# Capacity Limitation and Optimization Strategy for Flexible Point-to-Multi-
Point Optical Networks

Ji Zhou, Haide Wang, Liangchuan Li, Weiping Liu, Changyuan Yu, and Zhaohui Li
Manuscript received; revised. This work was supported in part by the National
Natural Science Foundation of China (62371207, 62005102); Hong Kong Scholars
Program (XJ2021018). (Corresponding authors: Ji Zhou and Liangchuan Li),
(e-mail<EMAIL_ADDRESS>liliangchuan@huawei.com).J. Zhou, H. Wang, and W.
Liu are with the Department of Electronic Engineering, College of Information
Science and Technology, Jinan University, Guangzhou 510632, China.Z. Xing and
L. Li are with Optical Research Department, Huawei Technologies Co Ltd,
Dongguan, 523808, China.C. Yu is with the Department of Electronic and
Information Engineering, The Hong Kong Polytechnic University, Hong Kong.Z. Li
is with the Guangdong Provincial Key Laboratory of Optoelectronic Information
Processing Chips and Systems, Sun Yat-sen University, Guangzhou 510275, China

###### Abstract

Point-to-multi-point (PtMP) optical networks become the main solutions for
network-edge applications such as passive optical networks and radio access
networks. Entropy-loading digital subcarrier multiplexing (DSCM) is the core
technology to achieve low latency and approach high capacity for flexible PtMP
optical networks. However, the high peak-to-average power ratio of the
entropy-loading DSCM signal limits the power budget and restricts the
capacity, which can be reduced effectively by clipping operation. In this
paper, we derive the theoretical capacity limitation of the flexible PtMP
optical networks based on the entropy-loading DSCM signal. Meanwhile, an
optimal clipping ratio for the clipping operation is acquired to approach the
highest capacity limitation. Based on an accurate clipping-noise model under
the optimal clipping ratio, we establish a three-dimensional look-up table for
bit-error ratio, spectral efficiency, and link loss. Based on the three-
dimensional look-up table, an optimization strategy is proposed to acquire
optimal spectral efficiencies for achieving a higher capacity of the flexible
PtMP optical networks.

###### Index Terms:

Flexible PtMP optical networks, theoretical capacity limitation, entropy-
loading DSCM, three-dimensional look-up table, optimization strategy.

##  I Introduction

Point-to-multi-point (PtMP) optical networks with hub-to-leaves architecture
become the main solutions for the network-edge applications such as optical
access networks and optical metro networks [1, 2, 3]. Compared to traditional
point-to-point (PtP) optical networks, PtMP optical networks require less
optical transceivers, thus have lower capital expense (CapEx) and operating
expense (OpEx) costs [4, 5, 6]. In current commercial PtMP optical access
networks, only time-division multiple access (TDMA) has been widely applied,
which is statistical multiple access to make full use of bandwidth [7, 8, 9].
However, the TDMA-based PtMP optical networks naturally have a high latency,
which faces the challenge in the latency-sensitive scenarios [10, 11, 12].
Particularly, the low-latency requirement becomes stricter and stricter in the
future optical networks[13, 14, 15].

Owing to the development of coherent optical techniques, entropy-loading
digital subcarrier multiplexing (DSCM) has been commercially applied in the
800Gb/s long-haul optical communications [16, 17, 18]. Furthermore, coherent
entropy-loading DSCM can implement the revolutionary frequency-division
multiple access (FDMA) for the PtMP optical networks [19, 20, 21]. For the
leaves of the PtMP optical networks, the FDMA can provide the independent
frequency channel to achieve low latency, and the entropy loading allocates
maximal spectral efficiency to approach high capacity [22, 23, 24]. Meanwhile,
the leaves of FDMA can use a relatively low-bandwidth transceiver matching the
subcarrier bandwidth but that of TDMA needs a whole-bandwidth transceiver
matching the signal bandwidth. Unfortunately, the high peak-to-average-power-
ratio (PAPR) of the entropy-loading DSCM signal causes a low optical power
budget, thus limiting the capacity in the peak-power-constrained (PPC) PtMP
optical networks [25, 26, 27].

In this paper, the entropy-loading digital subcarriers with well-matched
spectral efficiencies are assigned to the leaves depending on their different
link losses, which can approach the highest possible capacity for the flexible
PtMP optical networks. Meanwhile, the clipping operation is used to reduce the
PAPR of the entropy-loading DSCM signal, thus increasing the optical power
budget and gaining more capacity. Furthermore, the theoretical capacity
limitation and optimization strategy will be investigated, which can guide the
optimal parameter setting for the flexible PtMP optical networks. The main
contributions of this paper are as follows:

  * •

The theoretical capacity limitation is derived considering the clipping
operation for the flexible PtMP optical networks. Meanwhile, an optimal
clipping ratio is acquired based on the theoretical capacity limitation.

