Open Access
Add to BibSonomyAbstract
Experiments and numerical simulation demonstrate the validity of wavelength-interleaving (WI) transmission for reducing the penalty induced by polarization dependent loss (PDL) through the method of extreme value statistics. It is confirmed that applying the WI technique across n (n>1) channels can effectively reduce PDL-induced Q-penalty or outage probability.
©2012 Optical Society of America
1. Introduction
The use of polarization-division-multiplexing (PDM) transmission is one effective way of meeting the growing demand for optical link systems with increased spectral efficiency. However, PDM optical signals are more susceptible than single-polarized signals to degradation due to polarization mode dispersion (PMD) and polarization dependent loss (PDL) [1].
Digital equalization after coherent detection can effectively enhance tolerance to the impairments caused by PMD [2]. On the other hand, PDL causes temporal power fluctuation in each optical PDM-channel, which cannot be equalized even with the digital signal processing technique. As a result, PDL degrades system performance significantly [3].
The n-wavelength-interleaving (n-WI) technique was originally proposed in [4] as a way to reduce the net burst error length in optical links with PMD. Assuming the worst case of optical signals splitting into two principal states of polarization, they evaluated the effect of the WI technique theoretically. They found that WI can enhance forward error correction(FEC) performance by interspersing the bit errors in different wavelength channels. While their main conclusion is that this technique can on average improve the performance of a system with FEC, we emphasize here the ability of WI transmission to reduce the penalty or outage probability caused by PDL even under conditions without FEC.
In this paper, we describe the first-ever experimental demonstration of using WI to improve the robustness to PDL-induced impairments via a statistical approach. In addition, through numerical simulation we reveal the mechanism by which WI reduces the PDL-induced penalty or the outage probability when the number of interleaving-channels n is increased.
2. n-wavelength-interleaving transmission
Figure 1 illustrates an n-WI transmission system (for the case of n = 2) to explain the principle of n-WI technique: in the transmitter, given blocks (e.g. bits, bytes, or frames) are interleaved between n channels. Hereafter, we use the subscript i to denote parameters related to the i-th “original” channels and the subscript int to denote those related to interleaved channels. Note that in this context the “original” channels mean those to which WI has not yet been applied. Owing to the existence of differential group delay (DGD), the PDL values of n channels are statistically uncorrelated if the channels’ central frequencies are sufficiently separated [5]. Accordingly, signal PDL-induced degradations in the original channels are statistically independent. Since interleaved blocks are de-interleaved after demodulation at the receiver, the data of one channel is transmitted over n wavelength. Thus, the bit error rate (BER) of an interleaved channel is equivalently given by BERint = ΣBERi / n. Such averaging is expected to reduce the variance of the probability density functions (PDFs) of the BERs (or Q-factors), which reduces outage probability. This “thinning effect” is confirmed later (Fig. 3(c)).
Fig. 1 Schematic illustration of 2-WI transmission system. The numbers in blocks indicate the channel number and shadings mean the degradation difference of optical signals between 2 channels.
3. Experimental setup and results
To test the validity of WI transmission, we performed an experiment using DGD- and PDL-emulators as shown in Fig. 2 . Continuous-wave light of λ1 (1550.12 nm) and λ2 (1547.72 nm) is modulated by a 211-1 PRBS pattern with a 32 GHz clock to create 128 Gbps PDM-QPSK optical signals. Note that the two channels are separated by 300 GHz and modulated by the same optical modulator. Optical signals enter the DGD-PDL-emulated link nine sections, each of which has a 1-km single mode fiber (SMF), a PDL-emulator, and a polarization-maintaining fiber (which works as a DGD-emulator). Each DGD- and PDL-emulator has DGD of ~10 ps and PDL of ~1 dB, respectively. Under the assumption of the large number of emulators, the estimated link PMD was 27.6 ps and the link PDL was 2.8 dB. To increase the exploration speed of DGD and PDL, 1-km SMFs are set in an isothermal chamber whose temperature randomly fluctuates between 10 and 50 ◦C. Before entering the optical frontend, amplified spontaneous emission (ASE) noise is added to optical signals to decrease the OSNR to 17 dB. Optical signals are detected by balanced-PD at the same time for both channels. After analog-to-digital conversion, the signals are demodulated by an offline program to calculate the BER and Q-factor of channels 1 and 2. Here we assumed PMD-induced penalty was almost completely equalized by four butterfly 11-tap T/2-spaced adaptive finite impulse response (FIR) filter based on the constant modulus algorithm [2].
