Performance of a single-channel fiber-optic transmission system in which signal regenerators are periodically inserted is analyzed in terms of information rate (IR) considering channel memory. Limitations in using regenerators in a system having non-zero residual dispersion between the regenerators are discussed. It is shown that a type of signal impairment caused by the interaction between the transmission-fiber dispersion and the regenerator nonlinearity is pattern-dependent and will be mitigated by the use of sequence estimation after detection at the receiver.
© 2013 Optical Society of America
Optical signals propagating in transmission systems are impaired by various reasons. Chromatic and polarization-mode dispersion (CD and PMD) are linear and reversible effects, so that they can in principle be compensated for at the receiver. Even if the amount of dispersion, in the case of CD, or differential group delay and principal states of polarization, in the case of PMD, vary randomly in time, adaptive compensation of them is possible when the speed of their variation is much slower than the signal speed. Nonlinear effects related to self-phase modulation are also deterministic and can be compensated by the combined compensation of dispersion and nonlinearity such as mid-span optical phase conjugation and digital back propagation. Recent progress in digital signal processing (DSP) has made the electrical-domain compensation practical in conjunction with digital coherent technologies .
Signal impairments caused by amplifier noise are, on the other hand, fast, unpredictable, and irreversible, so that they cannot be compensated for. Nonlinear interaction between signals in different channels induced by cross-phase modulation and four-wave mixing also causes signal distortions that are noise-like when the data transmitted in the interfering channels are unknown. It is noted that the noise and signals are often interacted through the transmission fiber nonlinearities, which may enhance the impact of noise on the signal [2,3].
The noise-induced signal deterioration can be suppressed by signal regeneration applied before the noise accumulation becomes significant. Signal regeneration has been traditionally performed by an electrical regenerator, or a receiver-transmitter pair, in which signals are detected, and reshaped and retimed by electrical means. Anticipating great potential in high-speed operation with simple low-cost implementation, a number of studies have been devoted to realization of all-optical signal regenerators that avoid optical/electrical/optical signal conversion and power-hungry electrical signal processing [4,5]. Recently, all-optical regeneration of on-off keying (OOK) signals as fast as 640Gbit/s has been demonstrated using nonlinearities in a periodically poled lithium niobate waveguide . Regeneration of quadrature phase-shift keying (QPSK) signals at 40Gbit/s has also been reported that uses a fiber-based phase-sensitive amplifier (PSA) .
One fundamental issue in using all-optical regenerators in real transmission system environments is their adaptability to multiplexed and dispersively broadened signals. Almost all high-capacity transmission systems use wavelength multiplexing and, in some cases, polarization multiplexing. Because the regeneration operation is highly nonlinear, the regenerator requires that the signal pulses to be regenerated are well isolated in time from other pulses when they are nonlinearly processed inside the regenerator. Dispersion compensation and channel demultiplexing are therefore required before the noise removal is performed. If these requirements are not fulfilled, nonlinear crosstalk among neighboring pulses and wavelength and polarization channels appear. Although several studies have been reported for realization of asynchronous multi-wavelength channel all-optical signal regeneration, satisfactory suppression of interchannel nonlinear interaction with sufficiently high spectral efficiencies is yet to be achieved [8,9]. It is noted that 12-wavelength-channel 10Gb/s OOK signal regeneration has been recently reported with no observable degradation from the interchannel nonlinear effects but with a limited spectral efficiency of 5 percent . In current status of technologies, multi-channel signal regeneration will resort to channel demultiplexing and per-channel regeneration followed by multiplexing again. Photonic integration technologies  will be highly important for the multi-channel regenerator.
In this paper we address the issue of signal regenerator performance in the presence of residual dispersion before regenerator in single-channel operation. To know the achievable performance of the signal regenerators with simultaneous use of the electrical-domain processing at the receiver such as maximum likelihood sequence estimation (MLSE) , an analysis in terms of information rate (IR) considering channel memory [13–16] is performed for an OOK signal transmission where 2R (reamplification and reshaping) regenerators are periodically inserted. Here we note that “channel memory” refers to the property of the transmission channel that the received signal at a time is not determined solely by a single transmitted symbol but also is influenced by the symbols transmitted before and after it. We also show by numerical simulations that the MLSE can partially mitigate the regenerator-induced signal distortions appearing when imperfectly dispersion-compensated signals are launched into the regenerator. Although the analysis in this paper is restricted to OOK signal transmission and regeneration, it can be extended to other, multi-level, modulation formats such as QPSK . In the following, after describing the configuration of the simulated system in Section 2, we discuss the analysis of IR and present numerical examples in Sections 3 and 4, respectively. Numerical results of application of MLSE to the system are shown in Section 5. Finally, we conclude the paper in Section 6.
