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Abstract
We propose a probability mass function (PMF) optimization scheme for quadrature amplitude modulation (QAM) signals by considering the parametric characteristic of the training sequence. The training sequence for optimization is mapped in standard Maxwell-Boltzmann (M-B) distribution, and the considered characterizing parameters incorporate either the noise variance or the error matrix of the received symbols. The proposed PMF optimization is based on independent reallocation within each constellation ring, generating new distribution with almost the same entropy and transmitted power as the original distribution. This reallocation mechanism is model-free and iterative-free with very low computational complexity. By characterizing the channel in terms of constellation performance asymmetry, PMF reallocation can be effectively implemented to supplement the existing equalization algorithm. The effectiveness of this approach is experimentally verified in a 40-km transmission system with 24 Gbaud 64-QAM signals under three different scenarios. Through PMF reallocation, we achieve generalized mutual information (GMI) improvement of ∼0.06 and throughput improvement of ∼1.5 Gbit/s before forward error correction in comparison with the standard M-B distribution. The proposed mechanism provides a solution to obtain the optimal PMF in practical communication channels, which suffer from various types of noises and distortions.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Over the last decade, the advances in complex modulation formats and probabilistic shaping (PS) have led to significant growth in the optical transmission capacity [1]. Apart from the capability of controlling the information capacity, PS technology also offers shaping gain in the channel [2]. For an ideal additive white Gaussian noise (AWGN) channel, it is known that a transmitted signal that follows Maxwell-Boltzmann (M-B) distribution provides the theoretically optimal performance [3]. Previously published works have realized PS technology by arranging the frequency of occurrence of each constellation point to decrease exponentially with its amplitude [4,5]. It is verified that the signal transmitted in the M-B distribution can provide up to 1.53-dB signal-to-noise ratio (SNR) gain towards the Shannon limit [3,6].
However, the 1.53-dB shaping gain is the theoretically optimal value under an ideal AWGN channel. The AWGN approximation is not the optimal solution in practical communication channels which suffer from residual nonlinear distortion, transceiver noise, as well as amplifier spontaneous emission noise [7]. Moreover, the characteristics of these distortions might vary adaptively with time. Under this circumstance, the constellation performance appears to be asymmetrical, and the standard M-B distribution is no longer the optimal solution [8,9]. It is unfortunate that the asymmetric performance among the constellation points is difficult to model and characterize due to the complexity of its composition and the slightness of the asymmetry. Consequently, it is important to develop an approach to find a probability mass function (PMF) with performance superior to that of the M-B distribution. Here, PMF is a matrix that shows the probability of the transmitted signal at each constellation point, which can be used to generate QAM signals with the required distribution. The signal with new distribution should also cope with the asymmetrical degradation.
Researchers have been searching for an approach to generate better distribution and eliminate the performance penalty brought by the sub-optimal AWGN approximation in the practical communication channel. The method proposed in [10] generates a new PS distribution by modeling the nonlinear interference noise in the channel. The authors numerically verified the proposed distribution provided improved performance over the M-B distribution. Recent published work also reported PMF modification via modulation depth and clipping percentages optimization [8]. However, the above-mentioned schemes are built for static optimization and therefore are unable to adaptively cope with dynamic channel distortion. The method proposed in [11,12] successfully demonstrated adaptive PMF modification based on the detected errors at the receiver. Utilizing training sequences with uniformly shaped 16-QAM signal, the authors obtained performance improvement compared to the M-B distribution under certain scenarios. However, the channels were characterized simply by calculating the error number of each constellation point, which might appear to be stochastic under low optical signal-to-noise ratio (OSNR) conditions.
