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SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 1

Dual-polarization NFDM transmission with

continuous and discrete spectral modulation

F. Da Ros, S. Civelli, S. Gaiarin, E.P. da Silva, N. De Renzis, M. Secondini, and D. Zibar

Abstract—Nonlinear distortion experienced by signals dur-

ing their propagation through optical ﬁbers strongly limits

the throughput of optical communication systems. Recently,

a strong research focus has been dedicated to nonlinearity

mitigation and compensation techniques. At the same time,

a more disruptive approach, the nonlinear Fourier transform

(NFT), aims at designing signaling schemes more suited to

the nonlinear ﬁber channel. In a short period, impressive

results have been reported by modulating either the continuous

spectrum or the discrete spectrum. Additionally, very recent

works further introduced the opportunity to modulate both

spectra for single polarization transmission. Here, we extend

the joint modulation scheme to dual-polarization transmission

by introducing the framework to construct a dual-polarization

optical signal with the desired continuous and discrete spectra.

After a brief analysis of the numerical algorithms used to

implement the proposed scheme, the ﬁrst experimental demon-

stration of dual-polarization joint nonlinear frequency division

multiplexing (NFDM) modulation is reported for up to 3200 km

of low-loss transmission ﬁber. The proposed dual-polarization

joint modulation schemes enables to exploit all the degrees of

freedom for modulation (both polarizations and both spectra)

provided by a single-mode ﬁber (SMF).

Index Terms—nonlinear frequency division multiplexing,

Raman ampliﬁcation, nonlinear Fourier transform, inverse

scattering transform

I. Introduction

OPTICAL communication systems have experienced

an impressive growth over the past few decades with

ever increasing transmission rates. Such a growth has been

the results of key enabling technologies that have allowed

to counteract the several physical eﬀects hindering the

transmission. As linear eﬀects such as loss and dispersion

can be dealt with in the telecommunication C-band, one of

the current limiting factor preventing to further enhance

the throughput of SMFs is the impact of Kerr nonlinearity.

Whereas a signiﬁcant research eﬀort has been devoted

to counteract nonlinear distortion experienced by the

Manuscript received October 18th, 2018;

F. Da Ros, S. Gaiarin, N. De Renzis, and D. Zibar are

with the Department of Photonics Engineering, Technical Uni-

versity of Denmark, Kongens Lyngby, 2800 Denmark, e-mail:

{fdro,simga,nidre,dazi}@fotonik.dtu.dk

S. Civelli and M. Secondini are with the TeCIP

Institute, Scuola Superiore Sant’Anna, Pisa, Italy, email:

{stella.civelli,marco.secondini}@santannapisa.it

E.P. da Silva was with the Department of Photonics Engineering,

Technical University of Denmark, Kongens Lyngby, 2800 Denmark.

He is now with the Department of Electrical Engineering of the

Federal University of Campina Grande (UFCG), Paraba, Brazil,

email:edson.silva@dee.ufcg.edu.br.

F. Da Ros, S. Civelli and S. Gaiarin equally contributed to this

work.

signals during propagation, with solutions presented both

in the optical and digital domain [1], no clear practical

solution has yet been devised. Over the past few years,

a theoretical approach, which had led to soliton-based

communication, has been rediscovered [2], [3]. Soliton-

based communication was developed in the 1980’s [4]–[6],

and, in particular, eigenvalue based communication was

ﬁrst attempted in [7], before the advent of wavelength-

division multiplexing (WDM), and it was quickly aban-

doned due to its low spectral eﬃciency, challenges with

ﬁber loss, noise and soliton-soliton interaction and the

lack of coherent transceivers enabling access to the full

electrical ﬁeld. Recently, however, a generalization of

the mathematical theory behind soliton communication,

i.e., the inverse scattering transform (IST), has been

exploited to devise a new approach to transmit over single-

mode ﬁbers (SMFs). The IST, re-branded as nonlinear

Fourier transform (NFT) within the optical communica-

tion community, provides a transformation that enables

to eﬀectively linearize the nonlinear Schr¨odinger equation

(NLSE) describing the optical wave propagation through a

SMF. By using such a transformation, the impact of group

velocity dispersion and Kerr nonlinearity can be construc-

tively taken into account to design a novel signaling system

that aims at not being limited by signal-signal nonlinear

interaction. Furthermore, this transformation may provide

more ﬂexible modulation techniques compared to stan-

dard coherent approaches, as it associates two spectral

quantities to one time-domain waveform: a continuous

spectrum corresponding to dispersive waves and a discrete

spectrum representing the solitonic solutions [2]. Since

the NFT theory strictly requires loss-less, noise-less, and

dispersion-slope free transmission, a signiﬁcant research

eﬀort has been devoted into addressing these requirements

both numerically, e.g., by using the lossless path-averaged

(LPA) approximation [8]–[10], and experimentally, e.g.,

using distributed Raman ampliﬁcation [11]–[13]. This

has yield several impressive demonstrations by encoding

information in either the continuous [14] (up to 3.3 Gb/s

over 7344 km), or the discrete spectrum [15] (4 Gb/s

over 640 km) [16] (4 Gb/s over 1600 km) [13] (4 Gb/s

over 4900 km) and [17] (1.5 Gb/s over 1800 km), ﬁnally

leading to a recent report of a joint continuous and discrete

spectral modulation [18], [19] (up to 64 Gb/s over 976 km).

