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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 26 — Dec. 10, 2012
  • pp: B181–B196
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Nonlinear impairment compensation using expectation maximization for dispersion managed and unmanaged PDM 16-QAM transmission

Darko Zibar, Ole Winther, Niccolo Franceschi, Robert Borkowski, Antonio Caballero, Valeria Arlunno, Mikkel N. Schmidt, Neil Guerrero Gonzales, Bangning Mao, Yabin Ye, Knud J. Larsen, and Idelfonso Tafur Monroy  »View Author Affiliations


Optics Express, Vol. 20, Issue 26, pp. B181-B196 (2012)
http://dx.doi.org/10.1364/OE.20.00B181


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Abstract

In this paper, we show numerically and experimentally that expectation maximization (EM) algorithm is a powerful tool in combating system impairments such as fibre nonlinearities, inphase and quadrature (I/Q) modulator imperfections and laser linewidth. The EM algorithm is an iterative algorithm that can be used to compensate for the impairments which have an imprint on a signal constellation, i.e. rotation and distortion of the constellation points. The EM is especially effective for combating non-linear phase noise (NLPN). It is because NLPN severely distorts the signal constellation and this can be tracked by the EM. The gain in the nonlinear system tolerance for the system under consideration is shown to be dependent on the transmission scenario. We show experimentally that for a dispersion managed polarization multiplexed 16-QAM system at 14 Gbaud a gain in the nonlinear system tolerance of up to 3 dB can be obtained. For, a dispersion unmanaged system this gain reduces to 0.5 dB.

© 2012 OSA

1. Introduction

The application of digital signal processing (DSP) based coherent detection has allowed optical communication systems to operate closer to the nonlinear Shannon capacity limit by employing spectrally efficient modulation formats. Therefore, there is currently a lot of ongoing research on DSP based algorithms for signal detection and optical fibre channel impairment compensation. Linear signal processing algorithms can be effectively used to compensate for linear fibre channel impairments and have been demonstrated very successfully for higher order quadrature amplitude modulation (QAM) signaling [1

1. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron 16, 1164–1179 (2010). [CrossRef]

]. However, for long-haul systems employing higher order QAM, nonlinear optical fibre impairments can severely limit the transmission distance as well as the achievable total capacity [2

2. R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

]. Mitigation of optical fibre nonlinearities is therefore very crucial as it will allow launching more power into the fibre and thereby enhancing the transmission distance. Additionally, mitigation of fibre nonlinearities will help us reduce the nonlinear crosstalk from the neighboring channel in a multi-channel transmission system.

It has been shown that nonlinear fibre impairments can be compensated by various techniques: digital backpropagation (DBP), maximum-likelihood sequence estimation, nonlinear polarization crosstalk cancelation, nonlinear pre- and post-compensation, RF-pilot, etc, [3

3. A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol. 25, 3008–3016 (2007). [CrossRef]

7

7. D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun. E94-B, 1815–1822 (2011). [CrossRef]

] and references therein. Some of the mentioned methods suffer from complexity and, additionally, the achievable gain in the nonlinear tolerance is dependent on particular transmission scenarios. Therefore, efficient and widely applicable DSP algorithms for nonlinearity compensation are still open for research.

2. Numerical and experimental system set-up

In this section, a numerical and experimental set-up is presented. We first start by describing the numerical set-up used for simulations, and then we move to the experimental set-up. The section is concluded by a subsection describing the DSP algorithms used for signal equalization and demodulation.

2.1. Numerical set-up

The set-up used for the numerical investigations is shown in Fig. 1. All simulations are done using MATLAB (R2010a). For all numerical simulations, the baud rate is kept at 28 Gbaud resulting in the total bit rate of 224 Gb/s for the system under consideration. The transmitter and local oscillator (LO) laser phase noise is modeled as a random walk Wiener process. The output of the laser is then passed through an optical I/Q modulator. The I/Q modulator is driven by two four level pulse amplitude modulated (4-PAM) electrical signals, 4-PAM(1) and 4-PAM(2), in order to generate the optical 16-QAM signal. The module for generating 4-PAM signals is shown in Fig. 1. It consists of pseudo-random binary sequence (PRBS) generator (generates four independent sequences of length 215 – 1), signal mapping, upsampling, pulse shaping filter (raised cosine), digital-to-analog converter (DAC), attenuators and electrical amplifiers. The actual impulse response of the driving amplifiers is not taken into consideration. It is assumed that the electrical amplifiers have sufficient bandwidth such that they don’t induce any signal distortion. The method of 4-PAM signal generation is very similar to the one reported in [9

