Abstract

By using a generalized cost function, a modified constant modulus algorithm (CMA) that allows polarization demultiplexing and equalization of polarization-switched QPSK is found. An implementation that allows easy switching between the conventional and the modified CMA is described. Using numerical simulations, the suggested algorithm is shown to have similar performance for polarization-switched QPSK as CMA has for polarization-multiplexed QPSK.

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References

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  1. H. Bülow, “Polarization QAM modulation (POL-QAM) for coherent detection schemes,” in Optical Fiber Communication Conference (OFC) (2009), paper OWG2.
  2. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
    [CrossRef] [PubMed]
  3. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
    [CrossRef]
  4. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PS-QPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express 18, 11360–11371 (2010).
    [CrossRef] [PubMed]
  5. P. Serena, A. Vannucci, and A. Bononi, “The performance of polarization switched-QPSK (PS-QPSK) in dispersion managed WDM transmissions,” in European Conference on Optical Communication (ECOC) (2010), paper Th.10.E.2.
    [CrossRef]
  6. M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, and M. Karlsson, “Experimental comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express (submitted).
    [PubMed]
  7. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980).
    [CrossRef]
  8. A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
    [CrossRef]
  9. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008).
    [CrossRef] [PubMed]
  10. K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE/LEOS Summer Topical Meetings (LEOSST) (2008), paper MC2.2.
  11. I. Roudas, A. Vgenis, C. S. Petrou, D. Toumpakaris, J. Hurley, M. Sauer, J. Downie, Y. Mauro, and S. Raghavan, “Optimal polarization demultiplexing for coherent optical communications systems,” J. Lightwave Technol. 28, 1121–1134 (2010).
    [CrossRef]
  12. L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (OFC) (2009), paper OMT2.
  13. U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).
  14. T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarisation dependent loss on coherent POLMUX-NRZ-DQPSK,” in Optical Fiber Communication Conference (OFC) (2008), paper OThU5.
  15. M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27, 3614–3622 (2009).
    [CrossRef]

2010

2009

2008

U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008).
[CrossRef] [PubMed]

2007

A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
[CrossRef]

1980

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980).
[CrossRef]

Agrell, E.

Alfiad, M. S.

Bosco, G.

Carena, A.

Curri, V.

Downie, J.

Forghieri, F.

Gesbert, D.

A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
[CrossRef]

Godard, D. N.

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980).
[CrossRef]

Hauske, F. N.

Hjørungnes, A.

A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
[CrossRef]

Hurley, J.

Karlsson, M.

Kuschnerov, M.

Lankl, B.

Madhow, U.

U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).

Mauro, Y.

Napoli, A.

Petrou, C. S.

Piyawanno, K.

Poggiolini, P.

Raghavan, S.

Roudas, I.

Sauer, M.

Savory, S. J.

Spinnler, B.

Toumpakaris, D.

Vgenis, A.

IEEE Trans. Commun.

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980).
[CrossRef]

IEEE Trans. Signal Process.

A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Other

H. Bülow, “Polarization QAM modulation (POL-QAM) for coherent detection schemes,” in Optical Fiber Communication Conference (OFC) (2009), paper OWG2.

P. Serena, A. Vannucci, and A. Bononi, “The performance of polarization switched-QPSK (PS-QPSK) in dispersion managed WDM transmissions,” in European Conference on Optical Communication (ECOC) (2010), paper Th.10.E.2.
[CrossRef]

M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, and M. Karlsson, “Experimental comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s,” Opt. Express (submitted).
[PubMed]

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (OFC) (2009), paper OMT2.

U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).

T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarisation dependent loss on coherent POLMUX-NRZ-DQPSK,” in Optical Fiber Communication Conference (OFC) (2008), paper OThU5.

K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE/LEOS Summer Topical Meetings (LEOSST) (2008), paper MC2.2.

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Figures (3)

Fig. 1
Fig. 1

(a) PS-QPSK after a T 1 rotation. (b) PM-QPSK after a T 1 rotation. (c) PS-QPSK after a T 1 T 2 T 1 rotation. (d) PM-QPSK after a T 1 T 2 T 1 rotation.

Fig. 2
Fig. 2

(a) The probability to be above the SNR penalty limit of 1 dB as a function of the number of processed symbols. (b) The SNR penalty (in dB) as a function of the angular drift frequency of the fiber matrix. The step sizes are indicated close to each curve.

Fig. 3
Fig. 3

The averaged SNR penalty as a function of the amount of PDL. The PDL alignments leading to the maximum and minimum penalties, respectively, have been used to plot the dashed black lines.

Equations (16)

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T 1 = 1 2 ( 1 1 1 1 ) and T 2 = ( 1 0 0 e j φ ) .
J PM CMA = 𝔼 [ ( | E x | 2 P 0 ) 2 + ( | E y | 2 P 0 ) 2 ] ,
s 0 = | E x | 2 + | E y | 2 , s 1 = | E x | 2 | E y | 2 , s 2 = 2 Re ( E x E y * ) , s 3 = 2 Im ( E x E y * ) ,
J PM CMA = 𝔼 [ ( s 0 2 P 0 ) 2 2 + s 1 2 2 ] .
J = 𝔼 [ ( s 0 P ) 2 2 + Q s 2 2 + s 3 2 2 ] = 𝔼 [ ( | E x | 2 + | E y | 2 P ) 2 2 + 2 Q | E x | 2 | E y | 2 ] ,
y 1 = h 11 T x 1 + h 12 T x 2 , and y 2 = h 21 T x 1 + h 22 T x 2 .
h 11 ( k + 1 ) = h 11 ( k ) μ [ | y 1 | 2 + ( 1 + 2 Q ) | y 2 | 2 P ] y 1 x 1 * ,
h 12 ( k + 1 ) = h 12 ( k ) μ [ | y 1 | 2 + ( 1 + 2 Q ) | y 2 | 2 P ] y 1 x 2 * ,
h 21 ( k + 1 ) = h 21 ( k ) μ [ ( 1 + 2 Q ) | y 1 | 2 + | y 2 | 2 P ] y 2 x 1 * ,
h 22 ( k + 1 ) = h 22 ( k ) μ [ ( 1 + 2 Q ) | y 1 | 2 + | y 2 | 2 P ] y 2 x 2 * ,
M 1 = ( 1 0 1 0 ) and M 2 = ( 1 1 0 0 ) .
A = ( cos ϕ ( t ) sin ϕ ( t ) sin ϕ ( t ) cos ϕ ( t ) ) ,
( y 1 y 2 ) = ( C 11 C 12 C 21 C 22 ) ( a 1 a 2 ) + ( B 11 B 12 B 21 B 22 ) ( n 1 n 2 ) .
SNR 1 = 𝔼 [ | C 11 a 1 | 2 ] 𝔼 [ | C 11 n 1 + B 12 a 2 + B 12 n 2 | 2 ] = | C 11 | 2 P s | C 12 | 2 P s + 2 σ 2 ( | B 11 | 2 + | B 12 | 2 )
SNR k pen = SNR nom min ( SNR 1 , k , SNR 2 , k ) .
B MMSE = ( A ˜ H A ˜ + 2 σ 2 P s I ) 1 A ˜ H .

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