Modified constant modulus algorithm for polarization-switched QPSK

Pontus Johannisson; Martin Sjödin; Magnus Karlsson; Henk Wymeersch; Erik Agrell; Peter A. Andrekson

doi:10.1364/OE.19.007734

Modified constant modulus algorithm for polarization-switched QPSK

Pontus Johannisson, Martin Sjödin, Magnus Karlsson, Henk Wymeersch, Erik Agrell, and Peter A. Andrekson

Optics Express
Vol. 19,
Issue 8,
pp. 7734-7741
(2011)
•https://doi.org/10.1364/OE.19.007734

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Pontus Johannisson, Martin Sjödin, Magnus Karlsson, Henk Wymeersch, Erik Agrell, and Peter A. Andrekson, "Modified constant modulus algorithm for polarization-switched QPSK," Opt. Express 19, 7734-7741 (2011)

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Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Coherent communications (060.1660)
- Fiber optics communications (060.2330)
- Modulation (060.4080)

About this Article
History
- Original Manuscript: February 16, 2011
- Revised Manuscript: March 27, 2011
- Manuscript Accepted: March 27, 2011
- Published: April 6, 2011

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Open Access

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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Year
|
Author
|
Publication

H. Bülow, “Polarization QAM modulation (POL-QAM) for coherent detection schemes,” in Optical Fiber Communication Conference (OFC) (2009), paper OWG2.
M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
[Crossref] [PubMed]
E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
[Crossref]
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]
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]
D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28, 1867–1875 (1980).
[Crossref]
A. Hjørungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” IEEE Trans. Signal Process. 55, 2740–2746 (2007).
[Crossref]
S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008).
[Crossref] [PubMed]
K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in IEEE/LEOS Summer Topical Meetings (LEOSST) (2008), paper MC2.2.
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]
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.
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]

2007 (1)

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

1980 (1)

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.

M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
[Crossref] [PubMed]

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
[Crossref]

Alfiad, M. S.

Andrekson, P.

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]

Bononi, A.

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]

Bosco, G.

Bülow, H.

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

Carena, A.

Curri, V.

Downie, J.

Duthel, T.

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.

Fludger, C. R. S.

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.

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]

Geyer, J.

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.

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]

Hoshida, T.

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.

Hurley, J.

Johannisson, P.

Karlsson, M.

M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
[Crossref] [PubMed]

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
[Crossref]

Kikuchi, K.

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

Kuschnerov, M.

Lankl, B.

Liu, L.

Madhow, U.

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

Mauro, Y.

Napoli, A.

Oda, S.

Petrou, C. S.

Piyawanno, K.

Poggiolini, P.

Raghavan, S.

Rasmussen, J. C.

Roudas, I.

Sauer, M.

Savory, S. J.

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

Schulien, C.

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.

Serena, P.

Sjödin, M.

Spinnler, B.

Tao, Z.

Toumpakaris, D.

Vannucci, A.

Vgenis, A.

Wymeersch, H.

Yan, W.

IEEE Trans. Commun. (1)

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. (1)

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. (3)

E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. 27, 5115–5126 (2009).
[Crossref]

Opt. Express (3)

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

M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express 17, 10814–10819 (2009).
[Crossref] [PubMed]

Other (7)

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

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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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.

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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.

Download Full Size | PPT Slide | PDF

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.

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Equations (16)

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

T_{1} = \frac{1}{\sqrt{2}} (\begin{array}{r} 1 & 1 \\ - 1 & 1 \end{array}) and T_{2} = (\begin{matrix} 1 & 0 \\ 0 & e^{j φ} \end{matrix}) .

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 [\frac{{(s_{0} - 2 P_{0})}^{2}}{2} + \frac{s_{1}^{2}}{2}] .

J = 𝔼 [\frac{{(s_{0} - P)}^{2}}{2} + Q \frac{s_{2}^{2} + s_{3}^{2}}{2}] = 𝔼 [\frac{{(| 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} = (\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}) and M_{2} = (\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}) .

A = (\begin{array}{r} cos ϕ (t) & sin ϕ (t) \\ - sin ϕ (t) & cos ϕ (t) \end{array}),

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{matrix}) (\begin{matrix} a_{1} \\ a_{2} \end{matrix}) + (\begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix}) (\begin{matrix} n_{1} \\ n_{2} \end{matrix}) .

{SNR}_{1} = \frac{𝔼 [| C_{11} a_{1} |^{2}]}{𝔼 [| C_{11} n_{1} + B_{12} a_{2} + B_{12} n_{2} |^{2}]} = \frac{| 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} = {({\tilde{A}}^{H} \tilde{A} + \frac{2 σ^{2}}{P_{s}} I)}^{- 1} {\tilde{A}}^{H} .

Abstract

References

2010 (2)

2009 (3)

2008 (1)

2007 (1)

1980 (1)

Agrell, E.

Alfiad, M. S.

Andrekson, P.

Bononi, A.

Bosco, G.

Bülow, H.

Carena, A.

Curri, V.

Downie, J.

Duthel, T.

Fludger, C. R. S.

Forghieri, F.

Gesbert, D.

Geyer, J.

Godard, D. N.

Hauske, F. N.

Hjørungnes, A.

Hoshida, T.

Hurley, J.

Johannisson, P.

Karlsson, M.

Kikuchi, K.

Kuschnerov, M.

Lankl, B.

Liu, L.

Madhow, U.

Mauro, Y.

Napoli, A.

Oda, S.

Petrou, C. S.

Piyawanno, K.

Poggiolini, P.

Raghavan, S.

Rasmussen, J. C.

Roudas, I.

Sauer, M.

Savory, S. J.

Schulien, C.

Serena, P.

Sjödin, M.

Spinnler, B.

Tao, Z.

Toumpakaris, D.

Vannucci, A.

Vgenis, A.

Wymeersch, H.

Yan, W.

IEEE Trans. Commun. (1)

IEEE Trans. Signal Process. (1)

J. Lightwave Technol. (3)

Opt. Express (3)

Other (7)

Cited By

Figures (3)

Equations (16)

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