In this paper, we develop a reallocation mechanism to generate a new distribution through adaptive channel characterization utilizing a training sequence with standard M-B distribution. Two different characterizing parameters (noise variance and error number of each constellation point) are considered in the PMF reallocation. We propose reallocation constraints and generate the correction matrix based on the characterizing parameters, subsequently performing reallocation to the original M-B distribution. We obtained the new PMF that shows superior performance by adding the correction matrix to the existing M-B distribution. The proposed scheme is evaluated under three different practical scenarios during the experimental verification. In a 40-km transmission system, the signal with a newly generated distribution achieved generalized mutual information (GMI) improvement of ∼0.06 and throughput improvement of ∼1.5 Gbit/s before forward error correction (FEC) in comparison with standard M-B distribution under the three different scenarios. This reallocation mechanism is model-free and iterative-free, thus possessing extremely low computational complexity. Here, “Iterative-free” indicates that the optimization procedure of the PMF reallocation is an instantaneous input-output process, which does not contain any cycle process or iterative optimization. The proposed three constraints can generate the reallocated PMF directly, instead of through hundreds of optimization iterations until convergence. By characterizing the channel in terms of asymmetrical constellation performance, PMF reallocation provides an effective means to supplement the existing equalization algorithm. The proposed reallocation mechanism can work jointly with the conventional nonlinear compensation algorithms to cope with the asymmetric degradation in practical communication systems.
2. Principle
Conventional PS technology requires the transmitted signal to follow the standard M-B distribution, which is the optimal solution for an ideal AWGN channel [3]. The probability of a constellation point ${P^{MB}}({{a_i}} )$ decreases exponentially with its power, and we can describe the distribution by
Here, ${a_i}$ is the constellation point, A denotes the set of 64-QAM constellation points, and $\lambda $ is the shaping factor that determines the decreasing rate of the probability. PS signals transmitted in modern optical networks are loaded with adaptive entropy value $\mathrm{\mathbb{H}}$, which denotes the information carried by the symbols. The entropy is calculated from:
In our following demonstration, the entropy of the training signal is set as 4. The entropy of the signal after reallocation is almost the same as that of the training signal.
The principle of the PMF reallocation mechanism is illustrated in Fig. 1. The PS distribution before PMF reallocation is the standard M-B distribution, as shown in Fig. 1(a). The correction matrix that makes improvement to the M-B distribution is generated subject to the reallocation constraints stated in Fig. 1(b). The PS distribution after PMF reallocation is obtained by adding the correction matrix to the existing M-B distribution, as shown in Fig. 1(c). In the reallocation constraints, we denote ${P^{MB}}$ as the standard M-B distribution before reallocation, and ${P^A}$ as the PMF matrix after reallocation. ${P^{corr}}$ is the correction matrix and is obtained through $P_i^{corr} = \bar{V} - {V_i}$, where V denotes the parameter matrix containing the noise variance or error number of the constellation. ${V_i}$ denotes the parameter at each constellation point, and $\bar{V}$ represents the average of the parameter matrix elements. The reallocation constraints of ${P^A}$ can be represented as shown in Fig. 1, where $P_i^A$ is the probability of each constellation point after reallocation. The first constraint represents that ${P^A}$ is obtained through the standard M-B distribution ${P^{MB}}$ and the correction matrix ${P^{corr}}$. The parameter $\beta $ is used to adjust the correction level. In the process of reallocation, $\beta $ is maximized until ${P^A}$ meets the limit of the second constraint. The second constraint ensures that each probability is larger than zero after correction. The third constraint ensures that the sum of all the probabilities is equal to 1. By solving these equations, we can generate a new PMF after reallocation. For a clear illustration, we describe in Appendix A the detailed procedure of PMF reallocation as well as that of calculating the noise variance and number of errors based on the received symbols.
Fig. 1. (a) The concept and principle of PMF reallocation. (a) The PS distribution before PMF reallocation. (b) The generated correction matrix based on the reallocation constraints. (c) The PS distribution after PMF reallocation.
