1. Introduction

Electronic techniques for the mitigation of fiber propagation effects have been the subject of considerable research efforts over the last few years. One of the most promising mitigation techniques is Maximum Likelihood Sequence Estimation (MLSE). This technique has been addressed in several theoretical and experimental papers, among which [1]–[27].

In [1]–[9] the theoretical foundations of MLSE in the context of optical transmission were laid out, analytical performance evaluation results were derived and specifically optimized branch-metric solutions were proposed.

Through simulations, MLSE was shown by various groups to be effective in mitigating the impact of Chromatic Dispersion (CD), or a combination of CD and Polarization Mode Dispersion (PMD), with various modulation formats [1]–[14]. MLSE was also shown, again through simulations, to be potentially capable of supporting transmission over narrow-bandwidth channels, characterized by eiter optical or electrical tight filtering [15],[16].

Besides theory, a few experiments using MLSE have been carried out, too. They were either based on a commercial 4-state processor [17]–[23] or on the real-time sampling and off-line processing of the received signal [24]. In two cases, the experiments were run at a high launched power, to test the potential for MLSE to mitigate non-linear intra-channel effects as well [22],[23].

In this paper we report on two long-haul uncompensated 10 Gb/s IMDD transmission experiments carried out over a re-circulating loop, using G.652 Standard Single Mode Fiber (SSMF), with real-time sampling and off-line processing. They were conceived to probe the limits of the effectiveness of MLSE in dealing with very large accumulated CD, and a combination of large CD with substantial intra-channel non-linearity.

The theoretical possibility of long-haul uncompensated IMDD transmission at 10 Gb/s had been investigated up to 700 km of SSMF (11,700 ps/nm) in [7], [14]. Simulations predicted that, provided that enough complexity was allowed for in the MLSE processor, the OSNR penalty with respect to back-to-back would saturate at about 2 to 3 dB, depending on system set-up, at a distance between 300 and 400 km, and would not further grow for longer link lengths. Very recently, further theoretical and simulative evidence of the same prediction has appeared in [9].

Verifying this non-trivial predicted behavior was one of our main goals and we wanted to design an experiment which could be ‘proof-of-concept’ (in the sense of ‘proof-of-conjecture’). We therefore selected a distance large enough (1,040 km, 17,600 ps/nm) to prove or disprove this result with reasonable confidence. Over such link length the channel memory is between 13 and 15 bits and the commercial 4-state (2 bits of memory) MLSE processor could not be used. Thus, we resorted to real-time sampling and off-line processing.

It should be mentioned that some of the experiments using the commercial MLSE processor did reach long-haul distances, but they achieved it resorting to a clever combination of the 4-state MLSE processor with other mitigation technologies. In [19] low-dispersion fiber was used together with the duobinary format and with the aid of substantial pre-compensation. In [20], low-dispersion fiber was again used and the transmitter (TX) employed a CML (Chirp-Managed Laser).

An interesting off-line 10 Gb/s experiment reached 600 km of SSMF [24], using a relatively small MLSE processor memory of 6 bits (64 states). This result was obtained by curtailing the spectral width of the TX signal with a 12 GHz filter, thus mitigating some of the impact of dispersion. Due to the narrow filter, the transmitted signal eye diagram at the TX was completely closed and the modified format was called NF-OOK (Narrow-Filtered ON-Off Keying).

In accordance to the said proof-of-concept nature of our experiment, we instead wanted to use pure NRZ (Non-Return-to-Zero) IMDD over a substantially longer distance, while putting all the burden of dispersion compensation on the MLSE processor alone: no form of optical mitigation (including narrow optical filtering) or optical compensation was used.

Our first experiment was conducted at a low-enough launched power to ensure linearity. However, MLSE is known to have the theoretical potential for mitigating intra-channel non-linearity (INL) as well¹ Once again, we wanted to put ourselves in extreme conditions, for the purpose of probing the limits of MLSE when a combination of large CD and substantial non-linearity were simultaneously present. So we decided to operate over 800 km of SSMF (13,600 ps/nm) while increasing the launched power up to 12 dBm.

As mentioned, two recent papers have addressed INL mitigation by means of MLSE [22],[23]. However, in [22] the link length was limited to 214 km of SSMF, to allow using the 4-state commercial MLSE processor. Ref. [23] focused on the completely different scenario of optical in-line dispersion-managed systems, and again the limited-memory 4-state commercial MLSE processor was used.