In parallel, the use of SMFs for transmission provides an

additional degree of freedom to increase the transmission

throughput by exploiting two orthogonal ﬁeld polariza-

tions. The NFT for the NLSE has therefore been extended

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 2

to the Manakov system (MS) [20], where two signal

polarizations are transmitted under the assumption that

the state-of-polarization varies fast enough to be averaged

over the nonlinear and dispersion lengths, and polarization

mode dispersion can be neglected. The theory has been

applied to numerical and experimental demonstrations of

dual-polarization NFT-based transmission using either the

continuous [21]–[23] or the discrete spectrum [11], [24].

A dual-polarization transmission where both continuous

and discrete spectra are jointly modulated would therefore

represent the complete system where all the degrees of

freedom for modulation provided by a SMF are exploited.

In this work, we therefore demonstrate joint dual-

polarization nonlinear frequency division multiplexing

(NFDM) modulation for the ﬁrst time. First we dis-

cuss the framework to perform a joint inverse nonlinear

Fourier transform (INFT) operation, i.e., the operation

of constructing a dual-polarization time-domain waveform

with a desired dual-polarization continuous and discrete

spectrum. The steps required are described in details

and the numerical algorithms that can be employed to

implement the INFT are brieﬂy discussed and numerically

characterized. By applying such INFT, a dual-polarization

joint NFDM system is experimentally characterized. A

transmission distance of up to 3200 km is demonstrated

by jointly modulating the dual-polarization signal with a

10-GBd quadrature phase-shift keying (QPSK)-modulated

continuous spectrum and a 2-eigenvalue 125-MBd QPSK

discrete spectrum, for a total net line rate of 8.4 Gb/s (af-

ter forward error correction (FEC) overhead subtraction).

The rate is mainly limited by the wide guard intervals

required to fulﬁll the NFT vanishing boundary conditions

at the receiver, i.e., after the dispersion-induced signal

broadening.

The paper is organized as follows: ﬁrst in Section II the

theoretical framework for joint NFT and INFT operation

is provided. The discussion on the transmitter and receiver

digital signal processing (DSP) algorithms chosen for

implementing the dual-polarization joint NFDM system

and the achievable digital back-to-back performance follow

in Section III. The experimental setup used as testbed is

described in Section IV together with the characterization

of the optical back-to-back performance. The transmission

results are reported and discussed in Section V and the

conclusions drawn in Section VI.

II. Dual-polarization joint continuous and discrete

spectral modulation

In this Section, the framework for dual-polarization joint

NFDM modulation is described. First the channel model

will be brieﬂy outlined in Section II-A, followed by the

direct NFT in Section II-B. Finally Section II-C describes

in details one of the key contributions of this work,

i.e., the joint INFT using the Darboux transform (DT).

Throughout this section, bold symbol characters indicate

vector or matrices, while their components are indicated

with subscripts (without bold): e.g., φ= (φ1,...,φN)and

v= (v1,...,vN).

A. Channel model

Let us consider a SMF exhibiting random birefringence,

and whose dispersion and nonlinear lengths are much

longer than the birefringence correlation length [20]. Under

those conditions, the averaged MS describes the evolution

in the ﬁber of the two complex-envelope polarization

components of a signal Ej=Ej(τ, ℓ),j= 1,2[25],

∂E1

∂ℓ =−jβ2

2

∂2E1

∂τ 2+j8γ

9|E1|2+|E2|2E1

∂E2

∂ℓ =−jβ2

2

∂2E2

∂τ 2+j8γ

9|E1|2+|E2|2E2

(1)

where τis the retarded time, ℓthe distance, β2the

group velocity dispersion (GVD), and γthe Kerr nonlinear

coeﬃcient of the ﬁber.

The normalized MS [26] for the anomalous dispersion

regime (β2<0) is

j∂q1

∂z =∂2q1

∂t2+ 2 |q1|2+|q2|2q1

j∂q2

∂z =∂2q2

∂t2+ 2 |q1|2+|q2|2q2

(2)

where tis the normalized retarded time, and zthe

normalized distance. The equation is derived from (1)

through the change of variables

qj(t) = Ej(τ)

√P, t =τ

T0

, z =−ℓ

L,(3)

with P=|β2|/(8

9γT 2

0),L= 2T2

0/|β2|, and T0is the free

normalization parameter.

