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Abstract
A subcarrier-pairing entropy loading (SubP-EL) scheme with fairly low complexity is proposed for digital subcarrier-multiplexing (SCM) systems with colored signal-to-noise ratio (SNR) distributions. With the constraint of the target entropy, SubP-EL iteratively optimizes the entropy of subcarriers in pair. After convergence, SubP-EL can approach the optimal performance which is evaluated in simulations and experiments by comparison with the brute-force search method. Meanwhile the complexity of SubP-EL is significantly reduced compared with the brute-force search. In particular, in an 8-subcarrier system with five different SNRs, the complexity of SubP-EL is reduced by approximately a factor of 764, 95 and 13 with the entropy granularities of 0.05 bits/2D-symbol, 0.1 bits/2D-symbol and 0.2 bits/2D-symbol, respectively. The performance of SubP-EL is evaluated in simulations and experiments. In the experiments with 345 km fiber transmissions, the average NGMI gain of SubP-EL over the system without entropy loading is 0.0286 for different optical filtering cases.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, the increasing demand for network traffic has promoted the use of flexible and high-capacity meshed optical networks [1]. In flexible networks, light paths are routed by reconfigurable optical add-drop multiplexers (ROADMs) which are enabled by wavelength selective switches (WSSs) [2]. However, the application of cascaded WSSs distorts the edge signal spectrum and thereby degrades the quality of received signals. The performance degradation becomes severer as a signal occupies a larger ratio of channel spacing for a higher spectral efficiency (SE) [3].
On the other hand, digital subcarrier-multiplexing (SCM) systems have been well-known for their higher resilience to fiber nonlinearity compared with single-carrier systems in long-haul fiber transmissions [4–7]. As an advanced communication technique, digital SCM has been implemented in the design of application-specific integrated circuit (ASIC) for 800G digital signal processing (DSP) [8]. For SCM systems, the filtering effects induced by cascaded WSSs result in colored signal-to-noise ratio (SNR) distributions of subcarriers. In this case, applying the same modulation format to all subcarriers limits the overall system performance.
To address the issue of colored-SNR distributions in digital SCM systems, several bit loading schemes based on uniform quadrature amplitude modulation (QAM) were proposed in [9–12]. However, these schemes can only enable discrete adjustment of information entropy. In addition, the systems based on uniform QAM formats have a large gap to the channel capacity. Then adaptive bit loading schemes using time-domain hybrid QAM (TDHQ) were proposed to tune entropy more flexibly [13,14]. However, TDHQ is a combination of different uniform QAM formats, so the gap resulted from uniform QAM formats to the channel capacity still exists. Probabilistic shaping (PS) based on the Maxwell-Boltzmann distribution can approach the capacity limit and enable continuous entropy tuning by adjusting the probability distribution of transmitted symbols [15]. Therefore, entropy loading schemes based on PS were proposed [16,17]. In [16], mutual information (MI) of each subcarrier is firstly maximized through entropy loading based on the Shannon formula. Then the entropy values of all subcarriers are added/subtracted by a constant entropy in order to meet the target entropy. In this case, this entropy loading method does not guarantee the optimal performance. Recently, NGMI is used more often as the performance metric especially for PS signals since it can predict the bit error rate (BER) performance after forward error correction (FEC) with high accuracy [18]. In [17], the maximum entropy is assigned to each subcarrier to satisfy the normalized generalized mutual information (NGMI) limit based on the relations between NGMI, entropy and SNR. However, the scheme cannot be used for NGMI optimization given a fixed entropy.