The results from our long-haul uncompensated experiments seem to confirm that MLSE can in fact mitigate very large amounts of CD, both alone and in combination with considerable INL. The OSNR penalty-saturation effect seems to be confirmed as well.

Preliminary reports on such experiments were presented in [25]–[27]. Here we organize, expand and further elaborate on such results. Several new previously unpublished results, here displayed in Figs. 4, 5, 8, 10, 11 and 12, were included. We also show a statistical analysis of the error-event length that is substantially different and more detailed than the preliminary one shown in [27].

 

Fig. 1. Set-up used for the experiments.

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2. Experimental set-up

Both the linear and the non-linear experiment used the same set-up, which is shown in Fig. 1. A pulse pattern generator (Anritsu MP1763B) was employed to generate a 10 Gbit/s Pseudo-Random Bit Sequence (PRBS) of length 2²⁰-1. The tunable laser (Agilent 8164A) was set at 1551.7 nm. A conventional LiNbO₃ Mach-Zehnder modulator, biased at the -3 dB point, was used to generate the NRZ-IMDD format signal. Its extinction ratio was approximately 12 dB. A 4 GHz Bessel low-pass filter (LPF) was placed between the pulse pattern generator and the modulator driver. The driver worked in linearity. The bandwidth of the low-pass filter was optimized through computer simulations of the 1,040 km linear experiment and was not changed for the 800 km non-linear experiment. No optical filter was present at the TX. The signal eye diagram at the output of the TX is shown in Fig. 2.

The signal was then input to a re-circulating loop with 80-km SSMF span length. No Dispersion Compensating Fiber (DCF) was installed in the loop. The RX consisted of an EDFA pre-amplifier, followed by a 50 GHz (0.4 nm) Gaussian optical filter, followed by a photodetector and a 6.2 GHz Bessel post-detection (PD) filter. The optical power input to the photodetector was kept constant at -3 dBm. A noise-loading stage was placed immediately before the EDFA pre-amplifier to set the desired OSNR. The electrical RX signal was sampled using a Tektronix TDS6124C real-time oscilloscope, characterized by a 12 GHz analog bandwidth and approximately 5 resolution bits. Data runs consisted of two full cycles of the transmitted 2²⁰-1 PRBS, for a total of 2,097,150 contiguous bits. The corresponding RX signal was sampled and stored to disk for each OSNR value, using the maximum sampling rate allowed by the real-time oscilloscope (40 Gsamples/s, i.e., 4 samples per bit interval).

Since the oscilloscope did not have an external clock input, the PRBS generator and the scope ran on two different and independent 10 GHz clocks. Therefore, a clock skew was always present, which typically did not exceed half a bit over a full 2²⁰-1 PRBS. Though small, it caused a large penalty and had to be corrected for, using off-line clock extraction and re-sampling.

 

Fig. 2. Experimental eye diagram at the output of the transmitter.

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Besides re-sampling, we also wanted to be able to apply off-line PD re-filtering to the recorded signal, to study RX bandwidth optimization. To be able to perform both re-sampling and re-filtering accurately, without causing spurious signal degradation, we used a standard aliasing-free DSP technique (zero-padding in FFT frequency domain) to increase the signal sampling rate from 4 to 8 samples/bit. Then, re-sampling and re-filtering took place. Finally, subsampling followed and either 1, 2 or 4 samples/bit were passed on to the MLSE processor for decoding.

The off-line MLSE processor employed the SQRT branch metric, in the form proposed in [7]. Another form had been proposed in [29], as an approximation derived from [30] and from [28] Eq. (1). The two forms have a different channel estimation parameter since their theoretical derivation is different. The specific features of the SQRT metric used here are discussed in [31]. Recently, the effectiveness of the SQRT metric for ASE-noise dominated systems ha been confirmed in the context of a theoretical study on a wider class of approximated parametric branch metrics [8] based on the non-linear distortion of the photo-detected signal.

The advantage of the SQRT metric is its low computational complexity and ease of trellis instruction, combined with good performance. The alignment between trellis transitions and the received waveforms was optimized as proposed in [28]. The channel estimator related to the SQRT metric is described both in [31] and [8].

The trellis was instructed on the first PRBS cycle of each data run. BER evaluation was then carried out on the second contiguous PRBS cycle. This ensures noise independence between trellis instruction and BER evaluation, preventing a BER-underestimation effect which might otherwise occur. The number of trellis states was varied from 2¹⁰ (1,024) to 2¹⁴ (16,384). The 2²⁰-1 PRBS was therefore sufficiently long to ensure passing through all the states of the trellis at least about 2⁶ times per data run, even when using 2¹⁴ states.