It is worth noting that the generation of the correction matrix ${P^{corr}}$ is performed ring-by-ring in the PCS 64-QAM constellation, as shown in Fig. 2. From the innermost ring to the outermost ring, different numbers of constellation points are taken into account when performing the reallocation. By reallocating independently within each ring, the new PMF is of almost the same entropy and transmitted power as the original distribution. In a practical coherent transmission system, the entropy of the transmitted signal is usually determined by the user's needs and the required transmission length [13]. Therefore, it is undesirable if the entropy of the signal is unpredictable during the PMF optimization. The rationale behind is that if the PMF optimization leads to unavoidably entropy change, the new PMF cannot meet the same system requirement. Our mechanism solves this problem with the ring-by-ring design because the entropy will only have a slight deviation after the PMF reallocation. In addition, the ring-by-ring design ensures that the reallocating optimization will focus on overcoming the asymmetric degradation of the channel, instead of reaching for shaping gain by simply decreasing the entropy of the PMF. Furthermore, the reallocation mechanism is compatible with communication systems supporting multiple phase shift keying (M-PSK) signals or geometric constellation shaping (GCS) signals thanks to the ring-by-ring design. Another advantage of PMF reallocation is that the conventional communication system can adopt our scheme simply by replacing the PMF of the signal from the standard M-B distribution to the reallocated PMF at the PS mapping stage. Other algorithms remain the same as the ones based on the M-B distribution, thus avoiding reconstruction of the Rx DSP flow.
Fig. 2. The constellation of the PS 64-QAM signal and the number of constellation points that take into account on each ring. The PMF reallocation within each ring is performed independently.
The workflow of the PMF reallocation is split into a few steps. First, we utilize standard MB-distributed PS signal to conduct transmission and collect 20 sets of data at the receiver side (Rx-side). Based on the received signal, we generate a noise variance/error counting matrix from each set of data and calculate the average. The symbol error rate is obtained through a one-to-one, symbol-by-symbol comparison between the transmitted (Tx) signal and the demodulated Rx signal. Subsequently, we can determine exactly the number of errors at each constellation point. The error matrices are then used for the following parametrical characterization. We then perform PMF reallocation within each ring under the above-mentioned constraints. Subsequently, we conduct the transmission again and evaluate the performance of the received signal in terms of GMI and normalized GMI (NGMI). The reason we utilize both GMI and NGMI for the evaluation is because the entropy of the newly generated PMF is slightly smaller than that of the M-B distribution (3.99 compared to 4.00). Therefore, GMI will reflect the actual improvement more objectively.
3. Experimental results and discussion
We build a coherent detection system to evaluate the performance improvement of the PMF reallocation under three different scenarios as illustrated in Fig. 3. The first scenario only comprises a 24 Gbaud PS 64-QAM signal with −6.5 dBm transmitted power, which represents the most common channel environment in the coherent transmission system. Under the second scenario, we add a 7 dBm continuous wavelength (CW) pump at 50-GHz spacing for further verification, which exacerbates the asymmetrical performance of the constellation. The third scenario is a dual-carrier system with 50-GHz channel spacing, and both carriers are modulated with 24 Gbaud PS 64-QAM signal with −9.5 dBm transmitted power. The 3-dB power loss is induced by the coupler. The characteristic of the channel is different from the first scenario due to the crosstalk between these two channels. We maintain the transmission at 40 km single-mode fiber (SMF) for all the three scenarios.
Fig. 3. Schematic diagrams illustrating the three scenarios for experimental verification of the PMF reallocation mechanism.