9. P. Winzer, A. Gnauck, C. Doerr, M. Magarini, and L. Buhl, “Spectrally efficient long-haul optical networking using 112-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol. 28, 547–556 (2010). [CrossRef]

]. The output of the I/Q modulator is then passed through a polarization multiplexing stage with a delay of 10 symbols, and the output is then amplified (EDFA). For the back-to-back numerical investigations, the generated PDM 16-QAM signal is coherently detected in a 90 degrees optical hybrid, photodetected and sampled at twice the baud rate by the analog-to-digital converter. We assume that the sampling frequency and the phase is not synchronized to the incoming signal and that the clock recovery is thereby performed by the DSP. The response of the analog-to-digital converter is modeled as a fourth-order Butterworth filter with a 3 dB bandwidth corresponding to 75% of the signal symbol rate. The sampled signal is then sent to the DSP modules which are described in subsection 2.3.

Fig. 1 Schematic diagram of the set-up used for simulations and experiment. PD: photodiode, PBS: polarization beam splitter, A/D: analog-to-digital converter, LO: local oscillator

2.2. Experimental set-up

2.3. Digital signal processing algorithms

3. Theory

3.1. Statistical signal representation - mixture of Gaussians

In this section, we will first describe how a received signal can be modeled as a so called ”Mixture of Gaussians, (MoG)” and then we will move into basic principles of the EM algorithm. For a more detailed treatment of MoG see [13

13. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

], Chapter 9.2. Throughout, the entire section we will assume that the signal that is input to the EM algorithm contains one sample per symbol, and is obtained after polarization demultiplexing, frequency and phase recovery stage. We will refer to this signal as the demodulated signal.

The demodulated signal in x/y-polarization can be considered as a mixture of Gaussian densities (MoG) consisting of a number of components (clusters), where each of the components (clusters) can be described by a 2-D Gaussian distribution. For instance, in the case of 16-QAM, we have 16 clusters. For a 16-QAM signal constellation, in the absence of any impairment, we will have 16 distinct constellation points. However, in the presence of additive white Gaussian noise, around each of the 16 constellation points there will occur spread of symbols. The cluster is then defined as a grouping of the points/symbols around a mean value. Irrespective of the modulation format applied, (PSK or QAM), the demodulated signal can mathematically be expressed as a superposition of M Gaussian densities, where M is the number of clusters and corresponds to the number of constellation points. The probability density function of the demodulated signal is then expressed as:
p(x)=k=1MπkN(x|μk,Σk),
(1)
where k refers to each cluster in the constellation, πk is a mixing coefficient (for the considered case of a signal where symbols have uniform distribution πk = 1/M). x = [x1, x2] is a 2-D vector, corresponding to a detected symbol in the constellation (Inphase/Quadrature) plane and N(x|μk, Σk) is a 2-D Gaussian density with mean μk and a 2 × 2 covariance matrix Σk [13

13. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

]:
N(x|μk,Σk)=12π|Σk|1/2e12(xμk)TΣk1(xμk),
(2)
where |Σk| is the determinant of the covariance matrix and it expresses the area covered by the specific cluster k. The covariance matrix is defined as:
Σk=[var(x1)cov(x1,x2)cov(x1,x2)var(x2)][σ1,12σ1,22σ2,12σ2,22].
(3)

For the most general case, each cluster is described by its specific covariance matrix Σk. However, when the signal is mostly dominated by the additive Gaussian noise, the covariance matrices will be equal and diagonal, i.e. there is no correlation among symbols within the cluster. Additionally, the clusters will be circularly symmetric (equal variances) and the covariance matrix is then expressed as:
Σk=Σ=[σ200σ2].
(4)

An example of the demodulated 16-QAM signal dominated by additive Gaussian noise is shown in Fig. 2(a). It is observed in Fig. 2(a) that all the clusters look similar. An example of the demodulated signal strongly impaired by laser phase noise is shown in Fig. 2(b). It is observed in Fig. 2(b) that the clusters are not similar. Indeed, the clusters belonging to the outer ring are elliptical. Here, we will distinguish between two cases: (1) the covariance matrix is still diagonal and σ1,12σ2,22; the clusters are stretched in either vertical or horizontal direction, (2) the covariance matrix is non-diagonal and in this case the shape and orientation of the cluster is arbitrary, all depending if there is positive or negative correlation. Finally, let’s look at third case when the demodulated signal is severally impaired by non-linear phase noise, see Fig. 2(c). It is observed that in Fig. 2(c) not only outer cluster are affected but all the clusters experience distortion. It should also be noticed that the entire constellation is tilted (phase offset introduced), and the outer points have been compressed. This compression means that the mean values μk have been altered compared to the reference constellation. By reference constellation, it is meant the constellation which is free of any impairment.