3. Linear experiment results

In the linear experiment, the signal was re-circulated 13 times in the loop, for a total link length of 1,040 km (17,680 ps/nm). The average power launched into the loop was -3 dBm, small enough not to excite any substantial fiber non-linearity. The OSNR (over 0.1 nm) was varied between 13 and 23 dB in 1 dB steps.

As benchmark, the back-to-back performance of the system without MLSE was measured. To assess such benchmark fairly, the relatively narrow TX and RX electrical filters of the MLSE system were replaced by 7.5 GHz 5-pole Bessel filters. BER=10^-3 was achieved by the benchmark system, with hard-decision and optimized threshold, at an OSNR=11.5 dB. This is shown in Figs. 3 and 4 as a star-shaped marker. Not replacing the filters would have caused an excess OSNR penalty of about 1 dB, because of the slight Inter-Symbol-Interference (ISI) caused mainly by the MLSE system 4 GHz electrical TX filter, which is visible in the eye diagram of Fig. 2 as a small eye closure.

 

Fig. 3. BER vs. OSNR (over 0.1 nm) obtained using 2 samples/bit. Star: benchmark back-to-back sensitivity without MLSE.

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Fig. 3 shows the results obtained at 1,040 km with 2 samples/bit, for a number of states variable from 1,024 to 16,384. As general features, there appears to be no BER floor down to BER=3·10^-5. Also, curves tend to be steeper for a larger number of states. Even though we recorded BERs lower than 2·10^-5, we do not show them in the performance plots as their statistical significance is too low (less than 20 errors over the approximately one million bit PRBS).

The OSNR penalty with respect to the back-to-back benchmark system decreases as the number of states is increased, essentially reaching saturation at about 16,384 states. The gain from further doubling the number of states to 32,768 states was less than 0.1 dB (not shown in figure). In Fig. 3 the minimum penalty value at BER=10^-3 is approximately 2.9 dB.

We wanted to verify that this penalty could not be improved upon by increasing the number of samples per bit. A comparison of the results obtained using 2 and 4 samples per bit is shown in Fig. 4 for 2,048 and 4,096 states. At BER=10^-3, no performance gain was obtained when using 4 samples/bit, in general agreement with the available literature. However, at BERs lower than 10^-3, Fig. 4 does show some gain when using 4 samples/bit. This is possibly due to the fact that the system becomes more robust with respect to where in the bit time-slot the samples are taken. We did not perform any optimization of the sampling clock phase because we assumed that, with pulses spreading over 13–15 bit time-slots, the sensitivity to such optimization would be low. However, when using a number of states (such as 2,048 or 4,096) lower than the actual channel memory would require, some sensitivity to the sampling clock phase becomes in theory possible and is the probable cause of what is seen in Fig. 4.

3.1. Comparison with simulation results

We show in this section the performance results obtained through simulation, using the same system set-up as in the experiment. In order to calibrate our simulations to the experimental data, we simply used the nominal parameters of the system components (reported in Section 2), disregarding possible component impairments.

Doing this, we obtained an OSNR sensitivity discrepancy of approximately 1.5 dB in back-back, at BER=10^-3, between experimental and simulation results (10 dB simulations, 11.5 dB experimental). Such discrepancy could be ascribed to TX and RX imperfections, which were hard to identify, measure and correctly reproduce in simulations. For instance, the finite extinction ratio of the MZ modulator, or the fact that the level of the electrical output of the RX was close to the lower limits specified for the BERT, which may then have contributed non-negligible electrical noise.

 

Fig. 4. BER vs. OSNR (over 0.1 nm). Solid curves: 2 samples/bit. Dashed curves: 4 samples/ bit.

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Fig. 5. BER vs. OSNR (over 0.1 nm) at 1,040 km of SSMF. Results obtained through computer simulations.

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Like in the experiment, the PRBS length was (2²⁰-1) bits. The results obtained by simulating and processing 2²⁰ bits are shown in Fig. 5, for an MLSE processor with 2 samples per bit and up to 16384 states. The general behavior of the results in in agreement to the corresponding experimental curves shown in Fig. 4, though the lower-states ones have a somewhat higher penalty. The sensitivity discrepancy at BER=10^-3 between experiment and simulation at 1,040 km is still present and is essentially the same as the one obtained in back-to-back.

