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Abstract
A degenerated look-up table–based perturbative nonlinearity compensation (DLUT-PNC) algorithm is proposed to compensate for intra-channel fiber nonlinearity. It can flexibly optimize the implementation complexity for probabilistically shaped (PS) signals with different shaping rates. In addition, we propose a homomorphic DLUT-PNC (HDLUT-PNC) scheme to further reduce the complexity. In simulations with a transmission distance of 1200-km for 70-GBaud PS-16QAM signals, both the performances of the DLUT-PNC and HDLUT-PNC are investigated and compared. The HDLUT-PNC scheme significantly reduces the table’s input size, number of look-up operations and number of complex multipliers compared to the DLUT-PNC. Moreover, we also numerically investigate 56-Gbaud PS-32QAM signals with a transmission distance of 800-km, and the results are similar. Finally, the performance is verified in experiments for PS-16QAM signals with a transmission distance of 432-km.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The exponentially increasing capacity demands driven by new technologies such as big data, cloud computing, artificial intelligence (AI) and so forth push the capacity of optical communication systems towards the Shannon limit []. In recent years, probabilistic shaping has attracted many attentions since it enables optical communication systems to approach the Shannon capacity limit [2–5]. In the meantime, since fiber nonlinearity remains a major limiting factor for further increase of the capacity, various digital nonlinearity compensation schemes have been widely researched in last few years [6]. Well-known digital nonlinearity compensation schemes include digital backpropagation (DBP), perturbative nonlinearity compensation (PNC), Volterra series, and so forth [7–14]. Among them, PNC is considered as a promising method. It enables single-step fiber nonlinearity compensation by deriving an analytical solution of the nonlinear Schrödinger equation (NLSE). However, the large number of multipliers in calculating the perturbation terms limits the practical implementation of this method. Note that the complexity of a multiplier is about five to ten times higher than that of an adder in the ASIC implementation. Then a multiplier-free PNC is proposed based on the constant module characteristic of quadrature phase shift keying (QPSK) signals [13]. Since higher order modulation formats such as 16 quadrature amplitude modulation (16QAM) are required for next generation coherent optical systems, the multiplier-free PNC is extended to 16QAM and a degenerated PNC (DPNC) scheme was proposed in [15–17]. The DPNC degenerates 16QAM symbols into QPSK symbols to calculate the perturbation terms, and then the multiplier-free operation is applied. However, the compensation gain is significantly reduced by the degeneration. Moreover, in these works, probabilistic shaping, which affects the design and performance of DPNC, is not considered.
We proposed a degenerated look-up table (DLUT) based PNC, denoted as DLUT-PNC, to efficiently optimize the performance and complexity for signals with various shaping rates in [18]. We used a look-up table (LUT) method to obtain the products of three 16QAM symbols and degenerated the size of the table’s outputs. In this paper, we describe our work in [18] in more detail. In addition, we further propose a homomorphic DLUT-PNC (HDLUT-PNC). This method first reduces the size of the table’s input by adopting the homomorphic transformation. Then the perturbation coefficients are quantized and counters are used to decrease the number of the look-up operations. In simulations with a transmission distance of 1200-km for 70-GBaud probabilistically shaped 16QAM (PS-16QAM) signals, both the performances of the DLUT-PNC and HDLUT-PNC are investigated and compared. The HDLUT-PNC scheme can significantly reduce the table’s input size, number of look-up operations and number of complex multipliers compared to the DLUT-PNC scheme. Moreover, we also investigate 56-Gbaud probabilistically shaped 32 quadrature amplitude modulation (PS-32QAM) signals with a transmission distance of 800-km, and the obtained results are similar. Finally, we demonstrate the proposed schemes in experiments for PS-16QAM signals with a transmission distance of 432-km.
The rest of this paper is organized as follows. In Section 2, the principles, mathematical analysis and complexity analysis of the conventional PNC, DPNC and our proposed DLUT-PNC, HDLUT-PNC are provided. In Section 3, the performances of these schemes are investigated and discussed in simulations for 70-Gbaud PS-16QAM signals with a 1200-km transmission distance and 56-Gbaud PS-32QAM signals with an 800-km transmission distance. Moreover, we study these methods in experiments for 30-Gbaud DP-16QAM signals with a 432-km transmission distance in Section 4. Finally, in Section 5, conclusions are drawn.