Fig. 2 Impact of different impairments on signal constellation for a 16-QAM signal. (a) Constellation of a signal dominated by additive noise. (b) Constellation of a signal dominated by phase noise. (c) Constellation of a signal dominated by non-linear phase noise.

In general, different optical channel impairments will have a different imprint on the received signal constellation. This information can then be used to determine the impairment and make optimal signal detection as explained next.

The optimal signal detection in maximum likelihood sense is obtained by maximizing a posteriori probability of the received symbol x belonging to one of the clusters k, where k = 1,...,M:
k^=argmaxkp(k|x)
(5)
or in another words find a cluster k for which p(k|x) is maximized. The a posteriori probability p(k|x) is obtained from Bayes’ theorem [13

13. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

]:
p(k|x)=πkN(x|μk,Σk)l=1MπlN(x|μk,Σk).
(6)
Inserting the expression for the Gaussian distribution, the optimal decision, Eq. (5), reduces to a quadratic decision rule:
k^=argmaxk{12xTΣk1x+wkTx+wk0}
(7)
with wk=Σk1μk and wk0=logπklog|Σk|/2μkTΣk1μk/2. In the case, when the covariance matrices are equal, the quadratic term in optimal decision rule in Eq. (7) are the same for all k and the decision rule becomes linear:
k^=argmaxk{wkTx+wk0}.
(8)
However, in case when the signal constellation is distorted by nonlinear phase noise, laser phase noise, etc, Eq. (6) needs to be used in order to make optimum signal detection. In order to evaluate Eq. (6), M Gaussian densities, N(x|μk, Σk) and thereby parameters π ≡ {π1,...,πk}, μ ≡ {μ1,...,μk} and Σ ≡ {Σ1,..., Σk} describing Gaussian densities need to be determined. Next, we will show how to use a powerful method of EM in order to determine the parameters that generate the Gaussian mixture model. The EM will determine in a maximum likelihood sense the most likely parameters Ξ = [π,μ,Σ] that generated Gaussian densities.

3.2. Expectation maximization algorithm

In general, the EM is a numerical method of producing a solution to a maximum likelihood estimation for problems which can be simplified by introducing latent variables [13

13. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

, 14

14. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B. 39, 1–38 (1977).

]. In the mixture model context the latent variables are the assignments. The set of parameters Ξ to be estimated, in the maximum likelihood sense, from the demodulated signal X is governed by the following expression:
Ξ^=argmaxΞp(X|Ξ),
(9)
where X = [x1,..., xN], N is the length of the observation interval and the likelihood function of Ξ, p(X|Ξ), for independent identically distributed data is expressed as:
p(X|Ξ)=n=1Np(xn|Ξ)=n=1Nk=1MπkN(xn|μk,Σk).
(10)
No closed-form analytical solution for Eq. (9) is available. Therefore, the iterative EM framework can be used to find a solution. The EM is a two step iterative procedure which is guar-enteed to converge to the (local) maximum likelihood solution given in Eq. (9) [14

14. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B. 39, 1–38 (1977).

]. The two step procedure, so called expectation (E) step and maximization (M) step for the particular case considered in this section is as follows [13

13. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

]:
E-step:γnkp(k|xn)=πkN(xn|μk,Σk)l=1MπlN(xn|μl,Σl)forn=1,,Nandk=1,,M
(11)
M-step:Nk=n=1Nγnk
(12)
πk=NkN
(13)
μk=1Nkn=1Nγnkxn
(14)
Σk=1Nkn=1Nγnk(xnμk)(xnμk)Tfork=1,,M,
(15)
where γnk is called the responsibility and is nothing but a posteriori probability, Eq. (6), needed for optimal decisions.

The flow-chart describing the algorithm is shown in Fig. 3. To begin with, we initialize the EM with initial parameters for the means, covariance matrices and mixing coefficients and then the algorithms start to iterate in order to find most likely parameters. In the E-step, the current values of the parameters, Ξi at the iteration i, are used to evaluate the Eq. (11). The E-step expressed by Eq. (11) computes the probability of the received symbol belonging to one of the clusters, i.e. posterior probability. In the M-step we use those probabilities to re-estimate the parameters Ξ. In other words, in the M-step we are trying to find the parameters that maximize the probability that the data has been generated by a particular cluster. When making a parameter update resulting from the E step and followed by the M step, the likelihood function, P(Xi), on the parameters will increase and will flatten out when the algorithm has converged. The convergence properties of the EM strongly dependent on the initialization. For the considered cases throughout the paper, we found that the EM algorithm will converge after 3 iterations. Once the EM algorithms has converged (i > Niter), we use the results to perform the optimum signal detection governed by Eq. (5).