B. Direct NFT

The direct NFT computes the nonlinear spectrum of the

time-domain dual-polarization signal q(t) = (q1(t), q2(t)).

The spectrum is composed of a continuous (dispersive)

part and a ﬁnite number of discrete components, which

correspond to the solitonic components. The nonlinear

spectrum is deﬁned through the Zakharov-Shabat (Z-S)

problem [20], [22], [26]

L(q)v=λv(4)

where Lis a 3×3operator that depends on the signal

q(t), and is given by

L=

j∂

∂t −jq1(t)−jq2(t)

−jq∗

1(t)−j∂

∂t 0

−jq∗

2(t) 0 −j∂

∂t

.(5)

The canonical solutions of (4) are the solutions φ,¯

φ,

ψ, and ¯

ψdeﬁned by the boundary conditions [22], [26]

as t→ −∞

z}| {

φ(t, λ;q)∼

1

0

0

e−jλ t,¯

φ(t, λ;q)∼

0 0

1 0

0 1

ejλ t

(6)

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 3

λ

,

b

λ

Fourier

transform

Q

λSf

λπf

INFT

(solve GLME)

Calculate

canonical sol.

NIS mapper

Joint INFT

λ

ν

t

,

λ

st Sf Q

λ Q

λ qt

qt

λ

Step 1 Step 2 Step 3

Continuous Discrete

Darboux

Transform

Fig. 1. Schematic diagram of dual-polarization joint NFDM time-domain signal generation through a joint INFT operation. The information

is ﬁrst separately encoded into discrete and continuous spectrum, through b-modulation and NIS operation (via the signal Fourier transform

S(f)), respectively. Then the three steps of the joint INFT follow to obtain the time-domain waveform q(t).

ψ(t, λ;q)∼

0 0

1 0

0 1

ejλ t,¯

ψ(t, λ;q)∼

1

0

0

e−jλ t

|{z }

as t→+∞.(7)

The canonical solutions form two pairs of bases of the

same subspace, consequently there exist some coeﬃ-

cients—known as scattering data—a(λ)∈C, and b(λ)∈

C2×1such that

φ(t, λ) = ψ(t, λ )b(λ) + ¯

ψ(t, λ)a(λ).(8)

Such scattering data can be obtained as

a(λ) = lim

t→+∞φ1(t, λ)e+j λt ,(9a)

b1(λ) = lim

t→+∞φ2(t, λ)e−j λt ,(9b)

b2(λ) = lim

t→+∞φ3(t, λ)e−j λt ,(9c)

and are related to each other by

|a(λ)|2+|b1(λ)|2+|b2(λ)|2= 1.(10)

The continuous spectrum of the time domain signal q(t)

is deﬁned as

Qc(λ) = b(λ)/a(λ), λ ∈R.(11)

The discrete spectrum is made of a ﬁnite number NDS of

eigenvalues λi∈C+such that a(λi) = 0. Each eigenvalue

has an associated spectral amplitude (also referred to as

norming constant) deﬁned as

Qd(λi) = b(λi)/a′(λi), λi∈C+.(12)

with a′(λi) = da(λ)

dλ |λ=λi.

Several methods to perform the NFT numerically are

available for the single polarization scenario [2], [3], [27]

and for the dual polarization case [21], [22], [24]. In

this work, the trapezoidal discretization method was

used for computing both the continuous and the discrete

spectrum [28].

C. Inverse NFT

The INFT is the operation to generate a time domain

signal from a given nonlinear spectrum. Several methods

exist to numerically perform the INFT for the scalar (sin-

gle polarization) NLSE [2], [3], [9], [18], [29]. Additionally,

the theory can be extended to the MS [26] and a few

techniques reported leading to recent numerical analysis

and experimental demonstrations: the DT for the MS [24]

for a multi-soliton signal (no continuous spectrum), the

inverse Ablowitz-Ladik method for a signal with only

continuous spectrum [21], and the Nystr¨om conjugate

gradient (NCG) method—based on the solution of the

Gelfand-Levitan-Marchenko equation (GLME)—that can

be applied to a full spectrum [22], [30]. However, when

the overall energy of the signal, and so of the nonlinear

spectrum, is too high, the latter method may diverge

[22]. This issue becomes relevant in optical communication

when discrete eigenvalues are used for modulation [22]. In

this section, we describe an alternative method to generate

a time domain signal from a given continuous and discrete

nonlinear spectrum. Extending the concepts of [18], [28]