In this paper, we propose a low-complexity subcarrier-pairing entropy loading (SubP-EL) scheme to approach the optimal NGMI performance for a target entropy. SubP-EL firstly initializes the entropy of each subcarrier to the target entropy, which is calculated based on the required net data rate, and then iteratively selects the optimal pair of subcarriers for optimization. After convergence, SubP-EL can approach the optimal NGMI performance while the complexity is significantly reduced compared with the brute-force search method. The complexity of SubP-EL and the brute-force search are compared by counting the number of additions. Taking an 8-subcarrier SCM system with five different SNRs for example, the complexity of SubP-EL is reduced by approximately a factor of 764, 95 and 13 with three different entropy granularities of 0.05 bits/2D-symbol, 0.1 bits/2D-symbol and 0.2 bits/2D-symbol, respectively. The performance of SubP-EL is evaluated in both simulations and experiments. In the simulations, it is shown that SubP-EL can achieve the same performance as the brute-force search method. In the experiments with 345 km fiber transmissions, the average NGMI gain of SubP-EL over the system without entropy loading is 0.0286 for various optical filtering cases.
The rest of this paper is organized as follows. In Section 2, the principle of SubP-EL is described in detail and the complexity analysis of SubP-EL is provided. In Section 3, the performance of SubP-EL is investigated in a 4-subcarrier SCM system and an 8-subcarrier SCM system by simulations. In Section 4, the performance gain of SubP-EL is experimentally demonstrated in a 4-subcarrier system. Finally, the conclusion is drawn in Section 5.
2. Subcarrier-pairing entropy loading (SubP-EL)
For SCM systems with colored-SNR distributions induced by cascaded WSSs, allocating the same modulation format to each subcarrier can lead to a large performance penalty. In this paper, we propose a low-complexity entropy loading scheme, i.e. SubP-EL, to approach the optimal performance with the target entropy.
2.1 Performance metric
Defined as the maximum number of information bits per transmitted bit, NGMI can be calculated as [17]
where H is the entropy of the adopted modulation format. $m = lo{g_2}(M )$ and M is the QAM order. The $GMI$ denoting generalized mutual information is obtained through Monte-Carlo simulations and calculated as [15]For an SCM system in which each subcarrier has its own H, $GMI$, and m, the overall NGMI is defined as [17]
where $E({\cdot} )$ represents the expectation.2.2 Principle of the proposed SubP-EL scheme
Based on the relations between NGMI and SE for fixed SNRs, SubP-EL is proposed to approach the largest NGMI for a fixed entropy. Note that this NGMI can be achieved by the brute-force search method but with a high complexity. We reveal the basic principle of SubP-EL using an example of a 4-subcarrier system with two different SNRs of 10 dB and 16 dB. The target entropy $\bar{H}$ is 4.5 bits/2D-symbol and the tuning granularity is 0.05 bits/2D-symbol. As shown in Fig. 1(a), for a fixed entropy value, NGMI decreases more slowly for subcarriers with a higher SNR. For a fixed SNR, NGMI decreases more rapidly for a larger entropy value. Based on these results, we can assign an entropy value smaller than $\bar{H}$ to the subcarrier with a smaller SNR to get a larger NGMI increase, and a larger entropy value to the subcarrier with a larger SNR to get a smaller NGMI decrease. By doing so, a NGMI gain can be obtained.
Fig. 1. (a) Relations between NGMI, entropy and SNR for PS-64QAM signals. (b) NGMI changes versus the difference between the applied entropy and the target entropy.
The corresponding NGMI changes are shown in Fig. 1(b). $|{\Delta NGM{I_1}} |$ represents the NGMI increase after assigning entropy values smaller than $\bar{H}$ to the subcarrier with the SNR of 10 dB, and $|{\Delta NGM{I_2}} |$ represents the NGMI decrease after assigning entropy values larger than $\bar{H}$ to the subcarrier with the SNR of 16 dB. For this SCM system with two SNRs, the largest NGMI gain can be calculated by Eq. (5) and the optimal entropy loading results can be easily obtained with the largest NGMI gain of 0.0674.
