1. Introduction

Providing a practically-achievable up to ∼1-dB linear shaping gain, probabilistic amplitude shaping (PAS) has recently become a hot research topic in fibre-optic communications [1–3]. To enhance further the transmission performance, shaping with improved fibre nonlinearity tolerance [4–8], offering nonlinear benefit in addition to or at the expense of original linear shaping gain, is considered as one of the most promising solutions. Methods within this topic include managements of the excess kurtosis [4–5] and temporal property [6–7] for the transmitted data sequence, the latter of which appears to be more promising since it has been proved to provide substantial nonlinear benefit even over a dispersion-unmanaged link. Among the methods of temporal property management, finite-blocklength shaping has received lots of attention due to its feasible implementation [9–11]. Recently, based on the finite-blocklength shaping, a super-symbol method (SUP) is proposed for extra nonlinear benefit (∼0.15-dB over previous finite blocklength shaping) and a spectral model that explains its ability in mitigating both self-phase modulation (SPM) and cross-phase modulation (XPM) effects is provided [12–13].

From the spectral model in [13], the spectral dip of a signal’s intensity waveform, which is at near the direct-current (DC) frequency and is created by the unique feature of ∼constant energy of each super symbol, is the key to providing the nonlinear benefit of SUP. In general, a wider spectral dip, typically formed at the transmitter output, can avoid more fiber nonlinearities and thus offer a larger nonlinear gain. Such a wide dip, however, may not last long over distance due to the presence of accumulated chromatic dispersion (CD) that would broaden the SUP pulse and narrow the dip. Therefore, to understand the nonlinear benefit of SUP, the dip width as a function of distance (or accumulated CD) need to be analyzed, which has not been studied yet.

In this paper, for SUP systems we first derive the spectral dip width as a function of various system parameters and analyze its evolution with transmission distance, by which we then deduce the relative transmission performance between different baud rates and shaping blocklengths. Simulations with shell mapping are performed to verify our analysis and performance deduction, showing that the lower baud rate at 11.25 GHz (or longer blocklength at 120) can provide better performance than the higher baud rate at 45GHz (or shorter blocklength at 40), when the distance is relatively longer. This may suggest that the baud rate and the blocklength need be judiciously decided in order to access the optimum performance of SUP. Finally, a nonlinear noise study shows that the nonlinear benefit of SUP stems mainly from its relatively lower nonlinear phase noise (NLPN) power.

This paper is organized as follows. In Section 2 we will first derive the spectral dip and discuss the associated nonlinear performance as a function of distance; in Section 3 we describe the simulation setup and the adopted digital signal processing (DSP) algorithms, in Section 4 we discuss the simulation results and analyze the nonlinear noise, and finally in Section 5 we conclude this paper.

2. Principles

Shown in the top of Fig. 1(a) is the temporal 4-dimensional (4D) amplitude of a super symbol, with a SUP duration ${\textrm{T}_{\textrm{SUP}}}$, in the absence of CD. Since every SUP comprises exactly one shaped block from the amplitude shaper (AS) with all four tributaries (Xi, Xq, Yi, and Yq), the SUP duration is equal to ${\textrm{T}_{\textrm{SUP}}} = \textrm{N}/({4{\textrm{R}_\textrm{s}}} )$ where $\textrm{N}$ is the shaping blocklength and ${\textrm{R}_\textrm{s}}$ is the baud rate. For further discussion, we define the power spectrum of intensity waveform as $\rho (f )= |{{{\cal F}}\{{{P_{xy}}(t )} \}} |$² where ${{\cal F}}\{\cdot \}$ is Fourier transform operator and ${P_{xy}}(t )=|{{E_x}(t )} |$² + $|{{E_y}(t )} |$² is the intensity waveform of both polarization with ${E_x}(t )$ and ${E_y}(t )$ being the electric fields along the x and y polarization. Shown in the bottom of Fig. 1(a) is $\rho (f )$ of SUP. A distinctive feature of a SUP system is that a spectral dip at around the DC frequency exists, which never appears in a traditional PAS system (neither in any other regular uniformly-distributed-constellation system). As has been explained in [13], this spectral dip is created by the feature of (almost) constant energy of each SUP and its width is inversely proportional to the SUP duration i.e. $\textrm{W} \approx \textrm{C}/{\textrm{T}_{\textrm{SUP}}} = 4\textrm{C}{\textrm{R}_\textrm{s}}/\textrm{N}$ with $\textrm{C}$ being a constant. A wider dip is proven to be more effective in suppressing the self-phase modulation (SPM) and cross-phase modulation (XPM) effects. However, as we will see later, the dip width will typically reduce with the increase of distance (or the accumulated CD) which may diminish the SUP benefit.

