# Sphere Shaping with Dispersion-Aware Sequence Selection for Long-Haul
Transmission Systems

Jingtian Liu(1), Élie Awwad(1), Hartmut Hafermann(2), Yves Jaouën(1) E-mail of
corresponding author: jingtian.liu@telecom-paris.fr(1) Communication and
Electronics department, LTCI, Télécom Paris, Institut Polytechnique de Paris,
19 place Marguerite Perey, 91120 Palaiseau, France; (2) Optical Communication
Technology Lab, Paris Research Center, Huawei Technologies, France.

###### Abstract

We introduce a nonlinear-tolerant design approach to Enumerative Sphere
Shaping (ESS) for $M$-QAM signaling, utilizing sequence selection techniques
based on a sign-independent metric: the Energy Dispersion Index, EDI (a scheme
denoted E-ESS) and a novel sign-dependent metric: EDI of a dispersed sequence,
D-EDI (a scheme denoted D-E-ESS). These approaches are designed to minimize
rate loss and enhance transmission performance in nonlinear optical fiber
transmission systems, catering to both short-distance and long-haul scenarios.
Our simulation results reveal significant performance gains over conventional
ESS, with improvements up to $0.4$ bits/4D-symbol. These improvements were
observed over a 205 km single-span standard single mode fiber link in WDM
transmission, with five dual-polarization channels, each operating at a net
rate of $400$ Gbit/s. Furthermore, we demonstrate that D-E-ESS surpasses
conventional ESS by $0.03$ bits/4D-symbol in achievable information rate over
a $30$ span link with $80$ km span in a single-wavelength 8 discrete multi-
bands (DMB) transmission with an $880$ Gbit/s net rate per channel, achieving
similar performance to sequence selection based on a full split-step Fourier
method (SSFM) simulation. We also demonstrate that D-EDI exhibits a more
accurate negative correlation with the transmission performance compared to
EDI. Notably, our proposed D-E-ESS scheme maintains robust performance
improvements even when we consider reduced-complexity versions of it,
consistently delivering throughput enhancements across various block lengths
and selected sequence lengths.

###### Index Terms:

Probabilistic shaping, sequence selection, fiber non-linearity, coherent
transmission systems.

††publicationid: pubid:

##  I Introduction

Numerous studies have shown that achieving high throughput in optical fiber
transmission systems is feasible through the utilization of techniques like
Polarization Division Multiplexing (PDM), combined with $M$-ary Quadrature
Amplitude Modulation (QAM) formats and Forward Error Correction (FEC) codes
[1, 2, 3]. However, this increased efficiency often comes at the cost of
reduced transmission distances due to effects such as OSNR limitation and
Kerr-induced nonlinear distortions [4]. To approach the Shannon limit over an
Additive White Gaussian Noise (AWGN) channel, the concept of constellation
shaping has emerged as a promising strategy. Kschischang and Pasupathy’s work
demonstrated that the probability distribution that maximizes the achievable
information rate over an AWGN channel, while maintaining a fixed average
energy constraint, corresponds to the Maxwell-Boltzmann (MB) distribution [5,
6, 7].

To effectively implement the MB distribution, various Distribution Matching
(DM) techniques have been proposed for Probabilistic Constellation Shaping
(PCS). Notable methods include enumerative sphere shaping (ESS) [8] and
constant composition distribution matching (CCDM) [9]. CCDM [9] has emerged as
a competitive technique for implementing PCS with rectangular or square
$M$-QAM constellations by delivering near-optimal linear shaping gains when
large amplitude blocks are considered [10]. However, in the context of
mitigating Kerr-induced nonlinear interference (NLI) in optical fiber
transmission, long block length DM may exhibit a higher susceptibility to
performance degradation, as highlighted in previous analyses [11, 12, 13].
These works showed that short block length DM can effectively reduce NLI in
optical fiber transmission. Yet, it is crucial to consider that, as the block
length decreases, the rate loss increases. Hence, finding a high-performance
nonlinear tolerant shaping scheme involves striking a trade-off between
nonlinear gain and rate loss [14]. Besides, simulation results in [15, 16]
showed that carrier phase recovery (CPR) in the DSP chain at the receiver side
can exhibit comparable effectiveness to finite-length PCS in mitigating NLI.

To further enhance the nonlinear performance of PCS schemes, recent research
efforts focused on introducing constraints in the trellis grid tree of ESS.
For instance, in [17], the concept of kurtosis-limited ESS (K-ESS) was
introduced as an extension of ESS. K-ESS not only has a maximum energy limit
but also imposes a maximum kurtosis limit on the output sequence. This leads
to a reduced average one-dimensional (1D) amplitude kurtosis, which in turn
offers an additional nonlinear gain. This effect is underpinned by the EGN
model [18], which forecasts a more significant nonlinear penalty associated
with higher kurtosis levels. Subsequently, in [19], Band-Limited ESS (B-ESS)
was introduced, which uses a specific subset of the trellis tree to limit 1D
amplitude variance and kurtosis. B-ESS demonstrated superior performance
compared to both K-ESS and conventional ESS. While both K-ESS and B-ESS
implement sequences in four-dimensional (4D) space while applying energy
limitations over each separate dimension, we proposed in a previous work [20]
a band-limited version of ESS with energy limitations in 4D (two pairs of I
and Q signals), achieving further throughput enhancement. However, for these
three schemes, the rate loss was still significant.

Other design approaches of nonlinear tolerant signaling schemes consist in
manipulating the input signal to the DM to generate multiple candidate
sequences. Then, specific metrics are used to select “good” sequences, i.e.,
sequences generating low NLI. For example, the energy dispersion index (EDI),
introduced in [21], is a metric that quantifies energy changes of the signal
over a given temporal window. EDI showed a negative correlation with the
effective Signal-to-Noise Ratio (SNR) of the CCDM-based transmission; however,
this negative correlation is observed in a single-polarization transmission
scenario. As for ESS-based schemes, our study in [20] highlighted that the
EDI, when calculated over 4D-energy levels, demonstrates a notable negative
correlation for conventional ESS, B-ESS [19], and 4D-BL-ESS over $110$ Gbaud,
$400$ km, $5$ WDM transmission. However, this correlation appears to be
inaccurate in the case of K-ESS [17]. In a different work [22], EDI was
utilized for sequence selection in a scheme called List-Encoding CCDM
(L-CCDM), achieving nonlinear gains with respect to conventional CCDM. Yet,
for long-haul transmissions, CCDM is not the best distribution matcher, as
similar nonlinear gains can be achieved through conventional ESS with shorter
block lengths [14]. More recently, yet another approach, presented in [23],
used EDI and a perturbation-model-based low-pass-filtered symbol-amplitude
sequence (LSAS) metric [24] for sequence selection. These methods have
substantially improved the nonlinear tolerance of ESS or CCDM. However, it is
essential to highlight that most significant gains in ESS schemes have been
observed over short-distance single-span transmissions.