  * •

We establish a three-dimensional look-up table for bit-error ratio (BER),
spectral efficiency, and link loss, and propose an optimization strategy to
approach the capacity limitation of the flexible PtMP optical networks.

The remainder of this paper is organized as follows. The capacity limitation
for flexible PtMP optical networks is derived in Section II. In Section III,
an optimization strategy for flexible PtMP optical networks is introduced in
detail. Finally, the paper is concluded in Section IV.

##  II Capacity Limitation for Flexible PtMP Optical Networks

Figure 1 shows the schematic diagram of the DSCM-based flexible PtMP optical
networks. At the hub, the optical DSCM signal is generated and sent to the
leaves. The transmitted subcarriers are set to the same power. After the
optical distribution network (ODN), the leaves select and detect their
corresponding subcarriers by coherent optical detection. In the actual
situation, the optical signals to the different leaves pass the different
numbers of optical splitters and different lengths of fiber, leading to
different link losses. Based on the on-hand statistic of the deployed PON, the
maximum difference of the link losses among the leaves is probably larger than
10 dB [28]. Therefore, the received signal at the different leaves has
different power and signal-to-noise ratio (SNR). By considering the difference
of the leaves, we will investigate the theoretical capacity limitation for the
DSCM-based flexible PtMP optical networks, and simultaneously use the clipping
operation to increase the capacity.

Figure 1: The schematic diagram of the DSCM-based flexible PtMP optical
networks with different link losses.

At the hub, the DSCM signal is generated, of which the subcarriers are set to
the same average power $P_{\text{t}}$. After the ODN and fiber transmission,
the $i$-th subcarrier is selected and detected by the $i$-th leaf. The
received signal power of the $i$-th subcarrier can be defined as

$P_{\text{r-}i}=P_{\text{t}}/Loss_{i}=P_{\text{DSCM}}/(N\times Loss_{i})$ (1)

where ${P_{\text{DSCM}}}$ is the average power of the DSCM signal. $N$ is the
subcarrier number in the DSCM signal. $Loss_{i}$ is the link loss of the
$i$-th subcarrier.

Assuming that the devices for the leaves have good uniformity, all the
subcarriers suffer from the white noise with the same power. Therefore, the
SNR of the DSCM signal on $i$-th subcarrier is expressed as

$SNR_{i}=P_{\text{r-}i}/\sigma^{2}_{\text{n}}=P_{\text{DSCM}}/(N\times
Loss_{i}\times\sigma^{2}_{\text{n}})$ (2)

where $\sigma^{2}_{\text{n}}$ is the variance of the white noise on each
subcarrier. Obviously, the increase of ${P_{\text{DSCM}}}$ directly improves
the SNR. However, a high-PAPR DSCM signal has a low average power due to the
constrained peak power in the PPC PtMP optical networks. The clipping
operation can effectively reduce the PAPR of the DSCM signal to increase the
average power, which can be expressed as

$x_{\text{c}}=\left\\{\begin{array}[]{cc}A,&x~{}>~{}A\\\
x,&\left|x\right|\leq~{}A\\\ -A,&x<-A\end{array}\right.$ (3)

where $A$ is the clipping amplitude. The clipping ratio can be defined as
$20\times\text{log}_{10}(\eta)$ where $\eta$ is equal to
$A/\sqrt{P_{\text{DSCM}}}$. After the clipping operation, the frequency-domain
signal on each subcarrier is expressed as

$X_{\text{c}}=\alpha\times X+Noise_{\text{c}}$ (4)

where $X$ is the frequency-domain signal on each subcarrier before the
clipping operation. $Noise_{\text{c}}$ denotes the clipping noise. The
clipping attenuation $\alpha$ can be calculated by

$\alpha=\frac{E(x_{\text{c}}\cdot
x)}{E(x^{2})}=\frac{\int_{-\infty}^{\infty}x_{\text{c}}\cdot x\cdot
p(x)\text{d}x}{P_{\text{DSCM}}}=1-2Q(\eta)$ (5)

where $p(x)$ is the probability distribution function of the DSCM signal. Due
to the central limit theorem, the DSCM signal with enough subcarriers becomes
a Gaussian distribution with zero mean, which can be expressed as

$p(x)=\frac{1}{\sqrt{2\pi
P_{\text{DSCM}}}}e^{-\frac{x^{2}}{2P_{\text{DSCM}}}}.$ (6)

$Q(x)$ denotes the Q function, which can be defined as

$Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx.$ (7)

Figure 2: The simulation architecture of the flexible PtMP optical networks
based on the entropy-loading DSCM.