 

Fig. 1. Principle diagram of the temporal 4-dimentional (4D) amplitude of a super symbol and the corresponding power spectrum of intensity waveform as a function of chromatic dispersion (CD), (a) in the absence of CD and (b) in the presence of CD, where $\textrm{N}$ is the shaping blocklength and ${\textrm{R}_\textrm{s}}$ the baud rate.

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Shown in the top and bottom of Fig. 1(b) are the temporal 4D amplitude of a SUP and the corresponding $\rho (f )$, respectively, in the presence of CD. Each SUP is broadened by CD with a new duration $\textrm{T}^{\prime}_{\textrm{SUP}} = {\textrm{T}_{\textrm{SUP}}} + \Delta \textrm{T}$, where $\Delta \textrm{T} = \textrm{DB}L$ with $\textrm{D}$ being the fiber dispersion parameter, $\textrm{B}$ the signal bandwidth, and L the transmission distance. Since CD simply broadens a pulse but never changes its energy, each broadened SUP (with a new duration $\textrm{T}^{\prime}_{\textrm{SUP}}$) can still keep the ∼constant-energy feature. By applying this new duration to the above dip width formula, we can express the width as a function of distance as in (1):

To get a basic idea of how the dip width evolves with distance, we derive $\textrm{W}$ and $\Delta \textrm{W}$ for a small and large L, which represent respectively a negligible and significant amount of accumulated CD. When L is small (i.e. $\textrm{N} \gg 4\beta \textrm{D}{\textrm{R}_\textrm{s}}^2L$), earlier we have learnt that $\textrm{W}$₀$\; \approx 4\textrm{C}{\textrm{R}_\textrm{s}}/\textrm{N}$, and by ignoring the $4\beta \textrm{D}{\textrm{R}_\textrm{s}}^2L$ term in the denominator of (2) we can have $\Delta \textrm{W}$₀ ${\approx}{-} 16\textrm{C}\beta \textrm{D}{\textrm{R}_\textrm{s}}^3/{\textrm{N}^2}$, where the subscript “0” describes the condition of L approaching zero. When L is large (i.e. $\textrm{N} \ll 4\beta \textrm{D}{\textrm{R}_\textrm{s}}^2L$), by ignoring $\textrm{N}$ in (1) and (2) we can obtain $\textrm{W}$_∞${\approx} \textrm{C}/({\beta \textrm{D}{\textrm{R}_\textrm{s}}L} )$ and $\Delta \textrm{W}$_∞ ${\approx}{-} \textrm{C}/({\beta \textrm{D}{\textrm{R}_\textrm{s}}{L^2}} )$, respectively, where the subscript “$\infty $” describes the condition of L approaching infinity.

With (1) and (2) and their extreme cases of a small/large L, we then move further to the core port of this section. Next we will fix $\beta $ and $\textrm{D}$ and discuss how the width$\; \textrm{W}(L )$ as well as the nonlinear benefit will vary with the baud rate ${\textrm{R}_\textrm{s}}$ and the blocklength $\textrm{N}$.

To offer a better picture of the above discussion, in Fig. 2 we illustrate the dip width $\textrm{W}(L )$ in (1) with $\textrm{C}$ = 1, $\alpha $ = 0.1 and $D$ = 16.8 ps/(nm.km). For Fig. 2(a) we fix the blocklength at 40 and study three different baud rates at 11.25, 22.5, and 45 GBaud while for Fig. 2(b) we fix the baud rate at 45 GBaud and discuss two different blocklengths at 40 and 120. From Fig. 2(a), it is clear that in the beginning the higher baud rate has a wider dip which then reduces faster with distance, therefore leading to a crossing point at ${L_x}$ between the lower and higher baud rate. In this illustration we can see that the crossover distances ${L_x}$ occur at ∼67, 134, and 267 km, respectively, between (22.5, 45), (11.25, 45) and (11.25, 22.5) GBauds. Further increasing the distance, we’ve observed that all the widths reduce in a relatively-slower pace (compared with the beginning) and the width, though all three are rather small there, with the lower baud rate is wider. From Fig. 2(b), we can observe that the longer blocklength has a wider dip at first, and, similar to the higher baud in Fig. 2(a), it narrows more rapidly with increasing distance. However, there is no crossing point found between the short and long blocklengths, though their widths tend to converge with a further increase of distance. The inset shows a zoom-in for L between 800∼960 km.

