Backpropagation has been shown to be the most effective method for compensating intra-channel fiber nonlinearity in long-haul optical communications systems. However, effective compensation is computationally expensive, as it requires numerous steps and possibly increased sampling rates compared with the baud rate. This makes backpropagation difficult to implement in real-time. We propose: (i) low-pass filtering the compensation signal (the intensity waveform used to calculate the nonlinearity compensation) in each backpropagation step and (ii) optimizing the position of the nonlinear section in each step. With numerical simulations, we show that these modifications to backpropagation improve system performance, reducing the number of backpropagation steps and reducing the oversampling for a given system performance. Using our ‘filtered backpropagation’, with four backpropagation steps operating at the same sampling rate as that required for linear equalizers, the Q at the optimal launch power was improved by 2 dB and 1.6 dB for single wavelength CO-OFDM and CO-QPSK systems, respectively, in a 3200 km (40 × 80km) single-mode fiber link, with no optical dispersion compensation. With previously proposed backpropagation methods, 40 steps were required to achieve an equivalent performance. A doubling in the sampling rate of the OFDM system was also required. We estimate this is a reduction in computational complexity by a factor of around ten.
©2010 Optical Society of America
Recent experimental demonstrations have shown that coherent optical QPSK (CO-QPSK) and coherent optical OFDM (CO-OFDM) are the leading candidates for next-generation 100-Gbit/s Ethernet (100 GE) long-haul transmission [1–3]. Since both can compensate for linear impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD), fiber nonlinearity becomes the limiting factor of transmission distance in these systems.
Digital signal processing (DSP) can be used to partially compensate for fiber nonlinearities [4–9]. Single-step methods, where the compensating phase modulation is applied only once , assume that CD is negligible and thus compensate for all nonlinearity at either the transmitter or receiver . Whilst single-step methods are computationally efficient, they are only beneficial for low-dispersion fibers or when CD is periodically compensated as they assume the intensity waveform is similar at the beginning of each amplified span [5, 10, 11]. For greenfield deployments where dispersion compensation fiber (DCF) is not used and especially with high dispersion fibers such as standard single mode fiber (SMF), the signal waveform evolves from span to span due to frequency-dependent phase shifts induced by CD. Single-step compensation is, therefore, ineffective in such links .
In 1996, Pave et al.  proposed that propagating the signal through fiber with a negative nonlinearity and dispersion would effectively compensate for the nonlinearity of the transmission fiber. Although a negative nonlinearity is not available in a real fiber, it can be simulated numerically using split-step methods. For example, the ‘virtual’ inverse fiber can be placed before the transmitter , or after a coherent receiver [5, 6, 9]. Fiber nonlinearity can be compensated effectively by either method. More recently, split-step methods have been demonstrated experimentally at 100-Gb/s using offline processing [14, 15]. However, the computational cost of split-step methods was estimated to be over one-hundred times greater than for linear equalizers , thus making them difficult to implement in real-time.
We propose to: (i) band-limit the intensity waveform used for nonlinearity compensation; (ii) optimize the position of the nonlinear section within each step. We show that these modifications allow a lower oversampling factor and fewer steps to be used, which greatly reduces computational complexity. With only four steps for the entire link, the peak signal quality Q (at the optimal operational power) after 3200 km of SMF-only link can be improved by 2 dB and 1.6 dB for CO-OFDM and CO-QPSK respectively.
This paper is structured as follows. A review of multi-step nonlinearity compensation and previous work is presented in Section 2. In Section 3, the proposed improvements to multi-step nonlinearity compensation are presented and justified theoretically. Section 4 states the simulation properties and the process used to optimize the free parameters in the multi-step compensator. The benefits of the proposed improvements are presented in Section 5 and conclusions are drawn in Section 6.
2. Theory of digital backpropagation
2.1. The split-step Fourier method
In the receiver of a coherent optical communications system, the received photocurrents are linearly mapped to the optical field, so that both the optical amplitude and phase become available to the receiver’s digital processors. This received signal can be digitally propagated through an inverse fiber model to compensate for CD and fiber nonlinearity. This method has been previously referred to as backward propagation  or backpropagation (BP) . BP requires the inverse nonlinear Schrödinger equation (NLSE)  to be solved for the parameters of the optical link. For a single polarization and with the spatial domain negated, the NLSE is given by:17]. It is very important to note that the spatial domain negation means the optical link is modeled on a first-in-last-out principle, where the first fiber span is the last modeled span and the beginning of each fiber span is the end of each modeled span. The power in a two-span optical system and the corresponding inverse link model are plotted against the propagation distance in Fig. 1 . Note that the distance decreases as the signal is ‘virtually’ propagated through the inverse link model. In this paper, a reference to spans implies fiber spans in the real optical link.
