Abstract

We propose a reduced-complexity space-demultiplexing algorithm based on higher-order Poincaré spheres (HoPs) which is modulation format agnostic, free of training sequences and robust to the local oscillator phase fluctuations and frequency offsets. The signal tributaries are space-demultiplexed by calculating and realigning the best fit plane in the HoPs, with the inverse channel matrix being iteratively constructed by sequentially space-demultiplexing all pairs of tributaries. When compared with the previous proposed HoPs-based space-demultiplexing algorithm, results show a complexity reduction gain of 99% along with an improvement of 97% in terms of convergence speed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  8. F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
    [Crossref]
  9. T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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2017 (3)

2016 (3)

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

X. Pan, B. Liu, L. Li, and Q. Tian, “Low complexity MIMO method based on matrix transformation for few-mode multi-core optical transmission system,” Opt. Commun. 371, 238–242 (2016).
[Crossref]

2015 (3)

2014 (4)

2013 (2)

N. J. Muga and A. N. Pinto, “Digital PDL compensation in 3D Stokes space,” J. Lightwave Technol. 31(13), 2122–2130 (2013).
[Crossref]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

2012 (1)

N. Bai and G. Li, “Adaptive frequency-domain equalization for mode-division multiplexed transmission,” IEEE Photonics Technol. Lett. 24(21), 1918–1921 (2012).
[Crossref]

2011 (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
[Crossref] [PubMed]

2010 (2)

S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

B. Szafraniec, B. Nebendahl, and T. Marshall, “Polarization demultiplexing in Stokes space,” Opt. Express 18(17), 17928–17939 (2010).
[Crossref] [PubMed]

Aiello, A.

Alfano, R. R.

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Arik, S. Ö.

S. Ö. Arik, J. M. Kahn, and K.-P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Proc. Mag. 31(2), 25–34 (2014).
[Crossref]

S. Ö. Arik, D. Askarov, and J. M. Kahn, “Adaptive frequency-domain equalization in mode-division multiplexing systems,” J. Lightwave Technol. 32(10), 1841–1852 (2014).
[Crossref]

Askarov, D.

Awaji, Y.

Bai, N.

N. Bai and G. Li, “Adaptive frequency-domain equalization for mode-division multiplexed transmission,” IEEE Photonics Technol. Lett. 24(21), 1918–1921 (2012).
[Crossref]

Buchali, F.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling separation of channel compensation and polarization demultiplex,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th2A.15.
[Crossref]

Buelow, H.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling separation of channel compensation and polarization demultiplex,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th2A.15.
[Crossref]

Caballero, F. J. V.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Chen, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Du, C.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Dudley, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Essiambre, R. J.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Fernandes, G. M.

Ferreira, M. F. S.

M. F. S. Ferreira, Optical Fibers: Technology, Communications and Recent Advances (Nova Publishers, 2017), Chap.3.

Ferreira, R. M.

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Fontaine, N. K.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Forbes, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Gabriel, C.

Goeger, G.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Golub, G.

G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 2013), Chap. 2.

Guiomar, F. P.

Hayashi, T.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

He, Y.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Ho, K. P.

Ho, K.-P.

S. Ö. Arik, J. M. Kahn, and K.-P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Proc. Mag. 31(2), 25–34 (2014).
[Crossref]

Holleczek, A.

Hu, T.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Idler, W.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling separation of channel compensation and polarization demultiplex,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th2A.15.
[Crossref]

Kahn, J. M.

Kaminow, I. P.

I. P. Kaminow, T. Li, and A. E. Willner, Optical Fiber Telecommunications VIB: Systems and Networks, (Academic, 2013), Chap. 11.

Ke, Y.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Klaus, W.

Leuchs, G.

Li, G.

N. Bai and G. Li, “Adaptive frequency-domain equalization for mode-division multiplexed transmission,” IEEE Photonics Technol. Lett. 24(21), 1918–1921 (2012).
[Crossref]

Li, J.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Li, L.