 

Fig. 2. The derived spectral dip width W as a function of distance L. (a)$\; {\textrm{R}_\textrm{s}}$ = 11.25, 22.5, and 45 GBauds are compared with $\textrm{N}$ = 40, and (b) $\textrm{N}$ = 40 and 120 are compared with ${\textrm{R}_\textrm{s}}$ = 45 GBaud. Here we assume $\textrm{C}$ = 1, $\alpha $ = 0.1 and $\textrm{D}$ = 16.8 ps/(nm.km).

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Couples of notes are worth mentioning before we end this section. First, one clear advantage of SUP is that its duration ${\textrm{T}_{\textrm{SUP}}}$ is ∼$\textrm{N}/4$ times of the symbol duration which makes the spectral dip more tolerant to CD, on condition that $\textrm{N}$ ≫ 4 (which is usually the case). This allows the dip (and thus nonlinear benefit) remaining even in the presence of a moderate amount of CD as long as ${\textrm{T}_{\textrm{SUP}}}$ is still larger than $\Delta \textrm{T}$, which may explain why SUP can have nonlinear benefit in a dispersion-unmanaged link. To further enhance SUP performance, digital CD precompensation (CDPC) that reduces the maximum of crosstalk duration $\Delta \textrm{T}$ may be applied. Second, continuing the first note, $\Delta \textrm{T}$ may include not only accumulated CD but also other linear dispersive impairments such as polarization mode dispersion (PMD) and/or those caused by insufficient bandwidth of transponders/optical filters. This suggests that the SUP dip could be tolerable to all linear dispersive effects, supposing ${\textrm{T}_{\textrm{SUP}}}$ is still larger than the total $\Delta \textrm{T}$. More evidences are certainly required to confirm this in the future. Third, we have discussed the nonlinear benefit for different baud rates and blocklengths. To discuss the transmission performance, the linear benefit (i.e. linear shaping gain) need be considered as well. In general, the linear shaping gain is independent of the baud rate and is larger with a longer blocklength. This means that in the baud-rate discussion a better nonlinear benefit will directly lead to a better transmission performance. While in the blocklength discussion, the larger nonlinear benefit of the shorter blocklength will diminish with increasing distance which will eventually highlight its worse linear shaping gain. Therefore, it is expected that the longer blocklength can still achieve a higher performance in a longer-distant link, despite of its relatively worse nonlinear tolerance. We will discuss this in more detail in Section 4.

3. Simulation setup

For simulations we will conduct two performance comparisons to verify our discussion in Section 2: one for systems with different baud rates but equal shaping blocklength and one for systems with different blocklengths but equal baud rate.

In the first comparison, the WDM transmission performance are evaluated for the following three equal-capacity SUP systems with: (i) 12.5-GHz-spaced, 52 × 11.25-GBaud, (ii) 25-GHz-spaced, 26 × 22.5-GBaud, and (iii) 50-GHz-spaced, 13 × 45-GBaud dual-polarized PAS-64QAM signals. All systems assume a forward error correction (FEC) code rate at 0.9 (11% overhead) and use an AS rate (i.e. number of AS input bits/number of AS output bits) at 0.863. Therefore, each 11.25G-, 22.5G-, and 45GBaud channel carries a net data rate of ∼109, ∼218, and 436 Gb/s (net spectral efficiency = 4.85 bits/symbol per polarization), with a pre-FEC BER threshold at ∼2.1e-2. Regarding the AS we use shell mapping [15] with ($\textrm{K}$, $\textrm{N}$) = (69, 40) for all three systems, where $\textrm{K}$ is the number of input bits {0, 1} of every AS block and N is the number of output symbols {1, 3, 5, 7} of every AS block. The energy of each symbol is used as the cost when building the shell table, in order to maximize the linear shaping gain.

