Abstract

Polarization-division-multiplexed (PDM) four-level pulse amplitude modulation (PAM4) with coherent detection is a promising low cost solution for 80 km inter-datacenter transmissions at 100 Gb/s and beyond. In this paper, three modified adaptive equalization algorithms for the PDM-PAM4 optical coherent systems, i.e. signal-phase aid least-mean-square (SP-LMS) algorithm, training multi-modulus algorithm (TMMA) and cascaded four-modulus algorithm (CMMA-4), are proposed and compared. Based on the proposed algorithms, 112 Gb/s PDM-PAM4 transmission over 80 km standard single mode fiber (SSMF) in C-band for a bit error rate (BER) below 3.8e-3 is successfully demonstrated without optical amplifier, chromatic dispersion (CD) pre-compensation and extra carrier recovery operations.

© 2016 Optical Society of America

Full Article  |  PDF Article
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2016 (3)

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

L. Zhang, T. Zuo, Y. Mao, Q. Zhang, E. Zhou, G. N. Liu, and X. Xu, “Beyond 100-Gb/s Transmission Over 80-km SMF Using Direct-Detection SSB-DMT at C-Band,” J. Lightwave Technol. 34(2), 723–729 (2016).
[Crossref]

2015 (3)

2013 (3)

2010 (3)

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

J. T. Yuan and T. C. Lin, “Equalization and carrier phase recovery of CMA and MMA in blind adaptive receivers,” IEEE Trans. Signal Process. 58(6), 3206–3217 (2010).
[Crossref]

2009 (2)

2002 (1)

J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Comm. 20(5), 997–1015 (2002).
[Crossref]

1980 (1)

D. N. Godard, “Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

Alfiad, M. S.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

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(16), 3614–3622 (2009).
[Crossref]

Amann, M.

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Amann, M. C.

Bergmann, M.

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Borel, P. I.

Buhl, L. L.

Carlson, K.

Chandrasekhar, S.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

Che, D.

Chen, W.

Chen, X.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes Vector Direct Detection for Linear Complex Optical Channels,” J. Lightwave Technol. 33(3), 678–684 (2015).
[Crossref]

Chen, Y. K.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

Cheng, Q.

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

Chi, N.

Cunningham, D.

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

de Waardt, H.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

Dochhan, A.

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

Doerr, C. R.

Dong, P.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express 21(9), 11585–11589 (2013).
[Crossref] [PubMed]

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Dong, Z.

Dumont, G. A.

J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Comm. 20(5), 997–1015 (2002).
[Crossref]

Eiselt, M. H.

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

Eiselt, N.

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

Elbers, J. P.

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

Gao, Y.

Gnauck, A. H.

Godard, D. N.

D. N. Godard, “Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

Gréus, C.

Griesser, H.

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

Gui, T.

Hanik, N.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

Hauske, F. N.

Hu, Q.

Isaac, R.

Kim, K.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

Kögel, B.

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Kuschnerov, M.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

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(16), 3614–3622 (2009).
[Crossref]

Lankl, B.

Lau, A. P. K.

Li, A.

Li, X.

Lin, T. C.

J. T. Yuan and T. C. Lin, “Equalization and carrier phase recovery of CMA and MMA in blind adaptive receivers,” IEEE Trans. Signal Process. 58(6), 3206–3217 (2010).
[Crossref]

Liu, G. N.

Lu, C.

Machi, F.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

Magarini, M.

Magill, P.

Man, J.

Mao, Y.

Müller, M.

Napoli, A.

Nelson, L. E.

Neumeyr, C.

C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express 21(9), 11585–11589 (2013).
[Crossref] [PubMed]

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Ortsiefer, M.

Peckham, D. W.

Penty, R.

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

Piyawanno, K.

Shao, Y.

Shieh, W.

Sinsky, J. H.

P. Dong, X. Chen, K. Kim, S. Chandrasekhar, Y. K. Chen, and J. H. Sinsky, “128-Gb/s 100-km transmission with direct detection using silicon photonic Stokes vector receiver and I/Q modulator,” Opt. Express 34(13), 14208–14214 (2016).
[Crossref] [PubMed]

Spiga, S.

C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express 21(9), 11585–11589 (2013).
[Crossref] [PubMed]

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Spinnler, B.

Tao, L.

van den Borne, D.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

Wang, Y.

Wei, J.

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

Werner, J. J.

J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Comm. 20(5), 997–1015 (2002).
[Crossref]

White, I.

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

Winzer, P.

