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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26557–26567
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Weighting nonlinearities on future high aggregate data rate PONs

Jacklyn D. Reis, Darlene M. Neves, and António L. Teixeira  »View Author Affiliations


Optics Express, Vol. 19, Issue 27, pp. 26557-26567 (2011)
http://dx.doi.org/10.1364/OE.19.026557


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Abstract

In this paper we address high aggregate data rate coherent UDWDM-PONs (ultra-dense wavelength-division multiplexing passive optical networks) related physical impairments. Firstly, analog to digital converter resolution and laser linewidth are optimized giving the minimal signal to noise ratio penalty for UDWDM-PON systems at 10 Gb/s per user/wavelength. Secondly, inter-channel nonlinearities impact on high-order modulation formats at 1-2.5 Gb/s per channel and 100 km-reach is studied by means of Volterra series simulations.

© 2011 OSA

1. Introduction

2. Ultra-dense WDM-PON: system model

In access networks applications where the accumulated dispersion is not an issue for signals at 625 Mbaud/1250 Mbaud, sampling the signal at twice the baud rate may be sufficient, in most cases, for performing phase/carrier recovery and channel equalization (linear and nonlinear eventually). Thus ADC sampling rate is perfectly transparent to modulation formats at 625 Mbaud (1.25 GSa/s) or 1250 Mbaud (2.5 GSa/s) assuming electrical bandwidth broad enough to avoid distortion. On the other hand, different signal constellations require different ADC resolution (in bits) since higher order constellations are more sensitive to quantization error. In this paper we focus on the amplitude resolution of the ADC, remaining the time resolution subject for further studies.

3. Impact of nonlinear effects on UDWDM-PON systems

One of the challenges on current coherent optical systems is how to mitigate phase noises such that the information is recovered in a reliable way. There are two main sources of phase noise affecting the system’s performance: (i) phase noise coming from the laser sources both in the transmitter and local oscillator in the receiver; (ii) phase noise induced by linear and nonlinear effects during transmission.

A. Linewidth requirements

The advantage of using ONU, as a coherent receiver is that the phase noise is mitigated digitally using well-known carrier phase recovery techniques. In this section we firstly evaluate the phase noise in terms of the required linewidth to provide minimal SNR penalty with respect to BER = 10−3. We used the same UDWDM-PON system scenario described before operating at 625 Mbaud and 1.25 Gbaud per channel after 25 km of SSMF with the following input power per channel: −2 dBm/channel for QPSK, −4 dBm/channel for 8PSK and −6 dBm/channel for 16QAM. As a result, the system is entirely limited by inter-channel nonlinearities. In our simulations, 32 runs of 512 (625 Mbaud) and 1024 (1.25 Gbaud) symbols per channel were used for SNR estimation. After coherent detection at the ONU, we applied the feed forward M-th power block scheme described in [9

9. M. Seimetz, “High-Order Modulation for Optical Fiber Transmission,” in Optical Sciences, W.T. Rhodes, ed. (Springer, Atlanta, GA., 2009).

] (field averaging) for phase estimation considering a block size of 8 symbols for phase estimation. Figure 2 depicts the SNR penalty at BER = 10−3 of the received center channel as a function of linewidths per laser sources, i.e. the transmitter and local oscillator lasers have the same linewidths. The insets show the received constellations at 100 kHz-linewidth. Solid curves plus circles describe the system’s performance operating at 625 Mbaud per channel whereas dash curves plus squares describe the system operation at 1.25 Gbaud per channel.