2. Principles
2.1 Principles of the PNC and DPNC
The PNC is derived by solving the Manakov equation in the time domain based on the perturbation analysis. The obtained solution of x-polarization is given as [13]
Fig. 1. Two-dimensional plots of the amplitudes of the coefficients for system configurations as (a) 10-GBaud and 500-km, (b) 10-GBaud and 1000-km and (c) 50-Gbaud and 1000-km.
It reveals that the number of coefficients increases as the baud rate and transmission distance get larger, which implies that the computational complexity will become a severer issue for next generation systems with higher baud rates. Since the number of the perturbation terms should be sufficiently large for a high compensation gain, one possible solution to lower the complexity is to simplify the multiplications in calculating each term.
Then the multiplier-free PNC was proposed, which leveraged the constant module characteristic of QPSK signals [13]. Considering that the product of any QPSK symbols is still a QPSK symbol, multipliers are replaced by logical operators in this method. However, higher order modulation formats such as 16QAM are currently being developed and deployed. Since 16QAM signals have three different amplitude levels, the multiplier-free operation cannot be directly adopted. To address this issue, the degenerated PNC (DPNC) was proposed [15]. This method first degenerates the 16QAM symbols into QPSK symbols, followed by the multiplier-free operations for the calculation of perturbation terms.
2.2 Principle of the DLUT-PNC
In systems with probabilistic shaping, the conventional PNC is able to provide good performance. However, its complexity is always high regardless of the shaping rate, because the three-symbol products used to calculate the perturbation term are only determined by the base constellation. The complexity of the DPNC is significantly lower but it only performs well for signals with low shaping rates, in which case the symbols mainly consist of low power constellation points.
We propose a DLUT-PNC scheme to achieve a balance between performance and complexity for a specific shaping rate. In particular, it achieves a higher compensation gain than the DPNC and a lower complexity than the PNC. In this scheme, LUTs are used to replace the multipliers in calculating the products of these three-16QAM-symbol terms. Note that we define a three-symbol term as a triplet in the following discussions. As Fig. 2(a) shows, the product of any 16QAM triplet has 80 different values. It means that, in order to get an exact result, the input size and output size of the LUT should be 4096 and 80, respectively.
Since the complexity of the LUT mainly depends on the input size, output size and the number of look-up operations, we first propose to adopt logical operators in our scheme to reduce the table’s size. As Fig. 3 shows, we find all the triplets whose products locate at the first quadrant and set them as the input of the table. Then any other triplets whose products are not in the first quadrant are rotated to the table’s input terms. After the LUT operation, the inverse rotation is adopted. As a result, a table only contains the first quadrant values is built, and the size of the table’s input and output are 1024 and 20, respectively.
Fig. 3. (a) Block diagram of the logical operator aided LUT. (b) Illustration of the logical operations.
Afterwards, the table is degenerated to reduce the size of the table’s output. The examples of 1-DLUT and 4-DLUT at a shaping rate of 0.6 are shown in Fig. 2(b). Note that N-DLUT is defined as the degenerated LUT with a table’s output size of N. To satisfy the complexity requirement, we can degenerate the table’s output into any size, but the performance will be different.
In order to maximize the compensation gain of the DLUT-PNC, each element of the table’s output should be optimized. We define a performance metric $\eta $ to optimize the performance by calculating the summation of the Euclidean distance between the original output and degenerated output. This metric can be derived as follows. First, the perturbation term of x-polarization calculated by the degenerated symbols is given as
We propose a blind searching degeneration (BSD) method to find the minimal value of $\eta $, which corresponds to the optimal degeneration. This method scans all the possible degenerated points with a sufficiently small step size, so the minimal value of $\eta $ can always be obtained. As shown in Fig. 4, the performance metric of the optimal degeneration results for different shaping rates and table’s output sizes can be obtained. It should be noted that, in this paper, we define the relationship between the shaping rate and spectrum efficiency (SE) as
Since a lower $\eta $ means a better compensation performance, we observe that the compensation gain increases as the table’s output size gets larger, and the compensation gain decreases first then increases as the shaping rate gets higher.