The concept of sequence selection (SS) was first introduced in [25] and has
been further studied in [26]. This innovative signaling approach provided
valuable insights into the computation of a lower bound of the optical fiber
channel capacity. In particular, the analysis clearly showed that there is
considerable room for improvement in optical fiber transmission, especially
for long-haul links. A recent study [27] from the same authors emphasized the
importance of enhancing performance through sequence selection using sign-
dependent metrics in long-distance transmission scenarios. The authors
demonstrated that sign-independent metrics such as EDI, windowed kurtosis, or
LSAS do not yield significant throughput improvements over long-haul
transmissions. They also showed that a sequence selection based on the
computation of NLI through a numerical simulation (Split-Step Fourier Method
or SSFM) of a noiseless single-channel propagation of sequences achieved
throughput enhancements over long-distance transmissions even when an
optimized CPR is applied. However, this selection method is not feasible in
practice due to the high complexity of the SSFM.

Figure 1: Multi-block FEC-independent D-E-ESS sequence selection
implementations. The parity bits from a current FEC-independent D-E-ESS block
are allocated to a subset of the signs in the subsequent block.

In this paper, we introduce a novel sign-dependent metric, the EDI of
dispersed sequences, named D-EDI, which accounts for the influence of
chromatic dispersion throughout the transmission process. Additionally, we
show that D-EDI demonstrates a negative correlation with the effective SNR for
high-rate multi-span transmission scenarios. Then, we propose two new
sequence-selection ESS (SS-ESS) schemes designed for improving nonlinear
tolerance: E-ESS, which uses sign-independent EDI sequence selection, and D-E-
ESS111For an ESS system employing sequence selection based on either EDI or
D-EDI and comprising $n$ cascaded DMs, where each DM includes $\nu$ flipping
bits, we name the scheme $E\text{-}ESS_{n}^{\nu}$ for the EDI-based approach,
and $D\text{-}E\text{-}ESS_{n}^{\nu}$ for the D-EDI-based approach. The
schemes will be detailed in section III., which employs sequence selection
based on sign-dependent D-EDI. We thoroughly investigate and optimize the
performance of these schemes in both single-span and multi-span transmission
scenarios with optimized CPR. Our D-E-ESS scheme demonstrates superior
performance compared to conventional ESS across various block lengths and
complexity levels, in both single-channel and WDM transmission scenarios.
Remarkably, it performs on par with the ideal SSFM-based sequence selection
method, however with reduced complexity. The paper is structured as follows:
in Section II, we recall some building blocks from recent literature that we
reuse in our proposed scheme; then, in Section III, we define the new metric
D-EDI and outline the architecture of the E-ESS and D-E-ESS transmitter
schemes. Section IV focuses on the performance of a single-span transmission
system and gives insights into the optimization of the proposed methods in
this context. In Section V, we study the performance over multi-span single-
channel and WDM transmission systems with CPR at the receiver side. Finally,
Section VI summarizes key findings and conclusions drawn from our
investigations.

##  II Introduction to mapping strategy and Multi-block FEC-independent
sequence selection

###  II-A Mapping Strategy

Distribution matching can be applied across different dimensions, including
1D, 2D, 4D, or even more. Fig. 3 in [28] provides a visual representation of
the mapping strategies employed by ESS schemes up to four dimensions. In the
case of 1D mapping, 4D symbols are shaped using four independent amplitude
sequences, essentially employing one ESS per dimension. In 2D mapping, each
polarization, namely X and Y, utilizes a distinct ESS-shaped amplitude
sequence. In contrast, 4D mapping employs a single-shaped sequence for a dual-
polarization symbol sequence. For example, assuming the amplitude sequence is
$[a_{1},a_{2},\cdots,a_{8}]$, the dual-polarization symbol sequence over two
time slots employing 4D mapping will be:

$\begin{bmatrix}S_{Pol.X,T_{1}}&S_{Pol.X,T_{2}}\\\
S_{Pol.Y,T_{1}}&S_{Pol.Y,T_{2}}\end{bmatrix}=\begin{bmatrix}\pm a_{1}\pm
ja_{2}&\pm a_{5}\pm ja_{6}\\\ \pm a_{3}\pm ja_{4}&\pm a_{7}\pm
ja_{8}\end{bmatrix}$ (1)

This last approach offers the advantage of reducing the likelihood of having
equal amplitudes at the same time slot, resulting in less frequent occurrences
of high peak power values and leading to lower levels of non-linear
distortions. This advantage has been demonstrated in various studies [11, 17,
28]. Hence, in our work, we use 4D mapping of the amplitudes at the output of
the distribution matcher.

###  II-B Multi-Block FEC-Independent Sequence Selection

The goal of the multi-block FEC-independent SS scheme is to reduce the
complexity by moving the FEC from before to after the sequence selection. The
implementation of multi-block FEC-independent sequence selection was proposed
in [27]. As shown in Fig. 1, the parity bits of each FEC-independent D-E-ESS
block (a block that will be presented in Fig. 2 of the next section), are
assigned to a portion of the signs in the subsequent block. In the process of
selecting a specific sequence, these signs are predetermined by the parity
bits from the preceding block. These predetermined signs are required for the
calculation of a sign-dependent metric. It is important to note that, once the
candidate sequences have been generated, their bits become fixed and
unalterable. Consequently, if the FEC encoding is applied after the sequence
selection (SS) process, it becomes necessary to shift the parity bits to the
subsequent block.

##  III E-ESS and D-E-ESS design

Figure 2: Block diagram of FEC independent D-E-ESS in the PAS transmitter, EDI
of dispersed sequences (D-EDI) is determined by averaging the calculated EDI
at multiple locations along an ideal dispersive fiber. The dispersion is
applied at $1$ sample per symbol. EDI is a special case of D-EDI where
$m_{D}=0$. The sign bits are fixed through a multi-block FEC-independent
sequence selection process as shown in Fig. 1. Figure 3: Structure of cascaded
distribution matchers (DMs). The output amplitude sequences of $n$ DMs are
concatenated to form a longer sequence.

The fundamental concept behind E-ESS draws inspiration from the list-CCDM
[22]. Our approach allows for the use of various distribution matchers. In
this study, we specifically focus on ESS. In Fig. 2, we plot the encoding
process where input bits are fed into a 1D ESS distribution matcher (DM) with
block length $l$ that generates $l$ amplitudes. The input bit stream is
composed of two parts: $k-\nu$ information bits and $\nu$ prefix flipping
bits. Altering these $\nu$ prefix flipping bits can cause pronounced changes
in the sequences’ indices during the ESS encoding process. Such changes are
reflected as considerable alterations in the amplitudes across the block.
Conveniently, these flipping bits are discarded after the decoding process at
the receiver side.