BP involves calculating a numerical solution to Eq. (1), typically using the split-step Fourier method (SSFM), where the fiber is treated as a series of linear sections (where only D is considered) and dispersionless nonlinear sections (where only N is considered). The SSFM is a first-order approximation of the NLSE, meaning that a large number of steps are required for accurate modeling. However, the total number of computations required is proportional to the number of steps used. Therefore using a large number of steps implies a very large computational cost. A link can be modeled using the symmetrical SSFM , where the nonlinear section of each BP steps is placed in-between two linear sections, or using the asymmetrical SSFM , where the nonlinear section follows a single linear section in each step. Both have been proposed for BP. Figure 2 shows a transmission system with a P-span fiber plant and a Q-step BP equalizer, separated by a coherent receiver.
Each linear section in BP can be implemented with FIR filters, which are identical to those used for CD compensation . The required phase response of the FIR filter, for each linear section, in the frequency domain is:6], it is more computationally efficient to apply this phase response directly in the frequency domain by means of transform methods, aided by the FFT/IFFT algorithms. The attenuation factor, α, is only needed if the number of BP steps used is greater than the number of fiber spans, Q>P.
The nonlinear sections of BP are identical to the nonlinear section used in a single-step nonlinearity compensator [4, 10]. The phase shifts are proportional to the instantaneous power and are calculated for each sample using :5, 14, 19]. If each BP step only compensates for a fraction of a span (P< = Q), then the effective length becomes:20]. Each nonlinear section involves: (i) multiplying the complex field by its conjugate to calculate the instantaneous intensity; (ii) scaling the intensity waveform; (iii) phase modulating the input complex field by the scaled intensity waveform. A block diagram of a nonlinear section is shown in Fig. 3 .
2.2. Inter- and intra-channel nonlinearity compensation
Li et al. have shown that in WDM systems, all inter- and intra-channel nonlinear effects can be compensated with BP by sampling a band extending over all of the wavelength channels. This technique was demonstrated experimentally for a three-channel WDM system using a single coherent receiver and offline processing [17, 21]. These systems used symmetrical SSFM. They suggest that multiple BP steps are required to represent each fiber span (Q > P) in order to significantly mitigate fiber nonlinearity . If a large number of wavelengths were used, multi-channel BP would require sampling each wavelength channel, upsampling each digitally, then combining the upsampled representations of all of the channels, so all WDM channels are encapsulated by a single digital signal . In such a case, a complex lattice of high-speed inter-receiver connections and a very high sampling rate in the DSP would be required. Realization in a real-time system would therefore be very difficult.
The sampling rate within the DSP and the number of steps required would be greatly reduced, and the inter-receiver connections avoided, if compensation is limited to a single wavelength channel. This would limit compensation to intra-channel fiber nonlinearity only ; e.g. the compensation of Self-Phase Modulation (SPM) in QPSK systems. The narrower bandwidth of a single channel, compared with a WDM comb, means BP would require fewer steps. The complexity in the DSP can also be further reduced if the asymmetric SSFM was used . Single-channel BP, using the asymmetric SSFM, has been shown to significantly improve system performance when only one BP step is used to compensate for each fiber span (P = Q) [5, 19].
An experimental demonstration reported that BP, with one step per fiber span, provided a 46% increase in reach for a single wavelength system, a 24% increase in reach for a ten wavelength system with 100-GHz channel spacing, but only a 3% increase in reach for a ten wavelength system with 50-GHz channel spacing . This suggests that BP of a single wavelength channel produces only a very small benefit in WDM systems with a large number of channels if the 50-GHz ITU grid is used. This is because inter-channel nonlinearity effects dominate in such systems. Recently, less computationally intense methods of compensating for inter-channel nonlinear effects, such as Cross-Phase Modulation (XPM), have been proposed [22, 23]. These methods do not require the full optical field from the neighboring wavelength channels to be known, are less computationally intense and can be used in conjunction with intra-channel nonlinearity compensation techniques . We believe that BP, at least initially, will be used to compensate for intra-channel nonlinearity and used in conjunction with other inter-channel nonlinearity compensation methods.