2.3 Principle of the HDLUT-PNC
The DLUT-PNC scheme only degenerates the table’s output size. Although the output size could be degenerated into 4 or even less, the input size is still 1024. Moreover, the large number of look-up operations are not reduced. To address these two issues, we propose a HDLUT-PNC method. It decreases the table’s input size by adopting a homomorphic function. And the number of look-up operations is reduced by quantizing perturbation coefficients and using counters to group the triplets before the look-up operation.
The homomorphic function is adopted to replace multiplication by summation and its principle is explained as follows. As the block diagram in Fig. 5 shows, we first use a table whose input and output sizes are both 16 to obtain the logarithmic values of 16QAM symbols. Note that a logarithmic function is usually realized by a LUT in the design of a digital chip. Then the logarithmic value of any 16QAM triplets’ product can be obtained by summing up three logarithmic 16QAM symbols. At last, another LUT is used to get the exponential values of these summations, obtaining the product of any triplet.
The logical operators used at the input-side can be replaced by adding or subtracting one element of ${\{ \textrm{j}\pi }/2,{\; \textrm{j}\pi },{\; }3{\textrm{j}\pi }/2\} $. Since each symbol only needs one look-up operation to get the logarithmic value, the complexity of the HDLUT mainly depends on the second LUT operation. To conclude, we decrease the table’s input size from 1024 to 20 in this scheme.
Next, we use the perturbation coefficient quantization and counters to decrease the number of the look-up operations. The coefficient quantization was originally proposed to decrease the number of multiplications between the perturbation coefficients and the triplets in [8,20]. The perturbation term calculated with the quantized coefficients is given as
To avoid these repetitive operations, we propose to use counters for real-time implementation. By counting the numbers of different unique logarithmic values of these triplets, for each quantized coefficient it only needs 20 times of the look-up operations. Then the counted numbers are multiplied with the corresponding triplets. Finally, the perturbation term calculated by the HDLUT-PNC is given as
In this section, 16QAM signals are used as examples to introduce the principles of the DLUT-PNC and HDLUT-PNC schemes. It is explicit that these schemes can be extended to higher order modulation formats such as 32QAM and 64QAM. However, larger LUT’s are needed for them to achieve the same compensation performance.
2.4 Complexity analysis
In this section, we calculate and compare the complexity of the previously introduced schemes for 16QAM signals by counting the required complex multipliers, logical operators, LUT input and output sizes, and look-up operations for each symbol per polarization. The results are summarized in Table . Here we define the number of perturbation coefficients as M, the quantized levels as K and the degenerated table’s output size as N.
Table 1. Complexity analysis of PNC, N-DLUT-PNC and N-HDLUT-PNC
For high symbol rate long-haul transmission systems, the value of M is usually tens of thousands. And the value of K can be much smaller, e.g. tens or even less. Therefore, we can see that the complexity of the HDLUT-PNC is significantly lower than the PNC and DLUT-PNC. In Table 2, we show some examples of the complexity reduction for various values of M and K.
Table 2. Complexity comparison examples of PNC, N-DLUT-PNC and N-HDLUT-PNC
These examples show that the reduction of the table’s input size of the HDLUT-PNC compared to the DLUT-PNC only depends on the modulation format, regardless of the values of M and K. Moreover, the HDLUT-PNC can further decrease the number of complex multipliers and number of look-up operations compared to the DLUT-PNC as the value of M gets larger and the value of K gets smaller.