We may cascade the outputs of $n$ 1D-DMs sequentially to generate a longer
sequence, which can then undergo a selection process as shown in Fig. 3. This
resulting sequence will have a length of $l\times n$. Cascading DMs have two
advantages: it permits obtaining more candidate sequences while maintaining
the same rate loss, and at the same time, we maintain the use of short-block-
length DMs that show a clear advantage against long-block-length DM with
respect to non-linear effects. For an E-ESS (or a D-E-ESS) with $n$ cascaded
DMs, where each DM includes $\nu$ flipping bits, we can generate $2^{\nu n}$
different candidate sequences. Subsequently, the $l\times n$ amplitudes are
mapped to $(l\times n)/4$ 4D-QAM symbols. E-ESS employs EDI as a metric for
sequence selection, which is a sign-independent metric defined in Eq. (2)
below taken from [21].
$\mathbf{X}=\left[\cdots\mathbf{X}_{i-1},\mathbf{X}_{i},\mathbf{X}_{i+1},\cdots\right]$
represents a series of 4D complex symbols, and $w$ is the EDI averaging window
length, expressed as a number of symbols, and it is directly linked to the
two-sided channel memory, denoted $2M$ and given by
$2M=2\left\lfloor\pi|\beta_{2}|BR_{s}L\right\rceil$. In this formula,
$\beta_{2}$ stands for the group velocity dispersion, $B$ is the channel
spacing, $R_{s}$ is the symbol rate, and $L$ represents the transmission
distance. $\left\lfloor\cdot\right\rceil$ denotes the rounding operation to
the nearest integer. As observed in previous works such as in [21]
(specifically in their Fig. 9), the optimal window length $w$, defined as the
length where the correlation between EDI and SNR is maximized, is typically
less than $2M$. $G_{i}^{w}$ is the sum of the energies of $w+1$ symbols
centered on the symbol $\mathbf{X}_{i}$, as described by Eq. (3). The EDI,
denoted as $\Psi$, is calculated as the ratio of the variance of the energy
within the window (Eq. (4)) to the mean energy within the same window (Eq.
(5)). This calculation is averaged across
$i=\left\\{\frac{w}{2}+1,\cdots,L_{s}-\frac{w}{2}\right\\}$, where $L_{s}$ is
the sequence length. $L_{s}=(l\times n)/4$ in scenarios where 4D mapping is
applied in conjunction with cascaded DMs.

$\Psi\overset{\Delta}{=}\frac{\overline{Var}\left[G^{w}\right]}{\overline{E}\left[G^{w}\right]}$
(2)

$G_{i}^{w}\overset{\Delta}{=}\sum_{j=i-w/2}^{i+w/2}\left|X_{j}\right|^{2}$ (3)

$\overline{Var}\left[G^{w}\right]\overset{\Delta}{=}\frac{1}{L_{s}-w}\sum_{i=\frac{w}{2}+1}^{L_{s}-\frac{w}{2}}Var\left[G_{i}^{w}\right]$
(4)

$\overline{E}\left[G^{w}\right]\overset{\Delta}{=}\frac{1}{L_{s}-w}\sum_{i=\frac{w}{2}+1}^{L_{s}-\frac{w}{2}}E\left[G_{i}^{w}\right]$
(5)

Consequently, the assigned signs to the generated amplitudes are not necessary
for the evaluation of the metric. After 4D-mapping [28], the sequence selector
(EDI calculator) is presented with $2^{\nu n}$ candidate sequences, each with
a length of $(l\times n)/4$ 4D-QAM symbols. The sequence selector then
identifies and transmits the sequence with the lowest 4D-energy EDI.

To further boost the performance of sequence selection over long-haul links,
we introduce sequence selection ESS based on D-EDI. Within the sequence
selector, the EDI of dispersed sequences, abbreviated as D-EDI and denoted as
$\Psi_{D}$ in Eq. (6), is calculated by averaging the EDI values obtained at
multiple points along a linear dispersive fiber without attenuation, as
illustrated in Fig. 2. Here, $\mathbf{X}_{m}$ designates the symbol sequence
that has propagated over $m\times L_{D}$ kilometers of a linear dispersive
fiber without attenuation. $\mathfrak{D}_{D,x}$ in Eq. (7) represents the
chromatic dispersion operator of a fiber with a dispersion coefficient $D$ in
ps/nm/km and length $x$ km.

$\Psi_{D}\overset{\Delta}{=}\frac{1}{m_{D}+1}\sum_{m=0}^{m_{D}}\Psi\left[\mathbf{X}_{m}\right].$
(6)

$\mathbf{X}_{m}\overset{\Delta}{=}\mathfrak{D}_{D,m\times
L_{D}}\left[\mathbf{X}\right].$ (7)

Dispersion is digitally applied over the symbol sequence sampled at one sample
per symbol. D-EDI is a metric that requires knowledge of sign bits to compute
the form of a dispersed sequence. EDI is the special case of D-EDI where
$m_{D}=0$. The sign bits can be fixed through a multi-block FEC-independent
sequence selection process as illustrated in Sec. II-B. In this scenario, the
sign bits are chosen in two parts:a part from the information stream of the
current D-E-ESS block, denoted s, and a part from the FEC output parity bits
of the previous D-E-ESS block, denoted c. Once the QAM sequence is selected,
it is demapped into a bit sequence and sent into a FEC encoder after
discarding the parity bits from the previous D-E-ESS block, c, to generate
parity bits for the next D-E-ESS block.

For a single-span link, one may apply D-EDI by cutting the span into multiple
sections and adding up EDIs computed over the first sections given that
nonlinear distortions are more important at the beginning of each span, where
the signal power is still high. In what follows, to keep the computational
complexity low for a single-span link, we add up two EDIs, the one computed
over the sequences at the transmitter side and the one computed after
propagation over the effective length of the fiber span
$L_{D\mathrm{(single~{}span)}}=L_{eff}=\frac{1-\mathrm{exp}(-\alpha
L)}{\alpha}$ [29] where $\alpha$ is the fiber attenuation in
$\textrm{km}^{-1}$ and $L$ is the length of the fiber span in km. For a multi-
span link of $m$ spans of length $L$ km each, we may add up the EDIs computed
over sequences at the beginning of each span. For the sequence at the
beginning of the last span, we apply dispersion corresponding to
$m_{D}L_{D}=(m-1)L$ km.

Both E-ESS and D-E-ESS increase the rate loss in each dimension by
$\frac{\nu}{l}$ as in [23], which is an extra rate loss term that accounts for
the effect of the flipping bits. The total rate loss $R_{loss}$ in each
dimension is:

$R_{loss}=H(P_{a})-\frac{k-\nu}{l}.$ (8)

where $H(P_{a})$ is the entropy linked to the distribution $P_{a}$ and $P_{a}$
is the marginal distribution of the amplitudes of the PAM symbols. As the
value of $\nu$ increases, the number of candidate sequences also increases,
resulting in higher nonlinear shaping gain. However, this comes at the cost of
increased rate loss. As a result, an efficient sequence selection mechanism
should strike a balance between linear and nonlinear shaping gains. On another
note, the utilization of prefix flipping bits eliminates the need for look-up
tables (LUTs) and XOR operations proposed in the bit scrambling method in
[27], hence simplifying the implementation complexity and avoiding the
additional memory requirements of LUTs. Finally, D-EDI computation has clearly
a much lower complexity compared to the SSFM-based average NLI metric in [27].

