Hybrid Modulation Formats Enabling Elastic Fixed-Grid Optical Networks

Abstract

In this paper, we analyze hybrid modulation formats as an effective technology for the implementation of flexible transponders that are capable of trading off the delivered data rate by the lightpath quality of transmission with fine granularity. Flexible transponders are an enabling technology that can introduce the elastic paradigm in state-of-the-art networks while maintaining compatibility with the installed equipment, including fibers, mux-demux, and reconfigurable optical add–drop multiplexers, as required by telecom operators willing to exploit fixed-grid wavelength-division multiplexed (WDM) transmission. We consider two solutions achieving different levels of flexibility and employing different hybridization approaches: time-division (TDHMF) and quadrature-division (Flex-PAM) hybrid modulation formats. We introduce a comprehensive theoretical assessment of back-to-back performances, analyzing different transmitter operating conditions, and we provide an extensive simulation analysis on the propagation of a Nyquist-WDM channel comb over an uncompensated and amplified fiber link. After assessing the impact of non-linear propagation on the maximum signal reach, we present simple countermeasures for non-linear mitigation and discuss their effectiveness for both TDHMF and Flex-PAM.

Full text

Hybrid Modulation Formats Enabling
Elastic Fixed-Grid Optical Networks
Fernando P. Guiomar, Rixin Li, Chris R. S. Fludger, Andrea Carena, and Vittorio Curri
Abstract—In this paper, we analyze hybrid modulation
formats as an effective technology for the implementation
of flexible transponders that are capable of trading off
the delivered data rate by the lightpath quality of trans-
mission with fine granularity. Flexible transponders are
an enabling technology that can introduce the elastic
paradigm in state-of-the-art networks while maintaining
compatibility with the installed equipment, including
fibers, mux-demux, and reconfigurable optical add–drop
multiplexers, as required by telecom operators willing
to exploit fixed-grid wavelength-division multiplexed
(WDM) transmission. We consider two solutions achieving
different levels of flexibility and employing different
hybridization approaches: time-division (TDHMF) and
quadrature-division (Flex-PAM) hybrid modulation for-
mats. We introduce a comprehensive theoretical assess-
ment of back-to-back performances, analyzing different
transmitter operating conditions, and we provide an
extensive simulation analysis on the propagation of a
Nyquist-WDM channel comb over an uncompensated and
amplified fiber link. After assessing the impact of non-
linear propagation on the maximum signal reach, we present
simple countermeasures for non-linear mitigation and
discuss their effectiveness for both TDHMF and Flex-PAM.
Index Terms—Elastic networks; Flexible transponders;
GN-model; Hybrid formats.
I. INTRODUCTION
The exponential growth of IP traffic [1] together with
an increased level of traffic fluctuations drives the
need for highly efficient and flexible broadband networks,
starting from optical backbones. Optical networks are
evolving toward flexibility, with the purpose of maximizing
the data rate and reach by adapting their operation to the
traffic demand. From the point of view of telecommunica-
tions carriers, there is a firm requirement to have a return
on capex investments for the installed infrastructures,
aiming at solutions avoiding the replacement of equipment
and fibers [2]. Consequently, carriers plan to keep fixed-
grid dense wavelength-division multiplexing transmission
on the installed equipment, introducing network efficiency
and flexibility through the replacement of transponders.
This network evolution based on installed links envisions
to maintain the channel spacing, Δf, and the symbol rate,
Rs, as constant parameters, given by a state-of-the-art
transmitter (Tx) and receiver (Rx).
Thanks to the introduction of fast digital signal process-
ing (DSP) at the Tx/Rx, flexible transponders with the
capability to switch among polarization-multiplexed
(PM) multilevel modulation formats are now available
on the market. They enable network flexibility, allowing
users to trade off the data rate with the lightpath quality
of transmission (QoT), i.e., with the optical signal-to-noise
ratio (OSNR). But such data-rate flexibility is limited to the
cardinality of the selected modulation formats. For in-
stance, if we consider a transponder able to switch among
polarization-multiplexed quadrature phase-shift keying
(PM-QPSK), PM-16-quadrature amplitude modulation
(QAM), and PM-64QAM, delivering 4, 8, and 12 bits
per symbol (BpS), respectively, the available data rates,
given the symbol rate Rs, are only the three values
RbRs· BpS. An increased flexibility could be achieved
by using cross-QAM constellations, such as PM-8QAM
and PM-32QAM [3]. However, contrary to square QAM,
these non-rectangular constellations do not support perfect
Gray coding and also do not allow users to easily modulate
and demodulate the in-phase and quadrature components
as independent pulse-amplitude modulation (PAM) sig-
nals. This may be an important disadvantage for a flexible
transceiver supporting several modulation formats, as it
requires the coexistence of radically different modulation
and demodulation stages.
