Employing 100G polarization-multiplexed quaternary phase-shift keying (PM-QPSK) signals, we experimentally demonstrate a dual-polarization Volterra series nonlinear equalizer (VSNE) applied in frequency-domain, to mitigate intra-channel nonlinearities. The performance of the dual-polarization VSNE is assessed in both single-channel and in wavelength-division multiplexing (WDM) scenarios, providing direct comparisons with its single-polarization version and with the widely studied back-propagation split-step Fourier (SSF) approach. In single-channel transmission, the optimum power has been increased by about 1 dB, relatively to the single-polarization equalizers, and up to 3 dB over linear equalization, with a corresponding bit error rate (BER) reduction of up to 63% and 85%, respectively. Despite of the impact of inter-channel nonlinearities, we show that intra-channel nonlinear equalization is still able to provide approximately 1 dB improvement in the optimum power and a BER reduction of ∼33%, considering a 66 GHz WDM grid. By means of simulation, we demonstrate that the performance of nonlinear equalization can be substantially enhanced if both optical and electrical filtering are optimized, enabling the VSNE technique to outperform its SSF counterpart at high input powers.
© 2013 Optical Society of America
Thanks to coherent detection, digital equalization of linear fiber impairments can nowadays be performed with negligible penalty. The ultimate limits of fiber capacity are then set by nonlinearities and their interaction with noise . The development of efficient nonlinear compensation algorithms is therefore of utmost importance. The issue of nonlinear compensation has been most commonly approached through digital backward propagation (DBP) employing split-step Fourier (SSF) methods [2, 3]. Recently, new algorithms based on Volterra series theory have been reported either in time  or frequency-domain . In , an intra-channel Volterra series nonlinear equalizer (VSNE) applied in frequency-domain has been proposed and assessed through numerical simulations. The VSNE algorithm can be beneficial for real-time implementation due to its parallel structure, while maintaining very high equalization performance even at 2 samples per symbol .
Digital compensation of single-polarization inter-channel  and dual-polarization intra-channel  effects has been already experimentally demonstrated using SSF approaches. Time-domain Volterra series filtering has been demonstrated in single-polarization QPSK transmission  at 10 Gbaud, using 5 samples per symbol. A recent experimental demonstration of the single-polarization VSNE algorithm has been reported for a 28 Gbaud polarization-multiplexed (PM)-16QAM transmission over 250 km of ultra-large area fiber . However, these performance assessments of Volterra series equalization are limited by their underlying single-polarization model. Although a dual-polarization nonlinear equalizer based on the regular perturbation method has been recently proposed , its performance is only assessed through simulations.
In this paper, we extend our previous work  providing an experimental demonstration and validation of the dual-polarization VSNE in different PM-QPSK transmission links and data rates, considering both single-channel and WDM scenarios. A comprehensive simulation analysis has been also added to the work, from which we extrapolate the achievable performance of nonlinear equalization in near-optimum conditions. The remaining of this paper is organized as follows: a brief theoretical background for the VSNE algorithm is provided in section 2; experimental assessment and validation is presented in section 3, comprising both single-channel and WDM results; in section 4 we perform a simulation analysis of the experimental results, extrapolating the equalizer performance for ideal filtering conditions; finally, the most relevant conclusions drawn from this work are summarized in section 5.
2. Theoretical formulation
The propagation of polarization-multiplexed optical signals can be analytically modeled by the nonlinear Schrödinger (NLS) equation in its vectorial form. However, being a complex and computationally expensive model, the vectorial NLS equation is not adequate for DBP. To overcome this complexity issue, the vectorial model can be replaced by the Manakov equation. This simplified approach is based on the observation that, since the polarization scattering length is much shorter than the typical nonlinear interaction length, the resulting nonlinearities can be determined with small penalty by averaging the fast polarization changes along the fiber. Then, using the Manakov equation we may analytically describe DBP as 2]. The P(ωn1, ωn2) term refers to the coherent cross-polarization power transfer, being given by Equation (2) is applied span-by-span for each angular frequency, ωn, composing the signal spectrum. By scanning the spectrum of interest with the auxiliary angular frequencies, ωn1 and ωn2, the nonlinear equalizer is evaluated over a memory range of NFFT samples, corresponding to the length of each fast-Fourier transform (FFT) block. Finally, a phase correction can be applied to the output field in order to avoid energy divergence . Regarding numerical complexity, only an extra complex additions per sample are required, relatively to the single-polarization VSNE.
