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Abstract
We report the maximum probability directed blind phase search (MPD-BPS) solution to enhance the carrier phase estimation (CPE) capability for the probabilistically shaped-quadrature amplitude modulation (PS-QAM) with variable shaping factors. We first investigate the trade-off between the shaping factor (SF) and the noise rejection window of the BPS for the PS-64/256QAM at a fixed signal to noise ratio (SNR). Next, we carry out the joint optimization of the SF and the noise rejection window, in order to obtain the maximum achievable information rate (AIR) at the fixed SNR. Then, we numerically compare the performance of the MPD-BPS and the traditional BPS for the PS-64/256QAM with variable SFs. Finally, we conduct the experimental verification of 32 Gbaud PS-64QAM with the SFs of 0.02, 0.025, 0.03, and 0.035, under scenarios of back-to-back (B2B) transmission and the standard single mode fiber (SSMF) loop transmission. The experimental results indicate that the proposed MPD-BPS can obtain an average 0.13 dB SNR enhancement under the B2B transmission and the 1.67% reach enhancement for the SSMF transmission, in comparison with the use of traditional BPS.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Probabilistic shaping (PS) has attracted a worldwide research attention, due to its capability to approach the Shannon limit and realize the rate-adaptive transmission [1]. With the help of probabilistic amplitude shaping (PAS) architecture, the PS can be applied to the quadrature amplitude modulation (QAM) format. Since the occurrence probability of probabilistically shaped QAM (PS-QAM) constellation satisfies the Maxwell-Boltzmann (MB) distribution, up to 1.53 dB shaping gain can be obtained under the additive white Gaussian noise (AWGN) channel [2]. Due to the outstanding power-efficiency, the PS-QAM also outperforms other rate-adaptive transmission schemes such as the hybrid QAM [3]. Therefore, the PS-QAM has been widely used in current wavelength division multiplexing (WDM) fiber optical transmissions [4,5]. However, increasing the shaping factor (SF) of the PS-QAM reduces the occurrence probability of peripheral constellation points, which undermines the function of conventional digital signal processing (DSP) flow, including the clock recovery [6], the fast Fourier transform based frequency offset estimation (FFT-FOE) [7], and the polarization division de-multiplexing [8]. Previous investigation has verified that only after good carrier phase recovery (CPR), the shaping gain of PS can be obtained [9]. Blind phase search (BPS) as a successful CPR algorithm is essential for current coherent fiber optical transmission, due to its high performance and parallelism [10]. The peripheral constellation points are vital for the carrier phase estimation (CPE) of the BPS, but the number of peripheral constellation points within one single noise rejection window reduces when the SF is increased. In order to obtain a high shaping gain, the PS-QAM with a large SF is usually implemented under a lower signal-to-noise ratio (SNR), in comparison with the same order QAM format. Thus, more decision errors are likely to occur for the traditional BPS. It has been shown that both the probability of cycle slip (CS) and the mean square error (MSE) of the CPE enlarges with the increased SF of the PS-QAM and become the worst at the SF which is originally optimal under the AWGN channel [11,12]. Therefore, two solutions have been proposed to obtain a high generalized mutual information (GMI) during the implementation of traditional BPS. One is the SF detuning. The used SF must deviate from the originally optimal SF under the AWGN channel, when the BPS with a short noise rejection window is exploited as the CPR algorithm [13,14]. The other solution is to increase the length of noise rejection window [14]. There are two inherent problems of the BPS for the PS-QAM signal, including more decision errors aggravated by the PS technique and the improper metric by using the absolute distance in the cost function. Moreover, the information rate characterized by the GMI needs an ideal forward error correction (FEC). However, the practically used FEC is not ideal, indicating that the pre-FEC metric such as the normalized GMI (NGMI) is better to evaluate the transmission performance [15,16]. Only when the NGMI threshold can be reached, the used FEC is capable to realize an error-free transmission. Otherwise, the communication interruption happens. The SF detuning may decrease the net data rate, due to the entropy reduction. Therefore, both a comprehensive investigation and a joint optimization of the noise rejection window and the SF are important for the PS-QAM to obtain the highest AIR at the NGMI threshold.
