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Abstract
A split digital backpropagation (DBP) scheme for digital subcarrier-multiplexing (SCM) transmissions, denoted as SSDBP, is proposed and studied in both experiments and simulations. The implementation of the SSDBP is split at the transmitter and the receiver, leveraging existing chromatic dispersion (CD) compensation blocks to reduce complexity. We experimentally demonstrate that the SSDBP, with a complexity reduction up to 50% compared to the original receiver based SCM-DBP, can achieve a nonlinear compensation Q2 gain of 0.7-dB and 0.9-dB for 1920-km and 2880-km 34.94-GBd single channel PDM-16QAM transmissions, respectively. The maximum reach can be extended by 31.6% using 2-step SSDBP with only 27.5 complex multiplications per sample. Meanwhile, using 3-step SSDBP, the reach extension can be increased to 40.8%. The benefit of implementing part of SSDBP at the transmitter is experimentally validated with 0.1-dB Q2 improvement at 4-dBm launch power. We also numerically investigate the impact of the digital-to-analog converter (DAC) resolution and fiber parameter uncertainties on the nonlinear compensation performance of the SSDBP.
© 2017 Optical Society of America
1. Introduction
Fiber nonlinearity and its interplay with chromatic dispersion (CD) contribute to the primary bottleneck of current fiber optical long-haul transmissions. Therefore, driven by the ever-increasing transmission capacity demands, many efforts have been devoted to the compensation of the nonlinear interference (NLI) arising in single mode fiber transmissions. Digital nonlinearity compensation schemes such as digital backpropagation (DBP), perturbation based pre-distortion (PPD), and Volterra series have been demonstrated [1–4]. Among them, DBP is the most widely investigated technique. However, the conventional DBP algorithm has a prohibitively high complexity which prevents it from commercial implementation. In order to reduce the complexity, low pass filter assisted DBP (LDBP), correlative DBP (CDBP) and perturbation based on DBP [5–8] have been proposed with reduced number of DBP steps, each requiring a pair of fast Fourier transform (FFT) and inverse FFT (IFFT) for CD compensation in the frequency domain. Those schemes, exploiting the correlation of nonlinear noise between adjacent symbols, still impose a large computational load on long-haul transmission systems.
Recently, other methods have also been reported on reducing fiber nonlinearities. First, subcarrier-multiplexing (SCM) systems were proposed and experimentally demonstrated as an effective approach to mitigate the NLI [9–12]. By splitting a high symbol rate single carrier (SC) signal into several low symbol rate subcarriers, the transmission distance of single channel 24-GBd polarization-division-multiplexed 16QAM (PDM-16QAM) signals can be extended by 8% [12]. Second, based on the SCM, a novel low-complexity DBP scheme, denoted as SCM-DBP, was proposed and demonstrated to achieve sizable nonlinear compensation gain with only two steps for 2560-km 34.94-GBd PDM-16QAM standard single mode fiber (SSMF) transmission [13]. The SCM-DBP compensates self-subcarrier nonlinearity (SSN) and cross-subcarrier nonlinearity (CSN) separately based on a cross-phase modulation (XPM) model [14]. Moreover, a further complexity reduction of the SCM-DBP can be achieved by replacing the low pass filters of the XPM model with infinite impulse response (IIR) filters [15]. Third, it was shown by theoretical analysis that if we implement half of the DBP steps at transmitter (Tx) and the other half at receiver (Rx), denoted as split DBP, additional performance improvement can be achieved [16]. However, the performance gain was only validated by an ideal numerical simulation, assuming a very high complexity DBP scheme with 20 steps per span [16,17].The DBP implemented only at the transmitter-side has been experimentally demonstrated, which requires the fixed walk-off velocities among WDM channels [18]. Another relevant fact is that partial pre-CD compensation at the transmitter enables to improve the NLI tolerance as demonstrated in [19], and it has already been implemented in some commercial products [20].
In this paper, we propose a low complexity split SCM-DBP (SSDBP) for SCM based long-haul transmission systems. In addition to the existing nonlinear benefits from SCM and pre-CD compensation, the SSDBP minimizes the complexity of DBP by leveraging the existing CD compensation blocks. We experimentally demonstrate that a 2-step SSDBP with only 27.5 complex multiplications per sample can extend the transmission distance by 31.6% for single channel 34.94-GBd PDM-16QAM SCM signals over a SSMF link. The performance is comparable to the original receiver based 2-step SCM-DBP, but the complexity is halved. Over 1920-km and 2880-km transmission distances, 0.7-dB and 0.9-dB Q2 gains are achieved by the SSDBP, respectively. In addition, with a 3-step SSDBP the reach extension can be increased to 40.8%. Furthermore, we carry out numerical investigations on the requirement of digital-to-analog converter (DAC) resolution when one DBP step is implemented at the transmitter. Finally, the performance of SSDBP over various transmission distances and its sensitivity to the uncertainty of fiber parameters are numerically evaluated.