X. Pan, B. Liu, L. Li, and Q. Tian, “Low complexity MIMO method based on matrix transformation for few-mode multi-core optical transmission system,” Opt. Commun. 371, 238–242 (2016).
[Crossref]

Li, T.

I. P. Kaminow, T. Li, and A. E. Willner, Optical Fiber Telecommunications VIB: Systems and Networks, (Academic, 2013), Chap. 11.

Li, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Litvin, I.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Liu, B.

X. Pan, B. Liu, L. Li, and Q. Tian, “Low complexity MIMO method based on matrix transformation for few-mode multi-core optical transmission system,” Opt. Commun. 371, 238–242 (2016).
[Crossref]

Liu, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Luís, R. S.

Marquardt, C.

Marrucci, L.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Marshall, T.

Mendinueta, J.-M. D.

Milione, G.

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Mo, Q.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Monroy, I. T.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Muga, N. J.

Naidoo, D.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Nebendahl, B.

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Nolan, D. A.

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Pan, X.

X. Pan, B. Liu, L. Li, and Q. Tian, “Low complexity MIMO method based on matrix transformation for few-mode multi-core optical transmission system,” Opt. Commun. 371, 238–242 (2016).
[Crossref]

Piccirillo, B.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Pinto, A. N.

Pittalà, F.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Puttnam, B. J.

Randel, S.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Ren, F.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Richardson, D.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Roux, F. S.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Ryf, R.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Sakaguchi, J.

Sasaki, T.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Savory, S. J.

S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

Schuh, K.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling separation of channel compensation and polarization demultiplex,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th2A.15.
[Crossref]

Shahpari, A.

Stein, S. K.

S. K. Stein, Calculus and Analytic Geometry, (McGraw-Hill, 1982).

Szafraniec, B.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Tang, R.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Teixeira, A. L.

Tian, Q.

X. Pan, B. Liu, L. Li, and Q. Tian, “Low complexity MIMO method based on matrix transformation for few-mode multi-core optical transmission system,” Opt. Commun. 371, 238–242 (2016).
[Crossref]

Van Loan, C.

G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University, 2013), Chap. 2.

Wada, N.

Wang, M.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Willner, A. E.

I. P. Kaminow, T. Li, and A. E. Willner, Optical Fiber Telecommunications VIB: Systems and Networks, (Academic, 2013), Chap. 11.

Winzer, P. J.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Xia, C.

T. Hayashi, R. Ryf, N. K. Fontaine, C. Xia, S. Randel, R. J. Essiambre, P. J. Winzer, and T. Sasaki, “Coupled-core multi-core fibers: High-spatial-density optical transmission fibers with low differential modal properties,” in European Conference on Optical Communication (ECOC) (2015).

Ye, Y.

F. J. V. Caballero, F. Pittalà, G. Goeger, M. Wang, Y. Ye, and I. T. Monroy, “Novel equalization techniques for space-division multiplexing based on Stokes space update rule,” Photonics 4(1), 12 (2017).
[Crossref]

Yu, J.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of bidirectional PON based on mode-division multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201 (2016).
[Crossref]

Ziaie, S.

IEEE J. Sel. Top. Quantum Electron. (1)

S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

IEEE Photonics Technol. Lett. (2)

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Figures (8)