In current submission, we report a maximum probability directed blind phase search (MPD-BPS) algorithm for the PS-QAM. The proposed MPD-BPS does not need the maximum a posteriori (MAP) detection and enhances the role of peripheral constellation points during the identification of the optimal test angle. As a result, the proposed MPD-BPS can reduce phase estimation errors. We first investigate the trade-off between the SF and the noise rejection window of the BPS and optimize the noise rejection window of the BPS and the SF to obtain the maximum AIR. Then, we numerically compare the performance of the proposed MPD-BPS and the BPS for the PS-64/256QAM with variable SFs under the optimal noise rejection window. Both the SNR enhancement at the NGMI threshold and the enhanced linewidth tolerance of the proposed MPD-BPS can be secured. Finally, we conduct an experimental verification of 32 Gbaud PS-64QAM with the SFs of 0.02, 0.025, 0.03 and 0.035 under the scenarios of back-to-back (B2B) and standard single mode fiber (SSMF) transmission. The experimental results indicate that the MPD-BPS can obtain an average 0.13 dB SNR enhancement for the PS-64QAM under the B2B transmission and 1.67% reach enhancement after the SSMF transmission, in comparison with the traditional BPS.
2. Operation principle
2.1 Phase noise model
We treat the phase noise as a Wiener process [9], which can be denoted as
where ${\phi _n}(k)$ is the phase noise of kth symbol, $w(k)$ is a variable that satisfies the Gaussian distribution ${\mathbb N}(0,2\pi \Delta fT)$, where $\Delta f$ is the laser linewidth, and T is the symbol duration. The symbols before the CPR can be denoted as where r(k) is the received kth symbol, n(k) is the white noise on the kth symbol, and s(k) is the kth transmitted symbol with the MB distribution, which can be described by where x is the transmitted symbol, and $\lambda $ is the SF. Please note that the uniform QAM can be regarded as a specific PS-QAM with the SF of 0.2.2 MPD-BPS
The operation principle of traditional BPS is to identify the test angle that minimizes the average of the squared Euclidean distance [10]. The BPS implementation first multiplies the symbols with NTA test angles, whose range is from 0 to π/2 with an equal space of $\pi /(2{N_{TA}})$, as shown in Eq. (4) and (5)
where $\theta _t^{\;m}$ indicates the ${m^{th}}$ test angle. Then, the BPS calculates the squared Euclidean distance of the rotated symbols in relative to the decision symbols, as shown in Eq. (6), where $\hat{r}(k,m)$ is the corresponding decision symbol for ${r_{ro}}(k,m)$. For the traditional BPS, we must use the MAP decision to determine the decision symbols [17]. When we assume that the phase noise keeps constant over a short time period, a moving average window is helpful to mitigate the AWGN over NMA adjacent symbols. Therefore, the estimated phase noise can be obtained fromThe probability $P({r_{ro}}(k,m))$ is derived as
where x is the ideal constellation point. Equation (9) can be approximated with four neighbor x to ${r_{ro}}(k,m)$[17]. With the help of the AWGN auxiliary channel, we can rewrite the Eq. (9) asHere, we use the logarithmic operation to transform the multiplication into the addition, for the purpose of reducing the calculation complexity. As shown in Eq. (10), the final cost function is determined by both the probability of x and the squared Euclidean distance with x. For the ease of explanation, we assume that the probability P(rro(k,m)) is mainly determined by the decision symbol xD, while the impact of other three neighbor symbols can be neglected. Then, the metric of rro(k,m) in the cost function Eq. (11) can be changed as
We substitute Eq. (3) into Eq. (12) as
We can observe that the metric in Eq. (13) includes two terms. The first term is the squared absolute distance which is the metric used in the cost function of the traditional BPS. If we assume that the decision symbol xD is 7 + 7i for rro(k, m) and 5 + 7i for rro(k, m+1), |xD|2 in the second term of Eq. (13) for ln(P(rro(k, m))) and ln(P(rro(k, m+1))) are respectively |7 + 7i|2 = 98 and |5 + 7i|2 = 74, with a difference of 24. On the other hand, when we assume that the decision symbol xD is 3 + 3i for rro(k, m) and 1 + 3i for rro(k, m+1), |xD|2 will be respectively |3 + 3i|2 = 18 and |1 + 3i|2 = 10, with a difference of 8. Thus, the peripheral constellation points have a more evident impact on the variation of the cost function. Consequently, the MPD-BPS can exploit the outer constellation points effectively. The flow chart of the BPS and MPD-BPS is shown in Fig. 1. The traditional BPS first rotates the received signals with NTA test angles, performs the MAP decision, calculates the Euclidean distance between the rotated symbols and the corresponding decision symbols, averages the squared Euclidean distance from adjacent NMA symbols, and finally identifies the test angle to minimize the average Euclidean distance. In particular, the MAP decision in the BPS is realized by two steps [17]. The first step is to identify four neighbors to the rotated symbol from the alphabet of x by the ML detection. The second step is to identify the decision symbol from four neighbors by the MAP detection. However, our proposed MPD-BPS first rotates the received signals with NTA test angles, calculates the probability of the rotated symbols, multiplies the probability of the adjacent NMA rotated symbol to obtain the probability of the rotated symbol sequence, and finally identifies the test angle to maximize the probability of the rotated symbol sequence. In our proposed MPD-BPS, we only need to perform the first ML detection to identify four neighbors.