3. Simulation results and discussions
Figure 6 depicts the simulation setup. In the transmitter-side digital signal processing (DSP), single channel signals with probabilistic shaping are first generated. We study 70-Gbaud dual-polarization (DP)-16QAM signals and 56-Gbaud DP-32QAM signals. The evaluated shaping rate ranges from 0.1 to 0.9 for DP-16QAM and from 0.2 to 1.4 for DP-32QAM. After that, the signals are pulse shaped by a root-raised cosine (RRC) filter with a roll-off factor of 0.01. Then the signals are launched into a standard single mode fiber (SSMF) link with a span length of 80-km, and it is simulated by the Manakov equation based split-step Fourier method (SSFM) with a step size of 20-m. At the end of each span, an EDFA with a noise figure of 5-dB is used to compensate the fiber loss completely. The numbers of the spans are 15 for DP-16QAM and 10 for DP-32QAM, respectively. The receiver-side DSP includes chromatic dispersion compensation (CDC), frequency offset compensation (FOC), decision-directed least mean square (DD-LMS) equalization and carrier phase recovery (CPR). Then both of the received symbols and decided symbols are loaded to compensate fiber nonlinearity. Finally, the signal-to-noise ratio (SNR) is calculated based on the compensated symbols and transmitted symbols to evaluate the transmission performance.
Fig. 6. Simulation setup. ECL: external cavity laser; DP-IQ Mod.: dual-polarization IQ modulator; DAC: digital-to-analog converter; ADC: analog-to-digital converter.
First, the performance of the conventional PNC and DPNC schemes for 16QAM signals with different shaping rates is investigated to illustrate the impact of probabilistic shaping. Figure 7(a) shows the SNR versus launch power with a shaping rate of 0.7 for the PNC and DPNC. Note that we define the compensation gain as the difference in the maximum SNRs between the systems with and without nonlinearity compensation. Then the gains of the PNC and DPNC are 0.5-dB and 1.54-dB, respectively. Figure 7(b) shows the compensation gain of the PNC and DPNC as a function of shaping rate for 16QAM signals. As expected, the compensation gain of the PNC is nearly constant as the shaping rate changes, whereas the gain of the DPNC decreases and then remains similar as the shaping rate gets larger. In particular, the gain of the DPNC is 1.21-dB and 0.52-dB at the shaping rate of 0.1 and 0.9, respectively.
Fig. 7. (a) SNR vs. launch power for 16QAM signals with a shaping rate of 0.7. (b) Compensation gain of PNC and DPNC vs. shaping rate for 16QAM signals.
Then we investigate the performance of the DLUT-PNC with various table’s output sizes and shaping rates for 16QAM signals in Fig. 8(a). It should be noted that the scheme without degenerating the output size, denoted as LUT-PNC, is equivalent to the conventional PNC, since the table output is identical to the original multiplication result. Similarly, 1-DLUT-PNC is equivalent to the DPNC. Therefore, the compensation gains of the LUT-PNC and 1-DLUT-PNC are the upper and lower limits of the DLUT-PNC, respectively. We can find that the general trend of the curves in Fig. 8(a) is the same with the corresponding curves of the performance metrics in Fig. 4, validating the performance metric we define in Eq. (5) to find the optimal degeneration schemes. At a shaping rate of 0.7, the 4-DLUT-PNC achieves a 0.66-dB higher gain than the conventional DPNC.
Fig. 8. (a) Compensation gain of DLUT-PNC with various table’s output sizes vs. shaping rate for 16QAM signals. (b) Compensation gain of HDLUT-PNC with various table’s output sizes vs. shaping rate for 16QAM signals.
In Fig. 8(b), we show the compensation gain of the HDLUT-PNC for 16QAM signals. Here, we also note the scheme without table degeneration, denoted as HLUT-PNC, and the 1-HDLUT-PNC are the upper and lower limits of the HDLUT-PNC, respectively, in terms of compensation gain. The 4-HDLUT-PNC can achieve a 0.51-dB higher gain than the conventional DPNC for a shaping rate of 0.7. Note that the quantization level used in these schemes is 40. And the perturbation coefficient quantization used in the HDLUT scheme is the reason that the compensation performance of the HDLUT is slightly lower than the DLUT.