to the MS, the method uses an INFT algorithm (e.g.,

the NCG in this work) to obtain the time-domain signal

corresponding to a pre-modiﬁed continuous spectrum, and

then adds the discrete eigenvalues using the DT [24],

[31]. Using this approach, the aforementioned issues of

NCG can be relaxed as the energy of the input to the

NCG is decreased by the large amount carried by the

discrete spectrum, therefore shifting the energy barrier to

higher energy levels. A choice of initialization parameters

is further provided to ensure that the obtained signal

has the desired nonlinear spectrum, by following the

approach in [24], [28]. Whereas the algorithm for single

polarization in [18] is supported by the mathematical

framework demonstrated in [28], a rigorous mathematical

framework is beyond the scope of this work. The numerical

accuracy of the scheme is rather conﬁrmed a posteriori

with numerical simulations (see Section III) and relies on

following the single-polarization framework of [18], [28]

similarly to the approach of [24]. Similarly to the single-

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 4

polarization case, the three steps are illustrated below.

Assume that we want to digitally compute the time

domain signal q(t)corresponding to the continuous spec-

trum Qc(λ)and the discrete spectrum {λi,b(λi)}NDS

i=1 ,

adopting b-modulation on the discrete part as will be

discussed in Section III. The choice of b-modulation is

not a strict requirement for the scheme presented in the

following, and the approach can be easily extended to

mapping techniques other than b-modulation.

The algorithm consists of three steps (illustrated in Fig. 1):

1) Use the INFT to compute the time domain signal

e

q(t)corresponding to the pre-modiﬁed continuous

spectrum

e

Qc(λ) = Qc(λ)

NDS

Y

i=1

λ−λi

λ−λ∗

i

(13)

and with empty discrete spectrum. This can be

achieved by solving the GLME equation, for example

using the NCG method. [22].

2) For each eigenvalue λifor i= 1,...,NDS , obtain the

solution ν(t, λi)of the eigenvalue problem L(e

q)v=

λivwith boundary conditions

ν1(T, λi) = 1

ν2(−T, λi) = −b1(λi)

ν3(−T, λi) = −b2(λi)

(14)

where e

q(t) = 0 for t /∈[−T , T ]. The solution ν(t, λi)

can be obtained as

ν(t, λi) = φ(t, λi)

φ1(T, λi)−ψ(t, λi)ψ(2)(−T , λi)−1b(λi),

(15)

where φ(t, λi)and ψ(t, λi)are the canonical so-

lutions of L(e

q)v=λiv(see Section II-B) and

ψ(2)(t, λi)is the 2×2matrix made of the second

and the third rows of ψ(t, λi), i.e.,

ψ(2)(t, λi) = ψ21 ψ22

ψ31 ψ32.(16)

Note that Eq. (15) is a solution of the eigen-

value problem because it is a linear combination

of solutions. Furthermore, it veriﬁes the boundary

conditions Eq. (14). The canonical solutions φ(t, λi)

and ψ(t, λi)can be found with standard methods

used for the NFT (see Section II-B). Additionally, if

modulation techniques other than b-modulation are

chosen, a diﬀerent mapping can be easily derived

starting from Eq. (15).

3) Execute the DT for the MS, as in [24], [31], with in-

put parameters e

q(t),{λi}NDS

i=1 , and {ν(t, λi)}NDS

i=1 to

iteratively add the discrete spectrum {λi,b(λi)}NDS

i=1

to the spectrum of e

q(t). The ν(t, λi)are the generic

auxiliary solutions. The solution q(t)obtained in

this manner has continuous spectrum Qc(λ)and

discrete spectrum {λi,b(λi)}NDS

i=1 .

This scheme can be easily veriﬁed numerically by comput-

ing the direct NFT of the generated time-domain signal

and comparing the resulting spectra with the desired ones.

Transmitter DSP chain

Bit mapper

Continuous Discrete

PRBS generation

Bit mapper

NIS mapper

IQ-modulator predistortion

Denormalization

PRBS generation

a)

Receiver DSP chain

Continuous Discrete

Signal low-pass filtering

Frame synchronization

Signal amplitude rescaling

Phase estimation Equalization

Decision & BER

Normalization

CFO compensation

NIS demapper

Guardband

removal

Equalization Phase estimation

Decision & BER

b)

Add guardbands

INFT

Pulse shaper

Inverse channel

transfer func.

NFT

Fig. 2. (a) Transmitter and (b) receiver digital signal processing

chains, highlighting the key operations performed on the digital

waveforms.

III. Digital signal processing

The DSP chains implemented at the transmitter and re-

ceiver to properly encode the data into a digital waveform

(transmitter) and extract it back (receiver) are shown in

Fig. 2 (a) and (b), respectively.