In the second comparison, the WDM performance of two equal-capacity systems, both of which are with 50-GHz-spaced, 13 × 45-GBaud dual-polarized PAS-64QAM signals, are evaluated with different shaping blocklengths. Both systems still assume a FEC code rate at 0.9 and use an AS rate at 0.863, therefore resulting in a net data rate at 436 Gb/s with a pre-FEC BER at 2.1e-2. For the AS we use shell mapping with ($\textrm{K}$, $\textrm{N}) = ({69,\; 40} )$ and ($\textrm{K}$, $\textrm{N}) = ({207,120} )$ respectively for the two systems. Note that the system with $\textrm{N} = 40$ here is same to the 45-GBaud system in the first comparison.

It is worth mentioning that, although here shell mapping is applied for the shaped symbol production, other shaping algorithms, such as constant composition distribution matcher (CCDM) [1] and enumerative sphere shaping (ESS) [9], should not alter our conclusions provided the shaped blocks can exhibit ∼constant energy. In addition, the blocklengths at 40 and 120 are selected for comparison because they are considered feasible with acceptable performance. As a reference, an AS with a blocklength at 40 (120) is only ∼0.3 dB (0.15 dB) away from an ideal AS.

The transmission link consists of 1 to 12 span(s) of 80-km SSMF with fiber loss = 0.2 dB/km, fiber dispersion parameter D = 16.8 ps/(km.nm), and nonlinearity coefficient = 1.31 (1/W/km). The split-step Fourier method is used to model the fiber with an adaptive step size allowing a maximum nonlinear phase rotation equal to 4e-3 radian and the simulation bandwidth is fixed at 1.44 THz for all-baud systems. No PMD and no polarization dependent loss (PDL) is considered. No optical dispersion management is applied. The loss of each span is compensated by an ideal (noiseless) Erbium doped fiber amplifier (EDFA). The amplified spontaneous emission (ASE) noise is only loaded at the input of receiver and its amount is adjusted to achieve the desired optical-signal-to-noise ratio, (OSNR_d). The desired OSNR is calculated as OSNR_d$\; [{\textrm{dB}} ]= $ OSNR_r$\; [{\textrm{dB}} ]$ $- {\mathrm{\gamma }_{\textrm{ext}}}$, where OSNR_r is the received OSNR based on a typical OSNR budget formula [16]:

At the transmitter, an RRC filter is applied with $\mathrm{\alpha } = 0.1$. At the receiver, another RRC filter with $\mathrm{\alpha } = 0.1$ (i.e. serving the matched filter) as well as a perfect CD compensator is applied first. Then, an adaptive T/2-spaced, 21-tap 2 × 2 time-domain equalizer based on least mean square (LMS) algorithm, coupled with a pilot-assisted decision-directed phase locked loop (DD-PLL), is employed decoupling the x and y polarization signals as well as mitigating residual linear distortions. The working principle of the pilot-assisted DD-PLL is described below: 12 pilot symbols are used at the head of every ∼450 data symbols. Through the pilots we can obtain the more reliable phase estimation at the beginning of every 450 data symbols, which is then used to reset the estimated phase of DD-PLL [17] in order to correct the possible cycle slips. For the following 450 data symbols the regular DD-PLL is applied. Afterwards, bit error rate (BER) as a function of OSNR is derived by direct error counting and the OSNR margin to the BER threshold at 2.1e-2 is evaluated. Each BER is obtained with 3 runs of different noise seeds (different data patterns of each run and different data patterns across all channels) and each run involves 100 K symbols per tributary.