The experimental setup of the first two scenarios is shown in Fig. 4. The difference between the first scenario and the second one lies in the presence of a neighboring CW pump. In the transmitter digital signal processing (Tx-DSP) part, we generate a random signal sequence of 64000 symbols and apply PS mapping to the signal. After normalization, we apply the root-raised cosine (RRC) filter with a roll-off factor of 0.1. A 1550.6-nm optical carrier of 100-kHz linewidth is modulated with a 24 Gbaud PS 64-QAM signal produced from an arbitrary waveform generator (Keysight M8194A). After 40-km transmission in a standard SMF, a dispersion compensating fiber (DCF) is used to compensate for the dispersion. It is worth noting that this DCF can be replaced with Rx-DSP dispersion compensation algorithms without affecting the performance of the system. We add amplified spontaneous emission (ASE) noise to vary the OSNR of the channel and utilize a polarization controller to maintain a stable single-polarization state at the receiver. The detected photocurrent is captured by a real-time oscilloscope (Keysight UXR0592AP) operating at 256 GSa/s for the following Rx-DSP. The Rx-DSP includes the procedures of resampling, synchronization, RRC pulse shaping, frequency offset estimation (FOE), and power normalization. After blind phase search (BPS) phase noise compensation [14], we conduct decision-directed least mean square (DD-LMS) equalization. The system performance was evaluated in terms of GMI and NGMI after demodulation. The frequency offset is estimated through spectral analysis by considering the information related to the DC component [15] instead of simply utilizing the 4-power FOE algorithm. The performance penalty brought by shaping is thus avoided. It is worth noting that at low OSNR condition, BPS is not the optimal phase recovery algorithm. However, the additional error and phase noise caused by the BPS is symmetrical across the constellation diagram, which will be removed by the averaging operation based on 20 sets received symbols. Broadly speaking, the PMF reallocation is capable to provide improvement regarding the asymmetrical imperfection of the back-end DSP algorithm, while the symmetrically distributed distortion will not affect the performance improvement brought by PMF reallocation.
Fig. 4. Experimental setup for the 40-km coherent transmission system under scenarios 1 and 2. TL: Tunable laser; IQ Mod: IQ modulator; AWG: arbitrary waveform generator; SMF: single mode fiber; DCF: dispersion compensating fiber; EDFA: erbium-doped fiber amplifier; ASE: amplifier spontaneous emission; BPF: bandpass filter; VOA: variable optical attenuator; PC: polarization controller; LO: local oscillator. OSA: optical spectrum analyzer.
We present the 3-D bar graph of the correction matrix generated under scenario 1 in Fig. 5, where the height of each bar represents the actual probability value of the corresponding constellation point that needs to add on or subtracted from the standard M-B distribution. We can observe that though slightly different from each other, the general trend and distribution show some consistency across the 3-D bar graphs in Fig. 5. The absolute values of most constellation points in the correction matrix decreases as the OSNR increases. This indicates that the performance asymmetry across the constellation diagrams becomes milder under high OSNR condition. Figure 5 also shows general consistency between the two sets of correction matrices generated by considering the number of errors and from considering the noise variance of the received signal. This phenomenon reveals that both parameters are effective in characterizing the asymmetrical performance across the constellation diagrams. In the following discussion, we will compare the performances of the signals with reallocated PMF generated by these two characterizing parameters as well as the signals with standard M-B distribution in terms of NGMI and GMI.
Fig. 5. The 3-D bar graph of the correction matrix generated for scenario 1 (single channel) under OSNR of (a) 17 dB; (b) 23 dB; (c) 27 dB based on (i) number of errors and (ii) noise variance of the received signal, respectively.
We present the experimental results in Fig. 6 after conducting the 40-km transmission for scenario 1 under different OSNR conditions. Figure 6(a) and Fig. 6(b) show the NGMI and GMI performance comparison, respectively, of the standard M-B distributed signal and the reallocated signals under different OSNR conditions. Through reallocation, we have achieved an average pre-FEC GMI improvement of 0.0503 (noise variance) and 0.0498 (error counting). Considering the 24 Gbaud PCS 64-QAM signal, the average pre-FEC data rate improvement is 0.0503 × 24 = 1.2072 Gbit/s (noise variance) and 0.0498 × 24 = 1.1952 Gbit/s (error counting), respectively. The improvement is consistent with previously published simulation results [10]. It is worth mentioning that under low OSNR conditions, the improvement based on noise variance is larger than that of error counting. On the other hand, under high OSNR conditions, the improvement based on error counting is larger than that of the noise variance. This is because noise variance can better characterize the asymmetrical constellation performance when the ASE noise power is large, as reflected in the case of low OSNR. Figure 6(c) and (d) show respectively the PMF after reallocation based on the noise variance and the error number distribution on each constellation point of the received signals under 23 dB OSNR. Figure 6(e) and (f) show the constellations of the received signal with reallocated PMF. Our results reveal that the two proposed reallocation mechanisms can effectively improve the system performance compared to M-B distribution in practical communication systems.
