Abstract

In this paper, we propose a constant modulus algorithm (CMA) for mode division multiplexing (MDM) systems with improved convergence performance. In order to adapt to sparse channels with large differential mode group delay (DMGD) in MDM systems, the CMA adopts a variable step size similar to the improved proportionate normalized least-mean-square (IPNLMS) algorithm. In additional to that, when a singularity problem is encountered or the tap values fail to converge, it reinitializes the tap coefficients according to the tap vectors of the successfully de-multiplexed data tributaries. The proposed initialization approach is based on the fact that the channel matrix is unitary in the frequency domain in the absence of mode dependent loss (MDL), which means the channel coefficient vectors for each data tributary should be orthogonal to each other. By limiting the initial values of the taps within the null space of the complex conjugate vectors of the successfully de-multiplexed channels, singularity can be effectively avoided and the convergence of the taps is guaranteed. When the number of modes is two, the proposed algorithm becomes the constrained CMA, which has been commonly implemented in polarization division multiplexing (PDM) systems. Although the algorithm has been derived under zero MDL assumption, it is found that the proposed CMA can be quite resilient to MDL. No singularity/tap convergence failure problem occurs when the MDL is below 4 dB at both the input and the output ports.

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

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References

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2016 (2)

2015 (5)

K. Shi and B. C. Thomsen, “Sparse Adaptive Frequency Domain Equalizers for Mode-Group Division Multiplexing,” J. Lightwave Technol. 33(2), 311–317 (2015).
[Crossref]

I. Gasulla and J. M. Kahn, “Performance of Direct-Detection Mode-Group-Division Multiplexing using Fused Fiber Couplers,” J. Lightwave Technol. 33(9), 1748–1760 (2015).
[Crossref]

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

S. Ö. Arık, K.-P. Ho, and J. M. Kahn, “Delay Spread Reduction in Mode-Division Multiplexing: Mode Coupling Versus Delay Compensation,” J. Lightwave Technol. 33(21), 4504–4512 (2015).
[Crossref]

2014 (3)

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-Model of Fiber Non-Linear Propagation and its Applications,” J. Lightwave Technol. 32(4), 694–721 (2014).
[Crossref]

B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

2012 (2)

2011 (4)

2010 (1)

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

2005 (1)

1990 (1)

O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of non-minimum phase systems (channels),” IEEE Trans. Inf. Theory 36(2), 312–321 (1990).
[Crossref]

Agrawal, G. P.

Ahmed, N.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Al Amin, A.

Alfano, R. R.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

An Nguyen, T.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Arik, S. Ö.

Bai, N.

Bergmen, K.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Bosco, G.

Cao, Y.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Carena, A.

Chen, C. P.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Chen, S.

Chen, X.

Curri, V.

Cvijetic, M.

B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

Djordjevic, B.

B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

Essiambre, R.-J.

Ferreira, H. B.

Forghieri, F.

Fu, S.

Gabrielli, L. H.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Gao, G.

Gasulla, I.

He, Z.

Ho, K.-P.

Hu, G.

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

Hu, R.

Huang, H.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Ippen, E. P.

Jiang, Y.

Kahn, J. M.

Kawakami, H.

Kikuchi, K.

Kobayashi, T.

Lavery, M. P. J.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Lee, D.

Li, A.

Li, C.

Li, G.

Li, L.

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

Li, M. J.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Li, X.

Lin, C.

B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

Lipson, M.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Luo, L. W.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Luo, M.

Mello, D. A. A.

Milione, G.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Miyamoto, Y.

Mizuno, T.

Mumtaz, S.

Nakagawa, T.

Nolan, D. A.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Ophir, N.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Papadias, C.

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

Petrou, C.

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

Phua, P. B.

Poggiolini, P.

Poitras, C. B.

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Portela, T. F.

Qiu, Y.

Raptis, L.

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

Ren, Y.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Roudas, I.

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

Rozental, V. N.

Sano, A.

Shalvi, O.

O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of non-minimum phase systems (channels),” IEEE Trans. Inf. Theory 36(2), 312–321 (1990).
[Crossref]

Shi, K.

Shibahara, K.

Shieh, W.

Souto, D. V.

Takara, H.

Thomsen, B. C.

Tur, M.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Vgenis, A.

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

Wang, H.

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

Weinstein, E.

O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of non-minimum phase systems (channels),” IEEE Trans. Inf. Theory 36(2), 312–321 (1990).
[Crossref]

Willner, A. E.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Xia, C.

Xie, G.

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Yan, L.

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

Yang, Q.

Yu, S.

Zhao, L.