4. Results and discussion

We first inspect how the dip widths of different baud rates/blocklengths evolve with distance which can reveal directly their relative benefits. In Fig. 3, we illustrate the intensity waveform spectra $\rho (f )$ of a single 11.25-GBaud, 22.5-GBaud, and 45-GBaud SUP channel, which are taken from the respective WDM system in the first comparison, after transmitting over different distances of SSMF. The frequency range is limited to [$- $3, 3] GHz focusing on the details of the dip and the title above each plot indicates the corresponding distance. At 0 km, we can see that the 45-GBaud signal has the widest dip and the 11.25-GBaud the narrowest while the 22.5-GBaud in between. The dip width is found to be proportional to the baud rate which agrees well with $\textrm{W}$₀$\; \propto {\textrm{R}_\textrm{s}}$ in Section 2. With the increase of transmission distance, all three widths are expected to reduce at a rate proportional to the cube of baud rate, i.e. $\Delta \textrm{W}$₀$\; \propto {\textrm{R}_\textrm{s}}$³. At 80 km, all the widths are found reduced and as expected the reduction is indeed larger with the higher baud rate. The 45-GBaud width now is comparable with the 22.5-GBaud, while both of which are still wider than the 11.25-GBaud. At 160 km, the 22.5-GBaud width is found to be the widest, followed closely by the 11.25 GBaud which in turn is slightly wider than the 45 GBaud. The width reduction from 80 to 160 km is found much less intense than that from 0 to 80 km, especially for the 45 GBaud. At 240 km, the 11.25 GBaud is found to catch up with the 22.5-GBaud, both of which are now wider than the 45 GBaud. Further increasing the distance, at 480 km the 11.25-GBaud has the widest dip, 22.5-GBaud the second, and the 45-GBaud the third. This totally reverses their width ranking in the beginning (at 0 km). At last, the widths at 960 km, a very large L, are found to be inversely proportional to their baud rates, which agrees with $\textrm{W}$_∞$\; \propto (1/{\textrm{R}_\textrm{s}})$. The crossover distance ${L_x}$ (with a resolution of 80 km here), between (22.5, 45), (11.25, 45), and (11.25, 22.5)-GBauds, are found to be ∼80, 160, and 240 km, respectively, roughly matched to those derived by (3), which are ∼67, 133, and 267 km, respectively.

 

Fig. 3. The intensity waveform spectra $\rho (f )$ of a 11.25-GBaud, 22.5-GBaud, and 45-GBaud SUP signal after different spans of SSMF. (a) 0 km, (b) 80 km, (c) 160 km, (d) 240 km, (e) 480 km, and (f) 960 km. The spectral resolution is ∼10 MHz.

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In Fig. 4 we illustrate the intensity waveform spectra $\rho (f )$ of two 45-GBaud SUP signals, one with $\textrm{N}$ = 40 and one with $\textrm{N}$ = 120 each taken from the respective WDM system in the second comparison, after transmitting over different distances of SSMF. Same to Fig. 3, the frequency range is limited to [$- $3, 3] GHz and the title of each plot indicates the corresponding distance. At 0 km, the dip width with N = 40 is found to be ∼3x times wider than $\textrm{N}$ = 120 which aligns with $\textrm{W}$₀$\; \propto ({1/\textrm{N}} )$. Notably, we see that the dip depth with $\textrm{N}$ = 120 is slightly deeper than $\textrm{N}$ = 40. The reason behind this is that the dip depth is determined by the energy variance of the produced super symbols; typically, a smaller energy variance will result in a deeper dip. With shell mapping, SUP with $\textrm{N}$ = 120 will have a smaller energy variance than $\textrm{N}$ = 40, therefore leading to the observed deeper dip. However, we’ve found that the deeper depth with $\textrm{N}$ = 120 barely provides extra nonlinear benefit in addition to the “regular” depth with $\textrm{N}$ = 40, and therefore we will focus still on the width discussion. With the increase of distance, both the widths are expected to reduce with a rate inversely proportional to the square of the blocklength, i.e. $\Delta \textrm{W}$₀$\; \propto 1/\textrm{N}$². At 80 km, it is found the width with N = 40, which narrows more significantly, exhibits only slightly larger size as N = 120. At 160 km, the two widths are with very similar sizes. Further increasing the distances, ex. 240, 480, and 960 km, we can see that the two widths consistently exhibit nearly identical sizes with comparable reducing rates. Different than the results in Fig. 3, there is no crossover found between the two widths even after up to 960-km distance. Figs. 3 and 4 basically are in line with our discussions in Section 2.