Targeting an enhanced data-rate flexibility, a plethora
of enabling techniques for the implementation of
flexible transponders have been recently proposed, such
as four-dimensional optimized modulation formats [4,5],
coded modulation [6,7], and rate-adaptive modulation
[8]. Besides the aforementioned techniques, the use of
hybrid modulation formats has also been attracting signifi-
cant attention [9–20], introducing the ability to trade off
with fine granularity the data rate with the QoT. These
are based on mixing different constellations in order to
deliver fractional values of BpS. The hybridization can
be done with respect to time, generating the time-division
hybrid modulation formats (TDHMF), which have been
extensively assessed both in simulations [9–11] and
experimental works [12–18,21]. Alternatively, it is also
possible to hybridize with respect to the four quadratures
(Ix,Qx,Iy,Qy), generating the flexible pulse-amplitude
modulation (Flex-PAM) concept [19,20]. Although
http://dx.doi.org/10.1364/JOCN.8.000A92
Manuscript received January 4, 2016; revised April 18, 2016; accepted
May 10, 2016; published June 9, 2016 (Doc. ID 256637).
F. P. Guiomar (e-mail: fernando.guiomar@polito.it), R. Li, A. Carena, and
V. Curri are with the Department of Electronics and Telecommunications,
Politecnico di Torino, Torino, Italy.
C. R. S. Flugder is with Cisco Optical GmbH, Nuremberg 90411,
Germany.
A92 J. OPT. COMMUN. NETW./VOL. 8, NO. 7/JULY 2016 Guiomar et al.
1943-0620/16/070A92-09 Journal © 2016 Optical Society of America

providing limited bit-rate granularity with respect to
TDHMF, Flex-PAM resorts only to time-invariant
rectangular1QAM constellations, thereby facilitating the
modulation and demodulation stages, which are always
based on the assignment of independent PAM signals to
each quadrature.
In this paper, we provide a comprehensive theoretical
and numerical comparison of TDHMF and Flex-PAM, both
in terms of back-to-back (B2B) and signal propagation per-
formance, targeting a flexible but low-complexity transpon-
der architecture. The main performance-limiting aspects
of TDHMF and Flex-PAM due to fiber non-linearities are
discussed and appropriate countermeasures are presented
and numerically assessed in a Nyquist-wavelength-
division multiplexing (WDM) scenario comprising a wide
range of channel bit rates.
The remainder of this paper is organized as follows. In
Section II, we start by theoretically assessing the B2B per-
formances of both TDHMF and Flex-PAM modulation
formats, then reviewing different transmitter operation
modes and devising the main B2B aspects to be taken into
account for signal propagation. In Section III, we present a
comprehensive simulative analysis of non-linear propaga-
tion of a Nyquist-WDM channel comb over an uncompen-
sated amplified link, deriving the maximum reach for
a large set of conditions. To mitigate the non-linear
propagation impact, we assess two techniques, polarization
interleaving and power-ratio tuning, discussing their
effectiveness. A final comparison of maximum-reach per-
formances with the Gaussian noise (GN) model [22] predic-
tions, used here as a reference, shows that both TDHMF
and Flex-PAM present minor penalties when non-linear
mitigation techniques are used.
II. THEORETICAL FORMULATION
A. Time-Domain Hybrid Modulation Formats
The general structure of a dual-polarization TDHMF
frame is illustrated in Fig. 1, where for simplicity, we only
depict the configurations leading to an integer number of
bits per symbol. In order to reduce the complexity associ-
ated with the transmission and detection of TDHMF, in
this work, we will consider that each periodic frame is com-
posed of Nsymbols distributed among two neighboring
square QAM modulation formats with constellation sizes
M2and 2M2, where Mis the number of levels in each
quadrature. The TDHMF frame is also characterized by
a given frame ratio, κN1∕N, where N1represents the
number of symbols occupied by the modulation format with
the lowest constellation size, M2. In order to counteract the
detrimental impact of time-varying optical power on the
signal propagation performance [11], in the schematic
representation of Fig. 1, we also illustrate the application
of polarization interleaving (PI), which only changes the
frame layout in one polarization tributary, thus balancing
the optical power evolution over time. Nevertheless, from a
B2B perspective, note that PI produces no impact on
the performance of TDHMF, since the two polarization
tributaries are completely orthogonal.
