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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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