## Abstract

For polarization-division-multiplexing coherent optical orthogonal frequency division multiplexing (PDM-CO-OFDM) systems, we propose a per-symbol-based digital back-propagation (DBP) approach which, after cyclic prefix removal, conducts DBP for each OFDM symbol. Compared with previous DBP, this new proposal avoids the use of inefficient overlap-and-add operation and saves one fast Fourier transform (FFT) module, therefore simplifying the hardware implementation. Transmitting a 16-QAM, 42.8-Gb/s PDM-CO-OFDM signal over 960-km standard single mode fiber (SSMF), we compare the previous and the proposed DBP approaches with different receiver’s sampling rates and different step lengths in each DBP iteration, and found that the proposed DBP can achieve a similar performance as that of the previous DBP while enjoying a simpler implementation. We have also specifically introduced a small self-phase modulation (SPM) model for DBP and demonstrated its feasibility with the same experimental setup.

©2013 Optical Society of America

## 1. Introduction

Technology revolution in the field of optical communications began years ago with the first introduction of the digital signal processing (DSP) [1], which enabled the technologies of 1) dispersion-unmanaged link, 2) coherent detection, and 3) polarization division multiplexing (PDM). Since then coherent detection with DSP soon became almost invincible in the world of linear transmissions because of its great ability to compensate for both the chromatic dispersion (CD) and polarization mode dispersion (PMD). However, simply handling the linear impairments is insufficient for guaranteeing the receiving performance because the fiber nonlinearities would also play a significant role especially in a long-distance transport system. Therefore, one of the remaining issues in the coherent receiving would be the lack of an efficient nonlinear compensation scheme, preferably implemented with DSP.

Among the many DSP proposed measures against fiber nonlinearities, digital back-propagation (DBP), which reverses the signal back to the transmitter and removes the inline nonlinear noise through an iterative algorithm, might be the most powerful approach in terms of equalization ability [2–5]. Its effectiveness has been experimentally verified with offline processing for both the single- and multi-carrier transmissions [4,5]. In essence, DBP would require the following four modules to function: overlap-and-add-based fast Fourier transform (OA-FFT), CD compensator (CDC), OA-IFFT, and nonlinear compensator (NLC). In the single-carrier case, as depicted in Fig. 1(a) , DBP circuit could be merged into the frequency-domain CD equalizer which already utilizes one pair of OA-FFT/IFFT and one CDC so that the added hardware complexity with DBP would be only one NLC module per step. On the contrary, in the multi-carrier case (i.e. orthogonal frequency division multiplexing, OFDM), as shown in Fig. 1(b), the previous demonstration utilizes an extra dedicated DBP circuit that includes all the above-mentioned four modules, resulting in a dramatic increase in hardware complexity [3,5]. Therefore, to enhance the competition capability of OFDM, it would be advantageous if an alternate DBP approach could be provided with a hardware-efficient implementation.

In this paper, for PDM-CO-OFDM we experimentally demonstrate a per-symbol-based DBP approach, of which the iterative circuit is integrated into the inherent FFT module in an OFDM receiver. This proposal avoids the use of the inefficient OA-FFT/IFFT algorithms and saves one FFT module when compared with the previous DBP method. With a 16-QAM, 42.8-Gb/s PDM-CO-OFDM signal transmission over a 960-km SSMF link, we show that this hardware-efficient proposal exhibits similar performance to the previous approach with different sampling rates and different step lengths in each iteration. We also simplify the NLC. Essentially this paper extends our previous work [6] by more detailed descriptions and discussions.

The rest part of this paper is organized as follows: in Section 2 the working principle for the proposed pre-symbol-based DBP method is described and the small SPM model for NLC is introduced; in Section 3 the experimental setup is detailed and in Section 4 the results are presented and discussed. Finally, Section 5 concludes this paper.