C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express 21(9), 11585–11589 (2013).
[Crossref] [PubMed]

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Winzer, P. J.

Wuth, T.

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

Xie, C.

C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express 21(9), 11585–11589 (2013).
[Crossref] [PubMed]

C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “Generation and transmission of a 400-Gb/s PDM/WDM signal using a monolithic 2x4 VCSEL array and coherent detection,” in Optical Fiber Communication Conference (OFC), San Francisco, California, 2014, paper Th5C.9.
[Crossref]

Xu, X.

Yang, J.

J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Comm. 20(5), 997–1015 (2002).
[Crossref]

Yu, J.

Yuan, J. T.

J. T. Yuan and T. C. Lin, “Equalization and carrier phase recovery of CMA and MMA in blind adaptive receivers,” IEEE Trans. Signal Process. 58(6), 3206–3217 (2010).
[Crossref]

Zeng, L.

Zhang, J.

Zhang, L.

Zhang, Q.

Zhong, K.

Zhou, E.

Zhou, X.

Zhu, B.

Zuo, T.

IEEE Commun. Mag. (1)

J. Wei, Q. Cheng, R. Penty, I. White, and D. Cunningham, “400 Gigabit Ethernet using advanced modulation formats: Performance, complexity, and power dissipation,” IEEE Commun. Mag. 53(2), 182–189 (2015).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Comm. 20(5), 997–1015 (2002).
[Crossref]

IEEE Photonics Technol. Lett. (1)

F. Machi, M. S. Alfiad, M. Kuschnerov, T. Wuth, D. van den Borne, N. Hanik, and H. de Waardt, “111-Gb/s PolMux-quadrature duobinary for robust and bandwidth efficient transmission,” IEEE Photonics Technol. Lett. 22(11), 751–753 (2010).
[Crossref]

IEEE Trans. Commun. (1)

D. N. Godard, “Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[Crossref]

IEEE Trans. Signal Process. (1)

J. T. Yuan and T. C. Lin, “Equalization and carrier phase recovery of CMA and MMA in blind adaptive receivers,” IEEE Trans. Signal Process. 58(6), 3206–3217 (2010).
[Crossref]

J. Lightwave Technol. (8)

A. Dochhan, H. Griesser, N. Eiselt, M. H. Eiselt, and J. P. Elbers, “Solutions for 80km DWDM Systems,” J. Lightwave Technol. 24(2), 491–499 (2016).
[Crossref]

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(16), 3614–3622 (2009).
[Crossref]

X. Zhou and J. Yu, “Multi-level, multi-dimensional coding for high-speed and high-spectral- efficiency optical transmission,” J. Lightwave Technol. 27(16), 3641–3653 (2009).
[Crossref]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[Crossref]

X. Zhou, L. E. Nelson, P. Magill, R. Isaac, B. Zhu, D. W. Peckham, P. I. Borel, and K. Carlson, “High spectral efficiency 400 Gb/s transmission using PDM time-domain hybrid 32–64 QAM and training-assisted carrier recovery,” J. Lightwave Technol. 31(7), 999–1005 (2013).
[Crossref]

J. Zhang, J. Yu, N. Chi, Z. Dong, J. Yu, X. Li, L. Tao, and Y. Shao, “Multi-Modulus Blind Equalizations for Coherent Quadrature Duobinary Spectrum Shaped PM-QPSK Digital Signal Processing,” J. Lightwave Technol. 31(7), 1073–1078 (2013).
[Crossref]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes Vector Direct Detection for Linear Complex Optical Channels,” J. Lightwave Technol. 33(3), 678–684 (2015).
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L. Zhang, T. Zuo, Y. Mao, Q. Zhang, E. Zhou, G. N. Liu, and X. Xu, “Beyond 100-Gb/s Transmission Over 80-km SMF Using Direct-Detection SSB-DMT at C-Band,” J. Lightwave Technol. 34(2), 723–729 (2016).
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[Crossref]

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Figures (14)