Figure 4: Simulation in single channel, $8$ DMB subcarriers, $110$ GBaud
transmission over $30\times 80$ km SSMF link, with a net rate of $880$ Gbit/s.
Sequences A and B are two randomly generated sequences over the
$4^{\mathrm{th}}$ subcarrier, each consisting of $108$ 4D 64-QAM symbols.
Solid lines represent the evolution of EDI at the beginning of each span,
dashed lines represent D-EDI computed as an averaged EDI over all spans. D-EDI
gives a more accurate estimation of NLI generated by each sequence after
transmission.

In Fig. 4, we illustrate the evolution of the EDI metric at the beginning of
each span when transmitting two different random sequences over a $30\times
80$ km link with a single channel, $8$ DMB subcarriers, $110$ GBaud
transmission, and a net rate of $880$ Gbit/s. Each of these sequences
comprises $108$ random 4D 64-QAM symbols over the $4^{\mathrm{th}}$
subcarrier. EDI is calculated at the beginning of each span and both EDI and
D-EDI employ a window length of 2 symbols ($w=2$), that is $1$ symbol before
and $1$ symbol after the current symbol. Before transmission, sequence B
exhibits a lower EDI compared to sequence A. This implies that sequence B will
generate less NLI over the first span than sequence A. However, as the
sequences propagate through the dispersive fiber, their characteristics change
continuously due to the influence of dispersion. After only $4$ spans, the EDI
of sequence B surpasses that of sequence A. Ultimately, when the D-EDI is
computed by averaging EDI over all spans, sequence A yields a smaller D-EDI
even though it had a larger initial EDI. This suggests that sequence A
maintains a higher average nonlinear shaping gain throughout the entire
transmission. This outcome emphasizes the importance of considering sign-
dependent metrics, such as D-EDI, when assessing the nonlinear shaping gain
performance of sequences. Sign-independent metrics may not provide a complete
picture of NLI manifestation, as the influence of dispersion can cause
initially favorable sequences to show deteriorated performance after a certain
propagation distance, while initially unfavorable sequences may show an
improved performance. Hence, D-EDI allows for a more accurate estimation of
NLI generated by each sequence throughout a transmission.

##  IV Performance assessment: Single-span transmission

We study the performance of E-ESS and D-E-ESS in two scenarios: first, a
single-span transmission scenario similar to the one in [17, 19], and second,
a multi-span long-haul transmission scenario using digital-multi-band (DMB)
format. Through the two studies, we observe the effects of non-linear
distortions over various transmission distances. It is important to remind
that these distortions are dominant at higher power levels. Therefore, in a
single-span transmission, they primarily occur over the first kilometers of
the span. In contrast, in a multi-span scenario with discrete optical
amplification, NLI distortions are added at the start of each span.

Figure 5: Simulation of $50$ GBaud E-ESS transmission over $205$ km single
span SSMF link, $5$ WDM channels, $55$ GHz channel spacing, net rate per
channel $400$ Gbit/s. (a): Electrical SNR versus the EDI window length $w$
used in the sequence selector to evaluate the EDI of sequences. $\nu=3$
flipping bits and a single 1D-DM ($n=1$) is used. The power per channel is $9$
dBm. (b): Electrical SNR and GMI at $9$ dBm versus $\log_{2}(N_{S_{108}})$
when cascading 1, 2 or 4 DMs, where $N_{S_{108}}$ is the number of required
sequence selections to generate $108$ 4D-symbols. $E\text{-}ESS_{1}^{3}$
exhibits the highest performance in terms of GMI.

We conduct a performance evaluation of the E-ESS and D-E-ESS systems using PDM
64-QAM modulation with a block length of $l=108$ to align with the conditions
described in [17, 19]. Our transmitted signal comprised $5$ Wavelength-
Division Multiplexing (WDM) channels, each operating at $50$ GBaud, resulting
in a raw data rate of $600$ Gbit/s per channel. For all the tested ESS
schemes, we employed a Low-Density Parity-Check (LDPC) code with a length of
$64800$ bits, following the DVB-S2 standard, and a code rate of $r_{c}=5/6$.
Consequently, the achieved net bit rate amounted to $8$ bits per 4D symbol,
resulting in a net bit rate of $400$ Gbit/s per channel. The channel spacing
is set to $55$ GHz, and we apply root-raised cosine (RRC) pulse shaping with a
roll-off factor of $0.1$ to each channel. Standard single-mode fiber (SSMF) is
simulated with an attenuation coefficient $\alpha=0.2~{}\mathrm{dB/km}$,
chromatic dispersion coefficient $D=17~{}\mathrm{ps/nm/km}$, polarization mode
dispersion (PMD) of $0.04~{}\mathrm{ps/\sqrt{km}}$, and a nonlinear parameter
$\gamma=1.3~{}\mathrm{(W\cdot km)^{-1}}$ at $\lambda=1550~{}\mathrm{nm}$. Our
simulated channel consists in a single-span transmission over a distance of
$205~{}\mathrm{km}$, to perform a consistent comparison with [17, 19] in which
the Kurtosis-limited (K-ESS) and band-limited (B-ESS) sphere shaping were
introduced. We do not add any laser phase noise in the simulation. Following
propagation, the central channel underwent several signal processing steps,
including optical filtering, matched filtering, chromatic dispersion
compensation, and genie-aided multiple-input-multiple-output (MIMO) channel
equalization. Subsequently, we applied a fully-data-aided phase filter with a
window averaging over $64$ symbols to compensate for phase rotation induced by
cross-phase modulation (XPM) as carrier phase recovery (CPR), as elaborated in
[30]. Finally, we measured the electric SNR denoted $\mathrm{SNR_{elec}}$ from
the equalized constellations and computed the generalized mutual information
(GMI) using Eq. (10) from [31].

Figure 6: Simulation of conventional ESS, K-ESS, B-ESS,
$E\text{-}ESS_{1}^{3}$, and $D\text{-}E\text{-}ESS_{1}^{4}$ (best-performing
scheme) with D-EDI computed after applying a single $21$ km dispersion filter.
Same system configuration as for Fig. 5. Left: Electrical SNR versus power per
channel. Right: GMI versus power per channel. Both $E\text{-}ESS_{1}^{3}$ and
$D\text{-}E\text{-}ESS_{1}^{4}$ demonstrate superior performance when compared
to state-of-the art schemes.

