1. Introduction

Over the past few years, digital signal processing (DSP) techniques has been drawing much more attention as methods of mitigating optical transmission impairments, which stem from chromatic dispersion (CD) and polarization-mode dispersion (PMD). This is because they can offer the potential for significant cost savings compared to optical approaches. Among all electronic equalization techniques, finite-impulse-response (FIR) filtering and maximum-likelihood sequence estimation (MLSE) have attracted significant attention.

For direct-detection systems, MLSE is the most promising approach owing to its excellent compensation performance and fast adaptation to fiber-link conditions; on the other hand, FIR filters are not so effective because square-low detection kills the phase information of the optical signal. Several transmission experiments based on the direct-detection scheme have demonstrated that a commercial MLSE receiver can equalize CD and PMD [1–4].

In high-capacity and long-distance optical transmission systems, the increase in the optical power is effective for saving optical signal-to-noise ratio (OSNR); however, such increase in the optical power induces fiber nonlinearity, and the system performance is deteriorated by nonlinear impairments stemming from self-phase modulation (SPM) and cross-phase modulation (XPM). The mitigation the nonlinear impairments becomes a very important issue as the link distance or data rate increases [5], and MLSE is also playing an important role for this purpose [6–8].

On the other hand, the digital coherent receiver enables more sophisticated electronic signal processing because not only the amplitude but also the phase can be recovered after detection [9]. Linear transmission impairments due to CD and PMD have been equalized electrically by using FIR filters [10]. Since CD and PMD can be expressed as linear transfer functions operating on the complex amplitude of the modulated optical signal, we can compensate for them by linearly equalizing the homodyne-detected complex amplitude with FIR filters [11,12].

In digital coherent receivers, SPM and XPM can hardly be equalized even by FIR filters, because unlike CD and PMD, these cannot be expressed as linear transfer functions. On the other hand, differently from FIR filters, MLSE works well because the maximum likelihood data are determined from training patterns affected by the nonlinear impairments; and hence, it may play a superior role for the equalization of such nonlinear impairments also in the coherent receiver.

In this paper, we demonstrate optical transmission experiments, where we introduce the MLSE equalizer based on the Viterbi algorithm [13,14] together with the FIR filter into digital coherent receivers, aiming at equalization of both linear and nonlinear impairments. This experimental work deals only with SPM in a single-channel transmission system. XPM can usually be mitigated by the large amount of accumulated CD of standard single-mode fibers (SMFs) for transmission, and SPM remains the main origin of nonlinear impairments even for WDM transmission.

With the coherent receiver including the MLSE equalizer, we demonstrate unrepeated transmission of 20-Gbit/s optical quadrature phase-shift keying (QPSK) signals over a 200-km-long SMF. When the launched power is increased up to 10 dBm, SPM begins to deteriorate the bit-error rate (BER) performance, and it cannot be compensated for by adaptive FIR filters; however, the MLSE equalizer can equalizes SPM, improving the power penalty by 2 dB.

This paper is organized is as follows: Sec. 2 provides the experimental setup. In Sec. 3, we describe how we implement MLSE equalizers together with FIR filters into DSP circuits. In Sec. 4, MLSE performances are examined and compared with conventional adaptive FIR filters for both linear and nonlinear cases. In Sec. 5, we discuss effectiveness and limitation of our approach and finally Sec. 6 concludes our paper.

2. Experimental setup

Figure 1 shows the experimental setup for measuring QPSK transmission characteristics through 100- and 200-km-long SMFs. The laser for the transmitter was a distributed-feedback laser diode (DFB-LD) and its wavelength was 1552 nm. An arbitrary waveform generator (AWG) was used to generate two 10-Gbit/s data streams, which were pre-coded to be a 2⁹-1 pseudorandom pattern after decoding. An optical QPSK signal was created by a LiNbO₃ optical IQ modulator (IQM).

 

Fig. 1 Experimental setup for 20-Gbit/s QPSK transmission.