3. Improving digital backpropagation
3.1. Problem of computational complexity
Experimental demonstrations of BP have been limited to using offline processing [14, 17, 19, 21] because of their high computational requirements. One study estimated the number of computations required for single-channel BP compensating for a 25-span link to be around 100-times greater than for linear equalizers. This is because at least one BP step per span is required for effective nonlinearity compensation. In addition, it was suggested that an oversampling factor of three is needed to prevent aliasing of the nonlinear products .
FFT/IFFTs are required between the linear and nonlinear steps in order to convert from the frequency domain to the time domain and vice versa. The use of transform methods means that the computational complexity is, at large, decoupled from the duration of the impulse response compensated by each BP step. Therefore, the number of computations required is approximately linearly proportional to the number of steps used and proportional to the oversampling factor.
In order to reduce the computational cost of BP, we propose two improvements to BP, which reduce the number of steps required and the required oversampling factor. These are: (i) low-pass filtering the compensating intensity waveform in the nonlinear sections; (ii) optimizing the position of the nonlinear sections. Subsections 3.1 and 3.2 explain how these improvements decrease the required number of steps and oversampling.
3.2. Reducing the number of steps required for a given accuracy
The phenomena of CD and fiber nonlinearity occur simultaneously in fiber; therefore, the phase shifts induced by CD cause phase mismatching between the nonlinearity products generated at each point along the fiber, thereby reducing the intensity of their vector sum. The intensity of nonlinearity products resulting from distant frequency components is suppressed more greatly because the phase shifts are greater . Figure 4a shows the power along an 8 × 80 km periodically amplified link. Fiber nonlinearity is generated where the power is highest, which, is at the start of each span. Therefore, the phase-mismatching of the nonlinear products generated by subsequent amplified spans dominates over phase-mismatching of nonlinear products generated within in each span [24, 25].
In the SSFM, the linear sections attempt to model the effect of phase-mismatching from step to step. However, the nonlinearity of each step is approximated to be lumped at a single point, so the phase mismatch effect is ignored within each step. Therefore, fiber nonlinearity will be over-estimated, especially nonlinear phase modulation resulting from high-frequency intensity fluctuations. The asymmetric SSFM works well if one fiber span is compensated by one BP step because it models all the nonlinearities of each fiber span as a single lumped nonlinearity at the start of the respective span where the power is greatest, as shown in Fig. 4b. Figure 4c shows where the nonlinear mixing will be approximated if four spans are compensated with each BP step using the asymmetric SSFM. As can be seen, nonlinearities from multiple spans are lumped to a single location, which is inaccurate. Not accounting for the inter-span phase-mismatches between the four fiber spans results in overcompensation of fiber nonlinearity from high frequency intensity fluctuations. Reducing the compensation factor, without considering its frequency dependence, would result in under-compensation of the low-frequency terms, so would not help . In addition, CD causes the nonlinear products from subsequent spans to experience a phase rotation. This produces a net phase rotation in the vector sum of the components , so the compensation products will no longer be anti-phase to the nonlinearity produced along the fiber link.
Figure 5a shows the optical spectrum of a CO-OFDM signal after propagation through 40 × 80 km of simulated SMF with lumped amplifiers. ASE was not simulated in order to highlight the effects of CD and nonlinearity. The fiber model used very short steps and an adaptive algorithm which ensured the maximum phase shift per step was 0.05 radians. The nonlinearity in the received signal was then compensated by BP with either 40 steps (Fig. 5b) or 20 steps (Fig. 5c). The asymmetric SSFM and the optimal compensation factors, k (from Subsection 4.2), were used. BP clearly reduces the intensity of the in-band distortion; however, there is significantly more residual distortion when only a 20-step BP was used. In both cases, the intensity of the out-of-band distortion is higher after BP. The distortion is produced by high-frequency intensity fluctuations modulating the outer OFDM subcarriers in the nonlinear sections of the BP. These ‘compensation products’ become noise.