First, we focus on the E-ESS scheme. In Fig. 5 (a), we plot the
$\mathrm{SNR_{elec}}$ for various EDI window lengths $w$ used by the sequence
selector. For this test, we used a single 1D-DM ($n=1$) and $\nu=3$ flipping
bits and denote the scheme $E\text{-}ESS_{1}^{3}$ (or in general
$E\text{-}ESS_{n}^{\nu}$). This setup allows us to select the most suitable
sequence from eight possible candidates ($2^{3}=8$). We calculate
$\mathrm{SNR_{elec}}$ using equalized symbols after transmitting over a single
span with a channel power of 9 dBm, which is near the optimal power level. Our
findings show that as $w$ increases, the $\mathrm{SNR_{elec}}$ gradually
declines. We find that a window length $w=2$ yields the best performance, and
is much lower than the channel memory ($2M=76$ when considering the total
length of the span). Therefore, we proceed with $w=2$ in subsequent single-
span simulations. In Fig. 5 (b), we analyze the performance of E-ESS by
varying the number of sequentially cascaded DMs $n$, where each DM has $\nu$
flipping bits. As a complexity metric, we designate the number of tested
candidate sequences per$108$ 4D-symbols. We choose $4$ DMs as $4$ is the least
common multiple between the three studied values of
$n=\left\\{1,2,4\right\\}$. Moreover, having a block length $l=108$ per DM, 4
DMs will generate $108$ 4D symbols. We denote the number of tested sequences
as $N_{S_{108}}$ which is given by:

$N_{S_{108}}=\frac{4}{n}2^{\nu n}$ (9)

A single DM with a block length $l=108$ produces $27$ 4D-symbols after
4D-mapping. Consequently, when cascading $n=1,2,$ or $4$ DMs, the candidate
sequence lengths will be $27,54,$ and $108$ respectively. To achieve a total
sequence of $108$ 4D symbols, the E-ESS transmitter needs to be called
$\frac{4}{n}$ times, resulting in $4,2,$ and $1$ calls for $n=1,2,$ and $4$
cascaded DMs respectively. We see from Fig. 5 (b) that, as $N_{S_{108}}$
increases, $\mathrm{SNR_{elec}}$ also increases for
$n=\left\\{1,2,4\right\\}$. However, as the number of flipping bits increases,
the rate loss likewise increases. The maximum GMI is hence attained when
achieving a trade-off between rate loss and nonlinear gain. Specifically, the
largest GMI is achieved with a single 1D-DM at $\log_{2}(N_{S_{108}})=5$ ($32$
candidate sequences), corresponding to the $E\text{-}ESS_{1}^{3}$
configuration.

Next, we also evaluated the performance of the D-E-ESS scheme over the same
link by computing D-EDI as the sum of the EDI of the sequence at the
transmitter output and its EDI after propagating over a linear dispersive
fiber without attenuation of length $L_{eff}$ over which nonlinear effects
remain significant. $n=1$ and $\nu=4$ ($D\text{-}E\text{-}ESS_{1}^{4}$) yield
the best GMI gains through a similar optimization as the one shown in Fig.
5(b) for E-ESS. In Fig. 6 (a), we show $\mathrm{SNR_{elec}}$ versus the launch
power per channel for five different ESS schemes. The D-E-ESS, E-ESS, band-
limited ESS (B-ESS), and kurtosis-limited ESS (K-ESS) exhibit SNR gains of
approximately $0.8$ dB, $0.6$ dB, $1$ dB, and $0.3$ dB respectively, compared
to the conventional ESS at a power level of $10$ dBm. Additionally, they
demonstrate up to $1$ dB improvement in the optimal launch power. In Fig. 6
(b), we present the results in terms of GMI per 4D symbol. Notably, the D-E-
ESS scheme yields the highest GMI. Specifically, at a power level of $10$ dBm,
it achieves a GMI increase of $0.15$ bit/4D compared to B-ESS and a $0.3$
bit/4D improvement compared to conventional ESS. The E-ESS scheme also
delivers a GMI gain of $0.1$ bit/4D (respectively $0.2$ bit/4D) in comparison
to B-ESS (respectively conventional ESS). It is important to mention that both
EDI and D-EDI based shaping approaches exhibit nearly the same linear
performance as unconstrained ESS due to their small rate loss. E-ESS and D-E-
ESS exhibit similar performance largely because the initial shape of the
waveform at the start of the link plays a crucial role in the generation of
NLI for single span links. However, a slight performance improvement is
achievable by selecting sequences that also exhibit low EDI after the
effective length, $L_{eff}$. In terms of rate loss, D-E-ESS demonstrates
superiority over B-ESS and K-ESS, with a rate loss of $0.26$ bit/4D compared
to $0.5$ bit/4D and $0.27$ bit/4D respectively. The rate losses are not
discernible in Fig. 6 (b) between $5$ dBm and the optimal power level because
non-linear gains are already occurring within these power ranges. D-E-ESS also
outperforms E-ESS in the nonlinear regime. Overall, it outperforms B-ESS in
both linear and nonlinear regimes. Finally, we remind that D-E-ESS requires a
slightly higher complexity compared to E-ESS since it necessitates knowledge
of sign bits and computation of dispersed sequences at the transmitter side.

##  V Performance assessment: Multi-span transmission

We now evaluate the potential advantages of our schemes when applied to higher
baud rate long-haul links employing digital-multi-band (DMB) signals known for
their superior resistance to Kerr effects compared to single-carrier schemes
[32, 33]. The baud rate in this scenario is set at $110$ Gbaud, utilizing $8$
subcarriers in a single-wavelength configuration. The transmission channel is
$30\times 80$ km standard single-mode fiber (SSMF) spans with EDFA
amplification. The achieved net data rate per channel is $880$ Gbit/s. The
digital signal processing (DSP) blocks employed in this scenario are
consistent with those detailed in the preceding section.

###  V-A Optimization and Transmission Performance

In Fig. 7, we evaluate the performance of our D-E-ESS scheme for different
cascaded DMs and different number of flipping bits. In long-distance links,
accumulated chromatic dispersion becomes apparent as its induced inter-symbol
interference involves hundreds of symbols. For these links, it becomes more
important to cascade multiple 1D-DMs to generate extended candidate sequences
over which we will compute D-EDI. By doing so, we account for the impact of
the accumulated channel memory on the waveform, thus making a better selection
process. In addition, for the same complexity level as a single DM, this
approach helps in reducing rate loss. As we did in the previous section, the
complexity of the scheme is evaluated as $N_{S_{108}}$, the number of tested
candidate sequences for the generation of $108$ 4D-symbols. We calculate D-EDI
by summing up EDI values of the sequences at the input of each of the $30$
spans constituting the $30\times 80$ km link ($w=2$ corresponding to the
memory of a $13.75$ Gbaud transmission over a single $80$ km span). We observe
that, as $N_{S_{108}}$ increases, $\mathrm{SNR_{elec}}$ also increases.
However, as the number of flipping bits per DM increases, the rate loss
increases too. In the scenario of long-haul transmissions, the SNR gain is
less significant compared to single-span transmissions. Consequently, the rate
loss has a greater impact on the final performance. The trade-off between rate
loss and nonlinear gain shows that the largest GMI occurs in the configuration
where $n=4$ 1D-DMs are arranged in series, and each 4D-DM has $\nu=1$ flipping
bit, denoted $D\text{-}E\text{-}ESS_{4}^{1}$, and corresponding to
$\log_{2}(N_{S_{108}})=4$ (i.e. $16$ candidate sequences). The GMI curve for
1DM exhibits a significantly steeper slope compared to the one for 4DM, as the
rate loss increases at a slower pace in the case of cascading 4 DMs.