In order to analyze the B2B performance of TDHMF, we
make use of the analytical estimation of the bit error rate
(BER) for an M2-QAM modulation format [23], whose
expression is
ΨQAMSNR;M M−1
Mlog2Merfc0
@
3SNR
2M2
−1
s1
A;(1)
where SNR is the signal-to-noise ratio per symbol and
erfc·is the complementary error function, defined as
erfcx 2

π
pZ∞
x
e−t2dt: (2)
Taking into account the characteristic frame ratio, κ, and
constellation size, M2, the average BER of the TDHMF
frame can then be written as
ΨTDHMFSNRQAM;1;SNRQAM;2;M;κ
4
NBpS κlog2MΨQAMSNRQAM;1;M
1−κlog2M1ΨQAMSNRQAM;2;2M;(3)
where NBpS is the average number of bits per symbol en-
coded in the TDHMF frame,
NBpS 4log2M1−κ;(4)
and SNRQAM;1, SNRQAM;2represent the SNR perceived by
each QAM format, determining the average SNR for the
TDHMF frame, SNRTDHMF, which is given by
SNRTDHMF κSNRQAM;11−κSNRQAM;2:(5)
In general, we can define a given power ratio, PR,
between the two modulation formats such that,
Fig. 1. Structure of a dual-polarization TDHMF frame composed
of two square QAM modulation formats of sizes M2and 2M2.
Envisioning an improved non-linear propagation performance,
polarization interleaving can be applied to balance the distribution
of the optical power over time. For simplicity, only frame structures
that lead to an integer number of bits per symbol are illustrated.
1Note that, contrary to typical rectangular QAM, the Flex-PAM concept al-
lows for different Euclidean distances in the in-phase and quadrature com-
ponents. See Subsection II.C for further details.
Guiomar et al. VOL. 8, NO. 7/JULY 2016/J. OPT. COMMUN. NETW. A93

PR dB10 log10 SNRQAM;1
SNRQAM;2!:(6)
The power ratio will depend on the adopted transmitter
operation strategy and will set the B2B performance
of TDHMF.
B. Flex-PAM
As an alternative flexible modulation scheme, in Fig. 2
we show the proposed dual-polarization Flex-PAM frame
structure. As opposed to TDHMF, Flex-PAM does not
imply time-varying modulation. Instead, the flexibility is
achieved through the hybridization of PAM formats be-
tween the four orthogonal quadratures (in-phase and
quadrature in the two polarizations). Consequently, the
granularity of Flex-PAM is inherently limited to integer
numbers of bits per symbol. Similar to the TDHMF case,
in this work, we will consider that the Flex-PAM frame
is composed of the two nearest-sized PAM formats, whose
constellations include Mand 2Msymbols, respectively.
Under these assumptions, the Flex-PAM frame can be com-
pletely described by the number of levels of its lowest-sized
PAM format, M, and by the corresponding frame ratio, κ.
Similar to TDHMF, PI can also be applied to reduce the
optical power imbalance between the two polarizations.
However, PI is only possible in Flex-PAM for κ0.5,as
shown in Fig. 2. For the remaining frame ratios, it is not
possible to rearrange the frame structure in order to
balance the optical power between polarizations. Other ad-
vanced methodologies must be applied, as will be discussed
in Section III. In order to unequivocally identify the frame
structure when using PI, the Flex-PAM format can be
identified by a vector MM1;M
2;M
3;M
4, where Mi
represents the number of PAM levels in each quadrature,
such that MminM.
To analyze the B2B performance of Flex-PAM, we may
use the BER estimation expression of an M-PAM modula-
tion format [23],
ΨPAMSNR;M M−1
Mlog2Merfc
3SNR
M2
−1
r:(7)
Starting from Eq. (7), the overall BER of a Flex-PAM frame
can be estimated by individually accounting for the BER
incurred in each of its four orthogonal quadratures as
ΨFlex-PAMSNRPAM;1;SNRPAM;2;M;κ
4
NBpS κlog2MΨPAMSNRPAM;1;M
1−κlog2M1ΨPAM SNRPAM;2;2M;(8)
where NBpS is given by Eq. (4).
Similar to the TDHMF case, the average SNR is given by
SNRFlex-PAM κSNRPAM;11−κSNRPAM;2;(9)
with SNRPAM;1and SNRPAM;2now corresponding to the
SNR locally perceived by each PAM format in a single
quadrature, as opposed to Eq. (5), where SNRQAM;1and
SNRQAM;2are measured over both in-phase and quadra-
ture. Since the noise power is set by the propagation
channel but the signal power of M2-QAM is twice of that
of M-PAM, for the same M,
SNRQAM 2· SNRPAM:(10)
Note that the factor of 2 in Eq. (10) compensates for the
factor of 2 between denominators in the arguments of
the erfc·functions of Eqs. (1) and (7). Consequently, the
B2B performance of both strategies is exactly equivalent.
