far from the CUT.

They may be present or not depending on the value of both f

and cf . All three regions completely disappear when

2c sf R , for any value of f in the CUT band. This is

automatically accounted for in Eq. 4 which can hence be

considered a generalized complete formula for XCI, valid for

channels adjacent to the CUT but also for non-adjacent

channels, placed at any frequency interval from the CUT.

Even though the extra XCI D2-D4 regions appear only for the

two channels adjacent to the CUT, they may contribute

substantially to the overall NLI variance, depending on link and

system parameters, so that disregarding them may lead to

non-negligible error. This is due to the fact that these regions

are relatively close to the origin of the 1 2,f f , where the

integrand factors are maximum (see [6] for more details).

We investigated this matter by looking at the NLI normalized

variance XCI

defined as follows:

XCI XCI

/2

3

ch

/2

s

s

R

R

P G f

Eq. 5

with XCI

G f given by Eq. 4. This parameter collects the total

XCI noise spectrally located over the CUT, normalized so that

XCI itself does not depend on launch power. Note that for

simplicity we assume here:

INT CUTchP P P

We calculated XCI

for a system defined as follows:

PM-QPSK at sR =32 GBaud

roll-off 0.02

three channels with the CUT as the center channel

channel spacing 33.6 [GHz]f

SMF fiber with D =16.7 [ps/(nm km)], =1.3 [1/(W

km)], dB =0.22 dB/km

NZDSF fiber with D =3.8 [ps/(nm km)], =1.5

[1/(W km)], dB =0.22 dB/km

span length sL =100 [km]

For the same system we also calculated XPM

, defined as:

f 1

f 2

f 3 =f 1 + f 2

D2

D2

D1

D1 D3

D4

0

7

XPM XPM

/2

3

ch

/2

s

s

R

R

P G f

Eq. 6

with XPM

G f given by Eq. 3.

Finally, still for the same system, we simulatively estimated the

overall non-linearity, with single-channel effects removed. We

did this because we wanted to discuss the claim of [4] that the

XPM approximation can account for all of non-linearity

(except SCI) and that in this respect it represents a more

accurate estimator than the standard GN model. To remove SCI

from the simulation results, we simulated both the CUT alone

and the CUT with the two INT channels. Then we subtracted

the former simulation result from the latter at the field level,

thus ideally freeing the CUT completely from single-channel

effects while leaving in all other non-linearity (XCI and MCI).

Error! Reference source not found. shows the XPM

approximation XPM

of [4] provided by Eq. 3 as a magenta solid

line. The green solid line represents XCI

given by Eq. 4. The

red dashed curve represents the simulation result accounting for

all NLI except SCI. All curves are represented as a function of

the number of spans, up to 50. This may seem a large number

but in fact the reach of the simulated system, assuming SMF,

conventional EDFA amplification with realistic noise figure

(5-6 dB) and a realistic FEC BER threshold of about 210 , is

indeed on the order of 50 spans.

Fig. 4: Plot of normalized non-linearity coefficient vs. number of spans in

the link, assuming three PM-QPSK channels over SMF, with span length

100 km. The CUT is the center channel. The spacing is 1.05 times the

symbol rate. Red dashed line: simulation, with single-channel

non-linearity (SCI) removed. Blue solid line: standard coherent GN model

without SCI. Magenta solid line: the XPM approximation XPM

of [4] (Eq.

3). Green solid line: XCI

(Eq. 4).

The figure shows that the XPM approximation XPM

of [4]

underestimates XCI by about 1.3 dB. Our XCI result XCI

reduces such error to less than 0.3 dB throughout the plot.

Interestingly, at a distance that is comparable to the maximum

reach of the system, the error of XPM

is as large as the error of

the standard GN-model. The main difference is that the

GN-model overestimates NLI whereas XPM

underestimates it,

potentially leading to too optimistic system performance

predictions.

In Fig. 5, we show a similar plot, this time for NZDSF. Once

again, the gap between the XPM

of Eq. 3 and simulations is as

wide as the gap between simulation and the standard

GN-model, with XPM being optimistic (less NLI) and the GN

model conservative (more NLI). These gaps are almost 2 dB,

that is, they are substantially wider than in the SMF case. This

plot seems to show that the XPM approximation cannot be

considered a reliable overall NLI estimator, even in this

idealized case where SCI is removed.

Our XCI formula XCI

(Eq. 4) is more accurate, due to the

inclusion of the D2-D4 regions, but it shows some substantial

gap with simulations (about 1 dB). The presence of such gap is

interesting as it reveals that there must be some further

contribution to NLI which is not included Eq. 4 and which is

non-negligible in this case. Such contribution appears to be

MCI (see next section).