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Firstly, the input power launched on the SMF was fixed at 0 dBm to avoid nonlinear effects. Secondly, the input power was fixed to 10 dBm to induce nonlinear effects intentionally. The SPM effect along a fiber of length L_f becomes dominant when L_f is longer than the nonlinearity length L_NL defined by γP, where γ is the nonlinear coefficient of the fiber and P the launched power. When P = 0 dBm and 10 dBm, we calculate L_NL to be 670 km and 67 km, respectively, by using γ = 1.5/W/km. Since L_f = 100 km and 200 km in our case, the SPM effect may appear significantly when the input power is 10 dBm. On the other hand, we can almost neglect the SPM effect when the input power is 0 dBm. The SMF had the dispersion D = 17 ps/nm/km (β₂ = -22 ps²/km). We confirmed that the stimulated Brillouin scattering (SBS) never occurred when the input power was increased up to10 dBm.

At the receiver, the received average power P_in was controlled by a variable optical attenuator (VOA). Then, the signal was pre-amplified by an erbium-doped fiber amplifier (EDFA) and detected by a homodyne phase-diversity receiver. More details on homodyne phase-diversity coherent receivers can be found in [8]. Another DFB-LD having the characteristics same as the transmitter DFB-LD was used as a local oscillator (LO). The real and imaginary parts of the beat signal were simultaneously sampled at 20 Gsample/s with 8-bit analog-to-digital converters (ADCs). The sampled data were then processed offline by the digital signal processing (DSP) circuit, whose functions were FIR filtering, clock recovery, phase estimation and MLSE.

In our experiment, we used 15-tap and 21-tap fixed FIR filters to compensate for dispersion values of 1700 ps/nm of the 100-km-long fiber and 3400 ps/nm of the 200-km-long fiber, respectively. Although CD is compensated for to a significant degree by such FIR filters, power penalties still remain which may stem from non-ideal tap coefficients. Then, we used a 200-tap adaptive FIR filter for both lengths of the fibers. To show the effectiveness and performance comparison of MLSE, we replaced this adaptive FIR filter by MLSE which is based on the 16-state Viterbi algorithm.

3. MLSE implementation

If the level of modulation is M and the channel-memory length is L, the number of states in the trellis diagram (TD) is M^L and the number of state transitions is M^{L + 1.} In our experiment, we apply the QPSK (M = 4) signal with the channel memory L = 2. In such a case, each of 64 state transitions is given as a six-bit word consisting of a symbol together with a predecessor symbol and a post-cursor symbol.

Before the MLSE equalization starts, we need to prepare a look-up table during the training process, where we receive a known sequence of symbols that has been affected by impairments of the channel. We sort the received symbols surrounded by predecessor and post-cursor symbols according to prescribed state transitions, and calculate the mean complex amplitude p(i) of those sorted symbols, which belong to the i-th state transition. The complex amplitude p(i) thus obtained is stored in the look-up table.

In the MLSE equalization process, we compute the Branch metric (BM) given as BM_k(i) = |z(k)-p(i)|², which is the distance between a received complex amplitude z(k) of the k-th symbol and the average complex amplitude p(i) belonging to the i-th state transition. BM can be computed by either histogram or Euclidean-distance method, and we applied the later one because of its simplicity. Each BM is associated with one of the state transitions in TD. The path leading to a certain state accumulates all branch metrics within this path and thus leads a path metric (PM). Therefore, the path metric at the k-th state can be obtained as PM_k = PM_k-1 + BM_k. Each state is developed into a transition by adding a possible new symbol and each transition leads to a new state. This leads to a competition of paths prior to each consecutive state, where only the path with the maximum path metric survives. All other paths are omitted in the next steps. In summary, the operational principles of VA as well as of TD include the following steps:

The method of implementing the MLSE equalizer in our experiment is shown in Fig. 2 . The signal complex amplitude from the phase-diversity homodyne receiver is sampled and digitized by ADCs and then it is applied to a fixed FIR filter. After the clock is recovered by using the clock recovery unit, the signal is resampled so that each symbol has one sample. Then, the carrier phase is estimated through the Mth-power feed-forward scheme, and we restore the symbol z(k). Such symbol is now ready for equalization by an MLSE module consisting of the look up table, the BM computation block and the Viterbi algorithm block. The look up table stores p(i) and sends it to the BM computation block.