2. Operation principle
For DBP algorithms, the major complexity comes from the CD compensators (CDCs) which require FFTs and IFFTs. Therefore, to reduce the DBP complexity, we propose the SSDBP which takes advantage of existing transmitter-side pre-CDC and receiver-side post-CDC. Note that the pre-CDC has been implemented in some commercial products [20], as it can mitigate the NLI [18] and its implementation can be combined with pulse shaping for the purpose of complexity reduction. For the SSDBP, in addition to the complexity reduction, conducting part of the DBP at transmitter might also provide some performance benefit as demonstrated in previous numerical analysis [16,17]. However, in those works, ideal DBP compensation with 20 steps per span was assumed in the simulations. In this work, we experimentally demonstrate that such performance benefit can also be obtained with the low-complexity SSDBP.
The DSP flow with the 2-step SSDBP is shown in Fig. 1. At the transmitter, after constellation mapping and up-sampling to 2 samples per symbol, the root raised cosine (RRC) pulse shaping, CDC and nonlinear compensator (NLC) are applied before subcarrier multiplexing. At the receiver, subcarrier de-multiplexing is performed in the frequency domain after frequency offset compensation (FOC) [21]. After the receiver-side CDC and NLC, signals are processed with radius directed equalization (RDE) and carrier phase recovery (CPR) [22]. BER is estimated and averaged over all the subcarriers. At both the transmitter and receiver, CDC and RRC filters can be implemented together to share one stage FFT/IFFT for the purpose of complexity reduction.
The SSDBP is realized by splitting the SCM-DBP into the transmitter and receiver digital signal processing (DSP), as shown in Fig. 2(a). We use triangles to represent the accumulation of CD, where the ascending means the CD is being accumulated over the physical link, and the descending means the CD is being compensated in the DSP. For the original 2-step SCM-DBP in [15], the NLC calculates the NLI over its step length based on the signals at the middle of the step. The corresponding compensation sections of the physical link are marked with respect to the DBP steps in Fig. 2(a). For the proposed SSDBP, the NLC of the first step uses signals at the starting point of the transmission link to estimate the NLI over the step length. In comparison with conventional SCM-DBP, the modification of the first step for SSDBP may bring performance penalty [5]. However, it is expected that the performance gain brought by splitting the DBP at the transmitter and receiver will neutralize the penalty, resulting in a comparable performance with the SCM-DBP. The NLC of the second step of the SSDBP is based on the signals at the middle position with zero accumulated CD. Therefore, in both the transmitter and receiver DSP, the 2-step SSDBP requires only one CDC followed by a NLC. Longer transmission distances can be accommodated by 3-step SSDBP with one extra CDC and NLC at the receiver. The block diagram of the NLC in the SSDBP is shown in Fig. 2(b). First, the SSN of each subcarrier is compensated. Second, the CSN of all subcarriers is calculated and stored in a buffer, and then compensated based on a CSN compensation matrix. The details of the NLC block can be found in [15].
Fig. 2 (a) Illustration of NLC steps of 2-step SCM-DBP, 2-step SSDBP and 3-step SSDBP. (b) Block diagram of NLC in SSDBP.
3. Complexity of SSDBP
The complexity of the original M-step SCM-DBP evaluated by the number of complex multiplications per sample is expressed as [15]:
where is FFT size, and is CD compensation overhead based on the overlap-and-add method [13]. is the number of interfering subcarriers in the CSN calculation. The first term of Eq. (1) is the complexity of CDCs. For the M-step SCM-DBP, stages CDCs are required. The complexity of the first CDC is not included in Eq. (1), because the first stage CDC can be implemented with the RRC filter simultaneously to share the same FFT/IFFT operation. The overhead , which is determined by the memory length of CD compensation and walk-off effect between subcarriers, can be expressed as:where is the number of subcarriers, is the roll-off factor of RRC filter, is the dispersion coefficient, is the step length, and is the symbol rate of each subcarrier. Then, the FFT size is set as where K is the optimal integer in order to minimize the complexity of CD compensation. The complexity of CD compensation over different step lengths is presented in Table 1.