## 2. Working principle

DBP had been developed as a powerful equalization tool to jointly remove the SPM and CD effects of the transmission fiber [3], of which the idea actually originated from the numerical model of the fibers: the split-step Fourier method [7]. In the split-step Fourier method, a long distance fiber is cut into many short distance sections: in each section the nonlinear SPM is added at the start of the section in the time domain and the linear CD is introduced at the end of the section in the frequency domain. Therefore, to model the signal propagating along the long-distance fiber, the signal has to be Fourier transformed back and forth through the multiple fiber sections until it reaches the last one. This way, the joint effect of SPM and CD can be well modeled for a long-distance transmission fiber, as long as each section length is well-planned (which should be a function of the signal power and the fiber parameters). The DBP relies on a similar idea that, on the contrary, it propagates the signal from the receiver to the transmitter: it also splits the fiber link into many sections and tries to remove SPM and CD in a similar way as the split-step Fourier method. However, general issues of using DBP might exist in a practical system: 1) the noise would pollute the estimation of the power waveform and thus diminish the SPM compensation efficiency, and 2) the section length suffers a trade-off between the estimation accuracy and implementation complexity. In what follows we firstly discuss the conventional DBP in a CO-OFDM system and then introduce our hardware-efficient per-symbol-based DBP proposal.

Figure 2(a) depicts the previous DBP for PDM-CO-OFDM systems [5]. Throughout this paper we take the RF-pilot-based CO-OFDM [8] as examples while the proposed method should work as well for scattered-pilot-based CO-OFDM systems [9]. After synchronization, the signals are sent to the DBP iterative circuit which contains the four blocks of OA-FFT, CDC, OA-IFFT, and NLC. This iterative circuit will reverse the signal back to the transmitter via a digital virtual link which is in the reverse direction to the physical fiber link. The use of OA-FFT/IFFT is to switch the calculations in between the frequency and time domains, which are preferably for CDC and NLC, respectively. CDC mitigates the CD distortion with a parabolic phase profile [2,3] and NLC reduces the SPM noise via the following operation (here we define it as a regular SPM model to distinguish it from the later introduced small SPM model) [2–5]:

*X*(

*n*) and

*Y*(

*n*) are the NLC input waveforms in X and Y polarizations, $\overline{X}(n)$ and $\overline{Y}(n)$ are the NLC output waveforms in X and Y polarizations,

*j*is the imaginary unit,

*α*and

*β*represent the nonlinear coefficients which are functions of the fiber type (effective core area, nonlinear refractive index, and loss etc.), distance, and launch power. After the DBP circuit, a regular OFDM receiver is followed with RF-pilot-based phase noise compensation (PN), cyclic prefix (CP) removal, FFT, channel estimation with training symbols and multiple-input-multiple-output (MIMO) equalization. Notably, in this case the price of nonlinear compensation is the requirement for the extra four blocks of OA-FFT, OA-IFFT, CDC, and NLC.

In Fig. 2(b) we present the proposed per-symbol-based DBP method. After synchronization, PN compensation, and CP removal, DBP is implemented for each single OFDM symbol: the processed block size for DBP is exactly the same as that of one OFDM symbol, and also the block is taken synchronously with each OFDM symbol so that the inter-symbol interference would not occur during iteration. Following the DBP circuit, channel estimation and MIMO equalization are performed. This new method integrates the DBP circuit into the inherent FFT module and keeps the other parts of the processing the same as those in a regular OFDM receiver. Its uniqueness is that the iterative algorithm starts after CP removal and processes the signals in a symbol-by-symbol manner (which results in its surname of “per-symbol-based” where symbol means the OFDM symbol) so that the frequency- and time-domain conversions can be realized via the regular FFT and IFFT operations, thus avoiding the use of less-efficient OA-FFT and OA-IFFT (in which extra zero-padding and additions will be needed with a lower filtering efficiency due to the partial overlap between adjacent processed blocks [10]). In addition to the use of regular FFT/IFFT, this proposal also saves one FFT module when compared with the previous DBP method (see Fig. 2(a)). However, one potential issue with this proposal might be the need for a longer CP duration since the previous DBP uses OA-FFT/IFFT which can remove accumulated CD with minimum CP duration (it still needs some CP for the later MIMO processing). However, this CP overhead would become negligible when a larger FFT size is applied (since a FFT size of 1024 is used in our system, the CP overhead is only ~1% and could be considered negligible). In systems with a greater amount of accumulated CD, i.e. a longer transmission distance without optical dispersion management, the CP overhead would become an issue in the proposed DBP scheme.