In the following, we provide further details on possible
transmitter operation strategies for Flex-PAM, which
can also be equivalently applied to TDHMF. For the com-
pactness of notation, henceforth we will refer to SNRPAM;1,
SNRPAM;2, and SNRPAM simply as SNR1, SNR2, and SNR.
C. Transmitter Operation Strategies
In this section, we present different transmitter opera-
tion strategies analyzed on a per-channel basis, using
the average BER of the hybrid format as the performance
criterion.
1) Constant Power (PR 0dB): If the power is kept con-
stant for both PAM modulations composing the Flex-PAM,
then the power ratio between formats is PR 0dB and the
SNR perceived by each format is SNR1SNR2SNR.
Using Eq. (8), we obtain the following BER expression:
Ψ1
Mlog2M1−κ2
4κM−1erfc0
@
3SNR
M2
−1
s1
A
1−κM−
1
2erfc0
@
3SNR
4M2
−1
s1
A3
5:(11)
This is the simplest transmitter operation strategy, but, as
we are going to discuss in the following, it is inefficient both
in terms of performance (large B2B penalties) and receiver-
side complexity, since the two PAM/QAM formats operate
Fig. 2. Structure of a dual-polarization Flex-PAM frame com-
posed of PAM modulation formats of sizes Mand 2M.
Polarization interleaving can only be applied for κ0.5, enabling
users to balance the distribution of optical power between the two
polarizations. Note that Flex-PAM inherently only allows for an
integer number of bits per symbol.
A94 J. OPT. COMMUN. NETW./VOL. 8, NO. 7/JULY 2016 Guiomar et al.

at completely different BERs, thus requiring dedicated
forward-error correction (FEC) codes.
2) Same Euclidean Distance (d1d2): In order to guar-
antee the same Euclidean distance, d1and d2, between the
two modulation formats in the Flex-PAM frame, it can be
shown that SNR perceived by each of the PAM formats is
respectively given by
SNR1M2
−1
4−3κM2
−1SNR;(12)
and
SNR24M2
−1
4−3κM2
−1SNR:(13)
Substituting Eqs. (12) and (13) into the BER expression in
Eq. (8) and reducing all its terms as functions of SNR, M,
and κyields
ΨM−κ1
2
Mlog2M1−κerfc
3SNR
4−3κM2
−1
s:(14)
The associated power ratio between the two PAM formats
is then given by
PR 10 log104M2
−1
M2
−1;(15)
imposing a maximum value of ∼7dB for M2and asymp-
totically converging to ∼6dB when increasing M.
3) Same BER (Ψ1Ψ2): In order to impose that the two
modulation formats in the Flex-PAM frame must operate
at the same BER, we must determine the SNR required
by each format to achieve the target BER, Ψtarget, which
can be obtained by inverting the BER expression in Eq. (7),
SNRreq
1M2
−1
3erfc−1Mlog2M
M−1Ψtarget2
;(16)
and
SNRreq
24M2
−1
3erfc−12Mlog2M1
2M−1Ψtarget2
:(17)
The average BER of the entire frame is then obviously
given by Ψtarget, whereas the average SNR is obtained by
substituting SNRreq
1and SNRreq
2in Eq. (9). Finally, the
power ratio between PAM formats is given by
PR 10 log10SNRreq
1
SNRreq
2:(18)
4) Minimum BER (Ψmin ): To achieve the minimum global
BER requires finding the optimum SNR pair,
SNRopt
1;SNRopt
2, that minimizes the estimated BER of
Eq. (8). Taking into account that for any given SNR, the
value of SNR2is automatically determined by the value
of SNR1through Eq. (9), the following optimization pro-
cedure can be applied to determine SNRopt
1:
SNRopt
1SNR1:ΨFlex-PAMSNR1;SNR −κSNR1
1−κ
;M

min ΨFlex-PAMSNR1;SNR −κSNR1
1−κ
;M
;(19)
where as SNRopt
2is given by
SNRopt
2SNR −κSNRopt
1
1−κ
:(20)
The average BER of the entire frame is then given by
Ψ1
Mlog2M1−κ2
4κM−1erfc0
@
3SNRopt
1
M2
−1
s1
A
1−κM−
1
2erfc0
@
3SNRopt
2
4M2
−1
s1
A3
5:(21)
As for the Ψ1Ψ2strategy, the power ratio between PAM
formats is given by
PR 10 log10SNRopt
1
SNRopt
2:(22)
D. Back-to-Back Performance
Employing the previously described transmitter opera-
tion strategies and BER estimation formulas, the theoreti-
cal B2B sensitivity of TDHMF and Flex-PAM is shown in
Fig. 3, in terms of the required average SNR to achieve a
given number of bits per symbol, ranging from 4 (pure
PM-QPSK) up to 12 (pure PM-64QAM). Note that, as pre-
viously highlighted, TDHMF and Flex-PAM are equivalent
Fig. 3. Back-to-back sensitivity of TDHMF depending on the
transmitter operation strategy for a number of bits per symbol
ranging from 4 (PM-QPSK) up to 12 (PM-64QAM). Note that
the sensitivity of Flex-PAM coincides with that of TDHMF for
integer NBpS. The target BER is 2×10−2.