Based on the Eqs. (4) and (5), the average power $P_{\text{c}}$ of the
clipping noise $Noise_{\text{c}}$ can be calculated by

$\begin{split}P_{\text{c}}&=E(X_{\text{c}}^{2})-\alpha^{2}E(X^{2})\\\
&=\int_{-\infty}^{\infty}x_{\text{c}}^{2}\cdot
p_{\text{c}}(x)dx-\alpha^{2}P_{\text{DSCM}}\\\
&=2A^{2}Q(\eta)+\int_{-A}^{A}x^{2}p(x)dx-\alpha^{2}P_{\text{DSCM}}\\\
&=2P_{\text{DSCM}}\left\\{Q(\eta)[1+\eta^{2}-2Q(\eta)]-\frac{\eta}{\sqrt{2\pi}}e^{-\frac{\eta^{2}}{2}}\right\\}\\\
\end{split}$ (8)

where $p_{\text{c}}(x)$ is the probability distribution of the clipping DSCM
signal, which can be defined as

$p_{\text{c}}(x)=\left\\{\begin{array}[]{cc}p(x),&\left|x\right|\leq A\\\
Q(\eta)\delta(\left|x\right|-A),&\left|x\right|>A\end{array}\right.$ (9)

where $\delta(x)$ is the Dirac delta function with a unit impulse.

In the PPC PtMP optical networks, the signal peak is limited to constant value
$A_{\text{p}}$ due to the nonuse of expensive optical amplifier. The peak
amplitude $A$ of the clipping DSCM signal is matched to the $A_{\text{p}}$. We
define a matching coefficient $\beta$ as $A_{\text{p}}/A$ for the clipping
DSCM signal. The effective SNR on $i$-th subcarrier of the clipping DSCM
signal depends on the power of the clipping signal, clipping noise, and white
noise, which can be calculated by

$ESNR_{i}=\frac{\alpha^{2}\times\beta^{2}\times
P_{\text{DSCM}}}{\beta^{2}\times P_{\text{c}}+N\times
Loss_{i}\times\sigma_{\text{n}}^{2}}.$ (10)

Obviously, the $ESNR_{i}$ depends on the clipping ratio $\eta$ and the link
loss $Loss_{i}$. When the link loss of the $i$-th subcarrier is confirmed,
there is an optimal clipping ratio to maximize the effective SNR of the $i$-th
subcarrier. The optimal clipping ratios are different among the subcarriers
with different link losses. Considering the effective SNRs of all the
subcarriers, an optimal clipping ratio can be obtained to maximize the
capacity of the DSCM-based flexible P2MP network. The theoretical capacity
limitation of the DSCM-based flexible P2MP network can be defined as

$C=\sum_{i=1}^{N}C_{i}=B\sum_{i=1}^{N}\text{log}_{2}(1+ESNR_{i})$ (11)

where $B$ is the bandwidth of one subcarrier. The optimal clipping ratio to
maximize the capacity can be acquired by

$\eta_{\text{opt}}=\underset{\eta}{\arg\max}~{}B\sum_{i=1}^{N}\text{log}_{2}(1+ESNR_{i}).$
(12)

When the link losses of the leaves are measured, the theoretical results based
on Eq. (12) can get the optimal clipping ratio to achieve the highest capacity
for the DSCM-based flexible PtMP optical networks.

Figure 3: The effective SNR versus the clipping ratio for the subcarriers with
different link losses. The solid and dashed lines denote the simulation and
theoretical results, respectively.

Figure 2 shows the simulation architecture of the flexible PtMP optical
networks based on the entropy-loading DSCM. First of all, the SNR of each leaf
should be estimated to calculate the spectral efficiency for implementing
entropy loading. At the transmitter, the probabilistic-shaping 64-level
quadrature amplitude modulation (PS-64QAM) for each subcarrier is mapped
depending on the calculated spectral efficiency of each leaf. After the root-
raised-cosine-shaping (RRC) shaping and frequency upshift, the PS-64QAM-
modulated subcarriers are multiplexed to generate an entropy-loading DSCM
signal. The clipping operation is applied in the entropy-loading DSCM signal
to reduce the PAPR. After the normalization to the same peak amplitude
$A_{p}$, the different losses and the white noise with the same power are
added to the signals on the subcarriers for the different leaves. At the
receiver, the frequency downshift and matched filter are used to select the
subcarrier and recover the PS-64QAM signal for each leaf. Finally, the
recovered PS-64QAM signal is used to estimate the SNR and calculate the BER.