Fig. 6. Performances, distributions, and constellation diagrams of the PCS 64-QAM signal. (a) The NGMI and (b) GMI performance comparison of the standard M-B distributed signal and the reallocated signal under different OSNR conditions. The PMF after reallocation based on (c) the noise variance and (d) the error number of the standard M-B distributed signals under 23 dB OSNR. The constellation of the received signal with reallocated PMF based on (e) the noise variance and (f) the error number under 23 dB OSNR.
To further investigate the effectiveness of the reallocation mechanism, we add a neighboring CW pump to transmit along with the original channel. This will exacerbate the asymmetrical performance of the constellation. The wavelength of the CW pump and the transmitted signal are 1550.2 nm and 1550.6 nm, respectively. The frequency spectrum of the 24 Gbaud PS 64-QAM signal with the neighboring CW pump is shown in Fig. 7. Utilizing the ASE source, we vary the OSNR from 16 to 23 dB. The OSNR adjustment range is limited by the power of the ASE source as well as the existence of the CW pump.
The 3-D bar graphs of the correction matrices generated under scenario 2 is shown in Fig. 8. We can see that the absolute value in each constellation point of the correction matrices is generally larger than that in scenario 1. The larger absolute value indicates that the CW pump brings nonlinearity into the channel and subsequently exacerbates the asymmetry of the performance across the constellation points. Similar to what we have observed in scenario 1, consistency also exists across each 3-D bar graph in terms of the general trend and distribution. This phenomenon further verifies the effectiveness of both correction approaches based on error numbers and noise variance. We compare the performances of the signals with reallocated PMF generated by these two parameters as well as the signals with standard M-B distribution in terms of NGMI and GMI as shown in Fig. 9. Through reallocation, we have achieved an average pre-FEC GMI improvement of 0.0674 (noise variance) and 0.0676 (error counting). Considering the 24 Gbaud PCS 64-QAM signal, the average total data rate improvement before FEC is 1.6165 Gbit/s (noise variance) and 1.6224 Gbit/s (error counting), respectively. Our results reveal that in the presence of a neighboring CW pump, the performances of the signals following the reallocated PMF generated by both parameters shows superiority compared to the signals following standard M-B distributions. It is worth noting that we observe no distinct performance gap between the signals generated by considering the noise variance and by considering the number of errors in both low and high OSNR conditions. This is because the residual nonlinear noise originates mainly from the presence of the CW pump, rather than having ASE noise as the dominant source of constellation asymmetry.
Fig. 8. The correction matrix generated for scenario 2 (with neighboring CW pump) under OSNR of (a) 17 dB; (b) 19 dB; (c) 22 dB based on (i) number of errors and (ii) noise variance of the received signal, respectively.
Fig. 9. Performance comparison of the PCS 64-QAM signal with neighboring CW pump. (a) The NGMI and (b) GMI performance comparison of the standard M-B distributed signal and the reallocated signals under different OSNR conditions.
To further extend the applicability of the reallocation mechanism to multichannel transmission, we deploy another IQ modulator and transmit two data channels simultaneously, as shown in Fig. 10 (scenario 3). Under this scenario, the signal will suffer from wavelength division multiplexing (WDM) crosstalk in addition to the above-mentioned residual nonlinear distortion, transceiver noise, as well as amplifier spontaneous emission noise. Two optical carriers at 1550.2 nm and 1550.6 nm are modulated with 24 Gbaud PS 64-QAM signal generated from the AWG. Apart from the transmitter, the remaining part of the experimental setup remains the same as in scenarios 1 and 2. The frequency spectrum of the signal is shown in Fig. 11. By adjusting the output power of the ASE source, the OSNR is tuned from 27 dB to 18 dB. It is worth noting that some weak frequency tones appear outside the signal band for the case of 27 dB OSNR. This is a special phenomenon that exists while using the M8194A AWG, which will unavoidably generate such frequency tones after being enabled.