We recorded the Q-factor sequence of both channels every minute for 5.7 hours (accordingly, the total number of data sets was 340). Figure 3 shows the statistical properties of the recorded Q-factors. As mentioned above, the Qint (Q-factor of the interleaved channels) is equivalently obtained by averaging BER1 and BER2. Figure 3(a) shows the temporal change in Q-factor for channels 1 and 2. Fluctuations in the Q-factor can be caused by changes of the PDL itself and the relative angle between the polarization states of optical signals and PDL-axes [2]. The scatter plots of Q1 and Q2 for the two channels are shown in Fig. 3(b). They indicate that Q1 and Q2 are statistically uncorrelated: the calculated correlated coefficient was 0.102. Figure 3(c) represents the PDFs of Q1, Q2, and Qint. It is noteworthy that the calculated variance of Qint (0.052) is significantly smaller than that of Q1 (0.092) and Q2 (0.090).
Fig. 3 (a) Temporal change in observed Q-factor of ch1 and ch2. (b) Scatter plots of Q-factors for the two channels. (c) Probability density of ch1, ch2, and interleaved channel as a function of Q-factor.
4. Analysis by extreme value statistics
We apply extreme value statistics (EVS) to assess outage probability asymptotically, using the data sets obtained in the above experiment. EVS is usually used to model extremely rare events, such as cataclysmic earthquakes or torrential rain floods. It is demonstrated both experimentally and theoretically in [6] that the probability of the logarithm of BER in an optical link with PMD follows a Gumbel distribution based on EVS. Thus, in our analysis too, the application of EVS may reveal the validity of WI transmission more quantitatively.
We here use the peaks over threshold (POT) analysis found in EVS literature [7]. Note that POT analysis is designed for estimating maxima given a large number of data points; our analysis should deal with -Q values (i.e., Q-factors multiplied by −1) as stochastic variables in order to estimate minimum Q-factors.
Let the obtained Q-factor sequence be one of identically and independently random variables. If we set X = -Q-u for a given threshold u, EVS gives the conditional probability H(x) = Pr{x>X | X>0}. H(x) is the probability that X does not exceed x under the condition of X>0, i.e. –Q>u, which is referred to as the generalized Pareto distribution. Cumulative probability H(x) has the explicit form of
where σ and ξ are the scale parameter and the shape parameter, respectively.Parameters σ and ξ are estimated on the basis of the maxima likelihood method. Since there may still be uncertainty in setting appropriate threshold level u, we determine it by using the mean excess function, which is a familiar EVS technique (see [7] for more information). Fitted H(x) and the cumulative probabilities calculated from the experiment data are plotted in Fig. 4 ; they show excellent agreement. The estimated parameters are summarized in Table 1 .
Fig. 4 Cumulative probabilities from experimental data (symbols) and fitted H(x) obtained from Eq. (1) with estimated parameters (solid curves) for ch1 (a), ch2 (b), and interleaved channel (c).
Table 1. Estimated parameters for ch1, ch2, and interleaved channel.
Given the experimental system described above, we can now estimate the minimum Q-factor within m-observations, which is expressed as Qm below. From Eq. (1), Qm is approximated with estimated parameters after simple algebraic transformation as
where N and k are respectively the total number of experimental data elements and the number of data elements exceeding threshold u. On the basis of Eq. (2) and the estimated parameters (u, σ, and ξ), we can ascertain the expected minimum Q-factor Qm for a given m: for example, if we were to observe Q-factor m = 108 times for ch1 in this system, we can expect that we would, on average, witness the minimum Q-factor Qm of ~8.0 dB once. It can be also interpreted that this example corresponds to a link system in which the designed Q-limit is 8.0 dB and the outage probability is 10−8, because outage probability is given by 1/m.From this viewpoint of system design, we can draw the Q-limit curve as a function of outage probability (as in Fig. 5 ). This reveals the reduction effect of WI transmission for PDL-induced Q-penalty or outage probability, as in the following two scenarios: (1) For a fixed outage probability of 10−6, the Q-limit can be mitigated from ~8.1 to ~8.6 dB, which can also be explained as the Q-penalty being improved by 0.5 dB. (2) For a fixed Q-limit of 8.5 dB, outage probability can be decreased from 10−3 to 10−6. It should be noted that the Q-limit generally depends on which FEC code is applied and what corrected BER is required for the system, and that outage probability generally depends on the required system reliability.
Fig. 5 Estimated Q-limit from Eq. (2) as a function of outage probability for ch1, ch2 and interleaved channel. Note that error bars at outage probability of 10−2, 10−4, and 10−6 are estimated based on the 90% confidence intervals of σ and ξ.
In principle, there may be also another WIT’s effects: if Q-penalty and the outage probability are fixed, more total PDL in the optical link system can be tolerated. We cannot estimate here how much it may become, because the estimation will require another kind of analysis different from the one performed above.
Even though the above analysis considers only the case of two interleaving channels, it is reasonable to expect that WI transmission across interleaving channels n (n>2) will decrease PDL-induced impairments more effectively. In the next section, we perform a numerical simulation in which n is increased to 3 or more for the purpose of revealing the mechanism by which WI mitigates the PDL-induced penalty or outage probability.