A pseudo-random bit sequence is generated at the trans-

mitter side to be encoded in both the continuous and dis-

crete spectra. For the continuous spectrum, after mapping

the bits into 10-GBd QPSK symbols (16 samples/symbol),

guard intervals of 64 symbols are added for each 16-

symbol burst, leading to an overall NFDM symbol (burst)

length of 8 ns. Such guard intervals ensure that no inter-

burst interference takes place after the dispersion-induced

pulse broadening at the maximum transmission distance

considered in this work, i.e., 3200 km. Such long guard

intervals could be decreased by adding dispersion pre-

compensation at the transmitter side. This was avoided

for the experimental investigation since dispersion pre-

compensation would yield digital waveforms with diﬀerent

peak-to-average power ratio (PAPR) for each transmis-

sion distance, making it more challenging to compare

performance at diﬀerent distances. The symbols are pulse-

shaped with a raised cosine ﬁlter (roll-oﬀ of 1) and the NIS

is used to obtain the continuous spectrum as detailed in

Fig. 1: the Fourier transform of the pulse-shaped time-

domain signal is directly mapped into the continuous

spectrum Qc(λ)[10].

The data bits to be encoded onto the discrete spectrum

are also mapped to QPSK symbols. Then, the symbols

are associated to the b(λi)(b-modulation) corresponding

to the two purely-imaginary eigenvalues, λi∈ {0.3j, 0.6j}

that have been chosen. The free time normalization param-

eter deﬁned in (3) is chosen to be T0= 244 ps. The eigen-

values themselves are not modulated, i.e., in each symbol-

slot, both eigenvalues are transmitted. Additionally, sev-

eral recent works clearly showed that modulating directly

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 5

the b(λi)(b-modulation) of the discrete spectrum leads

to a lower correlation and thus better performance than

modulating Qd(λi)(Qd(λ)-modulation). In this work we

therefore focus on b-modulation for the discrete spectrum

[12], [32]. The radii of the two QPSK constellations have

been set to 5√2and 0.05√2, for the b(λi)associated to

λ1and λ2, respectively. This choice leads to a temporal

separation of the components of the time-domain signal

associated to the NFT coeﬃcients b(λi)corresponding to

diﬀerent eigenvalues. The separation is such that the dis-

crete spectral components (at the transmitter output) are

placed in time within the guard intervals of the continuous-

spectrum burst. The one corresponding to b(λ1)can be

seen in Fig. 3(a) after the burst encoded in the continuous

spectrum, whereas the waveform corresponding to b(λ2)is

located on the opposite end of the symbol slot. This choice

was made to avoid additional signal-dependent implemen-

tation penalties due to high PAPR at the transmitter side,

as well as to limit the time-frequency product of the multi-

soliton signal [33]. Nevertheless, continuous and discrete

time-components do interact during ﬁber propagation.

An INFT operation is then performed as described in

Section II-C to generate a time-domain waveform with

the desired continuous and discrete spectra. After proper

denormalization, the waveform shown in Fig. 3(a) is ob-

tained (signal power of -9.2 dBm with the ﬁber parameters

of Section IV). The ﬁgure clearly shows the two discrete

(solitonic) components with the continuous (dispersive)

components in between. Finally, the waveforms are pre-

distorted to account for the nonlinear transfer function of

the IQ modulator by applying an arcsin(·)function. Such

a digital waveform can then be encoded onto an optical

carrier using a standard IQ modulator, after digital-to-

analog conversion, as will be described in Section IV.

The net line rate of the generated signal is 8.4 Gb/s,

taking into account the 80%-guard intervals applied and

the 7%-hard-decision forward error correction (HD-FEC)

overhead [34], [35].