When the number of subcarriers with different SNRs increases, it becomes complicated to obtain the optimal entropy loading results. In this case, the SubP-EL scheme can solve the problem efficiently by iteratively selecting the optimal pair of subcarriers and adjusting their entropy values. The optimization process of SubP-EL is described as follows and illustrated in Fig. 2.
Step 1: Initialize entropy values of all subcarriers to the target entropy $\bar{H}$.
Step 2: For the $i$-th iteration, divide all subcarriers into two groups ${C_{1,i}}$ and ${C_{2,i}}$ based on their entropy values. Group ${C_{1,i}}$ (${C_{2,i}}$) consists of subcarriers of which entropy values are not greater (less) than the target entropy $\bar{H}$. It is noted that one subcarrier belongs to both sets if the entropy value equals to the target entropy $\bar{H}$.
Step 3: Determine the optimal pair of subcarriers $({{s_{1,i}},{s_{2,i}}} )$ and calculate the largest $NGM{I_{gain,i}}$ for the $i$-th iteration. ${s_{1,i}}$ is the subcarrier which obtains the largest NGMI increase after a step size $\mu $ is subtracted from the entropy values of ${C_{1,i}}$, and ${s_{2,i}}$ is the subcarrier which obtains the smallest NGMI decrease after a step size $\mu $ is added to the entropy values of ${C_{2,i}}$. The largest NGMI gain of the $i$-th iteration is then obtained by subtracting the smallest NGMI decrease from the largest NGMI increase.
The detailed calculation is as follows. First, for subcarriers in ${C_{1,i}}$ (${C_{2,i}}$), decrease (increase) their entropy values by one step $\mu $ and the NGMI increases (decreases) are denoted as the vector $|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{1,\boldsymbol{i}}}}}} |$ ($|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{2,\boldsymbol{i}}}}}} |$). Second, compute the maximum of $|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{1,\boldsymbol{i}}}}}} |$ and the minimum of $|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{2,\boldsymbol{i}}}}}} |$ and denote them as $|{\Delta NGM{I_{1,i}}} |$ and $|{\Delta NGM{I_{2,i}}} |$, respectively. Denote the subcarriers corresponding to $|{\Delta NGM{I_{1,i}}} |$ and $|{\Delta NGM{I_{2,i}}} |$ as ${s_{1,i}}$ and ${s_{2,i}}$, respectively. Third, calculate the largest $NGM{I_{gain,i}}$ of the $i$-th iteration by subtracting $|{\Delta NGM{I_{2,i}}} |$ from $|{\Delta NGM{I_{1,i}}} |$.
Step 4: If $NGM{I_{gain,i}}$ is greater than zero, update the entropy of subcarriers ${s_{1,i}}$ and ${s_{2,i}}$ to the values after adjustment in Step 3 and other subcarriers keep the entropy values before Step 3. Otherwise, end the algorithm.
Step 5: i is updated to $i + 1$, repeat Step 2 to Step 5.
As SubP-EL iterates, the NGMI gain of each iteration $NGM{I_{gain,i}}$ continues to decrease. Once $NGM{I_{gain,i}}$ becomes negative, the algorithm ends and the entropy loading result is obtained.