Figure 3 depicts the effective SNR versus clipping ratio for the subcarriers
with different link losses. The solid and dashed lines denote the simulation
and theoretical results, respectively. The subcarrier number $N$ is set to 8.
The losses of the leaves are set to the $\boldsymbol{Loss}$ of [1, 1.33, 1.74,
2.32, 3.05, 4.03, 5.25, 6.53]. The noise power $\sigma_{\text{n}}^{2}$ and the
peak value $A_{p}$ are set to 0.0237 and 2.579, respectively. The
theoretically effective SNR can be calculated by Eq. (10). Obviously, the
effective SNR obtained by the simulations coincides with the theoretically
effective SNR well. The effective SNR is improved with the decrease of
clipping ratio at the beginning owing to the increasing signal power and then
decreased with the decrease of clipping ratio due to the increasing clipping
noise. There is an optimal clipping ratio to maximize the effective SNR for
each subcarrier. However, the optimal clipping ratios are different among the
subcarriers with different losses.

Figure 4: The theoretical capacity limitation versus the clipping ratio for
the DSCM-based flexible PtMP optical networks when the symbol rate per
subcarrier is 8Gbaud.

The theoretical capacity limitation can be calculated by Eq. (11) for the
flexible PtMP network. Meanwhile, the optimal clipping ratio can be obtained
by Eq. (12) to achieve the highest capacity limitation. Fig. 4 shows the
theoretical capacity limitation versus clipping ratio for the DSCM-based
flexible PtMP optical networks when the symbol rate per subcarrier is 8Gbaud.
When the clipping operation is not used (i.e. a large clipping ratio is set
such as 13dB), the capacity limitation of the DSCM-based flexible PtMP optical
networks is approximately 263Gb/s. When the clipping operation is used and the
clipping ratio is set to the optimal value of $7$dB, the capacity limitation
of the DSCM-based flexible PtMP optical networks is approximately 361Gb/s. The
capacity limitation of the DSCM-based flexible PtMP optical networks with the
optimal clipping operation is approximately 98Gb/s higher than that of the
DSCM-based flexible PtMP optical networks without clipping operation.

##  III Optimization Strategy for flexible PtMP Optical Networks

In this section, we will investigate the optimization strategy to increase the
capacity of the DSCM-based flexible PtMP optical networks. A three-dimensional
look-up table for BER, spectral efficiency, and link loss will be established
to acquire the spectral efficiencies for the subcarriers with different losses
under the same targeted BER.

Figure 5 depicts the BERs of the subcarriers for the DSCM-based flexible PtMP
optical networks without and with the clipping operation. When the clipping
operation is not employed, the spectral efficiencies of the 8 subcarriers are
set to the $\boldsymbol{SE}$ of [4.80, 4.40, 4.00, 3.60, 3.20, 2.80, 2.40,
2.00] bits/symbol depending on the $\boldsymbol{Loss}$ of [1, 1.33, 1.74,
2.32, 3.05, 4.03, 5.25, 6.53]. Under the $\boldsymbol{SE}$, the BER of each
subcarrier is on the forward error correction (FEC) limit with 7% overhead.
When the clipping operation is used and the clipping ratio is set to optimal
7dB, the BER performance of each subcarrier is much improved, particularly the
subcarrier with low spectral efficiency. To obtain a higher capacity, the
spectral efficiency of each subcarrier can be increased until the BER
approaches the 7% FEC limit. In this section, we will establish a three-
dimensional look-up table for the BER, spectral efficiency, and link loss.
When the link losses of the leaves and targeted BERs of the subcarriers are
confirmed, the corresponding spectral efficiency for each subcarrier can be
obtained from the three-dimensional look-up table to achieve the highest
capacity for the DSCM-based flexible PtMP optical networks.

Figure 5: The BERs of the subcarriers for the DSCM-based flexible PtMP optical
networks without and with the clipping operation.