Firstly, we predict that if we separate high-frequency intensity fluctuations and low-frequency intensity fluctuations, it might be possible to provide optimal compensation for both. We propose improving BP by using a digital low pass filter (LPF) to limit the bandwidth of the compensating waveform so we can optimize the compensation for the low-frequency intensity fluctuations without overcompensating for the high-frequency intensity fluctuations. The block diagram of the filtered nonlinear section is shown in Fig. 7 with the LPF highlighted in blue. Note that the signal itself propagates from transmitter to receiver without filtering; only the waveform used to calculate the compensation is filtered.
Secondly, we predict that if the nonlinear mixing was modeled to occur near the middle of each step, as shown in Fig. 4d, the model’s accuracy should improve. If two linear sections of different lengths are used, as shown in Fig. 6 , the location of the nonlinear section can be adjusted to any location on the dispersion map. The second linear section of a step can be combined with the first linear section of a subsequent step to reduce the number of Fourier transforms. Compared to the asymmetric SSFM, there is only one additional linear section is needed for the entire BP process for any number of BP steps, as shown in bottom section of Fig. 6; the additional initial linear section is highlighted in blue.
3.3. Reduction of the required oversampling
It has been previously stated that an oversampling factor of at least three is required for BP because Kerr nonlinearity is a third-order effect . However, BP has been subsequently demonstrated to be effective using an oversampling factor of only two [15, 19]. Figure 8 shows the spectra of signals at different points in a nonlinear section of BP, both without (upper signal-flow diagram) and with (lower signal-flow diagram) a LPF. Consider the spectrum of a CO-OFDM input signal (Fig. 8a) and the square of its complex envelope, which is its intensity waveform (Fig. 8b). In the nonlinear sections of BP, the input signal is phase modulated by the intensity waveform. The output signal’s spectrum (Fig. 8d) will have three-times the bandwidth of the input signal’s spectrum. Therefore, three-times oversampling is required in the DSP to avoid aliasing. However, the high frequencies in the intensity waveform are significantly lower in amplitude than the low frequencies. This illustration suggests that using an oversampling factor of two should result in a small penalty from aliasing, thus a significant improvement from BP (without filtering) should be possible which is consistent with the results from [14, 17, 19]. However, the penalty from aliasing in BP (without filtering) will increase if the sampling rate is reduced.
If the intensity waveform is fed into a LPF, the bandwidth of the compensation signal is reduced to that of the pass-band of the LPF as shown in Fig. 12c . Thus, the output signal’s bandwidth (Fig. 10e ) will be reduced to the bandwidth of the input signal plus twice the bandwidth of the LPF. Therefore, filtering the intensity waveform will reduce the required oversampling factor. An OFDM signal was used for this explanation because its well-confined spectrum makes the illustration clearer. This reasoning can be generalized to CO-QPSK signals or indeed any arbitrary signal operating in the pseudo-linear regime.
4. Simulation details
4.1 Simulation properties
Numerical simulations were conducted using VPIsystems’ VPItransmissionMaker V8.3. The optical link comprised 40 × 80 km spans of SMF (3200 km). The SMF had an attenuation of 0.2 dB/km, dispersion of 16 ps/nm/km and nonlinearity factor of 1.3 /km/W. EDFAs with a noise figure of 5 dB were used to compensate for the loss in each span. No DCF was used.
The CO-OFDM transmitter used a 1024-point IFFT, with 880 of the subcarriers modulated with 4-QAM. Ten subcarriers either side of DC and 61 subcarriers either side of the Nyquist frequency were zeroed. Digital-to-Analog Converters (DAC) were modeled at 38 Gsample/s. A 128-sample cyclic prefix was used to give a bit rate of 58 Gb/s in a single polarization, which allowed for a 4% overhead for synchronization and training. This RF signal was then used to drive a complex Mach-Zehnder Modulator (MZM) which generated an optical spectrum 33-GHz wide.
The QPSK signal was generated by driving each arm of a complex MZM with a 28-Gb/s pseudorandom bit generator, giving a total bit rate of 56 Gb/s. A 45-GHz optical filter was then used to limit the bandwidth of the generated signal.
A typical coherent homodyne receiver, which contains an optical local oscillator, optical hybrid and two pairs of balanced photodiodes, was used. The analog to digital converters (ADC) for the OFDM and QPSK systems were modeled at 38 Gsample/s and 56 Gsample/s respectively. In order to isolate interactions between CD and fiber nonlinearity, the lasers’ phase noise and PMD were set to zero.