Table 1. Complexity of CD compensation over different step lengths
The second term of Eq. (1) is the complexity of NLC. The CSN is calculated from the interfering subcarriers, and each interfering subcarrier are filtered by three IIR filters in forward direction and backward direction, which cost 3 complex multiplications [15]. In addition, the compensation of SSN and CSN of the probe subcarrier requires 5.5 complex multiplications.
The number of complex multiplications of the M-step SSDBP is expressed as:
where N is the number of total subcarriers. Compared with the SCM-DBP, the complexity of the SSDBP is reduced mainly because the number of CDCs is reduced. For the M-step SSDBP, because the implementation of CDCs at both transmitter and receiver can be combined with RRC filters, only CDCs stages are required. Furthermore, a buffer structure is used during the CSN calculation, so that we can exploit the symmetric characteristic of CSN to slightly reduce the complexity of NLC. The symmetric characteristic refers to that the CSN from -th subcarrier to -th subcarrier is identical to the CSN from -th subcarrier to -th subcarrier. In particular, the CSN from the probe subcarrier to the neighboring subcarriers is calculated and stored. After all CSNs are calculated, the CSN compensation matrix is implemented for each subcarrier based on the results stored in the buffer.Based on Eq. (1) and (3), the complexity comparison between SCM-DBP and SSDBP with 2 and 3 steps are summarized in Table 2. Considering the trade-off between gain and complexity, NS is set to 4 for 8-subcarier PDM-16QAM [15]. The SSDBP can reduce the complexity by 50% and 38% for 2 and 3 steps, respectively, compared with the SCM-DBP.
Table 2. Complexity of SSDBP and SCM-DBP.
4. Experimental results and discussions
The performance of the proposed SSDBP is experimentally evaluated. The experimental setup is shown in Fig. 3. In the transmitter DSP, 8-subcarrier PDM-16QAM signals with an aggregate symbol rate of 34.94-GBd are generated after RRC pulse shaping with a roll-off factor of 0.1 and transmitter-side SSDBP processing. The insets of Fig. 3 show the spectrum of the signals, generated by the Tx DSP, with (upper) and without (lower) nonlinear compensation. The obtained waveforms are sent to a Ciena WaveLogic3 (WL3) transmitter which consists of a laser source and a dual-polarization IQ modulator driven by four 39.4-GSa/s DACs with 6 bit resolution. The laser source, with a linewidth of less than 100-kHz, is operated at 1550.2-nm. After a booster erbium-doped fiber amplifier (EDFA), the optical signals enter a recirculating loop which consists of four 80-km SSMF spans with four EDFAs for span loss compensation. An optical bandpass filter (OBPF) within the loop is used to remove out-of-band amplified spontaneous emission (ASE) noise. At the receiver, after optical amplification and filtering, an external cavity laser (ECL) is used as a local oscillator (OL) for coherent detection. The four electrical outputs are captured by an 80-GSa/s real-time oscilloscope followed by the offline DSP. The number of subcarriers is optimized to be 8 with the SSDBP applied to the PDM-16QAM SCM transmission. Therefore, we focus on the SCM systems with 8 subcarriers in the following discussions.
Fig. 3 Experiment setup and the spectrum of the generated signal of Tx DSP with (upper) and without (lower) nonlinear compensation. VOA: variable optical attenuator, SW: switch.
4.1 Back-to-back transmission results
Figure 4 shows the back-to-back (BTB) transmission results. We calculate the Q2 factor by the measured bit-error-rate (BER) at different optical signal-to-noised ratios (OSNR). The required OSNR at a target bit-error ratio (,) for single carrier (SC) signals is 19.1-dB, which has a 1.5-dB implementation penalty in comparison with the theory. For the 8-subcarrier signals, an additional 0.4-dB OSNR penalty is observed compared with the SC signals. The major cause of this is the limited bit resolution of the DACs in the presence of the increased peak-to-average power ratio (PAPR) in the SCM signals [12], which will be numerically investigated later.