Fig. 1
Fig. 1 Schematic representation of the reduced-complexity space-demultiplexing (SpDemux) algorithm: (a) SpDemux step, (b) SpDemux filter and (c) iterative process or reduced-complexity algorithm. Note that, dashed lines depict the path for the residuals while solid lines represent the path of the samples.
Fig. 2
Fig. 2 (a) Schematic representation of a SpDemux filter using the reduced-complexity space-demultiplexing algorithm. (b) Schematic representation of the space-demultiplexing algorithm proposed in [11]. The SpDemux step, which is common to both algorithms, is represented by a block with two sides (gray and white) corresponding to the two output signals obtained with n⃗p and −n⃗p, respectively.
Fig. 3
Fig. 3 (a) Average SNR penalty after applying 1, 2 and 3 SpDemux filters along with the proposed algorithm, considering εST = 10% and 1% for a PM-QPSK signal with a SNR of 17 dB. (b) Average number of SpDemux filters required by the proposed algorithm for εST = 10% and 1%. Insets show the received PM-QPSK signals after space-demultiplexing assuming εST < 10%.
Fig. 4
Fig. 4 Average SNR penalty (solid lines) and number of SpDemux filters (dashed lines) as function of εST for several numbers of samples considered in the calculations of the inverse channel matrix for: (a) PM-QPSK signal; (b) PM-16QAM signal; and (c) PM-64QAM.
Fig. 5
Fig. 5 Average value of SNR penalty (solid lines) and number of SpDemux filters (dashed lines) as function of εST for a SDM system based on 2-, 3-, 4-, 5- and 7-cores CC-MCF, with each core carrying a PM-QPSK signal. In the calculation of the inverse channel matrix are assumed 100 samples.
Fig. 6
Fig. 6 Average value of SNR penalty as function of εST for a SDM system with a 3-core CC-MCF, each core carrying a PM-QPSK signal, considering values of SNR equal to 10, 20 and 30 dB.
Fig. 7
Fig. 7 (a) Average value of SNR penalty as function of the number of samples computed after space-demultiplexing through the static, the reduced-complexity algorithm, assuming εST = 90, 50 and 5%, and with the NLMS, assuming μ = 0.1 and 0.01. (b) Zoom in of Fig. 7(a), with the SNR penalty in log scale and the threshold of Δ = 0.1 dB pointed out by a solid line. It is used a SDM system based on a 2-core CC-MCF transmitting PM-QPSK signals with a SNR of 17 dB.
Fig. 8
Fig. 8 Average value of SNR penalty for the reduced-complexity and the static algorithm as function of the SNR of the transmitted signal for a SDM system with a 2-core CC-MCF, each core carrying a PM-QPSK signal.

Equations (17)

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| ϕ = ( υ x 1 , υ y 1 , υ x n , υ y n ) T ,
| ϕ = ( υ 1 , υ 2 , υ h , υ 2 n ) T ,
Ψ ( f , g ) = ϕ | Λ ( f , g ) | ϕ ,
Λ ( f , g ) = ( Λ 1 ( f , g ) , Λ 2 ( f , g ) , Λ 3 ( f , g ) ) T ,
ζ = j = 1 g s | r j | ,
ε = ζ ( n + 1 ) ζ ( n ) ζ ( n + 1 ) × 100 % ,
| ψ out = M tot ( ω ) | ψ in ,
M tot ( ω ) = [ k = 1 n s M MD k ( ω ) ] e i 2 ω 2 β ¯ 2 L ,
M MD k ( ω ) = diag ( e g 1 k 2 e g 2 n k 2 ) l = 1 n step V k l Θ ( ω ) U k l H ,
Θ ( ω ) = diag ( e i ω τ 1 e i ω τ i e i ω τ 2 n ) ,
Δ = SNR out SNR in ,
N HoPs , ref = 2 g s g s ! ,
N HoPs = g = 2 g s g [ 2 ( g 1 ) ] = 2 g s 3 ( g s 2 1 ) ,
N m = 12 n s 2 + 5 n s + 198 ,
N s = 12 n s 2 + 5 n s + 196 ,
G CR ( dB ) = 10 log ( N f N HoPs N m N HoPs , ref N m ) .
F ( f , g ) ( k , l ) = { cos ( p ) e i q 2 if k = g , l = g sin ( p ) e i q 2 if k = g , l = f sin ( p ) e i q 2 if k = f , l = g cos ( p ) e i q 2 if k = f , l = f 1 if k = l , and k , l g , f 0 otherwise ,

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