In particular, using 16384 states, the penalty between the respective back-to-back and 1,040 km transmission is 2.8–2.9 dB, both for the experiment and for the simulated system.

3.2. Taking one sample per bit

We wanted to experimentally test the capability of MLSE to operate effectively on only 1 sample/ bit. Previous work showed this to be possible, with some excess penalty (among others, [5], [9], [24]). Through simulations, we found out that a PD filter bandwidth narrower than the 6.2 GHz of the hardware filter used in the experiment would be needed in order to minimize such excess penalty.

Therefore, we performed additional numerical PD filtering (five-pole, Bessel) on the sampled experimental data. We varied the additional filter bandwidth bewteen 2.5 and 8 GHz, in steps of 0.5 GHz. The results for 8,192 states are shown in Fig. 6, in terms of OSNR needed to obtain a BER=10^-3, as a function of the additional PD filter bandwidth (solid line). The resulting optimum bandwidth value was about 4 GHz.

We also tried using an additional PD filter when taking 2 samples/bit, even though our computer simulations of the experiment had predicted that no advantage would be obtained in this case. The results in Fig. 6 (dashed line) did in fact confirm that additional PD filtering could bring no improvement with 2 samples/bit.

One tentative explanation for the benefit of using a narrower PD filter with 1 sample/bit, is as follows. The use of 2 (or more) samples/bit enables the MLSE processor to effectively perform a sort of ‘digital’ post-detection (albeit non-linear) filtering, while it computes the branch metrics, by adding together 2 or more contributions for each branch. This ‘filtering’ by the MLSE processor tends to make the analog PD less needed. If the PD filter bandwidth is narrowed down, any possible benefit from noise reduction is overcome by the extra ISI introduced by the filter and performance degrades (Fig. 6, dashed line).

However, when only 1 sample/bit is used, the MLSE processor does not really perform any ‘digital filtering’. In that case a narrow analog post-detection filter makes up for the lack such ‘digital filtering’ by the MLSE processor and a narrower optimum bandwidth for the extra analog PD is found. This topic is worth further investigation but we deem it to be outside the scope of this experimental paper.

The optimum 4 GHz additional PD filter was used with 1 sample/bit to derive the solid curves in Fig. 7, where BER vs. OSNR is shown up to 8,192 states. For comparison, results for 2 samples/bit are shown as well.

Note that, when using 1 sample/bit, we found some non-negligible dependence of performance on the sampling time instant, which could be as high as 0.7–0.8 dB. Therefore, we optimized the sampling instant. Note also that such sensitivity tends to go down and become unimportant as the number of states is increased.

In Fig. 7, the two system configurations show the exact same performance down to BER=10^-4: 4,096 states with 2 samp/bit (henceforth configuration A); 8,192 states with 1 samp/bit (henceforth configuration B). At BER=10^-3, both have a penalty with respect to back-to-back of about 3.8 dB. Interestingly, these two configurations also have a very similar ‘complexity’. The trellis parameter-storage requirement is identical: 8,192 SQRT metric parameters for both. The (B) configuration requires 8,192 branch-metric computations per trellis step, whereas the (A) configuration only requires 4,096. However, in (A) each branch-metric computation requires two sub-computations, which exactly pairs off the effort. To the disadvantage of (A), the two sub-computations must be added up. To the disadvantage of (B), 8,192 comparisons must be performed between accumulated metrics for path grooming, rather than just 4,096 in (A).

The fact that ‘similar complexity’ gives similar performance cannot however be generalized. In Fig. 7, the shape of the markers indicates three pairs of ‘similar-complexity’ systems. In two such pairs, the 1 sample/bit system performs better than the similar-complexity 2 samples/bit system (for 1,024/2,048 states and 2,048/4,096 states). Only the 4,096/8,192 state pair has the same performance. We have verified that going up to the 8,192/16,384 state pair showed a reversal, with the 2 samples/bit system outperforming the 1 sample/bit system, although by just 0.3 dB.

 

Fig. 6. OSNR (over 0.1 nm) needed to achieve a BER=10^-3 vs. PD filter bandwidth. Solid curve: 1 sample/bit, 8,192 states. Dashed curve: 2 samples/bit, 4,096 states.

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Fig. 7. BER vs. OSNR (over 0.1 nm). Solid curves: 1 sample/bit. Dashed curves: 2 samples/ bit. Same-shape markers indicate similar-complexity systems.