 

Fig. 4. The intensity waveform spectra $\rho (f )$ of a 45-GBaud SUP channel with a shaping blocklength N = 40 and 120 after different distances of SSMF. (a) 0 km, (b) 80 km, (c) 160 km, (d) 240 km, (e) 480 km, and (f) 960 km. The spectral resolution is ∼10 MHz.

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We have seen the width evolution with distance and next we further quantify the benefit of this width in terms of OSNR margin gain. In Fig. 5, the transmission results of OSNR margin to the BER threshold at 2.1e-2, as a function of launch power, is presented with different distances for the first comparison. The launch power is presented with dBm per 50 GHz so that all the 11.25-GBaud, 22.5- and 45-GBaud SUP systems (denoted respectively as SUP-11.25G, SUP-22.5G, and SUP-45G) would have similar optimum launch power. The extra loss term ${\mathrm{\gamma }_{\textrm{ext}}}$, described in Section 3, that attempts to equalize the optimum OSNR margins across different span numbers is applied. The title of each plot indicates the transmission distance and the gain is defined as the difference between the optimum OSNR margins. The bandwidth of DD-PLL, which is claimed to be critical to the PAS performance [14, 18], is optimized at around the optimum launch power to maximize the margin.

 

Fig. 5. OSNR margin vs. launch power for the first comparison, after different spans of SSMF transmission: (a) 80 km (b) 160 km (b) 240 km (d) 400 km (e) 960 km. Here three 52x 11.25-GBaud, 26x 22.5-GBaud, and 13x 45-GBaud SUP systems with an equal blocklength N = 40, denoted respectively as SUP-11.25G, SUP-22.5G, and SUP-45G, are compared. Their traditional (TRA) counterparts, TRA-11.25G, TRA-22.5G, and TRA-45G, are also presented as a reference.

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Since the linear shaping gain is in principle independent of the baud rate, in the low power regime the performances of the three systems are found almost identical. After 80-km transmission, it is obvious that the higher baud rate performs better: the 45-GBaud SUP system shows ∼0.15-dB gain over the 22.5-GBaud, which is then ∼0.25-dB better than the 11.2 GBaud. This is owing to the wider dip of the higher baud rate as we have discussed. After 160 km, the 22.5-GBaud catches up with the 45 GBaud, while both of which show ∼0.2-dB gain over the 11.25 GBaud. Since the optimum margins of the 22.5-and 45-GBaud meet here, 160 km (i.e. 2 spans) can be considered as their crossover distance of margin ${L_{x,m}}$ after which the 22.5-GBaud is expected to perform better. After 240 km, the 22.5 GBaud outperforms both the others by ∼0.1 dB. Here the 11.25-GBaud and the 45-GBaud are found to exhibit nearly equal performance which defines 240 km as their ${L_{x,m}}$. After 400 km, the 11.25-GBaud pulls up with the 25-GBaud, both of which shows an ∼0.3-dB gain over the 45-GBaud. Therefore, 400 km (5 spans) marks ${L_{xm}}$ between the 11.25-G and 22.5-GBaud. Further increasing the distance to 960 km, it can be seen that the lower baud rate now performs better, the 11.25 GBaud is ∼0.1-dB better than the 22.5 GBaud, which is then ∼0.2-dB better than the 45 GBaud. The performance comeback of the lower baud rate with increasing distance aligns well with our analysis in Section 2. With the three baud rates discussed here, we may conclude that 45 GBaud is preferred for 1∼2 spans (i.e. smaller accumulated CD), the 22.5 GBaud is for 2∼5 spans, and the 11.25 Gbaud is for 5∼12 spans (i.e. larger accumulated CD).

Note that the crossover distances of margin ${L_{x,m}}$ (i.e. 160, 240, 400 km) are found larger than the crossover distance of width ${L_x}$ (i.e. 80, 160, 240 km). This is because more nonlinear benefit goes to the higher and lower baud rates respectively before and after ${L_x}$, and, therefore, to equalize their nonlinear benefit (which relates directly to the margin), ${L_{x,m}}$ has to be larger than ${L_x}$.