2. Transmission system model
Figure 1 shows the transmission system analyzed in this paper. The transmitter (TX) generates a non-return-to-zero (NRZ) OOK signal that is launched to the fiber without dispersion precompensation. Each span consists of a transmission fiber, a dispersion compensating fiber (DCF), an amplifier, and an optical bandpass filter (OBPF). The transmission fiber has a length of 100km, dispersion of 2ps/nm/km, loss and nonlinear coefficients of 0.2dB/km and 2/W/km, respectively. Nonlinearity in the DCF is neglected. The amplifier noise figure is 5dB. The OBPF defines the bandwidth of the amplified spontaneous emission to 1nm. Fiber launch power at each span is denoted as Psig in subsequent simulations in this paper, where Psig is the time-averaged channel power. 2R regenerators are periodically inserted in the system with an interval of Ns spans. The power transfer function of the regenerator is assumed to beFig. 1, where P0 is the signal power in mark at the transmitter, or P0 = 2Psig. We assume that the regenerator responds to the signal instantaneously and is transparent to the signal phase. Although detailed operation mechanism of the regenerator and its dynamic behavior should be specified for a more practical analysis, we employ the simplified model for rather qualitative analysis of the system including 2R regenerators. The data rate of the signal in this analysis is 40 Gbit/s. Total number of the span is 24, or the total transmission distance is 2,400km. When 24/Ns is an integer, the final regenerator located just before the receiver is omitted. At the receiver, the detected current, proportional to the signal power, is low-pass filtered with a cutoff frequency of 30 GHz. After that, the signal is sampled at a rate of one sample per symbol.
3. Method of information-rate analysis
Mutual information between the input and output of the system is expressed as13–16]. When the constellation of the input signal and occurrence probabilities of each signal point are optimized so that I becomes maximum, I gives the channel capacity. In the present study, the modulation format is fixed at OOK and the occurrence probabilities of Xi = 0 and 1 (i = 1,2,…,n) are assumed to be 0.5. Then the mutual information I is interpreted as an information rate. At the receiver, direct detection is assumed so that Yi(i = 1,2,…,n) takes a real continuous value.
When the channel is memoryless, that is, when each symbol is transmitted independently with each other, IR is evaluated in terms of per-symbol probability density functions as13–16].
In this approach, H(Y) is expressed by an expectation value of the logarithm of p(Y) as
Based on the Shannon-McMillan-Breiman theorem , the entropy rate (5) can be evaluated by a sample entropy rateFig. 2. αi(s) in (9) is the probability that the state is in s at a time instance i conditioned on having the sequence of the received signals yi-1 . and are related with each other by
In the numerical evaluation of IR, a long pseudo-random symbol sequence is firstly numerically transmitted over the system by the use of split-step Fourier method and large number of received samples are collected with which the conditional probability densities ’s are evaluated by the method of kernel estimation . Then ’s and ’s are recursively calculated according to (9) and (10). Finally, is given by (7) and (8). , on the other hand, is calculated by
4. Results of information-rate analysis
4.1 Cases with 100% per-span dispersion compensation
Here we analyze single-channel NRZ-OOK transmission at a speed of 40 Gbit/s in the system with 100-percent per-span dispersion compensation. For the transmitted data pattern, a De Bruijn bit sequence of length 4,098 is used. To collect samples sufficient for the calculation of the conditional probability densities ’s, the pulse train is numerically transmitted 256 times over the system with different noise realizations. These numbers are used after confirming convergence of the numerical results in increasing these numbers. The sampling phase at the receiver is optimized, that is, the sampling instance location is scanned over the symbol duration with an increment of 1.6 picoseconds while the sampling interval is fixed.