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In summary, MLSE retains its long-haul CD compensation effectiveness even when using 1 sample/bit. The incurred penalty is mild at or near the optimum number of states, and may turn into a gain for lower-than-optimum number of states, when comparing ‘similar-complexity’ systems. Our results are in general agreement with those presented in [5],[24], where the performance of MLSE RXs using either 1 or 2 samples per bit was discussed.

3.3. Penalty saturation

As shown in Fig. 3, the minimum recorded penalty in our linear 1,040 km experiment was 2.9 dB, with 16,384 states and 2 samples/bit.

Some of this penalty could be ascribed to the limited real-time oscilloscope A/D resolution. Factory calibration data for the specific experiment unit reported an effective resolution of 4.3, 4.7, 5.1 bits at 10, 5, 1 GHz, respectively. We performed simulations at 1,040 km with a frequency-flat resolution of 4 and 5 bits, finding 1.5 and 0.4 dB penalty, respectively. So we estimate that about 0.5 dB out of the experimental 2.9 dB minimum penalty may be due to the A/D finite resolution, still leaving 2.4 dB unaccounted for.

This remaining penalty is in line with the simulative and analytical findings in [7], [14] and, very recently, in [9]. The latter paper uses the powerful information rate analysis technique which was proposed in [36]. We found excellent agreement between the results of [9] and our own simulations, when we used the same parameters as [9]. Further comprehensive simulative evidence was shown in [31].

These findings predicted that in MLSE systems the OSNR penalty with respect to back-to-back would grow in the first 300–350 km of SSMF fiber (at 10 Gb/s) and then would level-off and remain constant for longer distances. For a sufficiently large number of trellis states and infinite A/D resolution, the constant saturated OSNR penalty would amount to between 2 and 3 dB, depending on system set-up optimization [31]. Our experimental results appear to be compatible with these theoretical predictions. We therefore deem that one of the main goals of our proof-of-concept experimental investigation, which consisted in proving or disproving such prediction, was essentially achieved.

4. Non-linear experiment results

To assess the capability of MLSE to deal with the challenging distortion caused by significant Kerr INL-induced chirp interacting with the overall-link uncompensated dispersion, the average power launched into each loop recirculation was increased up to 12 dBm. The total system length was reduced with respect to the linear experiment case, and the signal was re-circulated 10 times in the loop, for a total distance of 800 km (13,360 ps/nm). The OSNR (over 0.1 nm) at the input of the RX was set to 16 dB, through noise loading.

The reduction in total link length with respect to the linear experimentwas based on computer simulations that forecast that the combined effect of CD and non-linearity would increase the system memory. Since the 16,384 states required for minimum penalty in the linear experiment were close to the most states we could computationally handle, we decided to reduce the link length to cut back on the CD contribution to system memory, in order to make room for some extra-memory generated by the interplay of CD and non-linearity.

The same MLSE metric (the SQRT metric) and the same channel estimator and trellis instruction procedures were used for the non-linear experiment as were used for the linear one.

The number of MLSE trellis states was at first set to 2¹⁰ (1,024), yielding the experimental results shown in Fig. 8 (solid line). We then simulated the same link, obtaining the curve shown in Fig. 9 (solid line). The two curves are superimposed in Fig. 10 (squares: experiment; triangles: simulations). To get BER agreement at low power, the simulation OSNR was set to 14.5 dB (rather than 16 dB). Such OSNR discrepancy was also found in back-to-back operation and therefore could seemingly be ascribed to TX and RX imperfections which we could not entirely reproduce in the simulations.

 

Fig. 8. BER vs. TX power, 800 km uncompensated SSMF transmission experiment, OSNR=16 dB (0.1nm), MLSE RX with variable number of states.

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Fig. 9. BER vs. TX power, 800 km uncompensated SSMF transmission simulation, OSNR=14.5 dB (0.1nm), MLSE RX with variable number of states.

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Both the experiment and the simulations indicate that performance improves in the range 0 to 6 dBm. This could be ascribed to some chirp-compensating effect of SPM vs. anomalous dispersion, which leads to a decrease of system memory in that power range. The BER improvement comes from the fact that 1,024 states are somewhat suboptimum in linearity, where 2,048–4,096 states would be needed to fully saturate performance. The non-linearity-induced memory reduction between 0 and 6 dBm made 1,024 states closer to optimum, reducing the BER. This effect was found also in [22] and, notably, the shape of the plot in Fig.2b of [22] is qualitatively similar to Figs. 8, 9, with power levels shifted due to the much smaller link length (about 120 km for the specific plot in [22]).