The results for phase-modulated signals in red for QPSK (M = 4) and black for 8PSK (M = 8) show similar performance for the system operating either at 625 Mbaud or 1.25 Gbaud for lower linewidths. Allowing 1 dB penalty, QPSK is able to operate with linewidth per channel around 500 kHz whereas 8PSK should be around 100 kHz. On the other hand, 625 Mbaud-16QAM and 1.25 Gbaud-16QAM systems operate with 1.1 dB and 1.4 dB penalty at 100 kHz, respectively. In terms of the effective linewidth, defined as the linewidth per laser (Δf) times symbol duration (Ts), we obtained the following values within the 1 dB margin: Δf *Ts = 8x10−4 for QPSK, Δf *Ts = 1.6x10−4 for 8PSK and Δf *Ts = 8x10−5 for 16QAM. Note that the 16QAM transmission, on average only half of the 8 symbols is used for phase estimation, i.e. the inner and outer most symbols used as for QPSK phase estimation.

The challenge on the UDWDM-PON system is the increased symbol duration (decreased symbol rate), which effectively reduces significantly the maximum tolerable linewidth. With symbol durations around 800~1600 ps (symbol rates 1250~625 Mbaud), the tested UDWDM-PON should require linewidths ranging from only 500 kHz (625Mbaud) to 1 MHz (1.25Gbaud) if transporting QPSK (Δf*Ts≈8x10−4) for instance. On the other hand, higher symbol rates alleviate the needs for narrower linewidths as in [8

8. T. Pfau, S. Hoffmann, and R. Noé, “Hardware-Efficient Coherent Digital Receiver Concept With Feedforward Carrier Recovery for M-QAM Constellations,” J. Lightwave Technol. 27(8), 989–999 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=jlt-27-8-989. [CrossRef]

] where the authors found Δf*Ts≈4.1x10−4, which gives maximum linewidth of 4.1 MHz for the system at 10 Gbaud-QPSK. Although we used fixed block-symbol (8 symbols) for different symbol rates we obtained slightly lower penalty for the system at 1.25 Gbaud compared to 625 Mbaud, as would be expected. This can be seen on the red curves (QPSK) in Fig. 2 for linewidths higher than 100 kHz. For the other modulation formats, the penalty was prohibitively high to see the benefit of increased symbol rate. We point out that those results present the effects of the laser linewidth as well as its interaction with chromatic dispersion and Kerr nonlinearities. Furthermore, this tolerance can be improved by optimizing the block length [10

10. M. Seimetz, “Laser Linewidth Limitations for Optical Systems with High-Order Modulation Employing Feed Forward Digital Carrier Phase Estimation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuM2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OTuM2.

,11

11. I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010). [CrossRef]

]; by improving the field averaging filter for phase estimation in each block of symbols [12

12. A. Bononi, M. Bertolini, P. Serena, and G. Bellotti, “Cross-Phase Modulation Induced by OOK Channels on Higher-Rate DQPSK and Coherent QPSK Channels,” J. Lightwave Technol. 27(18), 3974–3983 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-18-3974. [CrossRef]

]; or by using more sophisticated schemes that take into account all symbols inside a block for phase estimation on QAM signals such as in [11

11. I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010). [CrossRef]

,13

13. F. Rice, B. Cowley, B. Moran, and M. Rice, “Cramér-Rao Lower Bounds for QAM Phase and Frequency Estimation,” IEEE Trans. Commun. 49(9), 1582–1591 (2001). [CrossRef]

].

B. Volterra theory

Using Volterra theory we are able to estimate the EVM of the recovered symbols associated with the most relevant fiber nonlinear effects: self-phase modulation (SPM), cross-phase modulation (XPM) and four-wave mixing (FWM). The signal propagation over the optical fiber can be simulated numerically using both Split-Step Fourier (SSF) and 3rd order Volterra Series Transfer Function (VSTF) methods. Both methods are employed to solve the generalized Nonlinear Schrödinger (NLS) equation, defined in (1) as

Az+β1Atjβ222At2β363At3+α2A=jγ|A|2A.
(1)

The authors in [14

14. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra Series Transfer Function of Single-Mode Fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