Here we also introduce a small SPM model for NLC to further simplify the DBP equalizer. Assuming SPM is small (low launch power with medium transmission distance), we could approximate Eq. (1) with a simpler form as follows (here we define it as a small SPM model):

We have emphasized the hardware efficiency of our proposal and now we further analyze the computational complexity of these two schemes. In the previous DBP method, if we denote the FFT/IFFT size as *N _{FFT}*, the useful size as

*N*(samples not discarded during overlap-and-add method), and the required iterative number as

_{U}*M*, then the total required multiplications for each OFDM symbol in each polarization could be expressed as (assuming

*α*=

*β*and the exponential function implemented with a lookup table):

*N*, and the required iterative number as

_{FFT}*M*, then the total required multiplications for each OFDM symbol in each polarization could be expressed as:

Since in Eq. (3) *N _{U}* is always smaller than

*N*[10], i.e.

_{FFT}*C*>

_{pre}*C*, we conclude that the proposed DBP method should demand for a lower computational complexity.

_{pro}## 3. Experimental setup

Figure 3 depicts the experimental setup of a 16-QAM, 42.8-Gb/s PDM-CO-OFDM system. A 10-kHz-linewidth fiber laser (FL) is used as the light source and its output is modulated with the electrical OFDM signal via a 16-GHz ˗3dB-bandwidth optical in-/quadrature-phase (I/Q) modulator. This OFDM signal is generated offline with Matlab software and composed of 200 OFDM symbols. For each OFDM symbol, a binary data sequence is firstly mapped to 16-QAM symbols, and then modulated onto 590 data subcarriers which later are zero-padded to a FFT size of 1024. After Inverse FFT, a 10-sample CP is added to the head of each OFDM symbol, resulting in 1034 samples per OFDM symbol. The OFDM waveform is then loaded into an arbitrary waveform generator (AWG) that has its “real” and “imaginary” outputs driving the IQ modulator with a 10-GS/s sampling rate. Hence, the output data rate is ~21.4 Gb/s occupying a bandwidth of around 6 GHz. Here the modulator is biased at the point slightly off the null to arrange an RF pilot at the center of the signal band for remote phase noise estimation and the pilot to sideband power ratio is controlled at around ̶ 12 dB. Later a PDM emulator, which makes one OFDM symbol delay between the output polarizations, is used to emulate a polarization-multiplexed transmitter with a data rate of 42.8 Gb/s. The output of the PDM emulator is then launched into a re-circulating fiber loop, which consists of three Erbium-doped fiber amplifiers (EDFAs) and three spans of 80-km SSMF. After 960-km (12-span) transmission, at the receiver the signal is optically pre-amplified, filtered, and heterodyne-detected with a 100-kHz-linewidth external cavity laser (ECL, output frequency at ~5 GHz away from the carrier) through a 90° optical hybrid. The optical “real” and “imaginary” components of both polarizations are recorded with the four channels of a 50-GS/s real-time scope (RTS). After down-conversion, the baseband signal is resampled with 10 GS/s (or 20 GS/s specifically in Fig. 4 ). Synchronization, carrier recovery, CP removal, FFT, channel estimation and equalization with MIMO processing are conducted offline with Matlab. Both previous and proposed DBP approaches are utilized directly after the synchronization block and CP removal block, respectively, as depicted in Fig. 2.