Figure 3 shows the IR versus the average signal power Psig launched to the transmission fiber when the fiber dispersion is completely compensated at each span. Two cases are studied: (1) no regenerators are used (blue dashed curves) and (2) the 2R regenerators are inserted every 4 spans (red solid curves). Channel memory considered in the calculation is either m = 0 (curves marked with circles) or m = 3 (curves marked with squares). The maximum achievable IR is 1 bit/symbol because binary modulation format is used. It is shown that the range of signal power Psig in which high IR~1 bit/symbol is obtained is extended by the use of regenerators when m = 0. The range extension is about 2dB both in low and high-power sides for the regenerator insertion period Ns of 4. The extension is because the regenerators suppress noise accumulation in the low-power region while they mitigate nonlinearity-induced signal distortions in the high-power region.
Eye diagrams of the received signals are shown in Fig. 4. Figures 4(a) and 4(b) are those at a low signal power of Psig = −7dBm with and without the use of the regenerators, respectively. Eye openings are shown in Fig. 4(a) but not in Fig. 4(b), showing the noise suppression by the regenerators. Figures 4(c) and 4(d) are the eye diagrams at a high signal power of Psig = 5dBm with and without the regenerators, respectively. As seen in these diagrams, the nonlinearity-induced signal distortions in the high-power regime are pattern-dependent, so that they can be mitigated by applying a sequence estimation algorithm at the receiver after detection. The potential effectiveness of the sequence estimation is indicated in Fig. 3 by the enhancement of IR in the range 2Psig10dBm as m is increased from 0 to 3, or the channel memory considered in the IR evaluation is increased. At much higher signal powers, signal spectrum is broadened beyond the bandwidth of the OBPF, which leads to the loss of the signal power. The IR is therefore decreased rapidly for Psig larger than about 10dBm.
Figure 5 shows upper and lower limits of the signal power range in which IR is high (IR>0.9) for various insertion intervals of the regenerator. 1/Ns = 0 corresponds to the case where no regenerators are used. The lower limit of usable Psig is decreased as Ns is decreased (1/Ns is increased). By inserting regenerators every span (Ns = 1), although it may be impractical in real systems considering complexities and costs, signal power smaller by about 6dB is allowed as compared to the case of no regenerators inserted. Frequent insertion of the regenerators into the system (large 1/Ns approaching unity) gives a higher upper limit of usable Psig by about 3dB when the detection neglecting channel memory (m = 0) is used. Frequent insertion of the regenerators, however, reduces the benefit of the detection considering channel memory, or reduces IR evaluated with m≥1. This is because the amplitude equalization by the regenerator rather erases the history how the signal pulses suffer from the pattern-dependent nonlinear interaction in the transmission fiber, which makes sequence detection at the receiver inefficient. For Ns larger than about 3 (1/Ns0.3), we can still improve the IR by increasing the length of memory in the detection algorithm. As for the lower limit of usable Psig, it cannot be further lowered by increasing m because the performance is not limited by the intersymbol interference (ISI) but by noise accumulation independent of signal data pattern.
4.2 Cases with residual dispersion
Now we discuss the IR when there is residual dispersion between the regenerators, i.e., the DCF in Fig. 1 imperfectly compensates the dispersion of the preceding transmission fiber in each span. The red curves in Fig. 6 show the IR versus the residual dispersion between the regenerators for the insertion period of regenerators of Ns = 4 spans. At the receiver, lumped dispersion compensation is given before detection that maximizes the IR. It is emphasized that no dispersion compensation for performance optimization is given in front of each regenerator. The memory depth m is varied between 0 and 3. The signal power launched into the fiber in each span is Psig = −8dBm, at which the system performance is almost limited by noise. The blue curves represent the IR when the regenerators are removed from the system. Figure 6 displays known facts about signal regeneration: (1) the regenerator can enhance signal quality by suppressing noise accumulation, which increases the IR [20, 21] when the dispersion does not disturb its effectiveness, and (2) the regenerated system behaves worse than that without regenerators when the signal launched into the regenerators is distorted by residual dispersion. It is noted that the IR deteriorated by the dispersion in the regenerated system is not appreciably recovered by the increase of m in evaluating the IR. This is because the performance degradation is mainly caused by the distortion of the individual signal shape, not by the pattern-dependent inter-symbol interaction.