At the receiver side, the digital waveforms are then

processed by the DSP highlighted in Fig 2(b). First,

carrier frequency oﬀset (CFO) compensation is performed

to remove any frequency shift due to frequency mistmatch

between signal and local oscillator (LO), followed by signal

amplitude rescaling, low-pass ﬁltering at twice the 20-

dB signal bandwidth, and frame synchronization. The

direct NFT described in Section II-B is then applied

to recover the continuous and discrete spectra from the

time-domain waveform. The inverse transfer function of

the channel exp4jλ2zis ﬁrst applied to the continuous

part of the spectrum, followed by NIS demapping with

the opposite transformation applied at the transmitter

side: ﬁrst Qc(λ)is mapped into the Fourier spectrum,

then a inverse Fourier transform is used to recover the

time-domain waveform [10]. The guard intervals are sub-

sequently removed and blind radius-directed equalization

is performed followed by phase estimation using digital

phase-lock loop. Finally, decisions on the symbols are

taken and the bit error rate (BER) is counted. The

steps for the demodulation of the discrete spectrum

consists of ﬁrst phase recovery using blind phase search

(BPS) independently on each constellation bj(λi), followed

by NFT-domain equalization [11], [12]. As the chosen

eigenvalues are purely imaginary, BPS inherently applies

the ideal inverse channel transfer function, which consists

of a constant phase rotation. After BPS, NFT-domain

equalization reduces the noise on the bj(λi)by exploiting

the correlation between the received eigenvalues and

the spectral amplitudes [11], [12]. This equalizer enables

to partially compensate for the rotation and re-scaling

experienced by b1(λi)and b2(λi)due to the displacement

of the eigenvalues. After equalization, decisions are taken

and BER counting is performed.

The DSP chains and numerical algorithm have been ﬁrst

benchmarked in a digital back-to-back scenario where the

digital waveforms before the IQ-modulator predistortion

are fed directly into the receiver DSP chain. This analysis

allows to ignore the impact of practical equipment lim-

itations, such as analog-to-digital converter (ADC) and

digital-to-analog converter (DAC) resolution, and electri-

cal/optical noise sources, to focus on the numerical algo-

rithms. The resulting performance is shown in Fig. 3(b),

as a function of the energy in the continuous spectrum,

comparing joint and continuous-only modulation. For the

joint modulation (top ﬁgure in Fig. 3(b)), the energy in the

discrete spectrum is kept constant to fulﬁll the duration-

amplitude relation [2]. The signal quality is evaluated by

calculating the error vector magnitude (EVM) as the BER

values are too low for reliable error counting [36].

In the case of joint modulation, as the energy in the

continuous spectrum increases, the performance of the

continuous spectrum improves (EVM decreases) with an

optimum at approx. 0.18 pJ (-9.2 dBm of launch power).

When the energy in the continuous spectrum approaches

zero, the discrete spectrum is dominant and worsens the

accuracy of the numerical algorithms for the continuous

part. In the case of continuous-only modulation (bottom

ﬁgure in Fig. 3(b)), the performance do not degrade as

the energy decreases as for the joint-modulation, thus

ruling out numerical errors of the NCG alone.

Beyond the optimum energy for the joint-modulation,

the performance worsens rather rapidly above -9.0 dBm.

A similar worsening of the performance is indeed re-

ﬂected when no discrete spectrum is present. Note that,

considering b-modulation also for the continuous spec-

trum (instead of modulating directly Qc(λ)) may provide

further improvement [27] even though using it in the

context of joint spectral modulation may present some

challenges [37]. Fig. 3(b) shows also the impact of the

energy in the continuous spectrum on the quality of

the discrete spectrum. As the energy in the continuous

spectrum is increased, the limited precision of the nu-

merical algorithms yields a loss of orthogonality between

continuous and discrete spectrum, thus decreasing the

performance of the latter when the continuous components

at high energy overlap with the solitons. The impact of the

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 6

0 2 4 6 8 10 12 14 16

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-polarization

Y-polarization

Error vector magnitude (dB)

-15

-12

-9

-6

-3

Launch power (dBm)

discrete

continuous

0.0 0.2 0.4 0.6 0.8

-25

-24

-23

-22

-21

-20

Energy continuous spectrum (pJ)

continuous

a)

-10.0 -9.1 -8.3 -7.6 -7.0

Joint modulation

Continuous modulation

b)

Fig. 3. (a) Time-domain waveforms (-9.2-dBm launch power,

2 NFDM symbols, X and Y polarizations) showing the discrete

(solitonic) components with the continuous components in between

and (b) digital back-to-back performance: EVM as a function of the

energy in the continuous spectrum for joint modulation (top) and

continuous-only modulation (bottom).

time-overlap is expected to worsen the performance during

transmission over a non-ideal (lossy and noisy) channel,

as discussed in [38]. Furthermore, the numerical precision

of the INFT is also expected to contribute to the overall

worsening of the performance of the discrete spectrum.

Regardless of these limitations and the consequent

energy balance between continuous and discrete spectrum,

the BER that can be estimated from the EVM is well be-

low 1×10−4, even for the highest power values considered.

IV. Experimental transmission setup

The experimental setup is shown in Fig. 4. The pre-

distorted digital waveforms generated as in Fig. 2(a) are

loaded into a 4-channel 64-GSamples/s arbitrary wave-

form generator (AWG) driving the IQ modulator, which

encodes the dual-polarization NFDM signal into an optical

carrier generated by a low-linewidth (≤1 kHz) ﬁber laser.

The same laser is used as LO at the receiver side.