2.3 Complexity analysis of SubP-EL
With the constraint of the target entropy, SubP-EL can approach the performance of the brute-force search method while the required complexity is significantly reduced. The complexity of SubP-EL ${C_S}$ is determined by the iteration times I and the number of additions per iteration. ${C_S}$ is calculated by
where ${n_{sub}}$ is the subcarrier number and $({4 \times {n_{sub}} - 1} )$ is the maximum number of additions per iteration including $2 \times {n_{sub}}$ additions to obtain $|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{1,\boldsymbol{i}}}}}} |$ and $|{{\boldsymbol{C}_{\Delta \boldsymbol{NGM}{\boldsymbol{I}_{2,\boldsymbol{i}}}}}} |$, $({2 \times {n_{sub}} - 2} )$ additions to obtain $|{\Delta NGM{I_{1,i}}} |$ and $|{\Delta NGM{I_{2,i}}} |$, and 1 addition to calculate the $NGM{I_{gain,i}}$.The iteration times of SubP-EL are related to the subcarrier number ${n_{sub}}$ and the number of entropy values used for entropy adjustment ${n_{SE}}$. For a fixed QAM order M, the entropy values used range from 2 to $lo{g_2}(M )$ with the granularity $\mu $ but do not include $lo{g_2}(M )$. The number of entropy values is calculated by
Taking PS-16QAM SCM signals with the target entropy $\bar{H}$ of 3 bits/2D-symbol for example, the iteration times of SubP-EL are shown in Table 1. It is noted that the required iteration times of SubP-EL can be slightly different for different SNR distributions. Therefore, we randomly generate 10000 SNR distributions in MATLAB and calculate the average iteration times. Note that the SNR distributions follow a fading shape to mimic optical filtering effects.
Table 1. Iteration times of SubP-EL
The complexity of the brute-force search method ${C_B}$ is determined by the number of searches and the number of additions for a search. For fixed conditions, the number of searches is about ${\left( {\frac{{{n_{SE}}}}{2}} \right)^{{n_{SNR}} - 1}}$ and the maximum number of additions for a search is ${n_{sub}}$ including $({{n_{sub}} - 1} )$ additions to calculate NGMI of the entropy allocation and 1 addition to determine whether NGMI of this search is larger than before or not. Therefore, the complexity of the brute-force search method ${C_B}$ is approximately calculated as
We compare the complexity of SubP-EL and the brute-force search method through the total number of additions ${C_{\boldsymbol{S}}}$ and ${C_{\boldsymbol{B}}}$ in an 8-subcarrier system, as shown in Table 2. The SCM signals are probabilistically shaped based on 16QAM with the target entropy $\bar{H}$ of 3 bits/2D-symbol. Compared with the brute-force search method, the complexity of SubP-EL is significantly reduced especially for a large number of SNRs. Taking the system with five SNRs for example, the complexity of SubP-EL is reduced by a factor of 764, 95 and 13 with the entropy granularities of 0.05 bits/2D-symbol, 0.1 bits/2D-symbol and 0.2 bits/2D-symbol, respectively.
Table 2. Complexity comparison of SubP-EL and the brute-force search through the number of additions
3. Simulation results
In this section, we firstly demonstrate that SubP-EL can approach the optimal performance through the comparison with the brute-force search method and then present the performance gain of SubP-EL over the systems without entropy loading in a 4-subcarrier SCM system and an 8-subcarrier SCM system.
The simulation setup of the SCM system is shown in Fig. 3. The total baud rate is 32 GBaud. At the transmitter-side, the transmitted symbols are probabilistically shaped by constant composition distribution matching (CCDM) [19]. The probabilistically shaped symbols of each subcarrier are firstly up-sampled to 2 samples per symbol, and then shaped by a 1024-tap root raised-cosine (RRC) filter with a roll-off factor of 0.01. Afterwards, the subcarriers are multiplexed. To emulate the transceiver SNR of 20 dB, we set the SNRs of both the transmitter and the receiver to 23 dB by loading additive white Gaussian noise (AWGN). Besides, laser phase noise, frequency offset (FO) and chromatic dispersion (CD) are also considered in our simulations. The laser linewidth is 20 kHz, the FO is set to be 0.5 GHz and the accumulated CD after each span of 80 km fiber is 1360 ps/nm at 1550 nm wavelength. In the transmission link, Erbium-doped fiber amplifiers (EDFAs) with a noise figure of 5 dB are inserted after each span. To emulate the colored-SNR distributions of SCM systems, ROADMs are inserted in the link. The ROADM architecture assumed in this paper is route-and-select, and a ROADM consists of two WSSs [20]. The WSS is modeled as follows [21]:
At the receiver-side, after coherent detection and CD compensation (CDC), the SCM signals are de-multiplexed and then the signal of each subcarrier is down-sampled to 2 samples per symbol. Afterwards, matched filtering and adaptive equalization are carried out. The adaptive equalization is implemented with decision-directed least-mean-square (DD-LMS) algorithm. Within the DD-LMS loop, a phase-locked loop (PLL) is applied for carrier phase recovery (CPR).