] presented the VSTF method as a solution to the NLS in the frequency domain. This solution retained up to the third order Volterra kernel expressed in (2) is the relationship between the input and output optical signals defined in the frequency domain through transfer functions as
A(ω,z)H1(ω,z)A(ω)+14π2H3(ω1,ω2,ωω1+ω2,z)×A(ω1)A(ω2)A(ωω1+ω2)dω1dω2,
(2)
where A(ω)=A(ω,0) is the Fourier transform of the input optical signal A (scalar version of the optical field), the operator (.)* stands for the complex conjugate, H1(ω,z) is the linear transfer function (1st order Volterra kernel) whereas H3(ω1,ω2,ωω1+ω2,z) is the nonlinear transfer function (3rd order Volterra kernel), defined in (3) and (4), respectively.
H1(ω,z)=e(α2jβ22ω2jβ36ω3)z,
(3)
H3(ω1,ω2,ω,z)=jγH1(ω,z)×1exp(αzjβ2(ω1ω)(ω1ω2)zjβ32(ω+ω2)(ω1ω)(ω1ω2)z)α+jβ2(ω1ω)(ω1ω2)+jβ32(ω+ω2)(ω1ω)(ω1ω2),
(4)
where z is the propagation distance with the moving coordinate system being considered equal to the group velocity; thus the group delay term β1 was simply dropped. The linear terms are the attenuation coefficient α, second order dispersion coefficient β2 and third order dispersion coefficient β3 whereas γ is the nonlinear coefficient (Kerr). Higher order nonlinearities such as self-steepening and stimulated Raman scattering are neglected and the only contribution to nonlinear phase noise will be from the Kerr effect and possibly with its interaction with amplified spontaneous emission (ASE). Therefore, the numerical evaluation of the double integral in (2) accounts for SPM (intra-channel), XPM and FWM (inter-channel) effects resulting in phase and intensity distortions. We point out that the modified version of the 3rd order VSTF in [15

15. B. Xu and M. B. Pearce, “Modified Volterra Series Transfer Function,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002). [CrossRef]

] has been used throughout this paper as to extend its accuracy, which is critical for the analysis of ultra-dense WDM systems.

Weighting independently fiber nonlinearities requires considering the interactions inside the double integral in (2) of the input modulated signals as Ai(ω1)Aj*(ω2)Ak(ωω1+ω2). i, j, and k correspond to the indices of any WDM channel then we have the following relations [16

16. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express 18(8), 8660–8670 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8660. [CrossRef] [PubMed]

]:

  • - If [i=j=k], then (2) accounts for SPM at ith channel as H3(ω1,ω2,ωω1+ω2,z)Ai(ω1)Ai*(ω2)Ai(ωω1+ω2);
  • - If [i=jkorij=k], then (2) accounts for XPM at kth channel or ith channel as H3(ω1,ω2,ωω1+ω2,z){Ak(ωω1+ω2)[ΣikAi(ω1)Ai*(ω2)]...+Ak(ω1)[ΣikAi*(ω2)Ai(ωω1+ω2)]};
  • - If [ikandjk], then (2) accounts for FWM at [ij+k] channel as Σi,j,kH3(ω1,ω2,ωω1+ω2,z)Ai(ω1)Ak*(ω2)Aj(ωω1+ω2).

To estimate the EVM associated with SPM, XPM and FWM effects we applied the matrix of WDM channels, being the columns representing the channels and the rows representing the time instant, prior multiplexing to the VSTF method described before. Each EVM value for VSTF, SSF, SPM and XPM represents a total of 16384 simulated symbols per channel (32 transmissions of 512 symbols). Calculating FWM effect requires a huge computation effort as the number of mixing products ((N3-N2)/2 with N the number of channel) for a 32-channel system exceeds 15000. To avoid extra calculations we evaluate the phase matching condition and efficiency of each new FWM component considering the CW (continuous wave) approach [17

17. N. Shibata, R. P. Braun, and R. G. Waarts, “Phase Mismatch Dependence of Efficiency of Wave Generation Through Four-Wave Mixing in a Single Mode Optical Fiber,” IEEE J. Quantum Electron. 23(7), 1205–1210 (1987). [CrossRef]

]. Thus any FWM component whose efficiency is below 20% of its maximum is not considered for calculation in the double integral of Eq. (2). Therefore, we evaluated 61 instead of 360 FWM products falling on the center channel under test. Our calculations indicated a penalty on the total FWM power below 1.2 dB after removing products whose efficiency is below the threshold.