The regenerator, or the nonlinear filter, has basins of attraction in the signal space each belonging to one of regenerated symbols. When the launched pulse to the regenerator is distorted and located close to the boundary of the basin, efficiency of regeneration is weakened and only small noise pushes the pulse away from the basin. Then the transmission performance of the regenerated system is severely degraded.
The IR gives maximum bits per symbol that are transmitted with arbitrarily small nonzero error probability. Figure 6 shows that although the IR with using regenerators is larger than that without regenerators and approaches to unity at small residual dispersion, the IR without regenerators itself is moderately high at about 0.92 bits/symbol. The difference between the IRs of ~1.0 and ~0.92 with and without using regenerators, respectively, seems small, which may indicate that the benefit of the regenerators is limited. To object to this notion, it is emphasized that the error-free (arbitrarily small nonzero error probability) data rate predicted by the IR is achieved only after optimum coding. To achieve ~0.92 bits/symbol from noisy signal such as that shown in Fig. 4(b) without using regenerators, very long codes and complicated coding/decoding procedures would be needed. According to numerical simulation, bit error rate (BER) without coding is about 0.02 for the case of IR ~0.92 while BER much less than 10−4 is obtained for the case of IR approaching to unity. The numerical evaluation of BER is discussed in Section 5.
Figure 7 shows the IR versus the residual dispersion between the regenerators at a larger signal power of Psig = 0dBm. Regenerators are inserted every Ns = 4 spans in Fig. 7(a) while they are removed in Fig. 7(b). Lumped dispersion compensation is again given before detection at the receiver that maximized the IR in each case. At this signal power, optical signal to noise ratio is sufficiently high so that the dispersion tolerance shown in Fig. 7(a) is larger than that in Fig. 6. High signal quality of IR~1 is obtained even without regenerators when the residual dispersion is small. When the absolute value of the residual dispersion is larger than about 40ps/nm, nonlinear interaction between neighboring pulses broadened by dispersion appears inside the transmission fiber and IR is degraded with small m. The degradation is more severe for the system using regenerators because of the same reason discussed in the previous paragraph. The impairments caused by the interaction between the fiber dispersion and nonlinearity under the large residual dispersion are pattern-dependent, so that the IR is recovered as the considered memory depth m is increased both for the cases with and without regenerators.
Figure 8(a) shows the IR behavior of the system including the regenerators every 4 spans (Ns = 4). Here the nonlinearity in the transmission fiber is turned off, which will reveal possible signal deterioration caused by the interaction between the residual dispersion before the regenerator and the regenerator nonlinearity. For |Dres| larger than about 60ps/nm, the IR becomes less than 1 bit/symbol. This is because the nonlinear filtering by the regenerator of the pulses broadened by dispersion leads to timing jitter of the pulse edges. The amount of pulse broadening, and so the timing jitter, depend on the data pattern for NRZ signals as shown in Fig. 8(b). The degraded IR, therefore, is restored by considering larger memory depth m in its evaluation.
5. Application of MLSE
In a data communication channel that suffers from ISI, a received signal at a time is not only determined by the symbol transmitted at the time slot but also influenced by the symbols transmitted in the neighboring time slots. If decision of the data is made without using the knowledge of the data transmitted in the neighboring time slots, the ISI may cause large error. Sequence detection that detects a series of symbols as a whole the length of which is determined by the extent of the ISI can avoid such pattern-dependent decision error. See  for illustrations of principles and examples of the sequence detection. This section analyzes the BER performance of the system when the sequence detection, i.e., MLSE, is applied. The BER analysis complements the IR analysis presented in the previous sections. While the IR predicts the maximum possible data rates after implementing optimum coding, the BER gives a more practical measure of the performance obtained before using coding.
In the IR computation presented in Section 4, probability density functions (pdf’s) of the received sample when symbol trains of length 2m + 1 are transmitted are numerically evaluated. MLSE can be performed at the receiver by using these pdf’s. Here we examine the reduction of decision error by the MLSE in the system using regenerators. In the MLSE, for the received data sequence y1, y2, …, yn, is calculated for all the possible state transition sequences along the trellis that is a series concatenation of the state transitions as shown in Fig. 2 (in the case of m = 1): when the 8 states (000), (001), (010), (011), (100), (101), (110), and (111) are named as 1, 2, 3, 4, 5, 6, 7, and 8, respectively, an input symbol sequence (0)01101000101110..., for example, is represented by a state transition sequence 2-4-7-6-3-5-1-2-3-6-4-8-7-.... .The input symbol sequence yielding the state transition sequence that gives maximum λ is chosen as the maximum likelihood transmitted symbol sequence. The Viterbi algorithm is used for the search of the sequence .
