Abstract

We address the issue of intra-channel nonlinear compensation using a Volterra series nonlinear equalizer based on an analytical closed-form solution for the 3rd order Volterra kernel in frequency-domain. The performance of the method is investigated through numerical simulations for a single-channel optical system using a 20 Gbaud NRZ-QPSK test signal propagated over 1600 km of both standard single-mode fiber and non-zero dispersion shifted fiber. We carry on performance and computational effort comparisons with the well-known backward propagation split-step Fourier (BP-SSF) method. The alias-free frequency-domain implementation of the Volterra series nonlinear equalizer makes it an attractive approach to work at low sampling rates, enabling to surpass the maximum performance of BP-SSF at 2× oversampling. Linear and nonlinear equalization can be treated independently, providing more flexibility to the equalization subsystem. The parallel structure of the algorithm is also a key advantage in terms of real-time implementation.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-26-20-3416
    [CrossRef]
  2. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-880
    [CrossRef] [PubMed]
  3. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).
  4. C. Xia and W. Rosenkranz, “Nonlinear electrical equalization for different modulation formats with optical filtering,” J. Lightwave Technol. 25(4), 996–1001 (2007). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-25-4-996
    [CrossRef]
  5. Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
    [CrossRef]
  6. Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThA040. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-JThA040
  7. R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuE3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OTuE3
  8. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. 15(12), 2232–2241 (1997).
    [CrossRef]
  9. 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]
  10. F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
    [CrossRef]
  11. B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002).
    [CrossRef]
  12. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17075
    [CrossRef] [PubMed]

2011 (1)

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

2010 (2)

2009 (1)

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

2008 (2)

2007 (1)

2002 (1)

B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002).
[CrossRef]

1997 (1)

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. 15(12), 2232–2241 (1997).
[CrossRef]

Brandt-Pearce, M.

B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002).
[CrossRef]

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. 15(12), 2232–2241 (1997).
[CrossRef]

Chen, X.

Chen, Z.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

Dou, L.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

Du, L. B.

Gao, Y.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

Goldfarb, G.

Guiomar, F. P.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

Ip, E.

Kahn, J. M.

Kim, I.

Li, G.

Li, X.

Lowery, A. J.

Mateo, E.

Peddanarappagari, K.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. 15(12), 2232–2241 (1997).
[CrossRef]

Pinto, A. N.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

Reis, J. D.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

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]

Rosenkranz, W.

Schetzen, M.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).

Teixeira, A. L.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

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]

Xia, C.

Xu, A.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

Xu, B.

B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002).
[CrossRef]

Yaman, F.

Zgang, F.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[CrossRef]

B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. 14(1), 47–49 (2002).
[CrossRef]

J. Light-wave Technol. (1)

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. 15(12), 2232–2241 (1997).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Commun. (1)

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[CrossRef]

Opt. Express (3)

Other (3)

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).

Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThA040. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-JThA040

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuE3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OTuE3

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

FD-VSNE implementation for each fiber span. Black solid lines represent the equalizer operations; Blue dashed lines represent the equalizer control plan.

Fig. 2
Fig. 2

Coherent NRZ-QPSK optical system model adopted in this work. DSP - Digital Signal Processor; ADC - Analog-to-Digital Converter; LPF - Low-Pass Filter.

Fig. 3
Fig. 3

Equalization results (in terms of EVM) obtained for a 20 Gbaud NRZ-QPSK signal transmitted over 20 × 80 km. a) SSMF link with Nsp = 3; b) SSMF link with Nsp = 2; c) NZDSF link with Nsp = 3; d) NZDSF link with Nsp = 2. For simplicity, both FD-CDE and FD-VSNE have been abbreviated to CDE and VSNE respectively.

Fig. 4
Fig. 4

EVM after nonlinear equalization as a function of the LPF cutoff frequency. a) Nsp = 3; b) Nsp = 2. Input optical power is 6 dBm.

Fig. 5
Fig. 5

Optical spectra of the propagated signal before and after the LPF and after down-sampling at 3 samples per symbol. The LPF cutoff frequency is placed at 18 GHz.

Fig. 6
Fig. 6

EVM as a function of the effective spectral support used to apply the FD-VSNE and LPF cutoff frequency (LPF3dB). The signal fed to the equalization block is sampled at 3 samples per symbol and the input power is 6 dBm.

Fig. 7
Fig. 7

Number of complex multiplies required by FD-VSNE and BP-SSF as a function of the FFT block-size.

Fig. 8
Fig. 8

Inter-block interference as a function of the FFT block-size. Red lines refer to the implementation of OS only at the link ends. Black lines are in respect with the span-by-span implementation of OS. Input power is 4 dBm; Nsp = 2.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

A z = α 2 A i β 2 2 2 A t 2 + i γ | A | 2 A ,
A ˜ in ( ω ) H 1 ( ω ) A ˜ out ( ω ) + H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 ) × A ˜ out ( ω 1 ) A ˜ out * ( ω 2 ) A ˜ out ( ω ω 1 + ω 2 ) d ω 1 d ω 2 ,
H 1 ( ω ) = exp ( α 2 L span i β 2 2 ω 2 L span ) ,
H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 ) = i γ H 1 ( ω ) 1 exp ( α L span i β 2 ( ω 1 ω ) ( ω 1 ω 2 ) L span ) α + i β 2 ( ω 1 ω ) ( ω 1 ω 2 ) .
A ˜ ( ω n ) = 1 N F F T k = 1 N F F T A ( t k ) exp ( i 2 π ( n 1 ) ( k 1 ) N F F T ) ,
A ˜ e q N L ( ω n ) = n 2 = 1 N F F T n 1 = 1 N F F T H 3 ( ω n 1 , ω n 2 , ω n ω n 1 + ω n 2 ) A ˜ out ( ω n 1 ) A ˜ out * ( ω n 2 ) A ˜ out ( ω n ω n 1 + ω n 2 ) ,
A e q = { A e q L I exp ( A e q N L A e q L I ) , if | A e q N L | < | A e q L I | A e q L I + A e q N L , otherwise ,

Metrics