C. Weighting nonlinearities

The evaluation of the system’s performance limited by fiber nonlinearities does not have a universal closed solution requiring brute-force simulations to have a comprehensive analysis [18

18. A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWO7. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-OWO7

]. In this section we weight independently the impact of the most relevant nonlinear effects and identify their ranges of dominancy over various bit rates per channel/user (fixed symbol rate). Therefore, we transmitted the 32 ultra-dense channels (laser sources without intensity noise) spaced by 3.125 GHz over 25 km, 60 km and 100 km of SSMF. Each channel carries 512 symbols via 625 Mb/s-29 pseudo random bit sequence (PRBS) as follows: (i) 1.25 Gb/s-QPSK encoding two 29-PRBS; (ii) 3.875 Gb/s-8PSK encoding three 29-PRBS; (iii) 2.5 Gb/s-16QAM encoding four 29-PRBS. Since we are interested in studying the phase noise induced by fiber nonlinearities, the laser linewidths of transmitters and local oscillators were set to zero. To measure the system’s performance we calculated the EVM over 512 transmitted symbols per channel. The resulting EVM is averaged over 32 transmissions with a total of 16384 simulated symbols or 32768 bits for QPSK, 49152 bits for 8PSK and 65536 bits for 16QAM.

The style blue curves in Fig. 3
Fig. 3 EVM of the received center channel versus input power per channel for 32x1.25 Gb/s-QPSK. Solid lines: 25 km; Dash lines: 60 km; Dash-dot lines: 100 km. Insets show constellations after 100 km of fiber with blue symbols from SSF simulations and the red ones from the Volterra XPM model.
-5
Fig. 5 EVM of the received center channel versus input power per channel for 32x2.5 Gb/s-16QAM. Solid lines: 25 km; Dash lines: 60 km; Dash-dot lines: 100 km. Insets show constellations after 100 km of fiber with blue symbols from SSF simulations and the red ones from the Volterra XPM model.
show the overall system’s performance using SSF simulations measured in terms of EVM in dB, which is mathematically equal to the inverse square root of SNR, of the received center channel (16th channel) of the transmitted comb as a function of input power per channel (average optical power). For both modulation formats, QPSK in Fig. 3, 8PSK in Fig. 4
Fig. 4 EVM of the received center channel versus input power per channel for 32x1.875 Gb/s-8PSK. Solid lines: 25 km; Dash lines: 60 km; Dash-dot lines: 100 km. Insets show constellations after 100 km of fiber with blue symbols from SSF simulations and the red ones from the Volterra XPM model.
and 16QAM in Fig. 5, VSTF method shows similar EVM results (blue circles) as the SSF (blue lines) for different transmission distances up to 100 km with power below −3 dBm/channel. Those results validate its application to model accurately the fiber propagation even for such a high density of WDM channels. The insets show the received constellations obtained from SSF simulations (in blue) with EVM close to BER = 10−3 (typical 7% overhead FEC limit) with input power below −2 dBm per channel, −4 dBm per channel and −6dBm per channel for QPSK (EVM ~-9.8 dB), 8PSK (EVM ~-15.3 dB) and 16QAM (EVM ~-16.53 dB), respectively.