The transmission link consists of a recirculating trans-

mission loop based on four 50-km transmission spans with

distributed Raman ampliﬁcation applied to each span

as in [11]. Backward pumping combined with low-loss

large eﬀective area ﬁber (SCUBA ﬁber) enables to achieve

maximum power variations of approximately 3 dB across

the full 200-km loop length. The power proﬁle measured

by optical time domain reﬂectometry is shown in inset (a)

of Fig. 4. Loss, dispersion, and nonlinear coeﬃcient of the

transmission ﬁber are 0.155 dB/km, 22 ps/nm/km, and

0.6 /W/km, respectively. These values have been used for

the (I)NFT (de)normalization as discussed in Section II-A.

In addiction to the transmission ﬁber, the loop consists of

acusto-optic modulators (AOMs) used as optical switches,

an optical band pass ﬁlter (OBPF) (0.5-nm bandwidth)

which suppresses out-of-band ampliﬁed spontaneous emis-

sion (ASE) noise, an isolator (ISO), and an erbium-doped

ﬁber ampliﬁer (EDFA) which compensates for the power

loss of all these components.

After the chosen number of recirculation turns, the

signal is received with a pre-ampliﬁed coherent receiver

using four balanced photodetectors (BPDs) and a 80-

GSamples/s digital storage oscilloscope (DSO) acting

as analog-to-digital converter. For simplicity, the signal

polarization is manually aligned at the receiver input

with a polarization controller (PC). However, demulti-

plexing schemes based on training sequences have already

been reported [23]. After analog-to-digital conversion, the

waveforms are processed oﬄine by the DSP discussed

in Section III and the performance are evaluated by

bit error counting performed on more than 106bits,

ensuring a reliable BER above 10−5. Only BER values

from direct error counting are reported for experimental

measurements. In the following the transmission reach

is evaluated considering the HD-FEC threshold (BER

of 3.8×10−3) [34], [35]. Remark that the frequency-

oﬀset estimation discussed in Section III is necessary due

to the frequency shift introduced by the AOMs which

results in self-heterodyne detection rather than homodyne.

An example of constellation diagrams for continuous and

discrete spectrum is shown in inset (b) of Fig. 4 after

2800-km transmission, illustrating the high quality of the

received signals.

Before discussing the transmission results in Section V,

the signal performance are evaluated in back-to-back

conﬁguration, i.e., connecting the receiver directly at

the output of the IQ modulator. The results are shown

in Fig. 5, distinguishing between the performance of

continuous and discrete spectral components as well as

showing the total BER. The optical signal-to-noise ratio

(OSNR) at the output of the transmitter was approx.

33.8 dB for all the launch powers considered.

For these back-to-back results, the total BER is domi-

nated by the BER of the continuous spectrum as more bits

are encoded in the continuous spectrum (64 bits/NFDM

symbol) compared to the discrete spectrum (8 bits/NFDM

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 7

Continuous Discrete

λ

1

λ

2

X Y

Transmitter Receiver

OBPFEDFA

Fiber laser

(

< 1 kHz)

PC

64 GSa/s

AWG

Transmission link

AOM

#1

AOM

#2

OBPF EDFAISO

Transmitter

DSP

LO BPD

BPD

BPD

BPD

90°- Hybrid

Receiver

DSP

80 GSa/s

DSO

0 50 100 150 200

-3

-2

-1

0

Normalized

power (dB)

Distance (km)

DP-IQ

Mod SCUBA

50 km

4

DRA

a)

b)

Fig. 4. Experimental setup for the transmission of the jointly modulated signal in a recirculating transmission loop. Insets: (a) power proﬁle

measured over a loop recirculation and (b) constellation diagrams after 2800-km transmission.

-10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0

1E-5

1E-4

0.001

0.01

BER

Launch power (dBm)

total

continuous

discrete

HD-FEC

Fig. 5. Optical back-to-back BER performance from direct error

counting as a function of the launch signal power.

symbol). Therefore, a higher error probability can be

tolerated on the discrete spectrum before it starts aﬀecting

the total BER. As discussed for the numerical results of

Fig. 3, the BER of the continuous spectrum improves with

its increased energy, reaches an optimum and worsens due

to numerical instabilities. Note that the optimum power is

the same for the digital back-to-back (see Fig. 3(b)). The

lack of variations in the optimum power hints that the

dominant limitation is currently related to the numerical

algorithms, whereas the impact of electrical/optical noise

at transmitter and receiver, as well as the AWG resolution

are rather negligible. The BER on the discrete spectrum is

also consistent with Fig. 3(b), as errors are only detected

at the highest launch power considered, -7 dBm. At such

a power level, the estimated BER in digital back-to-back

was estimated to almost the same value, showing that a

negligible penalty is introduced by the optical-frontends

(at both transmitter and receiver) also for the discrete

spectrum.