When power goes beyond 6 dBm, however, the increase of non-linearity-induced chirp together with the resulting spectral widening, interacting with 13,600 ps/nm of uncompensated CD, leads to a substantial increase of pulse spreading and, as a result, of system memory. To find out whether this was indeed the main cause for BER degradation at 9 dBm and higher power, we increased the number of MLSE states up to 16,384. The BER could be significantly improved both in the experiment (Fig. 8) and the simulations (Fig. 9), though this was more effective in the simulations, where a BER=10^-3 could be reached even at 12 dBm launched power with 16,384 states.

4.1. ODC simulated benchmark

As shown, MLSE can manage large combined levels of CD and INL, allowing high launched powers into the link. However, it is difficult to say how effective MLSE really is, unless a suitable benchmark is provided for comparison. Clearly such benchmark cannot be conventional 10 Gb/s IMDD because it would stop operating altogether, after only two loop recirculations.

To obtain an appropriate benchmark, a 10 Gb/s IMDD Optically Dispersion-Compensated (ODC) system was simulated. The analyzed system was modelled after the experimental setup, with the exception of the TX and RX electrical filters which were replaced by 7.5 GHz five-pole Bessel filters. This was done to eliminate the slight ISI due to the relatively narrow electrical filters of the MLSE system, as already explained in Sect. 3

Also, the OSNR was reduced to align the benchmark performance to that of the MLSE experiment and simulations at low launched power, at a common BER=2·10^-4. The OSNR needed by the simulated benchmark was 3.3 dB lower than that of the simulated MLSE system. This performance gap in linearity is due to the ‘saturated penalty’ of MLSE for large CD, already commented about in the context of the linear experiment (Sect. 3.3). Its somewhat larger value than the expected 2 to 3 dB was due to the use of 1,024 MLSE states which were slightly suboptimum, as previously pointed out.

Three different configurations of ODC were analyzed:

The obtained results are shown in Fig. 10. Despite the fact that the MLSE system operated under conditions that were similar to configuration (i), i.e., all compensation carried out at the RX, it greatly outperformed both (i) and (ii) and achieved a performance midway between (ii) and (iii). In fact, when the number of states is increased (see Figs. 8 and 9), MLSE results improve and come closer to (iii).

In summary, only the simulated benchmark using optimized dispersion maps (iii) proved to be better performing than MLSE, but it required careful, power-dependent optimization of in-line and RX DCUs. In addition, at high power (15 dBm), the optimum ranges for the DCU values were very narrow. The 0.5 dB OSNR-penalty tolerance for the in-line DCU was ±12 ps/nm and for the RX-DCU ±50 ps/nm.

 

Fig. 10. BER vs. TX power, 800km SMF, optical vs. electrical compensation; OSNR (0.1 nm) at RX: 16 dB (MLSE experiment), 14.5 dB (MLSE simulations), 11.2 dB (ODC simulations).

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5. Error events statistics

Recently, a real-time experimental investigation of the interplay between a commercial 4-state MLSE processor and two different commercial enhanced-FEC (e-FEC) codes was carried out [33]. The study has shown FEC performance degradation when used in conjunction with the MLSE processor. The extent of the degradation depended on the specific code used. The authors could not pinpoint the exact cause of the degradation, though they guessed it could be caused by some periodic anomaly in the distribution of errors at the output of the MLSE circuit.

MLSE as a general technique is known to possibly generate error bursts, i.e., to introduce error correlation, depending on channel characteristics. This behavior is typical of the Viterbi decoder that is used by MLSE [34],[35]. Error bursts may be detrimental to FECoperation, even though to different extents, depending on the burst-correction capability of a specific code.

Here we cannot reproduce the same level of detail as [33], as it would require real-time experimental equipment, rather than off-line. On the other hand, our MLSE processor, being ‘software’, is immune from possible specific implementation-related anomalous behaviors.

Since an error-burstiness analysis is within reach of an off-line study, we elected to carry it out. We must point out that finding no bursts does not necessarily mean that FECs would see no degradation. For instance, in [33] no direct evidence of anomalous error bursts was found, and yet FEC performance degradation was observed. On the other hand, if in our set-ups substantial error burstiness was found, this would positively signal a potentially harmful problem.