Fig. 5: Plot of normalized non-linearity coefficient vs. number of spans in

the link, assuming three PM-QPSK channels over NZDSF, with span

length 100 km. The CUT is the center channel. The spacing is 1.05 times

the symbol rate. Red dashed line: simulation, with single-channel

non-linearity (SCI) removed. Blue solid line: standard coherent GN model

(without SCI). Magenta solid line: the XPM approximation XPM

of [4]

(Eq. 3). Green solid line: XCI

(Eq. 4).

C. Multi-Channel Interference (MCI)

As mentioned, by definition MCI is NLI affecting the CUT

caused by any two or any three INT channels.

In general, MCI can be thought of as being weaker than SCI or

XCI, essentially because it arises on regions of the 1 2,f f

plane where the integrand factor , that appears in the model

8

equations, has substantially decayed (see FWM efficiency

factor [6] and Fig. 7 there).

However, it cannot be considered negligible when fiber

dispersion is relatively low (such as Truewave RS or even LS

fibers). Another condition that boosts the MCI contribution is

when NLI is evaluated at a frequency within the CUT which is

close to the CUT bandwidth edge ( / 2sf R ). In these

conditions, the error due to neglecting MCI can be substantial.

To provide an intuitive pictorial description of this effect, we

show in Fig. 6 the integration regions arising in the plane

1 2,f f when calculating the overall NLI PSD at the center of

CUT, i.e., NLI

(0)G , in the three-channel PM-QPSK example of

the previous section. The center region is SCI, the blue regions

are XCI and the red ones are MCI. All of these regions

contribute some NLI in the standard GN-model. In the EGN

model, each one of these region has a GN-model component

and one or more correction terms. They are weighed through

the factor that peaks at the origin and along the 1 2,f f

plane axes [6]. If fiber dispersion is large, the decay of away

from such maxima is fast and MCI is small. Otherwise, MCI

may become quite significant.

Fig. 6 explains the results of Fig. 5. Specifically, the XPM

magenta curve [4] is the lowest as it picks up only the D1

regions in Fig. 6. Then the green curve is XCI of Eq. 4, which

collects all of D1-D4 but neglects MCI. The simulation

intrinsically includes the red MCI regions too and hence is

higher than both XPM and XCI.

Fig. 6: Integration regions in the 1 2,f f plane needed to obtain the

power spectrum of NLI for f =0 , due to two adjacent INT channels with

spacing slightly higher than the symbol rate. The full XCI formula of Eq. 4

accounts for all D1-D4 regions. The XPM approximation [4] (Eq. 3)

considers the D1 regions only. SCI is the center region. MCI is the red

regions.

One of the main goals of this paper is that of proposing a

comprehensive enhanced GN model (EGN model) that takes

into account signal non-Gaussianity and comprises all relevant

NLI contributions. In view of the results shown in this section,

we believe that the XPM approximation [4] is not adequate as a

replacement of the GN model, as it may result in substantially

underestimating non-linearity. XCI of Eq. 4 is more accurate

but we believe that at least the main MCI contributions should

be considered as well.

The MCI formulas will be included in a later version of this

document, which will also include a more detailed quantitative

study of MCI strength.

III. COMMENTS AND CONCLUSION

This is the first version of this paper, which is still in partial

form. In the upcoming versions, as mentioned in the

Introduction, the following will be added:

the formulas for MCI

a detailed study of phase noise features in the system

a detailed study of the actual system impact of the use

of either the standard GN model, the EGN model or

other approximations such as the XPM of [4]

overall result extensions to LS fiber and to larger

number of channels in the system

The comments provided here derive in large part already from

the material presented in the current version. Some of them will

be fully justified by the material that is being prepared for the

next versions.

The standard GN model overestimates NLI and in this respect

is ‘safely’ conservative. The amount of overestimation is large

in the first spans (several dB’s) but it abates quickly along the

link. When looked at for a number of spans that is close to the

maximum reach, the error on NLI power estimation is typically

1 to 2 dB, depending on fiber type, modulation format and span

length, for realistic systems. Larger errors can be found by

pushing the system parameters outside of realism [4]-[5], such

as single-polarization, lossless fiber or perfect ideal

amplification, ultra-short spans, etc. In this paper we did not

explore unrealistic realms since our primary target was that of

investigating tools intended for practical system design

support.

The GN model errors in NLI power estimation in turn lead to

about 0.3-0.6 dB of error on the prediction of the maximum

reach or the optimum launch power, for typical realistic

systems. This error may or may not be acceptable, depending

on applications, but is guaranteed to be conservative for

PM-QAM systems.

When such error is not acceptable, the EGN model can be used,

which is capable of providing very accurate estimates of NLI

variance at any number of spans along the link, for any format

and system set of parameters. In this paper we have provided

the full set of formulas needed for a self-consistent complete

EGN model, derived using the procedure pioneered in [4] to

remove the signal Gaussianity assumption.