As a reference, the performance of traditional PAS systems with the herein discussed three baud rates (denoted as TRA-11.25G, TRA-22.5G and TRA-45G, respectively) are also presented. Each TRA channel is obtained by permuting randomly the shaped symbols of each tributary of its SUP counterpart. As we have mentioned previously [13], the TRA doesn’t have a spectral dip and thus its performance would not vary that considerately with baud rate as in a SUP system. The presented results here, from 1 to 12 spans, show that the performance difference across the three baud rates are found to be almost the same, with less than ∼0.1-dB difference at the peaks. This result agrees with previous literature [18] and clearly forms a sharp contrast to SUP.

If we focus on the SUP gain over TRA, it can be observed that the SUP-45G achieves up to an ∼1.0-dB gain at 80 km which later declines continuously with distance to ∼0.3 dB at 960 km; while the SUP-11.25G shows a relatively steady gain at between ∼0.6 and 0.7 dB from 80 km to 960 km. These behaviors are strongly related to their width evolution as discussed earlier. To further enhance the SUP gain, CDPC that can limit the maxima of pulse broadening could be adopted, possibly for scenarios where the link’s distance information is available at the transmitter.

In Fig. 6, the transmission results of OSNR margin to the BER threshold at 2.1e-2, as a function of launch power, is presented with different distances for the second comparison. The title of each plot indicates the transmission distance and ${\mathrm{\gamma }_{\textrm{ext}}}$ is also used for each distance. The bandwidth of DD-PLL is optimized the same way as in Fig. 5. The two systems with blocklengths at 40 and 120 are labelled respectively as SUP-40 and SUP-120. Since the linear shaping gain with $\textrm{N}$ = 120 is ∼0.15 dB larger than $\textrm{N}$ = 40, in the low power regime we always can find this performance gap between the two blocklengths. After 80 km, though with ∼0.15-dB worse linear performance, $\textrm{N}$ = 40 still shows ∼0.25-dB gain over $\textrm{N}$ = 120 due to its initially wider dip. Then, the width with $\textrm{N}$ = 40 narrows more rapidly than N = 120 which tends to equalize their nonlinear benefits. After 240 km, the optimum margins between the two become comparable, which defines 240 km (3 spans) as ${L_{x,m}}$ between the two blocklengths. Further increasing the distance to 480 km, the better linear performance with $\textrm{N}$ = 120 finally appear even at the optimum launch power, showing ∼0.1-dB gain over $\textrm{N}$ = 40. After 960 km, the curves of the two blocklenghts are almost parallel to each other with a gap caused by their linear performance difference. This shows that the nonlinear benefit of $\textrm{N}$ = 120 never exceeds $\textrm{N}$ = 40 for a distance up to 960 km, which agrees with our analysis. With the two blocklengths discussed here we may conclude that, $\textrm{N}$ = 40 is preferred for 1-3 spans (i.e. smaller accumulated CD) while $\textrm{N}$ = 120 is preferred for 3∼12 spans (i.e. larger accumulated CD).

 

Fig. 6. OSNR margin vs. launch power for the second comparison, after transmitting over different distances: (a) 80 km (b) 240 km (b) 480 km (d) 960 km. Here two 13x 45-GBaud SUP systems with different blocklengths at N = 40 and N = 120, denoted respectively as SUP-40 and SUP120, are compared. Their TRA counterparts, TRA-40 and TRA120, are also presented as a reference.

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Again, for reference the performance of TRA systems with the herein discussed two blocklengths, denoted as TRA-40 and TRA-120, respectively, are also presented. Each TRA channel is obtained by permuting randomly the shaped symbols of each tributary of its SUP counterpart. It can be seen that TRA-120 are always better than TRA-40 from low to high power levels for all distances. This simply shows that, contrary to the SUP, the TRA has similar nonlinearity tolerance regardless of the adopted blocklength so that its performance is mainly determined by its linear shaping gain.

Then we analyze the nonlinear interference noise (NLIN) for both the TRA and SUP systems. NLIN can be separated into a residual Gaussian (ResN) and a NLPN component. As in [19], we will first derive NLIN power with error vector magnitude (EVM) approach and ResN power with method of moments (MoMs), and then obtain NLPN power as the difference between NLIN and ResN power. Different than the adopted unit in [19], we will use a normalized quantity G to represent each noise power via G $= {\textrm{P}_\textrm{n}}{\textrm{R}_\textrm{s}}^2/{\textrm{P}_{\textrm{ch}}}^3$, where ${\textrm{P}_\textrm{n}}$ could be any nonlinear noise power discussed above. As discussed in [14], the benefit of this quantity is that it is independent of the per-channel launch power ${\textrm{P}_{\textrm{ch}}}$ and it can be used for a direct performance comparison between systems with different baud rates ${\textrm{R}_\textrm{s}}$.