3. Experimental demonstration
3.1. Experimental setup
In this work, we will consider two distinct major experimental setups, which are summarized in Table 1. A simplified schematic representation of all experimental scenarios is presented in Fig. 1. The differences between scenarios A and B lie on the transmitter side and on the propagation fiber. In experimental setup A, the transmitted signal is a Nyquist-shaped 30 Gbaud PM-QPSK, which is propagated in a recirculating loop consisting of a single 100 km NZDSF spooled span. In turn, setup B is based on a 25 Gbaud PM-QPSK signal recirculated over 63.6 km of installed SSMF (FASTWEB’s Turin Metro Plant). Additionally, within experimental setup A we will consider three different sub-scenarios: A.1, which is a single-channel experiment; A.2, which is based on a 66 GHz-spaced 10-channel WDM experiment; and A.3, which is a 10-channel ultra-dense WDM experiment (33 GHz interchannel spacing). In all scenarios, the central channel is optically generated by an external cavity laser (ECL) with 100 kHz linewidth. In scenarios A.2 and A.3, additional distributed feedback (DFB) lasers are used to generate the WDM signal. A pulse pattern generator (PPG) outputs the digital data, which is modulated onto the optical field by a nested Mach-Zehnder (NMZ) modulator. A WaveShaper™ (WS) is employed in setup A, providing a Nyquist pulse shaping to the transmitted optical signal . The −3 dB bandwidth of the WaveShaper™ filter has been set to 32 GHz, and a ∼2 dB pre-emphasis towards the filter sides has been applied to enhance high frequencies, partially compensating for the receiver bandwidth limitation. In setup A.3, an optical frequency doubler (OFD) is used to halve the inter-channel frequency spacing from 66 GHz down to 33 GHz . Finally, polarization-multiplexing (PolMux) is emulated by applying an optical delay line. The delay between the two polarization tributaries is of 882 symbols (29.4 ns) for scenario A and 916 symbols (36.64 ns) for scenario B, thus ensuring polarization de-correlation. The transmitted signal is then propagated in a recirculating fiber loop controlled by acousto-optic switches (AOS). Optical power at the fiber input and output is controlled by two pairs of erbium-doped fiber amplifiers (EDFA) cascaded with variable optical attenuators (VOA). A gain flattening filter is employed to equalize power level in all channels, compensating for EDFA unbalancing. After coherent detection, the four resulting electrical signals are finally sampled at 50 Gsample/s in a Tektronix DPO71604 and stored for subsequent offline processing. Given the different symbol rates, the sampling rates provided for the DSP system are 1.67 and 2 samples per symbol (SpS), for scenarios A and B, respectively.
3.2. DSP system
A block diagram representing the DSP system is presented in Fig. 2.
Offline processing for BER evaluation has been carried out on 217 bits. A digital upsampling stage is applied for scenario A, in order to provide 2 SpS for the remaining DSP subsystems. IQ imbalance is kept under control by careful back-to-back calibration and confirmed by analyzing the output constellations. Both chromatic dispersion and intra-channel nonlinearities are equalized within the static equalization block. Frequency-domain equalization is enabled by the overlap-save method, using the minimum FFT block length, NFFT, that captures the long-term memory effects of the signal (mainly due to chromatic dispersion), thus avoiding inter-block interference (32 samples for scenario A and 128 samples for scenario B). Linear equalization is performed by a frequency-domain chromatic dispersion equalizer (CDE). Digital nonlinear equalization is performed using both the single- and dual-polarization versions of the SSF and VSNE methods. The SSF method with N steps per span is denoted as SSFN. Subsequently, frequency estimation is achieved by a common feedforward spectral method . In the dynamic equalization block, polarization demultiplexing and residual dispersion compensation are performed by a 25 taps adaptive filter driven by the constant modulus algorithm (CMA). After downsampling to 1 SpS, phase estimation is implemented by the Viterbi-Viterbi algorithm (no differential coding) with an optimized block-length. Finally, BER calculation is carried out after symbol decoding.
3.3. Single-channel results
In order to find the best nonlinear compensation provided by the SSF and VSNE methods, we have performed an optimization procedure for each input power, which is based on measuring the nonlinear equalization performance for a set of predefined trial values of the ξ factor in Eq. (3). As a figure of merit for this optimization procedure we use the error vector magnitude (EVM) of the equalized signal, defined asFig. 3 we show the optimization of both single-and dual-polarization nonlinear equalizers at the optimum power (0.4 dBm for the 16 spans experiment and 0.2 dBm for the 31 spans experiment, as shown in Fig. 4) for scenario A.1, composed of 16 and 31 spans. The ξ parameter has been varied between 0.6 and 1, with a step-size of 0.05.