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (2)

A. Vgenis, C. Petrou, C. Papadias, I. Roudas, and L. Raptis, “Nonsingular Constant Modulus Equalizer for PDM-QPSK Coherent Optical Receivers,” IEEE Photonics Technol. Lett. 22(1), 45–47 (2010).
[Crossref]

L. Zhao, G. Hu, L. Yan, H. Wang, and L. Li, “Mode Demultiplexing Based on Frequency-Domain-Independent Component Analysis,” IEEE Photonics Technol. Lett. 27(2), 185–188 (2015).
[Crossref]

IEEE Signal Proc. (1)

B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

IEEE Trans. Inf. Theory (1)

O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of non-minimum phase systems (channels),” IEEE Trans. Inf. Theory 36(2), 312–321 (1990).
[Crossref]

J. Lightwave Technol. (7)

Nat. Commun. (1)

L. W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5(1), 3069 (2014).
[Crossref] [PubMed]

Opt. Express (6)

Sci. Rep. (1)

H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Other (7)

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 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

C. Xie and S. Chandrasekhar, “Two-Stage Constant Modulus Algorithm Equalizer for Singularity Free Operation and Optical Performance Monitoring in Optical Coherent Receiver,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMK3.
[Crossref]

K.-P. Ho and J. M. Kahn, “Mode Coupling and its Impact on Spatially Multiplexed Systems,” in Optical Fiber Telecommunications VI (Invited Chapter), I. P. Kaminow, T. Li and A. E. Willner, eds. (Elsevier, 2013).

X. Zhang, Y. Li, W. Du, F. Zhang, Y. Bao, W. Li, and J. Wu, “Compensation of Mode Coupling of Mode-Division Multiplexing Transmission System with MIMO CMA,” in Asia Communications and Photonics Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper AW3E.1.
[Crossref]

S. R. Hedemann, “Hyperspherical Parameterization of Unitary Matrices,” https://arXiv:1303.5904 (2013).

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[Crossref]

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Figures (10)

Fig. 1
Fig. 1 The standard flowchart of the proposed algorithm.
Fig. 2
Fig. 2 Singularity occurrence with respect to different θ1 and θ2 with θ3 = 0 degree (a) conventional CMA with MDL = 0dB (b) proposed CMA with MDL = 0dB (c) conventional CMA with MDL = 6dB (d) proposed CMA with MDL = 6dB (e) conventional CMA with MDL = 7dB (f) proposed CMA with MDL = 7dB.
Fig. 3
Fig. 3 Singularity occurrence with respect to different θ1 and θ2 with θ3 = 48 degree (a) conventional CMA with MDL = 0dB (b) proposed CMA with MDL = 0dB (c) conventional CMA with MDL = 6dB (d) proposed CMA with MDL = 6dB (e) conventional CMA with MDL = 7dB (f) proposed CMA with MDL = 7dB.
Fig. 4
Fig. 4 (a) The optical schematic of the simulation setup (b) the basic signal processing block.
Fig. 5
Fig. 5 Tap coefficients obtained by the standard CMA.
Fig. 6
Fig. 6 Tap coefficients obtained by the proposed upgraded CMA with null space initialization.
Fig. 7
Fig. 7 Constellation of the four de-multiplexed signals when the tap values reach the steady state by the standard CMA. Sub-figure (d) indicates that one of the data tributary has not been successfully de-multiplexed, while Sub-figures (a) and (b) correspond to the same data tributary.
Fig. 8
Fig. 8 Constellation of the four de-multiplexed signals when the tap values reach the steady state by the proposed upgraded CMA with null space initialization. Four signals have been successfully de-multiplexed.
Fig. 9
Fig. 9 Singularity probability of the standard CMA and the proposed algorithm under different OSNRs and MDLs.
Fig. 10
Fig. 10 BER Vs OSNR under different conditions.

Tables (1)

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Table 1 Probability to successfully de-multiplex different number of signals by the two CMAs

Equations (11)

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a out =T a in
T( ω )=( e jω τ 1 e jω τ 2 e jω τ N )
T=A( ω )U( ω )
MDL=20log | λ T | max | λ T | min
μ ij ( l )= ( 1α ) 2L + ( 1+α ) μ 0 | w ij ( l ) | 2 k | w ij ( k ) |+ε
w i H w j = δ ij δ ij ={ 1i=j 0ij i,j1tok
w i H w j =0 i1tok jk+1toN
T=BD B=( a bc bd b * e a * ce d * f * a * de+ c * f * b * f a * cf+ d * e * a * df c * e * ) D= R 1 ( 1 β γ )R MDL=max( 10log( β ),10log( γ ) )
| a | 2 + | b | 2 =1 | c | 2 + | d | 2 =1 | e | 2 + | f | 2 =1
a=cos( θ 1 ) e i δ 1 b=sin( θ 1 ) e i κ 1 c=cos( θ 2 ) e i δ 2 d=sin( θ 2 ) e i κ 2 e=cos( θ 3 ) e i δ 3 f=sin( θ 3 ) e i κ 3 θ i [ 0, π 2 ], δ i [ 0,π ],and κ i [ 0,π ]
T=( k U k D k V k ) D k =diag( exp( 1 2 g 1 jω τ 1 ),,exp( 1 2 g N jω τ N ) )

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