2.3 Computational complexity
For each symbol, 4NTA real-number multiplications and 2NTA additions are indispensable to calculate ${r_{ro}}(k,m)$ for both the BPS and the MPD-BPS. As for the BPS, one MAP decision circuit is necessary. Therefore, ${r_{ro}}(k,m)$ should be delayed for the calculation of D(k,m). Moreover, 2NTA real-number multiplications and 3NTA additions are required for the calculation of D(k,m). However, as for the proposed MPD-BPS, 8NTA real-number multiplications and 19NTA additions are required for the calculation of Eq. (10). In addition, two look up tables (LUTs) are required for both logarithmic and exponential operations. Only the ML detection is implemented to identify four neighbor constellation points, and the second step is unnecessary. For each block, both the BPS and the MPD-BPS require (NMA−1) additions to realize the sum operation and (NMA−1) comparators. Table 1 summarizes the complexity of the BPS and the MPD-BPS for each symbol, and the final (NMA−1) additions and (NMA−1) comparators are not included.
Table 1. Complexity of the BPS and the MPD-BPS
3. Numerical investigation
3.1 Optimization of noise rejection window at a fixed SNR
We first numerically investigate the trade-off between the SF and the noise rejection window of the BPS, in order to practically secure the maximum AIR. The CPE from symbols with longer modulus is easier due to the relative SNR gain [20,21]. For the PS-QAM, the growing SF will decrease the occurrence probabilities of peripheral constellation points. Enlarging the noise rejection window results in a better average of the AWGN, but at the cost of the tracking speed of phase noise. We first investigate the trade-off between the noise rejection window and the SF at the SNRs of 14 dB, 15 dB and 16 dB for the 40 Gbaud PS-64QAM, when the combined laser linewidth of a transmitter and a local oscillator (LO) is 200 kHz. The test angle number is 15 for the PS-64QAM. For each Monte-Carlo simulation, 65536 symbols are used. We conduct 100 Monte-Carlo simulations for each SNR condition and take the average result, as shown in Fig. 2. The joint optimization of both the noise rejection window and the SF aims to reach the NGMI threshold of 0.8105, which refers to a concatenated FEC rate of 0.7436 [16,22]. Such NGMI threshold is commonly-used to reliably predict the decoding performance of the PS-64QAM signal [16,22]. For a fixed FEC rate and a specific modulation format, the AIR can be denoted as
where H(x) is the entropy of transmitted signals, r is the FEC rate of 0.7436, and m = 6 stands for the PS-64QAM signal. As shown in Eq. (14), the AIR is improved by the entropy enhancement. Therefore, we can optimize the SF to maximize the entropy at the NGMI threshold. As shown in Fig. 2(a), when the noise rejection window is 200, the BPS function not well for the SF from 0.01 to 0.045. The λoptimal in Fig. 2 refers to the SF for obtaining the maximum GMI under the AWGN channel. The GMI reaches its maximum at the SF of 0.055. When the noise rejection window becomes longer, the GMI with respect to the SF from 0.01 to 0.045 increases as well. When the SF is first enlarged, more decision errors lead to the degradation of the CPE performance. At the SF range from 0.01 to 0.045, the noise tolerance enhanced by increasing SF is not enough to compensate for the PS-induced BPS penalty. Therefore, increasing the noise rejection window is helpful to achieve a better GMI performance. When the SF is more than 0.055, the noise tolerance is further enhanced. Now, the tracking speed of phase noise is dominant to the GMI performance. Consequently, increasing the noise rejection window results in a degradation of the GMI performance. Consequently, the GMI trend turns reversely. We can observe that the designated SF at the NGMI threshold becomes smaller when the noise rejection window increases from 200 to 300. And then, it becomes larger when the noise rejection window increases from 300 to 350, as shown in Fig. 2(b). Since the smallest SF at the NGMI threshold can maximize the entropy, leading to the achievement of maximum AIR, both the noise rejection window of 300 and the SF of 0.35 are optimal for PS-64QAM at the