The obtained results clearly show that the maximum SSF performance, for both the single-and dual-polarization models, has been achieved by applying only 1 step per span. This low DBP spatial resolution stems from the challenging filtering conditions, both at the transmitter and receiver. At the transmitter side, the employed Nyquist pulse shaping imposes an aggressive optical filtering to the signal spectrum, causing some inter-symbol interference that cannot be removed by the nonlinear equalizers. In turn, at the receiver side, the low ADC sampling rate (∼1.67 SpS) and analog bandwidth (∼13 GHz, corresponding to ∼43% of the symbol rate), reduce the available spectrum (before digital interpolation) for nonlinear equalization. Being limited by the same external factors, the VSNE optimization curves are also overlapped with those obtained for the SSF method. It is also worth mentioning that the dual-polarization versions of the SSF and VSNE methods enable to significantly increase the optimum fraction of nonlinear compensation, which is a consequence of taking into account the cross-polarization effects. The same ξ optimization procedure has been carried out for each input power, with similar conclusions.
In Fig. 4 we show the evolution of BER as a function of input power for experimental setup A.1, considering a total propagation distance of 1600 km (Fig. 4(a)) and 3100 km (Fig. 4(b)). Solid/dashed lines are obtained by third-order polynomial fittings of data.
The obtained results reemphasize the idea that the above mentioned experimental limitations are ceiling the maximum performance of the single- and dual-polarization VSNE’s, causing an overlap with the SSF curves and thus preventing to achieve the VSNE performance advantage that has been demonstrated by means of simulation in our previous work . Despite of these challenging experimental conditions, the results in Fig. 4 reveal that nonlinear equalization can still bring a significative improvement both in terms of BER and optimum power. For the 16×100 km scenario, the dual-polarization nonlinear equalizers provide a BER reduction by a factor of ∼5 (∼80% improvement), relatively to the CDE, leading to an increase in optimal power of about 1.7 dB. In addition, a BER gain of about 2.5× (∼60% reduction) is obtained over the single-polarization nonlinear equalizers, representing an increase in optimal power of about 0.7 dB. Regarding the 31×100 km loop, we may observe that the BER curves are roughly shifted up by one order of magnitude due to the accumulated noise and under-compensated nonlinearities. Although the BER gain is slightly decreased, the optimum powers for each curve remain almost unchanged.
In order to assess the performance of the nonlinear equalizers with less stringent filtering conditions, we will now analyze the alternative experimental setup B, which presents the following advantages: i) absence of tight optical filtering (no WS is used) at the transmitter side; ii) higher sampling rate (2 SpS) and higher ratio between analog ADC bandwidth and signal symbol rate, at the receiver side.
In Fig. 5(a), we show the optimization of the free parameter, ξ, as a function of the EVM. We may observe that due to the higher sampling rate (2 SpS) and higher accumulated dispersion per SSMF span, the best SSF performance in this scenario is now attained with 4 steps per span. As previously observed for scenario A.1, the optimum fraction of nonlinear compensation is significantly increased by the dual-polarization equalizers. However, the optimum ξ values have slightly decreased relatively to the previous scenario. This can be justified by the larger chromatic dispersion per span associated with accumulated noise, which tends to degrade the inversion of the nonlinear phase-shift. The BER results provided in Fig. 5(b) show that the dual-polarization nonlinear equalizers provide a ∼3 dB increase in the optimum power over CDE, with a corresponding BER gain of approximately 6.6× (∼85% reduction). Relatively to the single-polarization counterparts, the dual-polarization equalizers provide around 1 dB improvement in terms of optimal input power and a BER reduction of ∼2.7×, which corresponds to a ∼63% improvement. However, the negligible improvement of VSNE over SSF4 denotes that digital equalization remains strongly limited by the available electrical bandwidth, which is still only about 52% of the symbol rate. Indeed, besides limiting the performance of nonlinear equalization in general, a low ADC bandwidth also works as an analog low-pass anti-aliasing filter, reducing the VSNE performance gain over SSF, as reported in .
3.4. WDM results
With the aim to study the effectiveness of intra-channel nonlinear equalization in multi-channel signal propagation, we have also analyzed two 10-channel WDM scenarios, with 66 GHz and 33 GHz inter-channel spacing, corresponding to experimental setups A.2 and A.3, respectively. The performance of linear and nonlinear equalization in scenario A.2 is presented in Fig. 6.