Therefore, we analyzed the burstiness of the error events generated by the MLSE processor, under the extreme conditions of large CD and substantial intra-channel non-linearity in which our experimental set-ups operated.

A possible objection to our investigation is that operating over channels whose memory exceeds ten bits, with thousands of MLSE processor states needed to cope with it, may be unrealistic from a practical viewpoint and the results would be of little relevance to more practical scenarios.

However, the danger of MLSE generating bursts of correlated errors is more serious when there is a larger ‘memory’ (or ISI) present in the channel, correlatingwaveforms over many bits, rather than over channels with short memory. Therefore, we deem that if it can be shown that no or few bursts appear when a large channel memory is present, then this should reasonably guarantee that a better, or at least not worse, behavior should be found when a lower channel memory is present.

 

Fig. 11. Two-dimensional histogram (in linear and logarithmic scale) of the number of error events, vs. error event length L_e (from first erred bit to last erred bit, inclusive, within a preset window W=12 equal to the trellis memory) and vs. the number N_e of actually erred bits in the event. Linear 1,040 km experiment, OSNR=15 dB, 4,096 states, BER≈1.2·10^-3.

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Throughout this section, we always used 2 samples/bit and operated at BER≈10^-3. In such scenario, as previously discussed in Sect. 3 and shown in Fig. 4, the sensitivity to the sampling clock phase is small. As a result, we did not perform any sampling phase optimization.

In our analysis, we called error event of length L_e a sequence of L_e decoded bits such that:

Therefore, if two consecutive erred bits have a distance greater than W, then they belong to separate error events. Also, we call N_e the total number of erred bits in the error event, including the first and last.

As an example, if W=2, a decoded string of eleven bits such as rrrwwrrwrrr, where r means ‘right’ and w means ‘wrong’, contains one error event of length L_e=5, with N_e=3 errors in it. Instead, the decoded string of twelve bits rrrwwrrrwrrr contains two separate error events, one with L_e=N_e=2 and one with L_e=N_e=1.

In Fig. 11 we show the results of the error event analysis for the linear experiment described in Section 3 with OSNR=15 dB, 2¹²=4,096 states and 2 samples per bit, for which BER≈1.2·10^-3. Qualitatively similar results were obtained using 1 or 4 samples per bit and are not shown. The two-dimensional histograms display the number of occurrences of error events of length L_e with N_e erred bits in all, within a preset window W=12 equal to the trellis memory N=12. We chose W=N since two errors farther than the trellis memory can be essentially considered independent. This assumption was strongly confirmed by the fact that the percentage of error events whose distance ranged between W+1 and 2W was only 0.54%.

By far the most frequent error events (468) consisted of adjacent erred bits (L_e=N_e=2). Also, the error patterns were entirely made up of transposed bits, i.e., strings ‘01’ mistaken for ‘10’ and viceversa. No ‘11’ mistaken for ‘00’ (and viceversa) were found. Single-bit errors were also present (263), while the number of error events of length greater than 2 was much lower.

 

Fig. 12. Two-dimensional histogram (in linear and logarithmic scale) of the number of error events, vs. error event length L_e (from first erred bit to last erred bit, inclusive, within a preset window W=13 equal to the trellis memory) and vs. the number N_e of actually erred bits in the event. Linear 1,040 km experiment, OSNR=15 dB, 8,192 states, BER≈8·10^-4.

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When increasing the number of states to 8,192 with W=N=13 (Fig. 12), the most frequent error events became the single bit errors (430), whereas the adjacent erred bits events decreased to 124. The percentage of error events whose distance ranged between W+ and 2W was just 0.39%, confirming uncorrelation of error events beyond W bits. Fig. 12 clearly shows that increasing the number of states from 4,096 to 8,192 altered the sequences distance, causing a reversal of the ratio of single-bit vs. two-adjacent-bit error events. The total BER went down by about 30%.

As far as the impact of fiber non-linearity on the error events distribution is concerned, we show in Fig. 13 the results related to the non-linear experiment described in Section 4 with P_TX=11 dBm, 8,192 states (W=N=13), 2 samples per bit and OSNR=16 dB, for which BER was about 2.5·10^-3. As in the linear case, by far the most frequent error events were of length 1 or 2, but a larger number of longer error events were present as well, indicating that one of the effects of the interplay of strong nonlinearity with large CD might be that of making multiple-bit error events more likely. Note that no error events occurred outside of the figure bounds (i.e., for L_e>12 or N_e>8). Despite the presence of longer error events, the percentage of events whose distance ranged between W+1 and 2W remained low (0.42%), meaning that the non-linearity did not induce substantial correlation of error events beyond W=13 bits.