4.2 Transmission results
The performances of the 2-step SSDBP and SCM-DBP after 1920-km SSMF transmission are shown in Fig. 5(a). The step length of the two SSDBP steps is optimized as 640-km and 1280-km respectively, leading to 67% CD compensation at the transmitter. For the purpose of comparison, SCM and SC systems with linear compensation (LC) are also presented. The LC-SCM scheme outperforms the LC-SC scheme with 0.2-dB Q2 improvement. The 2-step SSDBP enhances the performance of the SCM signals with a 0.7-dB Q2 gain. At the optimal launch power, the 2-step SSDBP has only 0.06-dB Q2 penalty relative to the 2-step SCM-DBP. Note that the complexity of the 2-step SSDBP is only half of the 2-step SCM-DBP as described earlier. A total 0.9-dB Q2 improvement is observed by the SSDBP compared with the LC-SC system. Figure 5(b) shows the 3-step SSDBP and SCM-DBP performances after 2880-km transmission. The step length of the three SSDBP steps is optimized as 640-km, 1120-km and 1120-km, respectively, leading to 42% CD compensation at the transmitter. The 3-step SSDBP achieves 0.9-dB Q2 improvement, compared with the LC-SCM system. Combining the 0.4-dB Q2 gain of the LC-SCM relative to the LC-SC, the total Q2 improvement using the SSDBP is 1.3-dB relative to the LC-SC. For both 1920-km and 2880-km SSMF transmissions, the SSDBP and SCM-DBP have comparable performance.
Fig. 5 (a) Performance after 1920-km SSMF transmission. (b) Performance after 2880-km SSMF transmission. SC: single carrier. LC: linear compensation.
4.3 Performance gain by splitting SCM-DBP
To verify the performance gain resulting from splitting the DBP into transmitter and receiver DSP as predicted by the numerical analysis in [16,17], the receiver-side SSDBP, which conducts all the CDC and NLC at the receiver, is considered as a reference, as shown in Fig. 6(a). The Q2 gain of the SSDBP relative to the receiver-side SSDBP in both simulations and experiments is presented in Fig. 6(b). The configurations of the simulation system are the same as the experiments, except that the simulation system has ideal transmitter and receiver. The transmission distance of 2-step and 3-step SSDBP is 1920-km and 2880-km, respectively. The simulation results show a Q2 gain of 0.15~0.2-dB at 4-dBm launch power. However, the experimental results achieve a smaller gain of ~0.1-dB at 4-dBm, because there exists a Q2 penalty of ~0.05-dB at the linear transmission regime. We infer that this penalty is due to the limited bit resolution of DAC when the SSDBP at the transmitter leads to a larger PAPR.
Fig. 6 (a) Illustration of NLC steps of 2-step SSDBP and 2-step receiver-side SSDBP. (b) Q2 gain of SSDBP with respect to receiver-side SSDBP.
4.4 Maximum reach of SSDBP
Assuming a target BER of (), we evaluate the maximum transmission distance with different compensation schemes, as shown in Fig. 7. Compared to the LC-SC system, the 8-subcarrier SCM signals with LC can extend the transmission distance by 9.2%. Compared to the SC signals with linear compensation, the 2-step SSDBP achieves a transmission distance extension of 31.6%. The 3-step SSDBP can increase the reach by 40.8%, in comparison with the LC-SC system. Note that the transmission distance improvement is expected to be reduced for wavelength-division multiplexing (WDM) transmission [23].
5. Performance analysis by simulations
In this section, we provide more analysis on the performance of the proposed SSDBP by numerical simulations. First, the impact of the limited DAC resolution on the system performance when the SSDBP is implemented at the transmitter is evaluated. Then, we compare the SSDBP and SCM-DBP over distances longer than 1920-km. Further, the performance sensitivity of the SSDBP against the uncertainty of fiber parameters is assessed.
5.1 Impact of the DAC resolution
To verify that the limited DAC resolution results in the performance difference between the simulations and experiments in Fig. 6, we investigate the PAPR of the transmitted SC and SCM signals with CD and/or nonlinear pre-compensation, as shown in Fig. 8(a). The complementary cumulative distribution function (CCDF) shows the probability that the PAPR is larger than a certain value [24]. At a 0.001 probability, compared to the SC signals the SCM technique increases the PAPR by ~4.2-dB. The pre-CDC and SSDBP have similar results, which further increase the PAPR by ~0.7-dB to a total of 11.8-dB. Since the pre-CDC is part of the SSDBP, it implies that the additional one-stage NLC does not increase the PAPR.
Fig. 8 (a) PAPR of the transmitted signals in various schemes. (b) The required OSNR at versus DAC resolution bits.