Thirdly, XPM effect on 8PSK signals is a few dB higher than the XPM noise induced by QPSK signals as shown in Fig. 3,4. This can also be justified from the F/X analysis in which for QPSK this value ranged from 26.6 to 30 dB whereas for 8PSK the F/X ranged from 25 to 28.6 dB. In addition, Fig. 5 indicates that SPM effect on 16QAM is solely higher than the XPM effect on phase-modulated signals.

For instance, increasing the transmission distance by a factor of 2.4, 4 and 1.7 effectively increases the XPM effect on average by 2.9, 3.3 and 0.4 dB, respectively; whereas FWM is slightly reduced by 0.9, 1.1 and 0.1 dB. This analysis of nonlinearities and their dependence on transmission distances for 16QAM is summarized in Table 1.

4. Conclusion

Acknowledgements

This work was partially supported by the European Union within the EURO-FOS project, a Network of Excellence funded by the EU 7th ICT-FP. J.D. Reis also acknowledges his PhD grant from FCT (SFRH/43941/2008, “Fundação para a Ciência e a Tecnologia”).

References and links

1.

F. J. Effenberger, “The XG-PON System: Cost Effective 10 Gb/s Access,” J. Lightwave Technol. 29(4), 403–409 (2011), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-29-4-403. [CrossRef]

2.

S. Smolorz, H. Rohde, E. Gottwald, D. W. Smith, and A. Poustie, “Demonstration of a Coherent UDWDM-PON with Real-Time Processing,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPD4. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-PDPD4.

3.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital back propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-26-20-3416. [CrossRef]

4.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-28-4-662. [CrossRef]

5.

J. D. Reis, D. M. Neves, and A. Teixeira, “Weighting Nonlinearities on Future High Aggregate Data Rate PONs,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper We.10.P1.109. http://www.opticsinfobase.org/abstract.cfm?URI=ECOC-2011-We.10.P1.109.

6.

R. A. Shafik, Md.S. Rahman, A.R. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics,” in Proceedings of IEEE International Conference on Electrical and Computer Engineering (Institute of Electrical and Electronics Engineers, Dhaka, Bangladesh 2006), pp. 408–411.

7.

K.-P. Ho, “Phase-Modulated Optical Communication Systems,” (Springer, New York, NY., 2005).

8.

T. Pfau, S. Hoffmann, and R. Noé, “Hardware-Efficient Coherent Digital Receiver Concept With Feedforward Carrier Recovery for M-QAM Constellations,” J. Lightwave Technol. 27(8), 989–999 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=jlt-27-8-989. [CrossRef]

9.

M. Seimetz, “High-Order Modulation for Optical Fiber Transmission,” in Optical Sciences, W.T. Rhodes, ed. (Springer, Atlanta, GA., 2009).

10.

M. Seimetz, “Laser Linewidth Limitations for Optical Systems with High-Order Modulation Employing Feed Forward Digital Carrier Phase Estimation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuM2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OTuM2.

11.

I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photon. Technol. Lett. 22(9), 631–633 (2010). [CrossRef]

12.

A. Bononi, M. Bertolini, P. Serena, and G. Bellotti, “Cross-Phase Modulation Induced by OOK Channels on Higher-Rate DQPSK and Coherent QPSK Channels,” J. Lightwave Technol. 27(18), 3974–3983 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-18-3974. [CrossRef]

13.

F. Rice, B. Cowley, B. Moran, and M. Rice, “Cramér-Rao Lower Bounds for QAM Phase and Frequency Estimation,” IEEE Trans. Commun. 49(9), 1582–1591 (2001). [CrossRef]

14.

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra Series Transfer Function of Single-Mode Fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

15.

B. Xu and M. B. Pearce, “Modified Volterra Series Transfer Function,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002). [CrossRef]

16.

J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express 18(8), 8660–8670 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8660. [CrossRef] [PubMed]

17.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase Mismatch Dependence of Efficiency of Wave Generation Through Four-Wave Mixing in a Single Mode Optical Fiber,” IEEE J. Quantum Electron. 23(7), 1205–1210 (1987). [CrossRef]

18.