The SNR distributions with respect to the number of cascaded ROADMs are shown in Fig. 4(a) and Fig. 5(a) for PS-16QAM signals with $\bar{H}$ of 3 bits/2D-symbol in a 4-subcarrier system and an 8-subcarrier system, respectively. The SNR distributions are asymmetric as FO is considered in the simulations. Figure 4(b) and Fig. 5(b) show the NGMI performance of SubP-EL scheme and the brute-force search scheme with three entropy granularities of 0.05 bits/2D-symbol, 0.1 bits/2D-symbol and 0.2 bits/2D-symbol. It is obvious that SubP-EL can approach the NGMI performance of the brute-force search and better performance is obtained with a finer entropy granularity. In addition, the performance difference with different granularities is more distinct for severer filtering effects. This is because the SNRs of the edge subcarriers become lower as the filtering effects become severer, and the entropy granularity has a larger impact on subcarriers with lower SNRs.
Fig. 4. (a) SNR distributions versus the ROADM number in a 4-subcarrier system for PS-16QAM SCM signals. (b) NGMI performance comparison between SubP-EL and the brute-force search with different entropy granularities.
Fig. 5. (a) SNR distributions versus the ROADM number in an 8-subcarrier system for PS-16QAM SCM signals. (b) NGMI performance comparison between SubP-EL and the brute-force search with different entropy granularities.
The NGMI performance of SubP-EL, the brute-force search and the system without entropy loading, denoted as w/o-loading is shown in Fig. 6 and Fig. 7 for a 4-subcarrier system and an 8-subcarrier system, respectively. The transmitted signals are PS-16QAM SCM signals with $\bar{H}$ of 3 bits/2D-symbol and PS-64QAM SCM signals with $\bar{H}$ of 5 bits/2D-symbol. The tuning granularity of entropy is 0.05 bits/2D-symbol. The simulation results show that SubP-EL can approach the NGMI performance of the brute-force search under different transmission conditions for both PS-16QAM and PS-64QAM SCM signals. Meanwhile, they have a large gain over the system without entropy loading.
Fig. 6. NGMI performance versus the ROADM number in a 4-subcarrier system for (a) PS-16QAM SCM signals with $\bar{H}$ of 3 bits/2D-symbol; and (b) PS-64QAM SCM signals with $\bar{H}$ of 5 bits/2D-symbol.
Fig. 7. NGMI performance versus the ROADM number in an 8-subcarrier system for (a) PS-16QAM SCM signals with $\bar{H}$ of 3 bits/2D-symbol; and (b) PS-64QAM SCM signals with $\bar{H}$ of 5 bits/2D-symbol.
For a system with a fixed number of subcarriers, the NGMI gain of SubP-EL becomes more significant in the case of severer filtering circumstances. This is because the performance of a SCM system is mainly limited by the subcarrier with the worst SNR and the limitation becomes more obvious for severer filtering circumstances. For systems with different number of subcarriers, the NGMI performance is not necessarily related to the subcarrier number. This is because the NGMI performance is the average of all subcarriers in our simulations. For example, the edge subcarriers of an 8-subcarrier system have a larger performance penalty than those of a 4-subcarrier system. However, the performance penalty of an 8-subcarrier system is shared among 8 subcarriers while the performance penalty of a 4-subcarrier system is shared among 4 subcarriers. Therefore, the performance difference between systems with different number of subcarriers depends on specific conditions.