Figure 7: Simulation of D-E-ESS with D-EDI computed after applying dispersion
successively over $30\times 80$ km, single channel, $8$ DMB subcarriers, $110$
GBaud transmission, net rate per channel $880$ Gbit/s. Electrical SNR and GMI
versus $\log_{2}(N_{S_{108}})$ at the optimal power ($4.5$ dBm), where
$N_{S_{108}}$ is the number of tested sequences per $108~{}4$D-symbols.
$D\text{-}E\text{-}ESS_{4}^{1}$ exhibits the highest performance in terms of
GMI. Figure 8: Simulation of conventional ESS, $E\text{-}ESS_{4}^{1}$,
$D\text{-}E\text{-}ESS_{4}^{1}$ and $SSFM\text{-}ESS_{4}^{1}$ with $30\times
80$ km dispersion applier over $1$ channel $8$ DMB subcarriers, and
conventional ESS with $1$ channel $1$ carrier, in $110$ GBaud over $30\times
80$ km SSMF link, net rate per channel $880$ Gbit/s. (a): Electrical SNR
versus Power per channel in the non-linear channel. (b): GMI versus Power per
channel. $D\text{-}E\text{-}ESS_{4}^{1}$ exhibits the same performance as that
of an ideal SSFM-based sequence selection.

Next, in Fig. 8, we compare the performance of four single-wavelength
transmission schemes: conventional ESS over a single carrier (SC) serving as a
baseline, conventional ESS over DMB using $8$ subcarriers (8DMB),
$D\text{-}E\text{-}ESS_{4}^{1}$ (optimal case) schemes and
$E\text{-}ESS_{4}^{1}$, both in 8DMB mode. The $D\text{-}E\text{-}ESS_{4}^{1}$
shows a $0.2$ dB increase in $\mathrm{SNR}_{elec}$ and $0.03$ bit/4D-symbol
gain in GMI at a power of $4.5$ dBm compared to conventional ESS with 8DMB and
it shows a $0.66$ dB increase in $\mathrm{SNR}_{elec}$ and $0.16$ bit/4D in
GMI at the optimal power compared to conventional ESS over a single carrier.
We can also see that, even though the $E\text{-}ESS_{4}^{1}$ has a slight SNR
gain, it does not counterbalance the rate loss, and eventually there is no
gain in GMI with respect to conventional ESS. On the other hand, the sign-
dependent-based D-E-ESS scheme maintains the GMI advantage in both short-
distance and long-haul distance transmissions. Additionally, we conducted
simulations involving sequence selection based on the computation of the
actual NLI produced by each sequence using the Split-step Fourier method (SSFM
selection) to approximate an upper bound for the performance of a sequence
selection scheme. We kept the transmitter structure as 4 concatenated DMs,
each equipped with $1$ flipping bit. This approach is similar to the one
outlined in [27]. Each candidate sequence was transmitted over a $30\times 80$
km standard single mode fiber, single-wavelength, single-carrier system
operating at $13.75$ Gbaud, without the addition of ASE noise. After
transmission, we compare the NLI power of candidate sequences and select the
sequence that generates the lowest NLI. This process allows us to obtain the
optimal performance of sequence selection with $4$ serial DMs, and $1$
flipping bit when only considering self-phase modulation effects. From Fig. 8,
we observe that SSFM-ESS and D-E-ESS exhibit almost identical performance.
This observation indicates that within the analyzed system, selecting
sequences based on D-EDI can be a viable alternative to the more complex SSFM
approach. In other words, the variations of the waveform shaped by dispersion
effects primarily dictates nonlinear distortions in high-power regions.
Nonlinearity, in this context, acts as a small perturbation. Consequently, we
can feasibly substitute SSFM with the dispersion-aware metric, D-EDI, for
sequence selection.

Figure 9: Performance analysis of $D\text{-}E\text{-}ESS_{1}^{4}$ (a):
Electrical SNR and GMI versus $m_{D}$ at optimal power ($4.5$ dBm) where
$m_{D}$ is the maximum number of spans used by the dispersion applier during
sequence selection. Gains are maintained even when we reduce the complexity by
computing D-EDI over fewer spans. (b): D-EDI and EDI versus $m_{D}$. We add
the values for conventional ESS by horizontal dashed lines. Same system
configuration as in Fig.7. D-EDI demonstrates a negative correlation with
$\mathrm{SNR_{elec}}$.

###  V-B Complexity Reduction and Interpretation of the Results

In the above study, we aimed to improve sequence selection by calculating the
EDI at the beginning of each span along the entire transmission link. This
resulted in obtaining $30$ different EDI values, which were then averaged to
derive the final D-EDI. Now, we delve into the impact of the computational
complexity of D-EDI on the achieved performance. To reduce the computational
load of D-EDI, we explore two methods. First, we consider calculating D-EDI
for only the first $m_{D}=\left\\{1,\cdots,29\right\\}$ spans instead of the
entire transmission line. It is worth reminding that, when $m_{D}=0$, D-EDI
simplifies to original EDI. Second, we consider calculating D-EDI by summing
up EDIs after the signal propagates through intervals of $N_{D}>1$ spans
instead of computing it at the beginning of each span ($N_{D}=1$).

In Fig. 9, we explore the first method. In Fig. 9 (a), we present the
performance results in terms of $\mathrm{SNR_{elec}}$ and GMI. We include the
values for conventional ESS as horizontal dashed lines for comparison. SNR
exhibits a steady growth as $m_{D}$ increases. The peak value is achieved when
$m_{D}=29$, implying that D-EDI calculations across the entire link yield the
maximum benefit. After only three spans, the nonlinear gain brought by D-EDI-
based sequence selection overcomes the penalty in increased rate loss,
resulting in a higher GMI than the one obtained with conventional ESS. These
observations underscore the fact that in dispersion unmanaged transmission
systems, nonlinear penalties are equally distributed and independent across
homogeneous spans as predicted by the GN [34] and EGN models [18]. Hence, to
get the most out of sequence selection, it is crucial to calculate D-EDI for
the entire link. Thereafter, in Fig. 9 (b), we illustrate the EDI and D-EDI
values evaluated for the selected candidate sequences for different $m_{D}$
values. EDI is evaluated over $40$ symbols ($w=40$), which is an empirical
value lower than the total channel memory $2M=68$ (corresponding to the
channel memory of a $13.75$ Gbaud transmission over the entire link ($30\times
80$ km)) as explained in Sec. III; whereas D-EDI is assessed by averaging EDIs
computed with a memory of $2$ symbols ($w=2$), corresponding to the memory of
a $13.75$ Gbaud transmission over a single $80$ km span. This D-EDI used for
performance evaluation of the selected sequences is calculated along the
entire link. Notably, As $m_{D}$ increases, D-EDI continues to decrease and
hence exhibits a negative correlation with $\mathrm{SNR}_{elec}$, while EDI
struggles to do so, showing that D-EDI is a better estimator of the
performance of the selected sequences for multi-span transmissions.