Figure 7 depicts G value of NLIN, NLPN, and ResN, as a function of span count (from 1 to 12), for both TRA and SUP systems in the 1^st comparison. The launch power is fixed at 4 dBm/50GHz and no ASE is loaded to focus on nonlinear noise study. First we can observe that NLIN of SUP is indeed lower than TRA, by 1∼2 dB, for all the baud rates. In addition, NLIN diversity across different baud rates in SUP is found higher than that in TRA, especially at near the initial and final several spans, which may explain SUP’s higher sensitivity to the baud rate. Second, NLPN power of SUP is clearly lower than TRA for all baud rates, by 2∼4 dB. The NLPN diversity of SUP across different baud rates is also found higher than that of TRA. Third, ResN power of SUP is found slightly lower than TRA as well, by a negligible difference at initial spans and 0.3∼0.4 dB at final several spans. Different than NLPN, the ResN diversity of SUP across different baud rates is found quite similar to that of TRA. Conclusively, the power reduction and diversity (across different baud rates) of SUP in NLIN (total nonlinear noise) is mainly contributed by NLPN, for the baud rates and span counts under study.

 

Fig. 7. Normalized noise power vs. span count for the 1^st comparison for (a) nonlinear interference noise (NLIN), (b) nonlinear phase noise (NLPN), and (c) residual Gaussian noise (ResN). The insets in (a) and (c) show the zoom-in plots for span count = 10 to 12.

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We also obtain similar G values in Fig. 8 for the 2^nd comparison. Same power at 4 dBm/50GHz is launched with no ASE loading. First of all, we can see that all noise power (NLIN, NLPN, and ResN) of TRA is found almost same with different blocklengths. This simply shows the nonlinearity tolerance of TRA can hardly be affected by just switching the blocklengths. On the contrary, a clear power difference between SUP-40 and SUP-120 can be found in both NLIN and NLPN. This difference is larger at the initial spans and then diminished at the final several spans, which basically agrees well to our margin results in Fig. 6. Noticeably, the presented results in a recent study on the interplay between PAS and carrier phase estimation (CPE) [20], showing similar gain from the short-blocklength shaping could be obtained with blind phase search (BPS) method in either scenario of high received OSNR or using ideal digital back propagation (DBP), also implicitly indicates that the SUP gain over TRA mainly stems from the NLPN reduction.

 

Fig. 8. Normalized noise power vs. span count for the 2nd comparison for (a) NLIN (b) NLPN, and (c) ResN. The inset in (c) shows the zoom-in plot for span count from 10 to 12.

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At last, we stress several notes and discuss couples of potential research topics related to the extension of this work. First, unlike the TRA of which the performance will change remarkably with baud rate only when an ideal CPE is adopted [18], the SUP performance can vary easily with baud rate by the use of a simple and feasible CPE, i.e. DD-PLL in this paper. With a more advanced yet practical CPE, the presented results would be affected slightly, which, however, falls outside the scope of this paper and is left for future study. Second, according to (3), it is expected that the found crossover distances (${L_x}$ and ${L_{x,m}}$) in Figs. 5 and 6 are subject to change with a different set of system parameters. However, the principles in Section 2 should still apply. Third, an optimum baud rate may exist for SUP, which would be expected to vary with not only the link parameters but also the shaping blocklength. A related work on this would be needed as well, which is beyond the scope of this paper and is left open for future investigations.

5. Summary

For SUP transmission we have analyzed its unique spectral dip as a function of various system parameters and discussed the dip evolution as well as the associated nonlinear benefit with increasing distance. This analysis indicates that the SUP performance is a function of both the baud rate and blocklength. We then perform two simulative comparisons to verify the above indication, both of which deliver results in line with our analysis. This suggests that, given a link, the optimization of SUP performance relies on the judicious choice of both the baud rate and blocklength. Finally, a nonlinear noise study reveals that the nonlinear benefit of SUP mainly comes from its significant NLPN reduction.