In detail, we have provided for the first time single-channel

non-linearity formulas, which had not been addressed in [4].

f1

f2

f3 =f1+ f2

D2

D2

D1

D1D3

D4

D1

D1

D2

D2

D3

D4

SCI

shaded blue: XCIshaded red: MCID1 alone: XPM [4]

9

We have also shown that the ‘XPM’ formulas proposed in [4]

as an estimator for the overall NLI (except single-channel) can

in fact substantially underestimate NLI, especially in systems

with low-dispersion fibers. The XPM error can be as large in

underestimating NLI at the system maximum reach, thus

leading to too optimistic predictions, as the standard GN-model

overestimates NLI leading to conservative predictions. We

have shown why this happens (the neglect of part of XCI and all

of MCI in [4]). The EGN model formulas that we propose

include all contributions and achieve very good accuracy.

Looking at the final EGN model formulas, it is evident that the

price to pay for its increased accuracy is quite substantially

increased complexity. A key objective for research in the near

future it clearly that of trying to drastically reduce it, perhaps by

deriving from it suitable GN model correction terms which

permit to combine improved accuracy with reasonable

complexity.

IV. APPENDIX A

This appendix will contain the full detailed derivation of the

SCI formulas. It is in preparation and will be inserted here in a

subsequent version of this paper.

V. APPENDIX B

Here are the detailed expressions of the mn f contributions

for XCI appearing in Eq. 4. The formulas for 11 f and

12 f were already shown in Sect. II.A. The others are as

follows:

CUT CUT INT

/2 /2

2 3

21 1 2

/2 /2

2 2 2

2 1 2 1

2 2

1 2 1 2

32( )

27

( ) ( ) ( )

, , , ,

c s s

c s s

f R R

s

f R R

f R df df

s f s f f f s f

f f f f f f

INT CUT CUT CUT

CUT

/2 /2 /2

2 2

22 1 2 3

/2 /2 /2

2

1 2 3 1 2

1 3 1 2 1 3

1 2 1 3

1 2 1 3

80( )

81

( ) ( ) ( ) ( )

( ) , , , ,

, , , ,

, , , ,

c s s s

c s s s

f R R R

s

f R R R

f R df df df

s f s f s f s f f f

s f f f f f f f f f

f f f f f f

f f f f f f

CUT CUT INT

/2 /2

2 3

31 1 2

/2 /2

2 2 2

1 2 1 2

2 2

1 2 1 2

16( )

27

( ) ( ) ( )

, , , ,

s s

s s

R R

s

R R

f R df df

s f s f s f f f

f f f f f f

INT

CUT CUT CUT CUT

/2 /2 /2

2 2

32 1 2 3

/2 /2 /2

2

1 2

1 2 3 1 2 3

1 2 1 2 3 3

1 2 1 2 3 3

1 2 1 2 3 3

16( )

81

( )

( ) ( ) ( ) ( )

, , , ,

, , , ,

, , , ,

s s s

s s s

R R R

s

R R R

f R df df df

s f f f

s f s f s f s f f f

f f f f f f f f

f f f f f f f f

f f f f f f f f

INT INT INT

/2 /2

2 3

41 1 2

/2 /2

2 2 2

1 2 1 2

2 2

1 2 1 2

16( )

27

( ) ( ) ( )

, , , ,

c s c s

c s c s

f R f R

s

f R f R

f R df df

s f s f s f f f

f f f f f f

INT INT INT INT

INT

/2 /2 /2

2 2

42 1 2 3

/2 /2 /2

2

1 2 3 1 2

1 3 1 2 1 3

1 2 1 3

1 2 1 3

/2

2 2

1

/2 /2

80( )

81

( )

, , , ,

, , , ,

, , , ,

16

81

c s c s c s

c s c s c s

c s

c s c s

f R f R f R

s

f R f R f R

f R

s

f R f R

f R df df df

s f s f s f s f f f

s f f f f f f f f f

f f f f f f

f f f f f f

R df

INT

INT INT INT INT

/2 /2

2 3

/2

2

1 2

1 2 1 2 3 3

1 2 1 2 3 3

1 2 1 2 3 3

1 2 1 2 3 3

( )

, , , ,

, , , ,

, , , ,

c s c s

c s

f R f R

f R

df df

s f f f

s f s f s f f f s f

f f f f f f f f

f f f f f f f f

f f f f f f f f

INT INT INT INT INT

INT

/2 /2 /2 /2

2

43 1 2 3 4

/2 /2 /2 /2

1 2 1 2 3 4

3 4 1 2 3 4

1 2 3 4 1 2 3 4

16( )

81

( ) ( ) ( ) ( ) ( )

( ) , , , ,

, , , , , , , ,

c s c s c s c s

c s c s c s c s

f R f R f R f R

s

f R f R f R f R

f R df df df df

s f s f s f f f s f s f

s f f f f f f f f f

f f f f f f f f f f f f

10

VI. APPENDIX C

This appendix provides the full detailed derivation of the XCI

formulas shown in Appendix B. It is in preparation and will be

inserted here in a subsequent version of this paper.

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