Fig. 3 Flow-chart illustrating the steps performed by the EM algorithm and subsequent signal demodulation. Niter denotes the number of specified iteration.

4. Simulation results

4.1. Back-to-back investigation

First, we consider back-to-back case. It is investigated how the EM algorithm can be used to combat the combined effects of I/Q modulator nonlinearities and imperfection, and combined laser linewidth. We deliberately drive the I/Q modulator with large peak-to-peak amplitude, Vpp, of the electrical 4-PAM signal such that the constellation diagram of the 16-QAM signal is distorted. The resulting modulation depth of the modulator is then m = Vpp/Vπ =2.12. Furthermore, it is assumed that the phase shift between the inphase and quadrature branch of the I/Q modulator deviates from π/2 by 5%. In Fig. 4, −log[BER] is plotted as function of the combined laser linewidth for the optical signal to noise ratio (OSNR) of 25 dB. Figure 4 shows that compared to the case when no compensation is used (linear decision boundaries), the EM is very efficient in combating the combined impairments originating from I/Q modulator nonlinearities, non-ideal phase shift between the I and Q branches and combined laser linewidth.

Fig. 4 BER as a function of combined laser linewidth for the back-to-back-case. The modulator modulation depth, m = Vpp/Vπ = 2.12, I/Q imbalance: 5% and OSNR is 25 dB.

4.2. Dispersion managed link

Fig. 5 The total number of spans is 12 and the combined laser linewidth is 200 kHz. (a) BER as a function of span input power for NLPN dominated transmission link, (dispersion numerically set to zero). (b) BER as a function of span input power for dispersion managed link. Transmission link consists of SSMF and DCF.

4.3. Dispersion unmanaged link

Fig. 6 The total number of spans is 12. (a) BER as a function of span input power for dispersion unmanaged link. The combined laser linewidth is 200 kHz. (b) BER as a function of combined laser linewidth for dispersion unmanaged link.

5. Experimental results

5.1. Dispersion managed link

First, we will demonstrate how EM can be effectively used to extract information from a severely distorted constellation and use this information to mitigate the impairments. Figure 7, shows a constellation diagram of a signal impaired by nonlinear phase noise after one span transmission through the dispersion managed link. We plot the recovered constellation diagram of the x polarization after the carrier recovery stage. Together with the constellation, we have also plotted optimal (nonlinear) decision boundaries obtained by applying Eq. (6) in conjunction with the EM algorithm.

Fig. 7 Recovered constellation diagram impaired by nonlinear phase noise. Only a single transmission span is considered

It is observed from Fig. 7, that due to the nonlinear impairments, especially the outer constellation points are distorted. The inner constellation points are less affected due to lower power, however, all clusters experience a significant phase shift. In the case, no compensation is applied −log(BER) is 1.30. For the case when the k-means is used the respective −log(BER) is 2.04 while when the EM is applied −log(BER) gets down to 3. This example demonstrates the capabilities of EM of compensating distorted and phase shifted constellations.

In Fig. 8(a), we plot the demodulated signal constellation for the input power of Pin = 0 dBm and the corresponding optimal decision boundaries after 800 km of transmission. It is observed that the demodulated signal constellation shown in Fig. 8(a) is distorted and therefore the optimal decision boundaries are nonlinear. In Fig. 8(b), we plot −log(BER) as a function of span input power after 800 km of transmssion thorugh dispersion managed link. It is observed that there is an improvement in the nonlinear system tolerance by employing the EM algorithm which is in accordance with simulation results. We observe up to 3 dB of improvement in nonlinear tolerance compared to the case when no compensation is used. The reason why we get more improvement for the experimental data may be attributed to the fact that the EM is also effective in compensating residual distortion induced on the signal. It is observed from the figure that only very little improvement can be obtained by using the k-means algorithm, and this is also in good agreement with the simulation results.

Fig. 8 (a) Constellation diagram of the demodulated signal after 800 km of transmission through dispersion managed link. (b) BER as a function of span input power for dispersion managed link after 800 km of transmission.

5.2. Dispersion unmanaged link

Fig. 9 (a) Constellation diagram of the demodulated signal after 800 km of transmission through dispersion unmanaged link. (b) BER as a function of span input power for dispersion unmanaged link after 800 km of transmission.