For obtaining the three-dimensional look-up table, a theoretical BER should be
derived considering the white noise and clipping noise. The PS-64QAM signal
can be decomposed into independent in-phase and quadrature amplitudes, which
are PS 8-level pulse amplitude modulation (PS-8PAM) signals. Gray code is
applied to generate the PS-8PAM signals. Fig. 6 (a) shows the probability
density function for the in-phase PS-8PAM of the PS-64QAM in the clipping
entropy-loading DSCM signal. It demonstrates that the clipping noise does not
coincide with the Gaussian distribution. Figs. 6 (b)-(e) depict the
probability density functions of clipping noise on the amplitudes of 1, 3, 5,
and 7, respectively. Obviously, the distributions of the clipping noise on
different amplitudes are different. The piecewise power exponential model is
used to get the accurate fitting curves for the probability density function
of the clipping noise on the different amplitudes as the red lines in Figs. 6
(b)-(e) show. Based on the piecewise power exponential model, the theoretical
probability density function of the clipping noise can be defined as

$f_{Y_{\text{c}}}(y\mid k)=\left\\{\begin{array}[]{cc}A_{k,1}\times
e^{\frac{-\left|y-\mu_{k,1}\right|^{b_{k,1}}}{2\sigma_{k,1}^{2}}},{}&y\leq
D_{k}\\\ A_{k,2}\times
e^{\frac{-\left|y-\mu_{k,2}\right|^{b_{k,2}}}{2\sigma_{k,2}^{2}}},{}&y>D_{k}\end{array}\right.$
(13)

where $k$ is the amplitudes of [$-7$, $-5$, $-3$, $-1$, 1, 3, 5, 7] for the
in-phase amplitude of the PS-64QAM in the clipping entropy-loading DSCM
signal. $D_{k}$ is the value corresponding to the highest probability near the
amplitude of $k$. $A_{k,1}$, $\mu_{k,1}$, $b_{k,1}$ and $\sigma_{k,1}^{2}$ are
the fitting parameters of the signal at the left of $D_{k}$, while $A_{k,2}$,
$\mu_{k,2}$, $b_{k,2}$ and $\sigma_{k,2}^{2}$ are the those of the signal at
the right of $D_{k}$, respectively.

Figure 6: (a) Probability density function (PDF) of the in-phase PS-8PAM of
the PS-64QAM in the clipping entropy-loading DSCM signal. The Probability
density function of clipping noise on the amplitude of (b) 1, (c) 3, (d) 5,
and (e) 7, respectively. The red lines denote the fitting curves based on the
piecewise power exponential model.

Based on Eq. (13), the accurate fitting curves are used to acquire the fitting
parameters of $A_{k,1}$, $\mu_{k,1}$, $b_{k,1}$, $\sigma_{k,1}^{2}$,
$A_{k,2}$, $\mu_{k,2}$, $b_{k,2}$, $\sigma_{k,2}^{2}$, and $D_{k}$ for the
theoretical probability density function of the clipping noise. Meanwhile, to
make sure the integral value of $f_{Y_{\text{c}}}(y\mid k)$ is equal to $1$,
the $A_{k,i}$ should be updated by

$A_{k,i}=\frac{A_{k,i}}{\int_{-\infty}^{\infty}f_{Y_{\text{c}}}(y\mid
k)\text{d}y}$ (14)

where $i=1,2$ denotes the left and the right parts of the probability density
function, respectively.

The theoretical BER can be derived by the joint probability density function
based on the probability density functions of the white noise and clipping
noise. The white noise and clipping noise can be considered as two independent
continuous random variables $Y_{\text{n}}$ and $Y_{\text{c}}$. The white noise
coincides with Gaussian distribution, and its probability density functions
$f_{Y_{\text{n}}}$ is equal to
$1/(\sqrt{2\pi}\sigma_{\text{n}})e^{-y^{2}/(2\sigma_{\text{n}}^{2})}$. The
probability density functions of the clipping noise $f_{Y_{\text{c}}}$ is
equal to Eq. (13). The random variable of combined noise $Z$ is equal to
$Y_{\text{n}}+Y_{\text{c}}$, and its probability density functions $f_{Z}$ can
be calculated by

$f_{Z}(z)=\int_{-\infty}^{\infty}f_{Y_{\text{n}}}(y)f_{Y_{\text{c}}}(z-y)\text{d}y$
(15)

Substituting $f_{Y_{\text{n}}}$ and $f_{Y_{\text{c}}}$ into Eq. (15), the
probability density functions $f_{Z}$ can be expressed as