SNR of 14 dB. We also notice that the SF to maximize the GMI and the AIR are different, which means the theoretically optimal SF to maximize the GMI cannot practically obtain the maximum AIR. In addition, the CS probability, which is denoted as the number of CS occurrence over the total symbol number, can be significantly reduced to less than $2 \times {10^{ - 6}}$ by increasing the noise rejection window to 300, as shown in Fig. 2(c). Similarly, the optimal noise rejection windows are 200 and 150 at the SNRs of 15 dB and 16dB, respectively. When the SNR further increases, the gap between the maximum GMI and the Shannon limit becomes widen, as shown in Fig. 2(a), (d) and (g). The optimal SF at the NGMI threshold also decreases, as shown in Fig. 2(b), (e) and (h). Since the entropy of the PS-64QAM gradually saturates to its maximum value of 6 bit/symbol, the GMI also gradually saturates. The optimal SF at the NGMI threshold tends to be zero, leading to a trivial shaping gain for the PS-64QAM. Moreover, we notice that the offset between the optimal SF at the NGMI threshold and the λoptimal rapidly enlarges. When the SNR is 14 dB or 15 dB, the offset is trivial, while the offset becomes almost 0.02, when the SNR is 16 dB. This implies that the PS-64QAM with a SF of 0.02∼0.035 can be effectively exploited to approach the Shannon limit, when the SNR is between 14∼15 dB, with the help of the BPS. However, when the SNR increases to 16 dB or higher, PS-64QAM may have an AIR gap to the Shannon limit and the distribution of PS-64QAM tend to be uniform. This infers that a higher order modulation format is preferred. Following the same procedure, we can optimize the noise rejection window for the PS-256QAM signal, under the SNRs of 18 dB, 20 dB and 22 dB, respectively. The NGMI threshold of the PS-256QAM is 0.8456, due to the limitation of PAS on the lowest FEC rate of the PS-256QAM [1,16]. We increase the test angle to 20 for the PS-256QAM. As shown in Fig. 3, we can conclude that the noise rejection window should be 400, 250, 150 at the SNRs of 18 dB, 20 dB, and 22 dB, respectively. The corresponding SFs at the NGMI threshold are 0.02, 0.0125, and 0.005, respectively.
Fig. 2. Optimization results at the SNR of 14 dB (a) GMI performance, (b) NGMI performance, (c) cycle slip probability; 15 dB (d) GMI performance, (e) NGMI performance, (f) cycle slip probability; and 16 dB (g) GMI performance, (h) NGMI performance, (i) cycle slip probability;
3.2 Required SNR at the NGMI threshold for PS-QAM with variable SFs
We investigate the performance of the BPS and the MPD-BPS for the PS-64/256QAM under variable SNRs and compare the required SNR for the BPS and the MPD-BPS to reach the NGMI threshold. For the PS-64QAM, we choose the SFs of 0.02, 0.025, 0.03, and 0.035, and for PS-256QAM we choose the SFs of 0.005, 0.0075, 0.0125, 0.015 and 0.02. The SNR range is from 13.5 dB to 16 dB for the PS-64QAM and from 16.5 dB to 23 dB for the PS-256QAM. The noise rejection window is optimized according to the previous procedure. As shown in Fig. 4(a), the vertical dotted lines are the theoretically required SNR where the AIR achieved by each SF is equal to the Shannon limit [1]. Decreasing the SF results in a higher theoretically required SNR. Since the SF reduction raises the entropy, a higher SNR is necessary. On the other hand, decreasing the SF reduces the noise tolerance, therefore a higher SNR is required at the NGMI threshold. We can clearly observe that the gap between the practically required SNR and the theoretically required SNR includes two parts. Taking the SF of 0.035 as an example, one is the gap between the case with the phase noise and the case without, as denoted by the Gap1 in Fig. 4(a). The other gap comes from the practical FEC, which is almost a constant of about 1.7 dB, as denoted by the Gap2 in Fig. 4(a). The definition of Total Gap in Fig. 4(a) refers to the gap between the practically required SNR and the theoretically required SNR at the NGMI threshold. The proposed MPD-BPS can reduce the Gap1 from 0.5 dB to 0.4 dB for all four SFs. Although the 0.1 dB gain may seem modest by its absolute value, it is a solid step toward to the theoretical limit.