Figure 10: Electrical SNR and GMI versus $N_{D}$ for
$D\text{-}E\text{-}ESS_{1}^{4}$, where each $N_{D}$ value represents the
following span indices used to compute D-EDI:
$N_{D}=1\rightarrow[0:1:29];2\rightarrow[0,1:2:29];3\rightarrow[0,1,2:3:29];5\rightarrow[0,1,4:5:29];10\rightarrow[0,1,9:10:29];29\rightarrow[0,1,29]$.
We add the values for conventional ESS by horizontal dashed lines. Same system
configuration as in Fig. 7. Gains are maintained even when we reduce the
complexity by computing D-EDI over several spans.

Next, we explore the second reduced-complexity D-EDI computation method.
Instead of computing EDI at the start of each span, we choose to compute it
for fewer spans as explained in the caption of Fig. 10. The figure shows the
performance variations in terms of $\mathrm{SNR_{elec}}$ and GMI versus
$N_{D}$. We observe that, with increasing $N_{D}$, SNR experiences a gradual
decline. Significantly, even with $N_{D}=10$ (corresponding to EDI
calculations every $800$ km), the system still outperforms conventional ESS.
In this case, computing D-EDI necessitates merely $5$ evaluations, compared to
$30$ when $N_{D}=1$ and resulting in a substantial reduction in computational
complexity for D-EDI. This finding suggests that, for digital multi-band (DMB)
transmissions over long-haul links, we can still achieve gains while using a
reduced computational complexity metric for sequence selection.

To conclude this subsection, we highlight two key observations from the
analyses in Fig. 9 and Fig. 10 for DMB transmissions over long-haul links:
first, a sequence that performs well for D-EDI over fewer spans may not
necessarily maintain its effectiveness over longer links. This observation
underscores the complexity of sequence selection in optical communications. It
is not sufficient to assess a sequence’s suitability based solely on its
performance at a single point, such as the beginning of a link or elsewhere,
or at a part of the total distance. Second, identifying an optimal sequence
does not require exhaustive analysis at the start of every span. Although
dispersion-induced interference accumulates over longer distances, causing
significant power variations and nonlinear phase fluctuations, our goal is to
identify sequences that remain stable over distance. Since interference
typically builds up over an extended period, involving the overlap of several
symbols, it is practical to evaluate sequences at intervals spanning a few
spans, rather than continuously or at every span.

###  V-C The Impact of Block Length and Sequence Length Variations

Up to now, we have used a fixed block length of $108$ for the comparison of
our proposed schemes with state-of-the-art schemes in similar conditions. It
is important to note that in the case of ESS, the block length has a
significant impact on the performance (SNR and GMI). Decreasing the block
length leads to increased nonlinear gain but also results in greater rate
loss. Consequently, the optimal block length for ESS strikes a balance between
maximizing nonlinear gain and minimizing rate loss. In Fig. 11, we plot the
performance of ESS schemes for four different block lengths
$l=\left\\{60,108,200,300\right\\}$. The considered schemes are conventional
ESS alongside D-E-ESS with $16$ candidate sequences and D-E-ESS with $256$
candidate sequences. With the increase of cascaded DMs and flipping bits, the
number of candidate sequences can rapidly increase beyond $16$ or $256$. For
example, when $4$ serial DMs, each with $3$ flipping bits, are concatenated,
there are $2^{4\times 3}=4096$ candidate sequences. For these cases, we
randomly select $16$ or $256$ sequences for selection. In Fig. 11, for each
block length and each number of tested candidate sequences, we only show the
best D-E-ESS performance found after testing various configurations of
cascaded DMs and flipping bits. The main observations are:

  * •

For the conventional ESS, as the block length increases, the SNR gradually
decreases. Concurrently, the shortest block length of $60$ provides a
nonlinear gain that exceeds the relatively high rate loss, resulting in the
maximum achievable GMI.

  * •

For D-E-ESS, as the block length increases, the SNR slightly decreases. It has
its maximum value at a block length of $60$ and then levels off. At a block
length of $60$, the rate loss incurred by ESS itself, along with the
additional rate loss from flipping bits, becomes too substantial for the
achievable rate. Even though D-E-ESS exhibits a larger nonlinear gain, the
final GMI falls short of what ESS offers.

  * •

For longer block lengths, D-E-ESS consistently outperforms ESS due to its
ability to maintain a significant nonlinear gain while mitigating excessive
rate loss. Among these block lengths, D-E-ESS with a block length of $300$
achieves the highest GMI.

  * •

In particular, when the block length is equal to $300$, D-E-ESS with $256$
candidate sequences shows a $0.08$-bit/4D GMI advantage over ESS with the same
block length and a $0.04$-bit/4D advantage over ESS with a block length of
$60$. D-E-ESS with $16$ selections shows similar trends with slightly lower
gains.

Figure 11: Electrical SNR and GMI versus DM block length. We show the optimal
performance for each block length by changing the number of flipping bits and
the number of cascaded DMs. Same system configuration as in Fig.7. D-E-ESS
demonstrates superior performance over different block lengths. Figure 12: GMI
versus selected sequence length. We show the optimal performance versus
different sequence lengths, for various block lengths, by changing the number
of flipping bits and the number of cascaded DMs. Same system configuration as
in Fig.7. D-E-ESS demonstrates superior performance over different sequence
lengths.

For a thorough analysis of the performance of D-E-ESS, we vary the sequence
length under identical system configuration and report the performance in Fig.
12. We explore D-E-ESS configurations with three different block lengths and
vary the number of cascaded DMs, resulting in multiple sequence lengths. For
each sequence length, we report the optimal performance obtained by adjusting
the number of flipping bits and by performing the selection among $16$ (full
lines) or $256$ (dashed lines) candidate sequences. For ease of comparison,
the performance of ESS with four block lengths is also depicted on the graph
using horizontal dashed lines. When the block length is set to $300$, D-E-ESS
shows the best performance due to lower rate loss and comparable nonlinear
behavior with respect to shorter block lengths. $256$ selections tend to
perform better for longer sequences because longer sequences necessitate
cascading more DMs in series, hence providing more variability in the
candidates for selection. With $256$ selections, D-E-ESS can more effectively
explore and utilize these possibilities. D-E-ESS significantly outperforms ESS
across almost the entire range of tested sequence lengths for block lengths of
$200$ and $300$. D-E-ESS with optimal sequence-length range ($100$ to $200$ 4D
symbols) also surpasses the performance of ESS with block lengths of $60$ and
$108$. The optimal sequence length for all four block lengths falls within the
range of $100$ to $200$ 4D symbols. We explain the observed behavior by the
fact that overly long sequences capture more channel memory, yet they lead to
coarser selections (at a fixed number of selections to limit the complexity).
On the other hand, while very short sequences offer finer selections, they
only capture a small fraction of the channel memory. Consequently, for the
specific transmission scenario discussed in this article, the balance point
between channel memory and selection precision lies within the range of $100$
to $200$ 4D symbols.