Guiomar et al. VOL. 8, NO. 7/JULY 2016/J. OPT. COMMUN. NETW. A95

in terms of sensitivity for integer number of bits per sym-
bol. Complementarily to the sensitivity analysis of Fig. 3,
the frame structure, power ratio, and required SNR of
Flex-PAM are detailed in Table Ifor the same range of bits
per symbol. Since it can play a key role on the signal per-
formance after non-linear fiber propagation, besides the
power ratio between PAM formats, Table Ialso indicates
the corresponding power ratio between polarization tribu-
taries, PRpol. It is defined as the power ratio between the
y-pol (highest power polarization, according to Fig. 2) and
the x-pol. Note that, for square (κ1) and rectangular
(κ0.5) QAM constellations, the polarization power ratio
is always 0 dB, whereas it is highest for κ0.75.
It is also important to emphasize that, apart from the
PR 0dB strategy, which is clearly sub-optimal, the re-
maining operation strategies are nearly equivalent in
terms of sensitivity. Although the Ψmin operation provides
up to ∼0.25 dB improved sensitivity for the considered
range of bits per symbol, the Ψ1Ψ2strategy is the most
conservative approach: both formats operate at the same
BER, facilitating the FEC coding and decoding. Based on
this observation, we are going to focus on the same BER
transmitter operation strategy for the remainder of this
paper. However, this strategy implies that, whenever the
BERs of the two modulation formats are not exactly the
same (e.g., due to PR mismatch in B2B or due to non-linear
phenomena after the signal propagation in the fiber), the
system performance is limited by the highest BER.
To provide a more in-depth analysis of the consequences
of adopting the same BER operating strategy, in Fig. 4we
show the SNR penalty, ΔSNR, incurred by sweeping the
power ratio between formats. Indeed, the optimum perfor-
mance (ΔSNR 0dB) is obtained at the theoretical values
listed in Table I, 6.46 dB for M2and 5.72 dB for M4.
It is also important to notice that the slope of the curves in
Fig. 4strongly depends on the frame ratio, κ. For power ra-
tios lower than the optimum, the growth of ΔSNR evolves
slower with lower κ, whereas the opposite is true for power
ratios larger than the optimum. Setting a maximum SNR
penalty of 0.2 dB, note that a PR reduction of up to ∼2.5dB
is tolerated by κ0.25, whereas only ∼0.5dB of PR
reduction is tolerated by κ0.75.
Note that, although from the B2B perspective, it does not
make sense to tune the PR, for signal propagation in a fiber,
reducing the PR can be an effective countermeasure for the
mitigation of nonlinear effects, as it allows users to reduce
the power of the higher-cardinality PAM. With the same
objective of reducing non-linear impairments, it has also
been shown that PI can play an important role on the per-
formance of TDHMF [11], as it balances the distribution of
optical power over time (see Fig. 1). In contrast with
TDHMF, the distribution of optical power is inherently con-
stant over time for Flex-PAM. However, it suffers from
power imbalances between polarization tributaries, which
can only be solved through PI for the κ0.5case (see
Fig. 2). In the remaining cases (κ0.25 and κ0.75), it
is not possible to avoid optical power imbalance by simply
rearranging the PAM formats among the four quadratures.