Fig. 4. (a) NGMI and (b) MSE performance of the BPS and MPD-BPS for the PS-64QAM with various SF and SNR at the 200 kHz combined linewidth
The corresponding results of the PS-256QAM are presented in Fig. 5. Obviously, the PS-256QAM suffers more severely from the phase noise than the PS-64QAM. Thus, the Gap 1 becomes about 1.6 dB. Our proposed MPD-BPS can reduce the gap from 1.6 dB to 1.4 dB at the NGMI threshold. This attributes to the fact that the MPD-BPS can realize more accurate CPE. Since the MPD-BPS does not depend on the decision points to calculate the Euclidean distance, the errors arising in the decision can be mitigated. Moreover, the CPE MSE by using the MPD-BPS is smaller than that of the BPS for both PS-64QAM and PS-256QAM, as shown in Fig. 4(b) and Fig. 5(b). Our proposed MPD-BPS is superior to the BPS, no matter what the modulation format is chosen. As shown in Fig. 4(a) and 5(a), the gain of MPD-BPS increases with the SNR reduction, because the MPD-BPS does not fully depend on the decision symbols. Furthermore, the MPD-BPS can exploit the outer constellation points effectively, as mentioned in section 2. When the SNR is high enough, the gain of MPD-BPS is negligible because a high SNR is convenient for the BPS to realize the correct CPE. As shown in Fig. 4(b) and Fig. 5(b), the CPE MSE is very challenging to further reduce, when the SNR becomes quite high.
Fig. 5. (a)NGMI and (b)MSE performance of the BPS and MPD-BPS for the PS-256QAM with various SF and SNR at the 200 kHz combined linewidth
3.3 Linewidth tolerance for PS-QAM with variable SFs
We investigate the linewidth tolerance of the BPS and the MPD-BPS for the PS-64/256QAM under a fixed SNR, respectively. For the PS-64QAM we choose the SFs of 0.02, 0.025, 0.03, and 0.035, while for PS-256QAM, we choose the SFs of 0.005, 0.0075, 0.0125, 0.015 and 0.02. The SNR is 15 dB for the PS-64QAM and 21 dB for the PS-256QAM. The results are shown in Fig. 6. We can observe that the linewidth tolerance is enhanced by increasing the SF. Moreover, our proposed MPD-BPS is superior to the traditional BPS by about 50 kHz for the PS-64QAM at the NGMI threshold, as shown in Fig. 6(a). For the PS-256QAM with a SF of 0.005, the enhancement of linewidth tolerance is about 10 kHz at the NGMI threshold, but the enhancement increases to about 50 kHz, when the SF is more than 0.0125, as shown in Fig. 6(b). The CPR is severely affected by the white noise for the PS-256QAM with an SF of 0.005 and the linewidth tolerance is limited. When the SF increases, the noise tolerance is enhanced, leading to the successful CPR implementation. As a result, our proposed MPD-BPS only has a little improvement over the traditional BPS for the PS-256QAM with an SF of 0.005. However, our proposed MPD-BPS algorithm is always better than the BPS for all modulation formats with variable SFs.