To summarize, in our 1,040km ‘linear’ experiment, there were about 14 bits of channel memory (i.e., waveforms were correlated over 14 bits). Yet, the MLSE processor was capable to retrieve the original bit sequence without showing significantly correlated errors, or error bursts.

In the 800km non-linear experiment, where 10 bits of linear memory were compounded by extra-memory (further pulse spreading) caused by the interplay between violent chirp due to high Kerr non-linearity and the totally uncompensated fiber dispersion, more bursts than in the linear case were found. They did not exceed 8 erred bits over events 12-bit long.

The linear experiment result cannot per se guarantee that the performance of possible FECs would not be altered (see [33]) but they help to rule out a potentially harmful behavior, i.e., substantial burstiness, known to be possible with MLSE over certain pathological channels. A highly non-linear channel may however require some attention in this respect, as we found a greater number of error bursts, with longer length, in the non-linear experiment at 11 dBm.

FECs can be specifically designed to be able to handle long error bursts, such as Reed-Solomon codes, or concatenated codes with a proper error de-correlating interleaver [34]. A discussion of such codes is outside of the scope of this paper.

 

Fig. 13. Two-dimensional histogram (in linear and logarithmic scale) of the number of error events, vs. error event length L_e (from first erred bit to last erred bit, inclusive, within a preset window W=13 equal to the trellis memory) and vs. the number N_e of actually erred bits in the event. Non-linear 800 kmexperiment, P_TX=11 dBm, OSNR=16 dB 8,192 states, BER≈2.5·10^-3.

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6. Comments and conclusion

The experiments described in this paper were aimed at testing MLSE with NRZ-IMDD in extreme scenarios and attempting to prove or disprove certain simulative or theoretical predictions. They could be viewed as ‘proof-of-concept’, in the sense of ‘proof-of-conjecture’.

The first, non-trivial, prediction that we wanted to test was that MLSE could compensate for arbitrarily large values of CD, provided that the CD-induced channel memory be matched by the processor memory, while incurring a finite and asymptotically saturated penalty ranging between 2 and 3 dB. Our 1,040 km SSMF uncompensated experiment (13–15 bits of memory) that reached a minimum penalty of 2.9 dB, part of which certainly due to system imperfections, gave reasonable confirmation of such prediction.

Another predicted feature of MLSE was its capability to withstand and mitigate substantial amounts of intra-channel non-linearity, even when compounded with large amounts of CD. Our 800 km SSMF uncompensated experiment resulted in a remarkable 9 dBm non-linear threshold and showed that an increase in the number of MLSE trellis states could further push it up. Our MLSE experiment could not achieve the same non-linear threshold performance as our fully-optimized ‘benchmark’ optical dispersion management system (about 15 dBm) but it largely outperformed simpler optical compensation schemes. In addition, the benchmark was found by simulation, so it did not suffer from the inevitable imperfections of experimental set-ups.

Besides these main aspects, we also showed that, in our 1,040 km experiment set-up, 1 sample/ bit MLSE could perform nearly as well as 2 samples/bit MLSE, requiring only RX post-detection electrical filter bandwidth optimization.

Finally, we investigated the statistical distribution of errors at the output of the MLSE RX in the decoded experimental sequences. The linear experiment showed essentially no error bursts. The non-linear experiment did present error bursts, though of limited length, which may require suitable FECs to be handled properly.

Being designed to probe extreme scenarios, our experiments did not necessarily aim at mimicking or emulating any practically implementable system. Otherwise, the objection that the number of trellis states needed to successfully off-line process the data was far beyond today’s technological reach, would clearly be justified. However, the evidence from these proof-of-concept experiments confirmed theoretical predictions that imply that any progress in the number of states of practical MLSE processors would certainly pay off in terms of increased CD and intra-channel non-linearity compensation capabilities.

Research is also ongoing on potential techniques for drastically reducing the implementation complexity of large MLSE processors (such as [32]) which might possibly lead in the future to the availability of much more powerful processors than available today.

A possible context for further investigation is that of synergistically using MLSE and dispersion management. Preliminary results for NRZ/IMDD indicate that realistic-complexity MLSE RXs can greatly enhance the robustness and compensation margins of ODC systems, even in the presence of non-linearity [37]. This might lead to better cost-efficiency than either of the two approaches alone.