In that case, a possible way of counteracting polarization
imbalance is by reducing the PR, thereby incurring into
the B2B penalty shown in Fig. 4. The relationship between
the quadrature power ratio, PR, and the polarization power
ratio, PRpol, is illustrated in Fig. 5for the different Flex-
PAM configurations. Note that PRpol does not depend on
M, only on PR. It is also worth emphasizing that, similar
to the behavior depicted in Fig. 4, any changes in the polari-
zation power ratio are less sensitive for κ0.25. Note that,
TABLE I
FLEX-PAM FRAME STRUCTURE,POWER RATIO,AND REQUIRED SNR FOR DIFFERENT TRANSMITTER OPERATION STRATEGIES,
CONSIDERING A TARGET BER OF 2×10−2
d1d2Ψ1Ψ2Ψmin
SNRreq PR PRpol SNRreq PR PRpol SNRreq PR PRpol
NBpS log2MFormat Mκ[dB] [dB] [dB] [dB] [dB] [dB] [dB] [dB] [dB]
4 [1111] PM-QPSK 2 1 6.25 0 0 6.25 0 0 6.25 0 0
5 [1112] 2 0.75 9.07 7 4.78 8.94 6.46 4.33 8.76 4.87 3.08
6 [1212] PM-8QAMa2 0.5 10.69 7 0 10.59 6.46 0 10.46 5 0
7 [1222] 2 0.25 11.83 7 2.22 11.78 6.46 2.13 11.72 5.09 1.84
8 [2222] PM-16QAM 4 1 12.71 0 0 12.71 0 0 12.71 0 0
9 [2223] 4 0.75 15.11 6.23 4.15 14.98 5.72 3.74 14.72 3.69 2.23
10 [2323] PM-32QAMa4 0.5 16.57 6.23 0 16.45 5.72 0 16.25 3.9 0
11 [2333] 4 0.25 17.62 6.23 2.08 17.56 5.72 1.98 17.45 4.04 1.57
12 [3333] PM-64QAM 8 1 18.43 0 0 18.43 0 0 18.43 0 0
aNote that the PM-8QAM and PM-32QAM formats indicated in this table are non-square (rectangular) QAM formats.
Fig. 4. SNR penalty (ΔSNR) incurred in B2B by tuning the power
ratio around its optimum value using the Ψ1Ψ2transmitter op-
eration strategy. The target BER is 2×10−2.
A96 J. OPT. COMMUN. NETW./VOL. 8, NO. 7/JULY 2016 Guiomar et al.

while PRpol decreases with decreasing PR at a rate of
∼0.7dB∕dB for κ0.75, there is only ∼0.2dB of PRpol
variation for PR ∈56dB. Figure 4also shows that PI
can completely avoid the polarization power imbalance
for the case of κ0.5, regardless of the quadrature PR.
This B2B analysis will be particularly useful for the in-
terpretation of the signal propagation performance results
introduced in the following section.
III. RESULTS
In a previous work [11], we have shown that the propa-
gation performance of TDHMF can be significantly im-
proved by applying simple countermeasures, such as PI
and electronic pre-distortion, for the mitigation of non-
linear impairments. In this section, we aim to perform
an in-depth assessment of the non-linear propagation per-
formance of both TDHMF and Flex-PAM in uncompensated
amplified links for an extended set of channel net bit rates,
ranging from 100G up to 300G. It is important to remark
that, contrary to Ref. [11], no electronic pre-distortion is ap-
plied in this work, with the aim of avoiding extra process-
ing complexity both at the transmitter and receiver sides.
The simulation setup is based on the transmission of a
13-Nyquist-WDM channel comb propagated over a uniform
uncompensated and amplified multi-span link composed
of 100 km spans of standard single-mode fibers,
with attenuation α0.22 dB∕km, dispersion parameter
D16.7ps∕nm∕km, and non-linearity coefficient γ
1.3W−1km−1. At the end of each span, the fiber loss is fully
recovered by an erbium-doped fiber amplifier with a noise
figure of 5 dB. For simplicity, both the laser phase noise
and frequency offset are neglected, reducing the receiver-
side DSP to a simple least mean squares adaptive filter
with 51 taps. The net symbol rate per channel is
Rs25 Gbaud, corresponding to a gross symbol rate of
32 Gbaud, including the FEC overhead of 20% and the pro-
tocol overhead of 8%. The channel spectra are shaped using
a raised-cosine filter with roll-off factor of 0.2. A 50 GHz
channel spacing has been set in order to test a state-
of-the-art fixed-grid optical link. Targeting the lowest
implementation complexity and taking into account the
B2B discussion in Section II, we have adopted the same
BER (Ψ1Ψ2) transmitter operation strategy. The power
ratio has been set in accordance with Table I, for a
corresponding target BER of 2×10−2.
The first step of our analysis is to evaluate the maximum
transmission distance that can be achieved for each consid-
ered modulation strategy, TDHMF and Flex-PAM, without
applying any countermeasures for non-linear impairments.