As expected, due to the effect of inter-channel nonlinearities, we may observe that the gains achieved by nonlinear equalization in both optimum power and BER are now significantly reduced, relatively to the single-channel scenarios. A direct comparison between Figs. 6(b) and 4(a), which roughly correspond to the same propagation distance (1500/1600 km), reveals a BER degradation of about half an order of magnitude in terms of CDE performance. However, when the same comparison is performed in terms of nonlinear equalization, this BER degradation is now increased to approximately one order of magnitude, revealing the strong impact of uncompensated inter-channel nonlinearities.
To conclude the WDM analysis, we have tested nonlinear equalization in scenario A.3, which is based on a WDM experiment with ultra-narrow spacing of 1.1× the symbol rate. The experimental results presented in Fig. 7 clearly show the ineffectiveness of intra-channel nonlinear compensation in ultra-dense WDM scenarios. For the 5×100 km case (Fig. 7(a)) there is no visible improvement brought by either single- or dual-polarization nonlinear equalizers. Doubling the propagation distance (see Fig. 7(b)), a residual performance gain is still obtained, even though inter-channel effects are now clearly the dominant nonlinear distortions.
4. Simulation results
4.1. Simulation modeling of the experimental conditions
The previously discussed experimental results denote a severe equalization performance limitation due to narrow optical and electrical filtering. With the aim to provide a clearer picture on the impact of filtering on nonlinear equalization under the considered experimental conditions, we have performed a simulation analysis of the experimental results, from which we extrapolate the achievable performance of nonlinear equalization under near-optimum filtering conditions.
The performance of the simulation model obtained for scenario A.1 (16 spans trial) is presented in Fig. 8(a), showing a good agreement with the correspondent experimental results and thus enabling to accurately mimic the experimental conditions in simulation environment. Signal generation and propagation has been performed using VPItransmissionMaker8.6. The experimentally measured WS transfer function, with 3 dB bandwidth of approximately 32 GHz, has also been considered in the simulation setup. A 5 dB noise figure has been estimated for the EDFA and a noise loading stage has been added at the receiver side, in order to account for the equivalent noise figure characterization of the loop, as defined in . The entire DSP system is then implemented in MATLAB. A fifth-order Butterworth low-pass filter (LPF) with 3 dB cutoff frequency of 13 GHz has been used to model the ADC bandwidth limitation, providing a good agreement between simulated and experimental spectra, as shown in Fig. 8(b).
4.2. The impact of electrical and optical filtering on equalization performance
Making use of the previously obtained simulation model, in this section we intend to extrapolate the achievable performance of nonlinear equalization under near-optimum filtering conditions.
Considering a fixed input power of 1 dBm (the optimum power after nonlinear equalization, in the experimental conditions), we have gradually increased the bandwidth of the LPF from 13 GHz to 30 GHz, in order to remove the electrical filtering limitation at the receiver. The obtained results, presented in Fig. 9, show that both linear and nonlinear equalization approximate their maximum performance for an electrical bandwidth of 16 GHz, which is roughly matched with the equivalent baseband 3 dB cutoff frequency of the WS filter, but represents only approximately 53% of the symbol rate. Although the electrical bandwidth limitations have been removed, the effect of the Nyquist pulse shaping imposed by the WS filter still poses a severe impact on the equalization performance. As a consequence, there is no visible improvement obtained by increasing the sampling rate from 1.67 SpS to 2 SpS. Moreover, the maximum split-step spatial accuracy is still limited to 1 step per span, matching the performance of the VSNE technique. We highlight the impact of the WS filter on the SSF step-size, which can be easily understood by noticing that the WS employed in the experimental setup cuts approximately 65% of the ideal bandwidth of 3× the symbol rate, required by the SSF method . Due to the lack of spectral information, the nonlinear step is not capable to accurately invert the nonlinear phase-shift. Indeed, the power estimation error caused by aggressive filtering can be higher than that induced by the interplay between dispersion and nonlinearities over a given propagation distance, thereby limiting the maximum achievable spatial resolution. The same analysis has also been performed for higher input powers, from which similar conclusions were drawn.