In order to investigate how nonlinearity compensation by the EM scales with transmission distance, −log(BER) is plotted as a function of span input signal power for transmission distances of 240 km and 400 km, respectively, see Fig. 10. It is observed in Fig. 10(a) that an improvement in the nonlinear system tolerance of approximately 0.5 dB is observed for the entire range of the considered input optical power. For the transmission distance of 400 km, Fig. 10(b), a similar improvement of 0.5 is observed, however, the gain disappears for input power exceeding 2 dBm. One of the explanations could be that for input power exceeding 2 dBm the distortion is large and the EM cannot properly estimate the parameters.

Fig. 10 (a) BER as a function of span input power for dispersion unmanaged link after 240 km of transmission. (b) BER as a function of span input power for dispersion unmanaged link after 400 km of transmission.

6. Conclusion

In order to see what benefits EM brings we need to relate the performance of the EM to a digital backpropagation, which has become a benchmark for nonlinearity compensation techniques. For the dispersion managed links, it has been numerically shown that up to 4 dB of improvement can be obtained [19

19. E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

]. The gain in the nonlinear system tolerance will though depend on the implementation of the digital backpropagation. For instance this gain reduces to 2 dB if 1 step/fiber digital backpropagation is used [19

19. E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

]. For the dispersion unmanaged links, an improvement of 3 dB for PDM 16-QAM signal has been shown experimentally [20

20. S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112gbit/s pdm-qam-16 transmission with and without digital nonlinearity compensation,” Opt. Express 18, 12939–12947 (2010). [CrossRef] [PubMed]

]. As a rule of thumb, we may conclude that for dispersion unmanaged system digital backpropagation offers better performance, while for dispersion managed systems, depending on the implementation of digital backpropagation, the performance of the EM and digital backpropagation may be comparable. However, one technique does not have to exclude the other, as one may consider the combination of both techniques in combating optical fibre channel nonlinearities.

Acknowledgments

This work has been partly supported by the Danish Council for Independent Research, project CORESON and the CHRON (Cognitive Heterogeneous Reconfigurable Optical Network) project, with funding from the European Community’s Seventh Framework Programme [FP7/2007–2013] under grant agreement nº 258644, http://www.ict-chron.eu.

References and links

1.

S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron 16, 1164–1179 (2010). [CrossRef]

2.

R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]

3.

A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol. 25, 3008–3016 (2007). [CrossRef]

4.

E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef]

5.

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29, 2570–2576 (2011). [CrossRef]

6.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

7.

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun. E94-B, 1815–1822 (2011). [CrossRef]

8.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

9.

P. Winzer, A. Gnauck, C. Doerr, M. Magarini, and L. Buhl, “Spectrally efficient long-haul optical networking using 112-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol. 28, 547–556 (2010). [CrossRef]

10.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express 19, 429–437 (2011). [CrossRef]

11.

H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).

12.

J. Kurzweil, An Introduction to Digital Communications (John Wiley, 2000).

13.

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

14.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B. 39, 1–38 (1977).

15.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett. 22, 335–337 (2010). [CrossRef]

16.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]

17.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, L. Lorcy, J.-C. Antona, S. Bigo, and A. Bononi, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express 20, 1022–1032 (2012). [CrossRef] [PubMed]

18.

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).

19.

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

20.

S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112gbit/s pdm-qam-16 transmission with and without digital nonlinearity compensation,” Opt. Express 18, 12939–12947 (2010). [CrossRef] [PubMed]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.1660) Fiber optics and optical communications : Coherent communications

ToC Category:
Subsystems for Optical Networks

History
Original Manuscript: October 1, 2012
Revised Manuscript: November 4, 2012
Manuscript Accepted: November 5, 2012
Published: November 28, 2012

Virtual Issues
European Conference on Optical Communication 2012 (2012) Optics Express

Citation
Darko Zibar, Ole Winther, Niccolo Franceschi, Robert Borkowski, Antonio Caballero, Valeria Arlunno, Mikkel N. Schmidt, Neil Guerrero Gonzales, Bangning Mao, Yabin Ye, Knud J. Larsen, and Idelfonso Tafur Monroy, "Nonlinear impairment compensation using expectation maximization for dispersion managed and unmanaged PDM 16-QAM transmission," Opt. Express 20, B181-B196 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-26-B181


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References

  1. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron16, 1164–1179 (2010). [CrossRef]
  2. R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28, 662–701 (2010). [CrossRef]
  3. A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol.25, 3008–3016 (2007). [CrossRef]
  4. E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26, 3416–3425 (2008). [CrossRef]
  5. Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol.29, 2570–2576 (2011). [CrossRef]
  6. N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).
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