Figure 8(b) shows the required OSNR of a 34.94-dB PDM-16QAM signal at BER = 2 × 10−2 in the BTB transmission, when the DAC resolution ranges from 4 to 10 bits. For the transmitted signals with the SSDBP, the pre-distortion caused by the SSDBP is fully compensated at the receiver with opposite CD and nonlinear coefficients, so the penalty is only caused by the limited DAC resolution. With an ideal transmitter, the required OSNR is 17.6-dB. It is observed that the signals processed by the SSDBP require higher OSNR than that of SCM and SC signals. Since the PAPR is not increased by the one-stage NLC, the signals with the SSDBP and pre-CDC have similar required OSNRs, as observed in Fig. 8(b). The effective number of bits (ENOB) of the DACs used in the experiment is around 5 bits, and in this case the signals with the SSDBP has 0.1-dB OSNR penalty compared to the SCM signals according to Fig. 8(b). Describing the interference from the limited DAC resolution as an additive noise, we can theoretically estimate that the Q2 factor over 1920-km transmission at −2-dBm will be reduced by 0.05-dB. This is consistent with the measured Q2 penalty of the linear transmission regime in Fig. 6. However, as the DAC resolution improves, the penalty caused by the SSDBP will become negligible.
5.2 Performance sensitivity against link parameter uncertainties
In the previous experimental investigations, we show that the 2-step SSDBP and 2-step SCM-DBP have similar performances at a 1920-km distance, which implies that the low-complexity 2-step SSDBP is suitable for metro, regional and long-haul applications. However, for a longer transmission distance the 2-step SSDBP starts to become less effective as shown in Fig. 9(a), in which the nonlinear compensation Q2 gain versus transmission distance is numerically evaluated. The launch power is 3-dBm which is optimal over transmission distances beyond 2560-km. The performance difference is larger than 0.1-dB when the distance is 2560-km and it reaches 0.25-dB at 3840-km. This is because the SSDBP uses the signals at the starting point of the step, instead of the signals at the middle position of the step as in the SCM-DBP, to estimate the NLI of the first step. As the step length increases, the penalty caused by changing the position of signals for NLI estimation becomes larger. In this scenario, the 3-step SSDBP can be employed and it has a similar performance (within 0.05-dB) to the 3-step SCM-DBP for distances up to 3840-km.
Fig. 9 (a) Q2 gain of nonlinearity compensation over different transmission distances. (b) Contour of the Q2 gain of nonlinearity compensation against uncertainty of parameters.
Another disadvantage of the SSDBP is the reduced ability to adaptively adjust the DBP coefficients at the transmitter in the context of unknown or drifting link conditions compared with the purely receiver based SCM-DBP [25]. Therefore, in Fig. 9(b) we numerically evaluate the sensitivity of the 2-step SSDBP with respect to the provisioned CD coefficient and nonlinear coefficient after 1920-km transmission. The Q2 gain is measured at the optimal launch power of 2-dBm. The maximum Q2 factor of 9.52-dB is achieved when we have access to the actual parameters of fibers. Assuming the tolerable Q2 penalty of 0.1-dB because of the simultaneous variation of CD and nonlinear coefficient, the dynamic range of the coefficients relative to the true value is shown in Fig. 9(b). When the CD coefficient varies from −18% to 28% and nonlinear coefficient varies from −28% to 13%, the stable performance with a Q2 penalty less than 0.1-dB can be obtained. Therefore, the performance of the SSDBP is fairly stable against the uncertainty of fiber parameters.
6. Conclusions
We propose a low complexity SSDBP for SCM transmissions. With comparable performance, the 2-step SSDBP halves the complexity compared to the original receiver based 2-step SCM-DBP. For 1920-km and 2880-km PDM-16QAM SSMF transmissions, 0.7-dB and 0.9-dB Q2 gain can be achieved by the SSDBP, respectively. In addition, transmission reach extension of 31.6% and 40.8% are demonstrated by the 2-step and 3-step SSDBP, respectively. Under the condition of 4-dBm launch power, the experimental results show a 0.1-dB Q2 gain of implementing part of SSDBP at the transmitter. In numerical investigations, the impact of DAC resolution on the performance of SSDBP is investigated, which shows a 0.1-dB OSNR penalty. Moreover, the stable performance of the SSDBP against the uncertainty of fiber parameters is numerically investigated.
Funding
National Natural Science Foundation of China (NSFC) (61575071, 61377073); National Key Research and Development Program of China (2016YFE0121300).
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