$\displaystyle f_{Z}(z\mid k)$
$\displaystyle=\frac{A_{k,1}}{\sqrt{2\pi}\sigma_{\text{n}}}\int_{z-D_{k}}^{+\infty}e^{-\frac{y^{2}}{2\sigma_{\text{n}}^{2}}}e^{-\frac{\left|z-y-\mu_{k,1}\right|^{b_{k,1}}}{2\sigma_{k,1}^{2}}}\text{d}y$
(16)
$\displaystyle+\frac{A_{k,2}}{\sqrt{2\pi}\sigma_{\text{n}}}\int_{-\infty}^{z-D_{k}}e^{-\frac{y^{2}}{2\sigma_{\text{n}}^{2}}}e^{-\frac{\left|z-y-\mu_{k,2}\right|^{b_{k,2}}}{2\sigma_{k,2}^{2}}}\text{d}y$

The BER of the PS-64QAM signal is similar to that of the decomposed PS-8PAM
signal. For the PS-8PAM signal, the probabilities of the amplitudes are
different, which can be defined as $\boldsymbol{Pr}=[Pr_{\pm 1},Pr_{\pm
3},Pr_{\pm 5},Pr_{\pm 7}]$. The theoretical error ratio of the first bit for
the PS-PAM8 signal can be calculated by

$\displaystyle E_{\text{b,~{}1}}$
$\displaystyle=2\left[Pr_{1}\int_{-\infty}^{-d}f_{Z}(z\mid
1)\text{d}z+Pr_{3}\int_{-\infty}^{-3d}f_{Z}(z\mid 3)\text{d}z\right.$ (17)
$\displaystyle\left.+Pr_{5}\int_{-\infty}^{-5d}f_{Z}(z\mid
5)\text{d}z+Pr_{7}\int_{-\infty}^{-7d}f_{Z}(z\mid 7)\text{d}z\right]$

where $d$ is the Euclidean distance. Similarly, the theoretical error ratio of
the second bit can be calculated by

$\displaystyle E_{\text{b},~{}2}$
$\displaystyle=2Pr_{1}\left[\int_{3d}^{\infty}f_{Z}(z\mid
1)\text{d}z+\int_{-\infty}^{-5d}f_{Z}(z\mid 1)\text{d}z\right]$ (18)
$\displaystyle+2Pr_{3}\left[\int_{d}^{\infty}f_{Z}(z\mid
3)\text{d}z+\int_{-\infty}^{-7d}f_{Z}(z\mid 3)\text{d}z\right]$
$\displaystyle+2Pr_{5}\int_{-9d}^{-d}f_{Z}(z\mid
5)\text{d}z+2Pr_{7}\int_{-11d}^{3d}f_{Z}(z\mid 7)\text{d}z.$

The theoretical error ratio of the third bit can be calculated by Eq. (19).

$\displaystyle E_{\text{b},~{}3}$
$\displaystyle=2\left\\{Pr_{1}\left[\int_{d}^{5d}f_{Z}(z\mid
1)\text{d}z+\int_{-7d}^{-3d}f_{Z}(z\mid
1)\text{d}z\right]+Pr_{3}\left[\int_{3d}^{\infty}f_{Z}(z\mid
3)\text{d}z+\int_{-5d}^{-d}f_{Z}(z\mid
3)\text{d}z+\int_{-\infty}^{-9d}f_{Z}(z\mid 3)\text{d}z\right]\right.$ (19)
$\displaystyle\left.+Pr_{5}\left[\int_{d}^{\infty}f_{Z}(z\mid
5)\text{d}z+\int_{-7d}^{-3d}f_{Z}(z\mid
5)\text{d}z+\int_{-\infty}^{-11d}f_{Z}(z\mid
5)\text{d}z\right]+Pr_{7}\left[\int_{-5d}^{-d}f_{Z}(z\mid
7)\text{d}z+\int_{-13d}^{-9d}f_{Z}(z\mid 7)\text{d}z\right]\right\\}$

As shown in Eqs. (17)-(19), the $d$ should be confirmed before calculating the
theoretical error ratios of the three bits for the PS-8PAM signal. The
relationship between $d$ and the average power of the PS-64QAM signal can be
defined as

$d=\sqrt{\frac{P_{\text{PS-64QAM}}}{4\sum_{i=0}^{3}(2i+1)^{2}\times
Pr_{2i+1}}}.$ (20)

where $P_{\text{PS-64QAM}}$ is the average power of the PS-64QAM signal on the
subcarrier, which can be calculated by $\alpha^{2}\times\beta^{2}\times
P_{\text{DSCM}}/(N\times Loss_{i})$. Finally, the theoretical BER of the
PS-64QAM signal on the subcarrier can be calculated by

$E_{\text{b}}=(E_{\text{b, 1}}+E_{\text{b, 2}}+E_{\text{b, 3}})/3.$ (21)

Figure 7: Three-dimensional look-up table for BER, spectral efficiency (SE),
and link loss under the optimal clipping ratio.