5. Numerical simulation
In the following simulation, we increase the number of interleaving channels n from 2 to 8. As in the experiment (Section 3), the transmission line has nine sections, each of which has a 10ps-DGD- and a 1dB-PDL-emulator. However, instead of installing 1-km SMFs in an isothermal chamber, which changes the angles of the principal axes for both the DGD- and PDL-emulators between sections, we install DGD- and PDL-emulators with random angles in each section. We add ASE noises to the optical signals so that the Q-factor of the transmitted optical signal becomes 10.8 dB when the total PDL of the optical link is 0 dB, which was the case in the experiment. Note that BERs (i.e., Q-factors) of transmitted channels are statistically independent in the simulation, and that, as in the case of the experiment, we neglect the impact of PMD-induced penalty using four butterfly adaptive FIR equalizing filter. In each simulated case of different n, the number of data sets for Q-factor sequence is 500.
Figure 6(a) depicts the PDFs of Q-factors for a single channel, a 4-WI channel, and a 8-WI channel. Compared to Fig. 3(c), it is obvious that the variance of the PDF for an n-WI channel is likely to become smaller as the number of interleaved channels n increases. Therefore, it is expected that the effect of WI transmission on reducing PDL-induced impairments may be enhanced when n becomes larger.
Fig. 6 (a) Simulated probability densities of Q-factors for the channels without WI transmission (blue), with 4-WI transmission (magenta), and with 8-WI transmission (yellow). (b) The Q-penalty improvement in dB as a function of the number of interleaving wavelengths n for a fixed outage probability of 10−6.
As in Section 4, here too we apply EVS analysis to the simulation results. The parameters for the generalized Pareto distribution are estimated in the same way as in Section 4. Figure 6(b) summarizes the improvement achieved in Q-penalty reduction for a fixed outage probability of 10−6 when n is increased from 2 to 8. It can be seen that the reduction effect enhances monotonically; in particular, the Q-penalty for the 8-WI channel improves by about 1.8 dB.
Furthermore, it is noteworthy that the reduction effect has a tendency to saturate as n increases. We give an explanation for this saturation phenomenon here. As n becomes closer to infinity, the PDF of the Q-factor for an n-WI channel approaches a delta-function-like shape, which has its mean around 9.5 dB in this case (one may be able to visualize this with an analogy of the central limit theorem). Accordingly, the upper bound of the PDF integrated area that gives the fixed outage probability of 10−6 becomes larger monotonically towards the mean Q-factor of 9.5 dB, which establishes the fact that the reduction effect has an upper limit.
6. Conclusion
Subjecting experimental measured data to EVS analysis successfully demonstrated that WI transmission can decrease the Q-penalty or outage probability caused by PDL. Furthermore, numerical simulation revealed that this reduction effect tends to enhance as the number of interleaving channels n increases, and also that the amount of the reduction of PDL-induced impairments saturates towards an upper limit, which is explained by an analogy of the central limit theorem.
We believe that the use of WI may also be a valid approach to mitigating other impairments, such as signal degradation differing among different channels.
Acknowledgment
We would like to express our gratitude to K. Mori for his technical support in conducting our experiment.
References and links
1. M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16(18), 13918–13932 (2008). [CrossRef] [PubMed]
2. S. Yamamoto, T. Inui, H. Kawakami, S. Yamanaka, T. Kawai, T. Ono, K. Mori, M. Suzuki, A. Iwaki, T. Kataoka, M. Fukutoku, T. Nakagawa, T. Sakano, M. Tomizawa, Y. Miyamoto, S. Suzuki, K. Murata, T. Kotanigawa, and A. Maeda, “Hybrid 40-Gb/s and 100-Gb/s PDM-QPSK DWDM transmission using real-time DSP in field testbed,” Proc. OFC’12, JW2A.4 (2012).
3. O. Vassilieva, Inwoong Kim, and Takao Naito, “Systematic investigation of interplay between nonlinear and polarization dependent loss effects in coherent polarization multiplexed systems,” Proc. OFC’08, OThU6 (2008).
4. B. Xie, Y. Guan, Z. Li, and C. Lu, “FEC performance of optical communication systems with PMD and wavelength interleaving,” IEEE Photon. Technol. Lett. 16, 936–938 (2004). [CrossRef]
5. S. Hadjifaradji, S. Yang, L. Chen, and X. Bao, “PMD-PDL emulator designs for low interchannel correlation,” IEEE Photon. Technol. Lett. 18(22), 2362–2364 (2006). [CrossRef]
6. S. Savory, F. Payne, and A. Hadjifotiou, “Estimating outages due to polarization mode dispersion using extreme value statistics,” J. Lightwave Technol. 24(11), 3907–3913 (2006). [CrossRef]
7. R. D. Reiss and M. Thomas, Statistical analysis of extreme values (Cambridge, 1997.)