In Fig. 9(a), we show the compensation gain of the DLUT-PNC and HDLUT-PNC with various shaping rates as a function of the table’s output size for 16QAM signals. For a higher shaping rate, both of the DLUT-PNC and HDLUT-PNC need a larger table’s output size to achieve the same gain. And for the same shaping rate, the HDLUT-PNC needs a larger table’s output size than the DLUT-PNC to achieve the same gain. In Fig. 9(b), we show the performance penalty of the HDLUT-PNC relative to the DLUT-PNC. In a specific scene where the shaping rate is 0.7 and the table’s output size is 4, the performance penalty is only 0.21-dB. Nevertheless, the table’s input size, number of look-up operations and number of complex multipliers of the HDLUT-PNC are significantly reduced as described in the previous section.
Fig. 9. (a) Compensation gain of DLUT-PNC and HDLUT-PNC with various shaping rates vs. table’s output size for 16QAM signals. (b) Performance penalty of HDLUT-PNC relative to DLUT-PNC for 16QAM signals.
Next, we extend these studies to 32QAM signals. Comparing the results in Fig. 10(a) and that in Fig. 8(a), the DPNC performs slightly worse for 32QAM signals than 16QAM signals. The compensation gain is only 0.39-dB at a shaping rate of 1, whereas the compensation gain of the PNC is about 1.6-dB for any shaping rate. The larger performance gap between the PNC and DPNC implies that the compromise of the two methods is more necessary for 32QAM signals. And we can find that the 4-DLUT-PNC achieves 0.76-dB higher gain than the DPNC. As shown in Fig. 10(b), we also investigate the performance of the HDLUT-PNC. The 4-HDLUT-PNC can achieve 0.54-dB higher gain than the DPNC, which is only 0.34-dB, with the same coefficient quantization level.
Fig. 10. (a) Compensation gain of DLUT-PNC with various table’s output sizes vs. shaping rate for 32QAM signals. (b) Compensation gain of HDLUT-PNC with various table’s output sizes vs. shaping rate for 32QAM signals.
In Fig. 11(a), we show the compensation gain of the DLUT-PNC and HDLUT-PNC for 32QAM signals as a function of the table’s output size. The observations are similar to that of 16QAM signals. We further investigate the performance penalty of the HDLUT-PNC relative to the DLUT-PNC. In the specific scene where the shaping rate is 1 and the table’s output size is 4, the performance penalty is only 0.29-dB. Note that the HDLUT-PNC for 32QAM signals can also significantly reduce the complexity with respect to the DLUT-PNC.
Fig. 11. (a) Compensation gain of DLUT-PNC and HDLUT-PNC with various shaping rates vs. table’s output size for 32QAM signals. (b) Performance penalty of HDLUT-PNC relative to DLUT-PNC for 32QAM signals.
All the simulation results reveal that both of the DLUT-PNC and HDLUT-PNC can obtain a superior balance between the compensation gain and complexity for any specific shaping rate. More importantly, the HDLUT-PNC can significantly reduce the computational complexity compared to the DLUT-PNC.
4. Experimental results and discussions
Figure 12(a) depicts the experimental setup. In the transmitter-side DSP, a single channel 30-Gbaud DP-16QAM signal with RRC pulse shaping (roll-off factor = 0.1) and probabilistic shaping is first generated. The shaping rate ranges from 0.1 to 0.9. The transmitted waveforms are generated offline in MATLAB and then uploaded to an arbitrary waveform generator (AWG) with a sampling rate of 80-GSa/s. Afterwards, each output signal is driven by a radio frequency (RF) driver followed by a DP-I/Q modulator. The nominal linewidth of the tunable laser is 100-kHz and the central frequency is 193.41-THz. Then the modulated signal is amplified by an Erbium-doped fiber amplifier (EDFA). The launch power is adjusted by a variable optical attenuator (VOA). The total transmission distance is 432-km and the transmission link consists of 5 spans of SSMF and 5 inline EDFAs. After transmission, an EDFA is used to load extra amplified spontaneous emission (ASE) noise to adjust the optical signal-to-noise ratio (OSNR), and an optical filter is used to remove the out-of-band ASE noise. An optical spectrum analyzer (OSA) is used to measure the OSNR. The optical signal is input to an EDFA with a constant output power to obtain a fixed received optical power (ROP). Then a laser with a linewidth of 100-kHz is used as the local oscillator for coherent detection at the receiver. Afterwards, a 4-channel real-time digital storage oscilloscope (DSO) with a sampling rate of 100-GSa/s is used to digitize the signals. The receiver-side DSP is shown in Fig. 12(b). The in-phase/quadrature (IQ) errors are first compensated. Then the FOC and CDC are performed, followed by the DD-LMS equalization and CPR. After the nonlinearity compensation, the bit error ratio (BER) is calculated to evaluate the transmission performance.