For the PS-64QAM signal, the spectral efficiency corresponds to the specific
probabilities $\boldsymbol{Pr}$ of the amplitudes. As shown in Eqs. (17)-(19),
the $E_{\text{b},~{}i}$ is related to the $\boldsymbol{Pr}$ (i.e., spectral
efficiency) and the $d$ where $i$ is from 1 to 3. Meanwhile, as Eq. (20)
shows, the $d$ is relevant to the $Loss_{i}$ and the $\boldsymbol{Pr}$ (i.e.,
spectral efficiency) under a given clipping ratio. In conclusion, the
theoretical BER of the PS-64QAM signal is determined by the $Loss_{i}$ and
spectral efficiency. Fig. 7 depicts the three-dimensional look-up table for
BER, spectral efficiency, and link loss under the optimal clipping ratio for
the flexible PtMP optical networks. In Fig. 4, the optimal clipping ratio can
be confirmed depending on the link losses of the leaves to achieve the highest
capacity limitation of the flexible PtMP optical networks. The three-
dimensional look-up table is established by Eq. (21) under the optimal
clipping ratio. As Fig. 5 shows, the BER of each subcarrier for the DSCM-based
flexible PtMP optical networks with the clipping operation has a large margin
compared to the 7% FEC limit. Therefore, when the targeted BER is set to 7%
FEC limit, the spectral efficiency can be further improved for each subcarrier
based on the three-dimensional look-up table.

Algorithm 1 Optimization strategy for the capacity of flexible point-to-multi-
point optical networks

1:Subcarrier number $N$; link losses $\boldsymbol{Loss}$; Noise variance
$\sigma^{2}_{\text{n}}$; Signal power $P_{\text{DSCM}}$; Targeted BER
$BER_{\text{T}}$; Maximum spectral efficiency $SE_{\text{max}}$; Spectral-
efficiency updating step size $\Delta SE$; Three-dimensional look-up tables
$\boldsymbol{LUT}(BER,~{}SE,~{}Loss)$.

2:Optimal clipping ratio $\eta_{\text{opt}}$ and optimal spectral efficiencies
$\boldsymbol{SE}$.

3:Calculate the effective SNRs $\boldsymbol{ESNR}$ of the subcarriers by Eq.
(10) based on subcarrier number $N$, link losses $\boldsymbol{Loss}$, noise
variance $\sigma^{2}_{\text{n}}$, and signal power $P_{\text{DSCM}}$.

4:Calculate the optimal clipping ratio $\eta_{\text{opt}}$ by Eq. (12) based
on the $\boldsymbol{ESNR}$.

5:Choose the corresponding three-dimensional look-up table $LUT(BER,SE,Loss)$
based on the optimal clipping ratio $\eta_{\text{opt}}$. $\triangleright$
Next, the iterative algorithm is used to find the optimal $\boldsymbol{SE}$
based on $BER_{\text{T}}$ and $\boldsymbol{Loss}$ in $LUT$.

6:Initialize: $\boldsymbol{SE}:=SE_{\text{max}}\times\text{ones}(1,N)$

7:for $i=1$ to $N$ do

8: Initialize: $BER_{i}$ by $(SE_{i},Loss_{i})$ in $LUT$

9: while $BER_{i}>BER_{\text{T}}$ do

10: Update $SE_{i}\leftarrow SE_{i}-\Delta SE$

11: Update $BER_{i}$ by $(SE_{i},Loss_{i})$ in the $LUT$

12: end while

13:end for

14:return Optimal clipping ratio $\eta_{\text{opt}}$ and optimal spectral
efficiencies $\boldsymbol{SE}$