First, the MLSE is applied to the system with 100-percent per-span dispersion compensation. Bit error rates (BERs) are evaluated by counting errors in the estimated sequence, where a 16,384-bit random symbol train is transmitted. Figure 9 shows BER versus average signal power launched into each fiber span. Blue dashed curves show BERs when no regenerators are used while red solid curves show BERs when the 2R regenerators are inserted every 4 spans (Ns = 4). In both cases, BERs in the symbol-by-symbol decision (m = 0) are marked with circles and BERs in the MLSE with 128 states (m = 3) are marked with squares. The BER performance shown in Fig. 9 correlates with the result of the IR analysis of Fig. 3. It is found that low BERs are obtained when IR is close to 1 bit/symbol. It is shown in Fig. 8 that the MLSE is effective in suppressing BERs caused by the ISI induced by the transmission-fiber nonlinearity in the signal power range of 2Psig10 dBm in both cases with and without using regenerators.
Next, simulations are performed of the system having residual dispersion between the regenerators. The insertion period of the regenerators is Ns = 4 spans and the signal power is fixed at Psig = 0dBm. At this signal power, interaction between dispersive pulse broadening and transmission-fiber nonlinearity causes pattern-dependent signal distortion, which degrades transmission performance when the residual dispersion is large. Figure 10(a) shows the BER versus the residual dispersion between the regenerators. The number of transmitted symbols on which the BER calculation is based on is again 16,384. The curve denoted by m = 0 is the BER observed in the case of symbol-by-symbol decision while those denoted by m = 1,2, and 3 are BERs when the MLSE with 8, 32, and 128 states, respectively, are used. The MLSE simulation corresponds to the IR analysis of Fig. 7(a). It is seen that decision error caused by the interaction between residual dispersion and nonlinearity in transmission fibers in the regenerated system can be decreased by the use of MLSE.
Figure 10(b) shows the BER versus the residual dispersion when the transmission fiber nonlinearity is turned off. In this case the impairment is caused solely by the interaction between the residual dispersion and the nonlinearity of the regenerators as discussed with Fig. 8(a). The signal distortion again is pattern-dependent and smaller BERs are obtained by the use of MLSE.
Information-rate (IR) analysis is performed of a single-channel optical fiber transmission system including 2R signal regenerators. When the dispersion of the transmission fiber is perfectly compensated at each span, which is the ideal condition for the regenerator operation, the IR evaluated without considering channel memory is increased by the use of regenerators. This is because the regenerators suppress the accumulation of noise in the low-power regime and they reshape the signal against distortions induced by transmission-fiber nonlinearity in the high-power regime. The signal distortion caused by the transmission-fiber nonlinearity is pattern-dependent so that it is mitigated by the use of MLSE after detection at the receiver in both systems with and without using regenerators. Frequent insertion of the regenerators in the system, however, reduces the benefit of sequence estimation at the receiver, or reduces the IR evaluated with considering channel memory.
When the dispersion compensation is imperfect, broadened pulses are launched into the regenerator. When the distortion of each pulse is large, the effectiveness of the regenerator is weakened and lost. This deterioration, originating from single-pulse distortion, cannot be recovered by the sequence detection considering channel memory. Some type of regenerator-induced impairment, timing jitter caused by the interaction of dispersive pulse broadening and the regenerator nonlinearity, is pattern-dependent and can be mitigated by the MLSE.
One interesting issue that deserves further research is mitigation of interchannel crosstalk by joint processing of many channels using algorithms such as MLSE when multiple channels in a wavelength-division multiplexing system are simultaneously launched into a single regenerator. Although the ability of noise suppression by the regenerator is likely to be deteriorated in such a situation, some of severe interchannel crosstalk may be alleviated.
This work is supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B) under Grant 23360171.
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