Figures 13(a) and 13(b) show the required OSNR (ROSNR) of the DLUT-PNC and HDLUT-PNC schemes versus the launch power, respectively. The shaping rate for the 16QAM signals is 0.5 and the soft-decision forward error correction (SD-FEC) threshold is assumed to be 0.02. We evaluate the schemes at a launch power of 7-dBm where the nonlinearity penalty is 4.1-dB. The compensation gain is defined as the difference in the ROSNRs between the systems with and without nonlinearity compensation. In this case, the compensation gains of the 4-DLUT-PNC and 4-HDLUT-PNC are 0.98-dB and 0.97-dB, respectively. The performance penalty of 4-HDLUT-PNC is only 0.01-dB compared to the 4-DLUT-PNC, which implies the impact of the coefficient quantization is negligible in this scenario. This is because the number of perturbation coefficients is much smaller with such a low symbol rate and short transmission distance in our experiment.
Fig. 13. Measured ROSNR of (a) DLUT-PNC and (b) HDLUT-PNC vs. launch power for 16QAM signals with a shaping rate of 0.5.
Next, we further investigate the schemes for 16QAM signals with a shaping rate of 0.8. Since the linear performance is worse at a higher shaping rate, we evaluate the ROSNR at a larger SD-FEC threshold, which is 0.03. We test the schemes at a launch power of 5-dBm where the nonlinearity penalty is 3.2-dB. As Fig. 14 shows, the compensation gains of the 4-DLUT-PNC and 4-HDLUT-PNC are 1.02-dB and 1.01-dB, respectively. The performance penalty of the 4-HDLUT-PNC is again only 0.01-dB compared to the 4-DLUT-PNC method.
Fig. 14. Measured ROSNR of (a) DLUT-PNC and (b) HDLUT-PNC vs. launch power for 16QAM signals with a shaping rate of 0.8.
The experimental results verify the performance of both the DLUT-PNC and HDLUT-PNC schemes. We also observe that the 4-HDLUT-PNC scheme can improve the compensation performance compared to the 1-HDLUT-PNC at a shaping rate of 0.8, whereas the performance is nearly the same at a shaping rate of 0.5. This result well meets the expectation and shows the flexibility of the proposed approaches.
Moreover, since these compensation schemes are performed based on the decided symbols, their performances degrade as the BER increases. In the experiments, since the compensation gains are evaluated at the SD-FEC threshold, the impact of the decision errors has been included. As an alternative, the PNC schemes can be implemented at the transmitter without decision errors, and the proposed method can be applied to reduce complexity in that scenario as well.
5. Conclusion
In this paper, we first describe the DLUT-PNC scheme which we proposed in [18] in more detail. This method uses the LUT method to obtain the triplets’ products, and then degenerates the table’s output size to flexibly optimize the implementation complexity for PS signals. Moreover, we propose a HDLUT-PNC method which uses the homomorphic transformation to reduce the table’s input size, and adopts the coefficient quantization and counters to decrease the number of complex multipliers and number of look-up operations. Then we evaluate our schemes in simulations for PS-16QAM signals and PS-32QAM signals. All the results show that the HDLUT-PNC method can significantly reduce the implementation complexity with a small performance penalty. Finally, similar evaluations are conducted in experiments for PS-16QAM signals with a transmission distance of 432-km.
Funding
National Natural Science Foundation of China (61801291); Shanghai Rising-Star Program (19QA1404600); National Key Research and Development Program of China (2018YFB1801200).
Disclosures
The authors declare no conflicts of interest.
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