4. Experimental results
In this section, the performance of SubP-EL is experimentally demonstrated in a 4-subcarrier SCM system. Figure 8 plots the experimental setup. The transmission distance is 345 km. The target entropy is 4.5 bits/2D-symbol and the granularity of entropy adjustment is 0.05 bits/2D-symbol. At the transmitter-side, 32 GBaud dual-polarization (DP) PS-64QAM SCM signals are firstly generated offline in MATLAB. The SCM signals are then uploaded to an arbitrary waveform generator (AWG) with a sampling rate of 80 GSa/s. Afterwards, the output four electrical signals of AWG are driven by four radio frequency (RF) drivers followed by a DP In-phase/Quadrature (IQ) modulator. The wavelength of the laser is set to 1550 nm and the nominal linewidth is 100 kHz.
A WaveShaper (WS) is inserted in the middle of the transmission link to emulate two cascaded WSSs. The filtering shape of one WSS is based on the model in Eq. (9) and the filtering shape of the WaveShaper is determined by ${S^2}(f )$. The bandwidth of the WaveShaper is changed by varying ${B_0}$ of each WSS from 26 GHz to 38 GHz. ${B_{OTF}}$ is set to 12 GHz and the center frequency of the WaveShaper is aligned to that of the transmitter-side laser. After transmissions, the optical signals are input to an optical filter which is used to remove the out-of-band amplified spontaneous emission (ASE) noise. A variable optical attenuator (VOA) is used after the optical filter to obtain a fixed received optical power (ROP) of −3 dBm.
At the receiver-side, a laser with a nominal linewidth of 100 kHz is used as the local oscillator for coherent detection. Then signals are digitized by a 4-channel real-time digital storage oscilloscope (DSO) with a sampling rate of 100 GSa/s. The right inset of Fig. 8 presents the receiver DSP. IQ errors from the receiver side are firstly compensated by Gram-Schmidt orthogonalization procedure (GSOP) [22], followed by frequency offset compensation (FOC) and CDC. Then DD-LMS and CPR are carried out. Afterwards, in order to compensate the residual IQ errors from the transmitter which cause the crosstalk between subcarriers [23], an $8 \times 8$ real-valued post-equalizer is employed. Finally, NGMI is calculated.
QPSK signals are firstly transmitted to measure the SNR distributions which are shown in Fig. 9(a). For a smaller bandwidth of the WaveShaper, the SNR difference between the center and the edge subcarriers is larger. In addition, the SNR distributions are not exactly symmetric which is impacted by the non-ideal opto-electronic devices. The optimized entropy loading results of SubP-EL and the brute-force search are the same and shown in Table 3. Consistent with the analysis in Section 2.1, larger entropy values are allocated to subcarriers with larger SNRs. Figure 9(b) shows the same NGMI performance of SubP-EL and the brute-force search and that they achieve a large gain over the system without loading. The average NGMI gain of SubP-EL is 0.0286 over all bandwidth cases.
Table 3. Entropy loading results of SubP-EL
5. Conclusion
In this paper, we propose a low-complexity entropy loading SubP-EL scheme to optimize the performance with the target entropy. The performance of SubP-EL, the brute-force search and the system without entropy loading are compared in simulations and experiments. SubP-EL can approach the NGMI performance of the brute-force search while the complexity of SubP-EL is significantly reduced. In an 8-subcarrier SCM system with five different SNRs, the complexity of SubP-EL is reduced by approximately a factor of 764, 95 and 13 with the entropy granularities of 0.05 bits/2D-symbol, 0.1 bits/2D-symbol and 0.2 bits/2D-symbol, respectively. In the experiment with 345 km fiber transmissions, the average NGMI gain of SubP-EL over the system without loading is 0.0286 over all bandwidth cases ranging from 26 GHz to 38 GHz.
Funding
National Key Research and Development Program of China (2018YFB1801200); National Natural Science Foundation of China (61801291); Shanghai Rising-Star Program (19QA1404600).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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