4. Experimental verification
We carry out single polarization (SP) 32 Gbaud PS-64QAM under both B2B and 900 km SSMF transmission scenarios to verify the correct function of the proposed MPD-BPS. The SFs are 0.02, 0.025, 0.03 and 0.035, whose entropies are 5.84, 5.76, 5.67 and 5.57 bit/symbol, respectively. The experimental setup and corresponding DSP flow at the transmitter side (Tx) and the receiver side (Rx), are shown in Fig. 7. At the Tx, an external cavity laser (ECL) at 1549.35 nm with a linewidth of 100 kHz, is modulated with 32 Gbaud electrical signals by an in-phase/quadrature modulator (IQM), driven by two outputs of 92 GSa/s arbitrary waveform generator (AWG). As shown in Fig. 7(b), we generate PS-64QAM by the PAS architecture based on the MB distribution. 2048 4-QAM pilot symbols are loaded before the PS-64QAM symbols, for the ease of synchronization, first stage equalization, and the SNR monitoring at the receiver side. The SNR of the received pilots is calculated by
The NGMIs with respect to the estimated SNR under the B2B transmission are shown in Fig. 8. Since no fiber nonlinearity occurs, the approximation between the B2B transmission and the AWGN channel becomes reasonable. Enlarging the shaping factor can increase the Euclidean distance between adjacent constellation points, resulting in a robust noise tolerance. Therefore, a larger shaping factor requires a lower SNR to reach the NGMI threshold, as shown in Fig. 8(a). We can observe that the required SNR is increased from 13.4 dB to 15.1 dB, when the shaping factor decreases from 0.035 to 0.02. Since more entropy is introduced at the transmitter, a higher SNR is instinctively required. The average SNR enhancement at the NGMI threshold by our proposed MPD-BPS is about 0.13 dB, under the condition of B2B transmission. Thus, we attribute such SNR enhancement to our proposed MPD-BPS.
The NGMIs of PS-64QAM with a SF of 0.035 at different launched powers under 900 km fiber loop transmission are shown in Fig. 9(a). By varying the launched power, we can observe the transmission performance under variable fiber nonlinearities. The NGMIs of both the BPS and the proposed MPD-BPS first increases, when the launched power varies from −3 dBm to 0 dBm. In such condition, the linear transmission impairment arising in the SSMF is dominant. Further increasing the launched power can enhance the SNR of the received signal, resulting in a higher NGMI. Thus, it is still valid to approximate the SSMF transmission to the AWGN channel. The proposed MPD-BPS is superior to the BPS by about 0.01 NGMI improvement. Next, when the launched power is further increased, the NGMIs of both the BPS and the proposed MPD-BPS begin to reduce. In such region of the launched power, fiber nonlinearity becomes dominant, so that the NGMI performance has a penalty. It is no longer accurate to approximate the SSMF transmission to the AWGN channel. Therefore, we can observe the NGMI enhancement by the proposed MPD-BPS becomes trivial. The optimal launched power is 0 dBm for the PS-64QAM with an SF of 0.035 transmission over the 900 km SSMF, as shown in Fig. 9(a). We keep the launched power at 0 dBm for other SFs as well, because the optimal launched power at a fixed SSMF reach depends only mildly on the SF choice [16]. The NGMIs of various SFs at different SSMF reach are shown in Fig. 9(b). It is expected that the SSMF reach at the NGMI threshold can be extended by increasing the SF, because the required SNR is decreased as shown in the results of the B2B transmission. For all four SFs, the MPD-BPS can achieve about 1.67% reach enhancement, corresponding to an extension of 15 km SSMF. The results verify that it is still effective to approximate the SSMF transmission to the AWGN channel. We believe it is possible to further improve the performance of the proposed MPD-BPS by obtaining the practical channel transition probability. However, it unfortunately takes a huge complexity.
Fig. 9. Experimental results of SSMF loop transmission: (a) NGMI performance at various launched power, and (b) NGMI performance for variable SF
5. Conclusions
We propose an MPD-BPS algorithm for the PS-QAM with variable SFs. The proposed MPD-BPS does not require the MAP detection and enhances the role of peripheral constellation points during the CPE process. Thus, the CPE error is reduced, leading to further approaching the Shannon limit. We first investigate the trade-off between the SF and the noise rejection window of the traditional BPS for the PS-64/256QAM at a fixed SNR. Then, we carry out joint optimization of the SF and the noise rejection window, in order to obtain the maximum AIR at the fixed SNR. Next, we numerically compare the performance of the MPD-BPS and the traditional BPS for the PS-64/256QAM with variable SFs. Both the required SNR enhancement and the improved linewidth tolerance of our proposed MPD-BPS are numerically verified. In order to verify the proposed MPD-BPS, we experimentally conduct 32 Gbaud PS-64QAM with the variable SFs of 0.02, 0.025, 0.03 and 0.035 transmission over the SSMF. The experimental results indicate that the MPD-BPS can obtain an average 0.13 dB SNR enhancement under the B2B transmission and 1.67% reach enhancement after the SSMF transmission, in comparison with traditional BPS.
Funding
National Key Research and Development Program of China (2018YFB1801301); National Natural Science Foundation of China (61875061); Guangdong Guangxi Joint Science Key Foundation (2021GXNSFDA076001).
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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