4.2 Optimization of parameters in backpropagation
In this section, the optimal parameters of filtered BP (with both the improvements detailed in Subsection 3.2) and unfiltered BP (using asymmetrical SSFM)  were found for different numbers of BP steps; ranging from 1-step to 40-steps. To avoid aliasing in the system, the optimization simulations used an oversampling ratio of 4.0 for CO-QPSK and 2.4 for CO-OFDM. The launch power into each of the 40 spans was set to + 4 dBm, which is in the nonlinearity limited region of operation. The optimal filter bandwidths and the locations of the nonlinear sections were found to be identical for the CO-OFDM and CO-QPSK systems, which suggests that the parameters are independent of the signal format.
For filtered BP, the LPF bandwidth and the CD compensated in the initial linear section were swept over a range of values in order to fine-tune the optimal values. Figure 9a shows the optimal bandwidth for different numbers of BP steps. Figure 9b shows the characteristic of the filters. The filter characteristics where chosen for ease of simulation and optimization of the filter characteristics may lead to further performance improvements. The optimum filter bandwidth is very narrow when very few BP steps are used. When a greater number of steps is used, the optimal bandwidth increases. This is in agreement with the theoretical analysis presented in Subsection 3.2.
Figure 9c shows the ratio between the dispersion compensated in the initial linear section and every subsequent linear section, Dini/Dstep, for different numbers of BP steps. This ratio governs the positions of the nonlinear sections in the dispersion map. When multiple fiber spans were compensated with each BP step, the optimal location of the nonlinear section within each BP step was found to be just beyond the start of the middle fiber span. When one step per span was used, the optimal value was very close to the start of the span.
The compensation factor, k, was swept for both BP methods. Figure 10 shows the optimal k for different numbers of BP steps. The optimal value of k for unfiltered BP is close to zero when very few steps are used because a large portion of compensation products were inaccurate and contributed to phase noise. Increasing the number of steps increased the optimal value of k because more of the compensation products mitigated accumulated nonlinear products from the fiber. For filtered BP, however, the optimum k remained relatively constant when the optimal filter was used because most of the excess compensation products were removed.
From here on in, the optimal values for k, LPF bandwidth, and DIni will be used for both filtered and unfiltered BP.
5. Benefits of the proposed improvements
5.1 Reduced required sampling rate for BP
In order to investigate the penalty from aliasing, 40-step BP was tested using two different sampling rates for each system: at the ADC sampling rate and at twice this rate. The oversampling factors, for CO-OFDM were 1.2 and 2.4; and for CO-QPSK were 2.0 and 4.0. The oversampling factor refers to the Nyquist frequency (determined by the sampling rate) divided by the signal’s bandwidth. A launch power of + 4 dBm was used. Figure 11 plots the Q after equalization using 40-step filtered BP, 40-step unfiltered BP and linear equalization, for both CO-QPSK and CO-OFDM, against the oversampling factor.
For 40-step unfiltered BP, doubling the sampling rate improved the performance of the CO-OFDM system by 1.5 dB (from 7.7 dB to 9.2 dB). This improvement is because the performance of unfiltered BP is degraded by aliasing in the nonlinear sections if an oversampling factor of 1.2 is used. However, doubling the sampling rate only resulted in a 0.2 dB (9.9 dB to 10.1 dB) increase in performance for the CO-QPSK system. This suggests that the penalty from aliasing is small for an oversampling factor of 2.0, which agrees with previous experimental results [14, 19].
For 40-step filtered BP, improvements of 9.1 dB (3.0 dB to 12.1 dB) and 7.0 dB (5.6 dB to 12.6 dB) in Q were produced for all the sampling rates used. This shows that an oversampling factor as low as 1.2 is sufficient for filtered BP.
5.2. Reduction in the number of steps required
The received Q values, after filtered and unfiltered BP, for both CO-OFDM and CO-QPSK, are plotted against the number of BP steps in Fig. 12. The oversampling factor in the BP DSP was 1.2 for CO-OFDM and 2.0 for CO-QPSK. Filtered BP (solid) provides 2-dB more benefit than unfiltered BP (hollow) for CO-QPSK systems (■□) and 4.5-dB more benefit than unfiltered BP for CO-OFDM systems (●○) if 40 steps were used. This shows that filtering the intensity waveform improves performance when operating at 1-span/step; suggesting that filtered BP is a more accurate model than unfiltered BP.