V. Transmission performance

After having evaluated the system performance for

both digital and optical back-to-back, the transmission

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Power (a.u.)

Time (ns)

X-pol

Y-pol

!! " #$!! "

$!!! " $%!! "

a)

c)

b)

d)

e) f)

Fig. 6. Examples of time-domain waveforms showing one 8-ns NFDM

symbols (including guard intervals) at a ﬁxed launch power of -

9.2 dBm: (a) digital back-to-back, (b) optical back-to-back and after

(c) 400-km, (d) 1200-km, (e) 2000-km, (f) and 2800-km transmission.

The bit patter is not the same for the same for the diﬀerent

waveforms.

performance is reported in this section. Fig. 6 shows

the evolution along the ﬁber of one NFDM symbol at

the optimum launch power of -9.2 dBm. The waveforms

clearly show the interaction in time between the discrete

and continuous spectral components, whereas the guard

interval size is more than suﬃcient to guarantee the

vanishing boundary conditions required by the NFT also

at the longest transmission distances. The guard interval

size could actually be reduced by pre-dispersing the wave-

forms at the transmitter side by half of the transmission

length, i.e., by applying the inverse of the channel transfer

function [39]. Additionally, by tailoring the guard intervals

to the desired transmission distance, the transmission rate

can be maximized. The total BER results as a function

of the launch power for diﬀerent transmission distances

are shown in Fig. 7(a). The curves show an optimum

launch power (minimum BER) consistent with the digital

and optical back-to-back performance, i.e., −9.2dBm.

The BER values after a 400-km transmission are actually

SUBMITTED TO IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY 8

0 400 800 1200 1600 2000 2400 2800 3200

1E-5

1E-4

0.001

0.01

0.1

BER

Distance (km)

BER total

BER continuous

BER discrete

HD-FEC

a)

b)

-9.5 -9.0 -8.5 -8.0 -7.5 -7.0

1E-4

0.001

0.01

0.1

BER total

Launch power (dBm)

3200 km

2800 km

2000 km

1200 km

400 km

HD-FEC

Fig. 7. (a) Total BER performance as a function of the launch

power for diﬀerent transmission distances, and (b) total BER and

contributions from continuous and discrete spectrum as a function of

the transmission distance for the optimum launch power of -9.2 dBm

(shaded area in (a)).

in close agreement with the optical back-to-back ones

in Figs. 3(b). These results further conﬁrm that the

dominant performance limitation is currently linked to

the numerical precision in performing joint INFT and

NFT at high power values. By increasing the numerical

precision of the diﬀerent steps (Section II) using improved

numerical algorithms, we believe the performance could be

signiﬁcantly improved, potentially shifting the optimum

launch power to higher values [40].

The BER as a function of the transmission distance is

shown in Fig. 7(b), for a ﬁxed -9.2-dBm launch power. The

ﬁgure highlights that BER below the HD-FEC threshold

can be achieved for up to 3200-km transmission. Beyond

2800 km, the dominant contribution to the total BER

comes from the discrete spectral components and in

particular from the largest eigenvalue λ2= 0.6j. Improved

modulation schemes for the discrete spectral components,

such as the diﬀerential modulation proposed in [41] or

soliton detection based on matched ﬁlter [18], are thus

expected to increase the transmission reach. Furthermore,

as far as the continuous spectrum is concerned using

improved detection strategies, may also provide additional

performance gain [42], [43]. Finally, the net transmission

rate may be increased by reducing the guard intervals, as

mentioned above, and pre-dispersing the signal by half of

the transmission length at the transmitter side [19], [39],

[44].

VI. Conclusion

We have introduced a framework for dual-polarization

NFDM systems which allows encoding data on both

continuous and discrete spectral components. The steps

to perform the joint INFT at the transmitter side are

described and numerically implemented, evaluating its

performance ﬁrst in a transmission-free scenario without

(digital back-to-back) and with the optical front-ends

(optical back-to-back) considered. The dual-polarization

joint NFDM system has then been experimentally demon-

strated in a transmission scenario using distributed Raman

ampliﬁcation. A transmission reach of 3200 km is achieved

for a 8.4-Gb/s net rate NFDM signal, mainly limited

by the numerical implementation of the joint INFT and

NFT, which will need to be further improved. This work

demonstrates the use of all the degrees of freedom available

for NFDM-based transmission over SMFs.

Acknowledgment

This work is supported by the European Research

Council through the ERC-CoG FRECOM project (grant

agreement no. 771878) and the National Council for

Scientiﬁc and Technological Development (CNPq), Brazil,

grant 432214/2018-6. We thank OFS ﬁtel Denmark for

providing the SCUBA ﬁber used in the experiment and

the anonymous reviewers for their constructive feedback.

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