Fig. 10. Experimental setup for the dual-channel 40-km coherent transmission system under scenario 3. TL: Tunable laser; IQ Mod: IQ modulator; AWG: arbitrary waveform generator; SMF: single mode fiber; DCF: dispersion compensating fiber; EDFA: erbium-doped fiber amplifier; BPF: bandpass filter; VOA: variable optical attenuator; PC: polarization controller; LO: local oscillator. OSA: optical spectrum analyzer.
The 3-D bar graphs of the correction matrices generated under scenario 3 are presented in Fig. 12. Similar to the previous two scenarios, consistency is observed in terms of the general trend and distribution under different characterizing parameters or OSNR conditions. The absolute value of each constellation point in the correction matrices is smaller than that in scenario 2, indicating that the asymmetry of the performance brought by a neighboring channel is weaker than that of a CW pump, which has a larger power. From Fig. 12, we can imply that the consistency of the correction matrices across different OSNR conditions is higher than that between different characterizing parameters. We present the constellation diagrams recovered from the signals generated by these correction matrices in Fig. 13. Figure 14 compares the performances of the signals with reallocated PMF generated by these two parameters as well as the signals with standard M-B distribution in terms of NGMI and GMI. In the presence of a neighboring channel, we have achieved an average pre-FEC GMI improvement of 0.0605 (noise variance) and 0.0613 (error counting) through the reallocation mechanism. Considering the 24 Gbaud PCS 64-QAM signal, the average total data rate improvement before FEC is 1.4529 Gbit/s (noise variance) and 1.4703 Gbit/s (error counting). The results reveal that the proposed scheme of adopting a correction matrix is effective in WDM application scenarios.
Fig. 12. The correction matrices generated for scenario 3 (with neighboring channel) under OSNR of (a) 18 dB; (b) 22 dB; (c) 25 dB based on (i) number of errors and (ii) noise variance of the received signal, respectively.
Fig. 13. Constellation diagram of the recovered signal with reallocated PMF for scenario 3 (with neighboring channel) under OSNR of (a) 18 dB; (b) 22 dB; (c) 25 dB based on (i) number of errors and (ii) noise variance of the received signal, respectively.
Fig. 14. Performance comparison of the PCS 64-QAM signal with neighboring data channel. (a) The NGMI and (b) GMI performance comparison of the standard M-B distributed signal and the reallocated signals under different OSNR conditions.
To analyze the general trend of the correction matrices at different OSNRs under the three scenarios, we plot the average absolute value of the matrix elements as shown in Fig. 15. Due to the limited power of our ASE source, the OSNR ranges used in the three scenarios are not identical. We observe that the average correction value generated by the noise variance method is generally larger than that by error counting. Also, as the OSNR increases, the average correction value gradually decreases under all scenarios. In addition to OSNR, other parameters of the communication system can have different effects on the performance of reallocation. For example, transmission at a higher entropy may lead to relatively severe nonlinear distortion, especially for the outside ring of the constellation. The overall distribution of the asymmetrical imperfection may vary with the entropy value, leading to different performance improvements of PMF reallocation. Other factors such as constellation shape/synchronization/throughput are not likely to affect the characteristics of the noise distribution but the overall transmission quality. Therefore, the effectiveness of parametric characterization will not depend strongly on these factors.
Fig. 15. Average absolute value of the correction matrix elements at different OSNRs. The values obtained with different characterizing parameters under the three scenarios are presented.
4. Simulation results and discussion
To evaluate the performance gain of this scheme in long distance transmission scenario, we conduct simulations based on VPI TransmissionMaker 11.3. In the setup, the baud rate and the sampling rates of DAC and ADC remain consistent with the values used in our experiments. The 3-D bar graphs of the correction matrices generated under long-haul transmission are presented in Fig. 16. Similar to the experimental results, consistency is observed in terms of the general trend and distribution under different characterizing parameters or transmission lengths. The absolute value of each constellation point in the correction matrices is larger than that in experimental results due to stronger asymmetry of the performance brought by longer transmission length. From Fig. 16, we can also infer that the consistency of the correction matrices across different transmission lengths is higher than that between different characterizing parameters.