Fig. 1
Fig. 1 Digital equalizer based on different adaptive equalization algorithms.
Fig. 2
Fig. 2 (a) Input signal distributions for 28Gbaud PAM4 and (e) with the influence of carrier phase distortions. (b, f) after LMS-based equalization, (c, g) after CMA-based equalization, (d, h) after MMA-based equalization. (By simulation).
Fig. 3
Fig. 3 (a) Input signal constellations for 28 Gbaud PAM4 in the presence of carrier phase distortions and (d) with 200 ps/nm of CD. (b, e) after RDA-based equalization; (c, f) after CMMA-4 - based equalization. (By simulation).
Fig. 4
Fig. 4 Block diagram of the SP-LMS-based equalization.
Fig. 5
Fig. 5 Block diagram of the TMMA-based equalization.
Fig. 6
Fig. 6 Illustration of CMMA-4 for PAM4 signal.
Fig. 7
Fig. 7 (a) Experimental setup for 112Gbit/s PDM-PAM4 system, (b) offline DSP. DAC: digital-to-analog converter; PC: polarization controller; PBS: polarization beam splitter; PBC: polarization beam combiner; TS: training sequence.
Fig. 8
Fig. 8 Timing metric of TS synchronization for (a) BTB transmission at ROP = −20dBm, and (b) 80km SSMF transmission at ROP = −20dBm.
Fig. 9
Fig. 9 BER as a function of the number of taps (a) by using SP-LMS algorithm, and (b) by using TMMA algorithm (ROP = −20dBm).
Fig. 10
Fig. 10 Convergence properties with different step parameter - µ using (a) SP-LMS and (b) TMMA. MSE: mean square error. (After 80km transmission, ROP = −20dBm).
Fig. 11
Fig. 11 Constellations of X-pol for (a) the input of equalizer, (b) the output of equalizer, and the absolute amplitude of the output of equalizer. (After 80km transmission, ROP = −20dBm).
Fig. 12
Fig. 12 BER as a function of the number of training symbols using (a) SP-LMS and (b) TMMA. (ROP = −20dBm).
Fig. 13
Fig. 13 BER as a function of the number of training symbols using (a) SP-LMS and (b) TMMA. (ROP = −20dBm).
Fig. 14
Fig. 14 BER as a function of ROP by using two different equalization algorithms.

Tables (1)

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Table 1 Proposed PDM-PAM4 Algorithms Classified for the Different Convergence Stages

Equations (9)

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ε x ( n ) = exp ( j ϕ x ( n ) ) d x ( n ) x o u t ( n ) , ε y ( n ) = exp ( j ϕ y ( n ) ) d y ( n ) y o u t ( n )
ϕ x ( n ) = arg ( x o u t ( n ) ) , ϕ y ( n ) = arg ( y o u t ( n ) )
h x x ( n + 1 ) = h x x ( n ) + μ ε x ( n ) x * i n ( n ) , h x y ( n + 1 ) = h x y ( n ) + μ ε x ( n ) y * i n ( n ) h y x ( n + 1 ) = h y x ( n ) + μ ε y ( n ) x * i n ( n ) , h y y ( n + 1 ) = h y y ( n ) + μ ε y ( n ) y * i n ( n )
d x ( n ) = Decision ( | x o u t ( n ) | ) , d y ( n ) = Decision ( | y o u t ( n ) | )
ε x ( n ) = ( d x ( n ) ) 2 | x o u t ( n ) | 2 , ε y ( n ) = ( d y ( n ) ) 2 | y o u t ( n ) | 2
h x x ( n + 1 ) = h x x ( n ) + μ ε x ( n ) x o u t ( n ) x * i n ( n ) , h x y ( n + 1 ) = h x y ( n ) + μ ε x ( n ) x o u t ( n ) y * i n ( n ) h y x ( n + 1 ) = h y x ( n ) + μ ε y ( n ) y o u t ( n ) x * i n ( n ) , h y y ( n + 1 ) = h y y ( n ) + μ ε y ( n ) y o u t ( n ) y * i n ( n )
ε x ( n ) = A 4 | A 3 | A 2 | A 1 | x o u t ( n ) | | | | ε y ( n ) = A 4 | A 3 | A 2 | A 1 | y o u t ( n ) | | | |
e x , y ( n ) = sign ( x , y o u t ( n ) ) sign ( A 1 | x , y o u t ( n ) | ) sign ( A 2 | A 1 | x , y o u t ( n ) | | ) sign ( A 3 | A 2 | A 1 | x , y o u t ( n ) | | | )
h x x ( n + 1 ) = h x x ( n ) + μ ε x ( n ) e x ( n ) x * i n ( n ) , h x y ( n + 1 ) = h x y ( n ) + μ ε x ( n ) e x ( n ) y * i n ( n ) h y x ( n + 1 ) = h y x ( n ) + μ ε y ( n ) e y ( n ) x * i n ( n ) , h y y ( n + 1 ) = h y y ( n ) + μ ε y ( n ) e y ( n ) y * i n ( n )

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