 

Fig. 2 MLSE implementation in the DSP circuit.

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4. Experimental results

We performed offline measurements of BER from recorded 100-k-symbol data after unrepeated transmission through 100- and 200-km-long SMFs.

We measured BERs of the 20-Gbit/s QPSK signal for the following four cases:

 

Fig. 3 BERs measured as a function for received power after transmission through a 100-km-long SMF.

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Fig. 4 BERs measured as a function for received power after transmission through a 200-km-long SMF.

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From BERs for Case 1 and Case 2, we can observe a significant improvement of BER characteristics by the fixed FIR filter, which proves that GVD can be compensated for by such filter in the digital coherent receiver. However, there still remain some penalties due to non-ideal tap coefficients of the filter which can be equalized by adaptive FIR filters especially when the input power is low enough. Case 3 describes the BER improvement owing to such adaptive equalization. Since the performance of adaptive filters depends on the number of taps, we have optimized the number of taps to get the best BER performance.

To examine the MLSE performance, we compare BERs of Case 3 and Case 4 for both fiber lengths. In the linear region, where the input power is 0 dBm, BER curves obtained by using adaptive FIR filters are 1-dB and 2-dB worse than the theoretical limit for 100-km and 200-km transmission, respectively. MLSE gives us small improvement for both fiber lengths in the linear region. However, when the input power is increased up to 10 dBm, BERs obtained with adaptive FIR filters show higher penalties. On the other hand, introducing MLSE instead of adaptive FIR filters, we can achieve significant receiver sensitivity improvement, which is about 1 dB and 2 dB at BER = 10⁻⁴ for 100-km and 200-km transmission, respectively.

5. Discussion

Figure 5 presents the power-penalty analysis for the MLSE receiver in the 200-km-long system. We measured power penalties at BER = 10⁻⁴ with respect to the theoretical limit. Green curves show the BER performance in the linear case. We can see that GVD can be perfectly compensated for by fixed and adaptive FIR filters, and the sensitivity can be improved by very small amount (less than 1 dB) with the MLSE scheme. Although either FIR filters or MLSE can be the perfect solution to mitigate linear inter-symbol interference (ISI), the FIR-filter-based scheme should be implemented in the digital coherent receiver because of its smaller computational complexity.

 

Fig. 5 Power penalty analysis for the MLSE receiver. BERs are measured as a function of the received power in the 200-km-long 20-Gbit/s QPSK transmission system.

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Blue curves of the Fig. 5, on the other hand, represent the BER performance in the nonlinear region, where the input power is 10 dBm. In such region, the sensitivity of the MLSE receiver is improved by as large as 2 dB compared to the FIR-filter-based receiver, because MLSE can mitigate SPM in our receiver. However, we still have 3.6 dB power penalties from the theoretical limit even with MLSE. This power penalty can further be reduced by using a greater memory length in the MLSE scheme; however, the increase in the memory length results in the exponential-order increase in the computational complexity, making it much more difficult to implement MLSE in real systems.

We thus find that the FIR equalizer in the coherent receiver can compensate for CD nearly perfectly in the linear region, but that the MLSE equalizer is more effective for nonlinearity mitigation. Indeed, the combination of the MLSE equalizer and the fixed FIR filter shows the best BER performance.

6. Conclusion

We have successfully introduced MLSE into digital coherent receivers together with fixed FIR filters in order to equalize both linear and nonlinear fiber impairments. 20-Gbit/s QPSK signals are transmitted through a 200-km-long SMF, and the MLSE receiver can equalize SPM which becomes significant when the launched power is increased up to 10 dBm. The combination of MLSE and fixed FIR filtering in the digital coherent receiver is considered to be the most powerful technique for signal equalization in optical communication systems.