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWO7. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-OWO7

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4250) Fiber optics and optical communications : Networks
(060.1155) Fiber optics and optical communications : All-optical networks

ToC Category:
Access Networks and LAN

History
Original Manuscript: October 3, 2011
Revised Manuscript: November 4, 2011
Manuscript Accepted: November 22, 2011
Published: December 14, 2011

Virtual Issues
European Conference on Optical Communication 2011 (2011) Optics Express

Citation
Jacklyn D. Reis, Darlene M. Neves, and António L. Teixeira, "Weighting nonlinearities on future high aggregate data rate PONs," Opt. Express 19, 26557-26567 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26557


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References

  1. F. J. Effenberger, “The XG-PON System: Cost Effective 10 Gb/s Access,” J. Lightwave Technol.29(4), 403–409 (2011), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-29-4-403 . [CrossRef]
  2. S. Smolorz, H. Rohde, E. Gottwald, D. W. Smith, and A. Poustie, “Demonstration of a Coherent UDWDM-PON with Real-Time Processing,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPD4. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-PDPD4 .
  3. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital back propagation,” J. Lightwave Technol.26(20), 3416–3425 (2008), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-26-20-3416 . [CrossRef]
  4. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-28-4-662 . [CrossRef]
  5. J. D. Reis, D. M. Neves, and A. Teixeira, “Weighting Nonlinearities on Future High Aggregate Data Rate PONs,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper We.10.P1.109. http://www.opticsinfobase.org/abstract.cfm?URI=ECOC-2011-We.10.P1.109 .
  6. R. A. Shafik, Md.S. Rahman, A.R. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics,” in Proceedings of IEEE International Conference on Electrical and Computer Engineering (Institute of Electrical and Electronics Engineers, Dhaka, Bangladesh 2006), pp. 408–411.
  7. K.-P. Ho, “Phase-Modulated Optical Communication Systems,” (Springer, New York, NY., 2005).
  8. T. Pfau, S. Hoffmann, and R. Noé, “Hardware-Efficient Coherent Digital Receiver Concept With Feedforward Carrier Recovery for M-QAM Constellations,” J. Lightwave Technol.27(8), 989–999 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=jlt-27-8-989 . [CrossRef]
  9. M. Seimetz, “High-Order Modulation for Optical Fiber Transmission,” in Optical Sciences, W.T. Rhodes, ed. (Springer, Atlanta, GA., 2009).
  10. M. Seimetz, “Laser Linewidth Limitations for Optical Systems with High-Order Modulation Employing Feed Forward Digital Carrier Phase Estimation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuM2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OTuM2 .
  11. I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photon. Technol. Lett.22(9), 631–633 (2010). [CrossRef]
  12. A. Bononi, M. Bertolini, P. Serena, and G. Bellotti, “Cross-Phase Modulation Induced by OOK Channels on Higher-Rate DQPSK and Coherent QPSK Channels,” J. Lightwave Technol.27(18), 3974–3983 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-18-3974 . [CrossRef]
  13. F. Rice, B. Cowley, B. Moran, and M. Rice, “Cramér-Rao Lower Bounds for QAM Phase and Frequency Estimation,” IEEE Trans. Commun.49(9), 1582–1591 (2001). [CrossRef]
  14. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra Series Transfer Function of Single-Mode Fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997). [CrossRef]
  15. B. Xu and M. B. Pearce, “Modified Volterra Series Transfer Function,” IEEE Photon. Technol. Lett.14(1), 47–49 (2002). [CrossRef]
  16. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express18(8), 8660–8670 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8660 . [CrossRef] [PubMed]
  17. N. Shibata, R. P. Braun, and R. G. Waarts, “Phase Mismatch Dependence of Efficiency of Wave Generation Through Four-Wave Mixing in a Single Mode Optical Fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987). [CrossRef]
  18. A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWO7. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-OWO7

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