## 4. Results and discussions

Throughout this paper we use *α* = *β* which yields the optimum performance according to our observations and the previous reports [5], and present the signal quality in terms of Q factor which is derived from the calculated bit error rate (BER). Only the regular SPM model, i.e. Eq. (1), is used for NLC in Figs. 3 and 4, and both the regular and small SPM models are used in Fig. 5
to compare their performance. Since the proposed method has some hardware benefits over the conventional method, questions might arise such as whether similar performance to that of the conventional one could be maintained with different system parameters. Therefore in the following we investigate performance sensitivities against the receiver sampling rate and used step length per step (required iterations in DBP).

In Fig. 4 we compare the signal quality in terms of Q factor for the cases with no DBP, with previous DBP and with our proposed DBP. In particular, two different sampling rates with 10- and 20-GS/s (corresponding to 1.7 and 3.4 oversampling ratios, respectively) are compared to investigate whether the sampling rate would impose any threat to the compensation efficiency [3], especially for the new proposal. It can be observed that both DBP methods have very similar performance and, at the optimum launch power, exhibit an ~1.1-dB benefit over the case without DBP (similar to the previous reports [4,5]). Also we found an oversampling ratio of ~1.7 is sufficient for both DBP methods since an even-higher ratio of 3.4 simply brings a very-limited benefit within the high launch power regime.

In Fig. 5 we discuss the effect of span length per step on Q factor. The previous and proposed DBP methods both exhibit very similar performance. In particular, we’ve found the step length can be extended to ~480 km with almost no penalty compared with a step length = 80 km (1 span per step). Further increase in the step length (960-km step length) results in a 0.3-dB Q penalty while still having an ~0.7-dB gain over the case with no DBP, which are compared at their optimum launch power points. This indicates that the required iterations in DBP could be greatly reduced in OFDM systems [5], thus relieving the receiver’s complexity.

In Figs. 4 and 5 in the high nonlinear regime we’ve found that the previous DBP performs slightly better than the proposed DBP. This is because the per-symbol-based approach uses the regular FFT and IFFT operations which assume each received OFDM symbol is still periodic after transmission. However, since SPM would broaden the signal spectrum which in turn would increase the CD-induced pulse broadening, the assigned CP duration would fail to cover both the signal and the high-frequency SPM components, resulting in some inter-symbol interference. This phenomenon would not occur with the previous DBP method which does not rely on the periodicity of the OFDM symbols.

In Fig. 6 we examine the appropriateness of the small SPM model given in Eq. (2). The optimum Q (at the optimum launch power of ~0 dBm) with the regular and the small SPM model, i.e. with Eqs. (1) and (2), respectively, as a function of the step length are presented for the proposed DBP method. With reasonable step lengths, from 80~480 km per step, the small SPM model is found to be able to yield comparable performance to the regular model, which demonstrates the feasibility of the small SPM model Eq. (2) with current experimental conditions (960 km with EDFA only amplification). However, it is expected that, also verified by our observations, a higher launch power would destroy the low power assumption for Eq. (2) and cause this small SPM model to fail to accurately estimate the real SPM, therefore diminishing its compensation efficiency. Therefore, the use of the small SPM model would be more suitable for medium transmission distance.

## 5. Conclusion

We have proposed and experimentally demonstrated a per-symbol-based DBP approach for PDM-CO-OFDM systems, of which the iterative DBP circuit is integrated into the inherent FFT module in a regular OFDM receiver. This new proposal avoids the use of the inefficient overlap-and-add algorithms and saves one FFT module when compared with the previous DBP method. With a 16-QAM, 42.8-Gb/s PDM-CO-OFDM signal transmission over a 960-km SSMF link, we have showed that this hardware-efficient proposal exhibits similar performance to that of the previous approach with different sampling rates and different step lengths in each iteration. We have also simplified the NLC using a small SPM model and demonstrated its feasibility with the same experimental setup.

## Acknowledgment

This work was partly supported by the National Institute of Information and Communications Technology (NICT), Japan.

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