More importantly, the majority of the benefit of filtered BP can be obtained with only a few BP steps. For example, filtered BP improved the CO-OFDM system by: 8.0 dB with eight steps, 6.1 dB with four steps and 4.1 dB with only two steps. Unfiltered BP produces a 4.7-dB improvement with 40 steps; only slightly better than 2-step filtered BP. However, the improvement drops to only 2.3 dB with 20-step unfiltered BP. Similarly, for CO-QPSK, 40-step unfiltered BP produced an improvement of 4.3 dB; filtered BP produced improvements of 5.3 dB and 4.0 dB for four steps and two steps respectively. In this case, filtered BP enabled the number of steps to be reduced by a factor of 10-20 times for similar performance improvements. We estimate the computational saving of around a factor of ten given an effective filter design.
5.3. Transmission performance
To show the absolute benefit of filtered BP in a transmission system, the launch power into each fiber span was swept from −12 dBm to 6 dBm. The corresponding Q, for linear equalization and nonlinear equalization using filtered BP with different numbers of steps, are shown in Fig. 13a and Fig. 13b for CO-OFDM and CO-QPSK, respectively. At powers below −6 dBm, both systems were found to be in the linear regime, where the performance is dominated by noise. Therefore, no improvement was produced by nonlinearity compensation at low powers. At powers higher than −6 dBm, a penalty from fiber nonlinearity is experienced in both systems, thus filtered BP improved performance. For powers above −1 dBm, the dominant impairment of linearly equalized systems was fiber nonlinearity. The optimal operating power for linearly equalized systems was −4 dBm. If filtered BP was used, the performance continues to improve at a rate of 1 dB/dB, as the power is increased, until −2 dBm: a penalty from fiber nonlinearity is only experienced at powers higher than −2 dBm. The optimal power is increased to just under 0 dBm for both CO-OFDM and CO-QPSK if filtered BP, with four or more steps, is used.
For the CO-OFDM system, the peak Q (at the optimal power) was 14 dB using linear equalization; using filtered BP, the peak Q was improved to 16 dB with only four propagation steps, to 16.5 dB with 8 steps and to 17 dB with 40 steps. With linear equalization, the CO-QPSK system had a peak Q of 15 dB. This was improved to 16.5 dB with four steps, to 16.7 dB with 8 steps and 16.8 dB with 40 steps. These results confirm that the majority of the performance benefit can be achieved with only a few steps. Note that an increase in performance in the nonlinear regime widens the range of possible input powers. Also, a Q of 15.9 dB will produce a BER of 2 × 10−3 for 16-QAM. Thus, by using 4-step filtered BP, 3200-km transmission using 16-QAM is possible.
Because filtered BP only requires a small oversampling factor (~1.2 × ), and the number of computations per bit is linearly proportional to the oversampling factor, systems using small oversampling factors will be more computationally efficient. For CO-OFDM systems, the oversampling factor is determined by the subcarrier plan: an arbitrary oversampling factor can be used. This allows the computational efficiency of BP to be maximized. CO-QPSK systems (and other single-carrier systems) require the use of fractionally-spaced equalizers. Typically, a factor of two is used , as in this paper. The higher oversampling factor will mean the number of computations per bit is significantly increased.
We have shown that the computational complexity of BP can be reduced with the addition of an electrical LPF and tuning the position of the nonlinear sections. The savings result from a reduction in both the required sampling rate and the number of steps. Using only 4-steps for a 40-span link, more benefit was possible than that achieved with 40-steps using previously proposed methods of BP. The Q at the optimal power was increased by 2 dB and 1.6 dB for single wavelength CO-OFDM and CO-QPSK systems respectively with only four steps. This widens the possible operating power and enables higher constellations to be used. For similar performance, we estimate our improvements will reduce the number of computations needed by a factor of around ten times.
We would like to thank VPIphotonics (www.vpiphotonics.com) for the use of their simulator, VPItransmissionMakerWDM V8.3. This work is supported under the Australian Research Council’s Discovery funding scheme (DP1096782).
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