Figure 13: Electrical SNR and GMI versus DM block length. We show the optimal
point versus block length by changing the number of flipping bits and the
number of cascaded DMs. Same system configuration as in Fig.7, however with
$5$ WDM channels. D-E-ESS demonstrates superior performance in WDM
transmission scenario.

###  V-D WDM Transmission Performance

Finally, we maintain the same system configuration as before but extend it to
a $5$-channel WDM transmission. The combined net transmission rate for all
five channels is $4.4$ Tbits/s. In our earlier investigation, we observed poor
performance for D-E-ESS with a block length of $60$. Consequently, we assess
the performance starting with a block length of $108$. In Fig. 13, we see
that, as the block length increases, the $\mathrm{SNR_{elec}}$ for ESS
gradually decreases. However, the increase in GMI due to limited rate loss
compensates for the reduction in nonlinear gain, yielding an optimal block
length of $300$ for ESS. Conversely, D-E-ESS better maintains nonlinear gains
as the block length increases, and its rate loss continues to decrease. This
results in D-E-ESS achieving the maximum GMI at a block length of $300$.
Particularly, with $16$ selections, D-E-ESS with a block length of $300$
outperforms ESS by $0.02$ bits/4D-symbol, and it achieves a $0.06$
bits/4D-symbol improvement compared to ESS with a block length of $60$.
Lastly, it is important to note that $256$ selections, which require more
flipping bits, lead to higher rate loss. The additional nonlinear gain does
not fully compensate for the performance degradation caused by the increased
rate loss, resulting in a lower GMI than with $16$ selections. In a WDM
configuration, the gains are lower than the ones observed in a single-channel
system as in WDM, a high number of Cross-Phase Modulation components arise and
our selection metric does not account for the energy dispersion of the
neighboring channels.

##  VI Discussion and Conclusion

We have proposed two reduced-complexity shaping schemes based on sequence
selection and ESS. The first is based on a sign-independent metric (EDI), and
the second is based on a sign-dependent metric (D-EDI). These new schemes
achieved higher gains than state-of-the-art ESS schemes over short
transmission distances. Noticably, D-E-ESS showed gains even for longer
transmission distances and higher baud rates over single-channel and WDM
systems even when CPR is used to reduce the impact of non-linear phase
rotations. Concurrently, we showed that D-EDI can serve as a prediction metric
of the generation of NLI because it showed a negative correlation with the
measured performance of long-distance transmission systems. Moreover, D-E-ESS
retained a robust performance at moderate complexity (lower number of digital
dispersion operations to compute the metric) and outperformed conventional ESS
across different block lengths. These results pave the way for further
optimizations of shaping schemes for long-distance transmissions and confirm
the necessity of using sign-dependent and channel-aware metrics for
effectively mitigating nonlinear distortions using sequence selection.
Nevertheless, these findings tell us that there is still room for improvement.
The current probability shaping approach, based on sequence selection,
revolves around three primary considerations:

  1. 1.

Low-complexity candidate sequence generator: the method aims to maintain a low
level of complexity for the generation of candidate sequences, allowing for a
wider range of performance fluctuations. This enables a higher probability of
selecting the optimal sequence and helps to improve the upper limit of
sequence-selection performance.

  2. 2.

Accurate and low-complexity measurement metric: precise measurements are
employed to assess the nonlinear performance of each candidate sequence. These
measurements play a crucial role in determining which sequence is selected for
transmission.

  3. 3.

Matching of fixed sign bits and FEC encoding process in PCS schemes: another
key consideration is to minimize complexity and reduce latency when matching
fixed signs to FEC codewords. This optimization helps in improving the overall
efficiency and applicability of the scheme.

In this paper, we improved the first two considerations. First, we used
flipping bits to generate candidate sequences, instead of using LUTs and XOR-
based bit scrambling methods as in [27]. This approach reduces the complexity
of the scheme and avoids the additional memory requirements of LUTs. However,
our D-EDI-based sequence selection exhibited almost the same performance
improvement as the single-channel SSFM-based selection. This implies that we
have reached the single-channel performance limit of sequence selection by
flipping bits to generate sequences. To achieve further gains, we need to find
better methods for the generation of sequences that exhibit larger possible
performance variations, thus improving the upper limit of sequence-selection
performance.

Second, we proposed a novel metric D-EDI which shows a negative correlation
with the nonlinear performance of the candidate sequence, and helps in
achieving the same performance as the SSFM-based selection [27], however at a
much lower complexity. Yet, for DMB-based transmissions in which the
transmission rate per subcarrier is lower than single-carrier transmissions,
the impact of channel memory is limited to a few symbols per span or even less
than 1 symbol per span, hence only a few dozen symbols for all spans. Thus,
the impact of channel memory can be accounted for by concatenating several DMs
to minimize the mutual influence of adjacent selected sequences. As the
transmission rates continue to increase, for both multi-carrier or single-
carrier transmissions, the channel memory can spread over hundreds or
thousands of symbols. At the same time, increasing the length of candidate
sequences also increases the complexity of the scheme. Hence, modifications
and optimizations of D-EDI to achieve excellent performance predictions under
high-rate long-distance transmission while maintaining a reduced complexity is
still an open question. Other improvements of the metric may address different
dispersion management schemes or take into account inter-channel non-linear
effects. In this work, we only considered long-haul dispersion-unmanaged
systems where the NLI contributions from different spans can be considered
independent and D-EDI gives a good performance prediction by summing up EDIs
of the sequence at the start of each span.

Third, in this work, we used the multi-block FEC-independent structure
proposed by [27], which fixes the signs of symbols and combines the fixed-
signs sequence selection with FEC. However, the delay in this structure tends
to be significant due to the encoding process involving multiple blocks,
leading in this case to severe latency as the receiver should wait for the
last block to be received before decoding. A lower-latency and lower-
complexity matching method between fixed signs and FEC codewords is still to
be discovered.

Ultimately, an extended goal involves conducting an in-depth analysis of the
selected “good” sequences. By comprehensively understanding their properties,
we aim to design simpler sequence generation solutions.

## Acknowledgments

This work was funded by Huawei Technologies France. We thank Yann Frignac for
fruitful discussions.

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