Fig. 16. Simulated results of the correction matrices generated under long-haul transmission length of (a) 400 km; (b) 800 km; (c) 1200 km based on (i) number of errors and (ii) noise variance of the received signal, respectively.
Figure 17 compares the performances of the signals with reallocated PMF generated by these two parameters as well as the signals with standard M-B distribution in terms of NGMI and GMI. In the simulation of long-haul transmission, we have achieved an average pre-FEC GMI improvement of 0.0918 (noise variance) and 0.0853 (error counting) through the reallocation mechanism. Considering the 24 Gbaud PCS 64-QAM signal, the average total data rate improvement before FEC is 2.2035 Gbit/s (noise variance) and 2.0479 Gbit/s (error counting). It is worth mentioning that as the transmission length increases, a slightly larger improvement can be obtained based on both characterizing parameters. The result is expected as a longer transmission length will lead to a higher asymmetry. Another point to note is that the performance improvement is relatively large compared to the experimental results we obtained under 40-km transmission. We therefore conclude that the proposed scheme is effective in long-haul transmission scenarios. It is worth mentioning that our PMF reallocation scheme can also be applied in the presence of complex and dynamic polarization impairments in the communication system. For dual-polarization transmission, the performance improvement brought by the reallocation may be slightly affected.
Fig. 17. Performance comparison of the PCS 64-QAM signal for long-haul transmission (a) The NGMI and (b) GMI performance comparison of the M-B distributed signal and the reallocated signals under different transmission lengths.
5. Conclusion
We optimize the probabilistic shaping distribution by reallocating the symbol probability based on the performance of each constellation point in the training sequence. By characterizing the channel asymmetry in terms of noise variance and error matrix, we adaptively generate the correction matrix under the proposed constraints. The new PMF that shows superior performance is obtained by adding the correction matrix to the standard M-B distribution. The effectiveness of the proposed mechanism is experimentally verified in a 40-km transmission system with 24 Gbaud PCS 64-QAM signals under three different scenarios. We compare the performances of the signals with reallocated PMF generated by these two parameters as well as that of the standard M-B distribution. Through reallocation, the signal with a newly generated distribution achieved pre-FEC GMI improvement of ∼0.06 and throughput improvement of ∼1.5Gbit/s in comparison with the standard M-B distribution under three different scenarios. The characteristics of the correction matrix generated under different OSNR conditions, parameters, and scenarios are also analyzed. Our experimental results show that the reallocation mechanism can effectively improve the system performance compared to that of the M-B distribution in practical communication systems. The proposed mechanism is model-free and iterative-free, maintaining extremely low computational complexity. By characterizing the channel in terms of asymmetrical constellation performance, PMF reallocation provides an effective supplement to the existing equalization algorithm and copes with the asymmetric degradation in practical communication systems. The unique ring-by-ring design makes the mechanism compatible with M-PSK and GCS signals after slight modification. The proposed mechanism provides a valid solution to achieve optimal PMF in the presence of asymmetrical distortions for non-AWGN communication channels. In practical utilization, if the system is very robust and the condition of all the components incorporated by the system remains unchanged over time, PMF reallocation will not be frequently needed. However, we would suggest running the reallocation whenever there is any physical change in the system. The reallocation can also be run in a regular schedule even without any component replacement. After all, PMF reallocation is a lite scheme that consumes very little computational resources while supporting full utilization of each constellation point and at the same time reflecting the real-time system status.
Appendix A. Pseudo-code and annotation describing the calculation of the two parameters (noise variance and number of errors) and the PMF reallocation
Funding
National Natural Science Foundation of China (RGC Joint Research Scheme N_CUHK444/22); University Grants Committee (14210419, 14211120, 14221322).
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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