Algorithm 1 shows the proposed optimization strategy for increasing the
capacity of the flexible PtMP optical networks. The input of the algorithm
includes the subcarrier number $N$, link losses $\boldsymbol{Loss}$, noise
variance $\sigma^{2}_{\text{n}}$, signal power $P_{\text{DSCM}}$, targeted BER
$BER_{\text{T}}$, maximum spectral efficiency $SE_{\text{max}}$, spectral-
efficiency updating step size $\Delta SE$, and three-dimensional look-up
tables $\boldsymbol{LUT}(BER,~{}SE,~{}Loss)$. The first step of the proposed
algorithm is to calculate the effective SNRs $\boldsymbol{ESNR}$ of the
subcarriers by Eq. (10) based on subcarrier number $N$, link losses
$\boldsymbol{Loss}$, noise variance $\sigma^{2}_{\text{n}}$, and signal power
$P_{\text{DSCM}}$. In the second step, the optimal clipping ratio
$\eta_{\text{opt}}$ is calculated by Eq. (12) based on the
$\boldsymbol{ESNR}$. In the third step, the corresponding three-dimensional
look-up table $LUT(BER,SE,Loss)$ is chosen based on the optimal clipping ratio
$\eta_{\text{opt}}$. The last step uses an iterative algorithm to find the
optimal $\boldsymbol{SE}$ based on $BER_{\text{T}}$ and $\boldsymbol{Loss}$
based on the $LUT$. In the iterative algorithm, the spectral efficiency of all
subcarriers is initialized as the maximum spectral efficiency
$SE_{\text{max}}$ (i.e., 6bit/symbol for 64QAM). The outer iterations traverse
all the subcarriers to find the optimized spectral efficiencies
$\boldsymbol{SE}$. In the inner iterations, the spectral efficiency $SE_{i}$
is reduced by the step size $\Delta SE$ each iteration until the BER updated
by $(SE_{i},Loss_{i})$ using the $LUT$ is lower than the targeted BER
$BER_{\text{T}}$. After outer and inner iterations, the optimal spectral
efficiencies $\boldsymbol{SE}$ are obtained for all the subcarriers of the
flexible PtMP optical network. Finally, the optimal clipping ratio
$\eta_{\text{opt}}$ and the optimal spectral efficiencies $\boldsymbol{SE}$
are returned.

Figure 8: The BERs of the subcarriers for the DSCM-based flexible PtMP optical
networks after improving spectral efficiency by using a conventional look-up
table (LUT) and a new look-up table.

Figure 8 depicts the BERs of the subcarriers for the DSCM-based flexible PtMP
optical networks after improving spectral efficiency by using a conventional
look-up table and a new look-up table. For the conventional look-up table, the
clipping noise is assumed to coincide with Gaussian distribution. Based on the
conventional look-up table, the spectral efficiencies for the 8 subcarriers
are set to [5.90, 5.72, 5.49, 5.19, 4.87, 4.54, 4.20, 3.90] bits/symbol,
respectively. However, the BERs of all the subcarriers cannot be under the
targeted 7% FEC limit. Thus, the assumption of Gaussian distribution is not
exact for the clipping noise. For the new look-up table, the clipping noise
coincides with the distribution based on Eq. (13). When the new look-up table
in Fig. 7 is used, the BERs of all the subcarriers are below and close to the
targeted 7% FEC limit. Therefore, the new look-up table is more precise than
the conventional look-up table.

When the clipping operation is not used, the spectral efficiencies for the 8
subcarriers are set to [4.80, 4.40, 4.00, 3.60, 3.20, 2.80, 2.40, 2.00]
bits/symbol, respectively. The corresponding capacity is approximately
217.6Gb/s (i.e. 203.3GB/s excluding the 7% FEC overhead) for the flexible PtMP
optical networks based on 8$\times$8Gbaud DSCM signal. When the new look-up
table is employed, the spectral efficiencies for the 8 subcarriers are set to
[5.65, 5.51, 5.27, 5.04, 4.75, 4.38, 4.08, 3.80] bits/symbol, respectively.
The corresponding capacity is approximately 307.84Gb/s (i.e. 287.7GB/s
excluding the 7% FEC overhead). Using the optimal clipping operation and new
look-up table, the capacity of the DSCM-based flexible PtMP optical networks
is approximately 90.4Gb/s higher than that without clipping operation, which
approaches the theoretical result in Section II. Thus, the optimal clipping
operation and new look-up table can be used to achieve 41% higher capacity.

##  IV Conclusion

In this paper, we derive the theoretical capacity limitation versus the link
losses for the flexible PtMP optical networks based on the entropy-loading
DSCM signal. Meanwhile, the clipping operation with the optimal clipping ratio
reduces the PAPR of the entropy-loading DSCM signal to improve the effective
SNR for approaching the highest capacity limit. Based on the piecewise power
exponential model for the clipping noise, we establish a three-dimensional
look-up table for bit-error ratio, spectral efficiency, and link loss under
the optimal clipping ratio. Based on the three-dimensional look-up table, an
optimization strategy is proposed to acquire optimal spectral efficiencies for
achieving a higher capacity of the flexible PtMP optical networks.

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