In order to remove the optical filtering limitations, we have replaced the WS filter by a second-order Gaussian filter with a 3 dB bandwidth of 90 GHz. We have then repeated the previous analysis, adjusting the range of input powers of interest in order to observe the behavior of nonlinear equalization within the optimum power region (4 dBm) and for strong nonlinearities (6 and 8 dBm). The results presented in Fig. 10 reveal a substantial performance improvement relatively to the previous simulation scenario. Removing the tight optical filtering imposed by the WS, DBP is now able to operate with an enhanced spatial and temporal resolution. The maximum SSF accuracy is now obtained with 4 steps per span. The use of an increased temporal resolution also begins to pay off as the input power increases. At 8 dBm input power, a sampling rate of 60 Gsample/s (2 SpS at the ADC output) provides approximately 1 dB performance improvement over the experimental conditions (50 Gsample/s, 1.67 SpS), using VSNE. Additionally, we may also observe that the optimum LPF cutoff frequency tends to increase with the input power, reaching ∼24 GHz at 8 dBm, corresponding to 80% of the symbol rate. Under these near-optimum simulation conditions, the previously reported VSNE advantage over SSF  is finally achieved, leading to a 1.5 dB EVM improvement at 8 dBm.
Finally, in Fig. 11 we provide a direct comparison between the achievable equalization performance under the different filtering conditions discussed above. The highest EVM reference curves correspond to the performance in the experimental conditions, as depicted in Fig. 8(a). To represent the wide electrical filtering and Nyquist pulse shaping scenario, we have applied an LPF cutoff frequency of 20 GHz, which is shown in Fig. 9 to provide the optimum performance at 1 dBm, with a safe bandwidth margin for higher input powers. An improved setup is obtained without the WS filter and with an LPF cutoff frequency of 24 GHz. We may observe that, although an EVM improvement of approximately 2 dB can be obtained by solely removing the electrical filtering limitations, this leads to a negligible improvement in terms of optimum power. In fact, part of this 2 dB EVM gain is transversal to linear and nonlinear equalization, being obtained as a direct consequence of relaxing the electrical filtering at the receiver, which contributes to reduce the inter-symbol interference. Since optical filtering still affects a significant part of the baseband spectrum, nonlinear equalization performance remains limited, leading to a small gain in optimum power (∼1 dB). A further 2 dB improvement in the maximum performance has been obtained by relaxing the optical filtering at the transmitter. Noticeably, a ∼3 dB increase on the optimum input power is obtained relatively to the experimental conditions. With near-optimum electrical and optical filtering, the full spectrum at 2× over-sampling is available for nonlinear equalization, enabling to more accurately calculate the nonlinear phase-shift up to significantly higher launched powers. In those conditions, it can also be observed that the VSNE advantage over SSF tends to increase with the input power, confirming its robustness to highly nonlinear regimes.
Considering PM-QPSK single-channel and WDM propagation scenarios, we have experimentally demonstrated a dual-polarization VSNE applied in frequency-domain. Due to the tight optical filtering and the ADC bandwidth limitations, the dual-polarization VSNE and SSF methods were found to be approximately equivalent in performance. In single-channel transmission, we have demonstrated a BER improvement at the optimum power of about 60%, relatively to the previous single-polarization approach, and over 80%, relatively to linear equalization. The obtained WDM results reveal that intra-channel nonlinear equalization is still able to provide a moderate performance improvement over linear equalization (above 30%) for a channel spacing of 2.2× the symbol rate. In turn, a negligible performance improvement has been obtained when the WDM frequency grid is further reduced to only 1.1× the symbol rate, due to the limiting effect of uncompensated inter-channel nonlinearities.
Simulation results obtained for less stringent optical and electrical filtering conditions have confirmed that the assessed experimental equalization performance can be substantially optimized. Although the performance of nonlinear equalization can be enhanced by about 2 dB (in terms of EVM) by solely optimizing the electrical receiver bandwidth, we have found a strong impact of the Nyquist pulse shaping on equalization performance. Considering near-optimum optical/electrical filtering, we have shown that the dual-polarization VSNE is able to outperform the back-propagation SSF method in the nonlinear regime, confirming the previously reported advantage of the single-polarization version of the algorithm . Being intrinsically single-step, the VSNE is most beneficial in terms of computational complexity for scenarios with wide optical and electrical filtering, where the maximum SSF performance is attained at several steps per span. It is worth mentioning that, despite of the VSNE parallel structure, the total number of operations per FFT block depends quadratically on the FFT block-size, requiring very short FFT blocks in order to keep the algorithm in a tolerable region of complexity. Simplification of the algorithm structure and reduction of the overall computational complexity are topics currently under investigation.
This work was supported in part by the FCT - Fundação para a Ciência e a Tecnologia, through the Ph.D. Grant SFRH/BD/74049/2010, by EURO-FOS, a Network of Excellence funded by the European Union through the 7th ICT-Framework Programme, by PT Inovação, SA, through the project “AdaptDig” and by the FCT and the Instituto de Telecomunicações, under the PEst-OE/EEI/LA0008/2011 program, project NG-COS.
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