The maximum reach is defined in terms of the maximum
number of fiber spans, Nspans, along which the signal can
propagate, while still guaranteeing operation below the
defined target BER. Note that the maximum reach is
obtained by the linear interpolation of the BER results ob-
tained after propagation over a set of transmission distan-
ces, thus yielding fractional values of the maximum reach
in terms of the number of spans. In this context, rather
than a meaningful propagation distance, these fractional
values shall be interpreted as an OSNR margin over the
correspondent integer number of spans. For all cases,
the maximum reach is determined after optimizing the
transmitted power per channel, thus ensuring that the op-
tical signal is being transmitted in the optimal propagation
regime. The results of this simulation campaign are shown
in Fig. 6, in terms of the maximum reach versus the net
channel bit rate. The GN model [22] predictions are re-
ported as a reference and are based on the B2B SNR re-
quirements derived for each case. As expected, without
the implementation of any non-linear mitigation tech-
niques, both TDHMF and Flex-PAM perform worse than
the GN model predictions due to the power imbalance in
time (TDHMF) and in polarization (Flex-PAM). It is also
worth noting that Flex-PAM tends to be significantly more
penalized, especially for the case of κ0.5, where the
polarization power imbalance is highest, as shown in
Fig. 5. The higher power level needed by the higher cardi-
nality format generates an extra non-linear impairment in
fiber propagation, compared to a transmission technique
based on constant power. Consequently, the maximum
reach is significantly reduced with respect to the GN model
Fig. 5. Flex-PAM: impact of PR tuning on the power imbalance
between polarization tributaries (or equivalently, the polarization
power ratio PRpol).
Fig. 6. Maximum reach achieved by TDHMF and Flex-PAM for
different bit rates ranging from 100G to 300G without any non-
linear mitigation countermeasures. Note that both PM-8QAM
and PM-32QAM are indicated in the figure only as equivalent spec-
tral efficiency solutions for 150G and 250G.
Guiomar et al. VOL. 8, NO. 7/JULY 2016/J. OPT. COMMUN. NETW. A97

predictions. It is important to recall that TDHMF has
been tested with a frame composed of only 4 symbols (or
4 time slots), thus approximately reaching its maximum
performance. Note that in Ref. [11], it was shown that
TDHMF tends to significantly degrade its non-linear
propagation performance with the increasing frame size.
Focusing on the Flex-PAM results and considering
M2, we observe a shortening of the maximum reach
of about 25% (1.25 dB) for κ0.75, whereas a smaller pen-
alty of approximately 11.5% (0.5 dB) is found for the case of
κ0.25. This can be justified by the fact that the polari-
zation power imbalance of Flex-PAM is inherently lower
for κ0.25: approximately 2 dB, as shown in Table I.
The same exact pattern can be observed for the case of
M4, but with a reduced penalty: 24% (1.2 dB) and
9.5% (0.4 dB) for κ0.75 and κ0.25, respectively.
Again, this slight penalty reduction is well matched with
the correspondent reduction of PRpol in Table Ifrom
M2to M4. It therefore becomes apparent that the
polarization power imbalance plays a critical role in the
non-linear propagation performance of Flex-PAM.
In order to counteract the polarization imbalance, we
have then applied the PI technique, which has been firstly
proposed for TDHMF [13,18] to mitigate the optical power
imbalance over time. The impact of PI on Flex-PAM can be
observed in Fig. 7, which shows the maximum reach versus
the transmitted power per channel, Ptx, at a 250G net bit
rate. The advantage of PI is clearly visible, enabling ap-
proximately 1 dB (∼20%) improvement both in terms of
the reach extension and optimal power reduction. Such a
behavior confirms that a significant mitigation of the
non-linear propagation effects can be achieved by simply
balancing the power on the two polarizations. Similar re-
sults were also obtained for the 150G net bit rate, which
shares the property of κ0.5and is required to apply
PI in Flex-PAM.
However, as previously stated, for all other bit rates at
which the Flex-PAM is set to operate, where κ≠0.5,PI
cannot be applied. For those specific cases, we assessed
an alternative approach to mitigate the extra non-linear
impairments caused by the polarization power imbalance,
which is based on a fine tuning of the PR [21] with respect
to the theoretical B2B prescriptions, taking advantage of
the PR versus PRpol dependence shown in Fig. 5.
Although its B2B behavior is easy to predict, the effective-
ness of this technique for the reach extension must be
numerically tested, as the power balancing benefits must
overcome the associated SNR penalty experienced by
moving from the optimal working point, as shown in
Fig. 4. The corresponding maximum reach results obtained
for κ0.75 and κ0.25 after PR tuning are depicted
in Fig. 8, in which we can identify two clearly distinct
operation regions:
i) For the κ0.75 case (125G and 225G), an improve-
ment of ∼0.25–0.3dB in the maximum reach has been
achieved by reducing the PR up to ∼0.4–0.5dB. However,
further reducing the PR generates an increasing penalty.
This trade-off is tightly intertwined with the PR depend-
ence of the B2B performance and polarization power ratio,
as previously discussed when commenting on Figs. 4and 5.
The benefit of non-linear mitigation triggered by reducing
PRpol tends to be the dominant effect for small PR reduc-
tion, whereas the fast increase of the B2B SNR penalty
becomes the leading effect for larger PR reduction.
ii) For the κ0.25 case (175G and 275G), there is no sig-
nificant improvement in the maximum reach brought by
PR tuning. As shown in Figs. 4and 5, the benefit of the
slowly decreasing PRpol tends to cancel out with the slowly
increasing SNR penalty in B2B. In practice, improved non-
linear performance is being traded off for the higher B2B
SNR penalty, in such a way that the system performance
tends to remain constant.
Fig. 7. Maximum reach versus launched power per channel for
Flex-PAM at 250G (κ0.5), with and without PI.
(a)
(b)
Fig. 8. Power-ratio tuning of Flex-PAM for (a) κ0.75 (125G and
225G) and (b) κ0.25 (175G and 275G).
A98 J. OPT. COMMUN. NETW./VOL. 8, NO. 7/JULY 2016 Guiomar et al.

The overall effectiveness of the considered non-linear
mitigation countermeasures is summarized in Fig. 9, which
shows the maximum reach versus the net bit rate for
the two hybrid modulation techniques, including the non-
linear impairments’countermeasures. The GN model pre-
dictions are again plotted as a performance benchmark. A
direct comparison with the results of Fig. 6reveals that
all cases based on hybrid modulation formats get an im-
provement in the maximum reach due to the reduction
of non-linear effects, thanks to either PI or PR tuning. For
TDHMF, PI allows users to almost completely nullify the
maximum reach, shortening it with respect to the GN
model predictions for all tested bit rates. It is also clear that
PI is similarly effective for Flex-PAM when applicable, i.e.,
for κ0.5. Finally, for Flex-PAM with κ0.75 and
κ0.25, it is shown that PR tuning can partially shorten
the gap to the GN model predictions. It is worth emphasiz-
ing that, even if PR tuning is less effective for κ0.25,
its associated reach reduction is significantly lower than
that of κ0.75, regardless of the constellation size, due
to a lower polarization power imbalance. Consequently,
Flex-PAM is generally less impaired by non-linearities for
κ0.25. After applying PI and PR tuning to Flex-PAM, the
obtained reach reduction compared to the GN model pre-
dictions is less than 3% for κ0.5, 10% for κ0.25, and
20% for κ0.75.
Finally, it is worth mentioning that although further
performance improvement could be obtained by digital
non-linear compensation, in this paper, we have focused
on the design of simple techniques that do not require
any DSP overhead for non-linear mitigation.
IV. CONCLUSIONS
Bit-rate flexibility is a key feature for future optical tran-
sponders, in order to allow an efficient use of spectral re-
sources in meshed networks with fast-varying traffic
demand. We have theoretically and numerically compared
back-to-back and signal propagation performances of two
promising flexible modulation techniques: TDHMF and
Flex-PAM. Building upon the well-known TDHMF tech-
nique, which exploits time-varying modulation, we propose
and numerically assess the Flex-PAM concept, which al-
lows integer bit-per-symbol granularity resorting to hybrid
PAM modulation among the four orthogonal quadratures
in dual-polarization optical signals.
The numerical simulation analysis has revealed that,
whereas from the B2B perspective, TDHMF and Flex-
PAM are shown to be equivalent for integer bit-per-symbol
granularity, their signal propagation performance is
dominated by rather different non-linear phenomena,
associated with the specificities of their different frame
structures. Polarization interleaving is shown to be a sim-
ple and effective non-linear impairment countermeasure
for TDHMF and also for Flex-PAM with a 50% frame ratio.
Given the impossibility of applying PI to Flex-PAM with
25% and 75% frame ratios, we assessed a simple power-
ratio tuning procedure, which was shown to reduce the
power imbalance between polarizations, thereby mitigat-
ing non-linearities. Although the PR tuning was only found
to be partially effective for the case of the 75% frame ratio,
it was also shown that the non-linear penalty incurred
at the 25% frame ratio is inherently smaller, owing to its
correspondent Flex-PAM frame structure. From a general
system performance perspective, both TDHMF and
Flex-PAM were shown to approach the GN-model predic-
tions in terms of maximum signal reach, incurring small
penalties, provided that the appropriate non-linear propa-
gation countermeasures are applied. Flex-PAM is shown
to be an interesting alternative to the widely studied
TDHMF, providing bit-rate flexibility without requiring
time-dependent modulation.
ACKNOWLEDGMENT
This work was partially supported by the European
Commission through a Marie Skłodowska-Curie
Individual Fellowship, Project Flex-ON (653412), and by
the GARR Consortium through the “Orio Carlini”2014
Grant. We would like to thank the anonymous reviewers
for their insightful comments, which were an important
contribution to improving the paper.
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