Abstract

This paper proposes a novel two tributaries heterogeneous neural network (TTHnet) based channel emulator, which is suitable for both estimating single-carrier and multi-carrier modulated channels of underwater visible light communication (UVLC). Compared to traditional neural networks, the TTHnet channel emulator has only 1932 trainable parameters, which is only 0.8% of multilayer perceptron (MLP) based channel emulator and 1% of a convolutional neural network (CNN) based channel emulator. Furthermore, it provides a more accurate estimation of the UVLC channel and greater interpretability than MLP and CNN. The experiments in this paper use carrier-less amplitude/phase modulation (CAP) and discrete multi-tone modulation (DMT) as representative examples of single-carrier and multi-carrier modulation, respectively. The experiment proves that the TTHnet based channel emulator could effectively emulate the channel response of UVLC systems both in time and frequency domain. To the best of our knowledge, this is the first time that the single-carrier and multi-carrier modulated UVLC channel is emulated by the deep neural networks based channel emulator, which will effectively accelerate the research progress of UVLC and reduce research costs of UVLC systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. F. Miramirkhani and M. Uysal, “Visible Light Communication Channel Modeling for Underwater Environments with Blocking and Shadowing,” IEEE Access 6, 1082–1090 (2017).
  2. Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “8-Gb/s RGBY LED-Based WDM VLC System Employing High-Order CAP Modulation and Hybrid Post Equalizer,” IEEE Photonics J. 7(6), 17904507 (2015).
    [Crossref]
  3. N. Chi, Y. Zhou, J. Shi, Y. Wang, and X. Huang, “Enabling technologies for high-speed visible light communication employing CAP modulation,” J. Lightwave Technol. 36(2), 510–518 (2018).
    [Crossref]
  4. X. Ma, F. Yang, S. Liu, and J. Song, “Channel estimation for wideband underwater visible light communication: a compressive sensing perspective,” Opt. Express 26(1), 311–321 (2018).
    [Crossref] [PubMed]
  5. F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
    [Crossref]
  6. N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
    [Crossref]
  7. N. Chi, M. Zhang, Y. Zhou, and J. Zhao, “3.375-Gb/s RGB-LED based WDM visible light communication system employing PAM-8 modulation with phase shifted Manchester coding,” Opt. Express 24(19), 21663–21673 (2016).
    [Crossref] [PubMed]
  8. Y. Wang, X. Huang, L. Tao, J. Shi, and N. Chi, “4.5-Gb/s RGB-LED based WDM visible light communication system employing CAP modulation and RLS based adaptive equalization,” Opt. Express 23(10), 13626–13633 (2015).
    [Crossref] [PubMed]
  9. H. Wang, Y. Huang, W. Wang, C. Tsai, C. Cheng, Y. Chi, and G. Lin, “Seawater Communication with Blue Laser Carried 16-QAM OFDM at 3.7 GBaud,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), paper Tu2I.1.
    [Crossref]
  10. C. Shen, Y. Guo, H. M. Oubei, T. K. Ng, G. Liu, K.-H. Park, K.-T. Ho, M.-S. Alouini, and B. S. Ooi, “20-meter underwater wireless optical communication link with 1.5 Gbps data rate,” Opt. Express 24(22), 25502–25509 (2016).
    [Crossref] [PubMed]
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    [Crossref]
  12. S. Jaruwatanadilok, “Channel Modeling and Performance Evaluation using Vector Radiative Transfer Theory,” IEEE J. Sel. Areas Comm. 26(9), 1620–1627 (2008).
    [Crossref]
  13. C. Gabriel, M. Khalighi, S. Bourennane, P. Leon, V. Rigaud, I. Fresnel, and U. M. R. Cnrs, “Channel Modeling for Underwater Optical Communication,” in Globecom Workshops, (2011), pp. 833–837.
  14. H. M. Oubei, E. Zedini, R. T. ElAfandy, A. Kammoun, M. Abdallah, T. K. Ng, M. Hamdi, M.-S. Alouini, and B. S. Ooi, “Simple statistical channel model for weak temperature-induced turbulence in underwater wireless optical communication systems,” Opt. Lett. 42(13), 2455–2458 (2017).
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  15. A. Huang, L. Tao, C. Wang, and L. Zhang, “Error performance of underwater wireless optical communications with spatial diversity under turbulence channels,” Appl. Opt. 57(26), 7600–7608 (2018).
    [Crossref] [PubMed]
  16. N. Chi and M. Shi, “Advanced modulation formats for underwater visible light communications [Invited],” Chin. Opt. Lett. 16(12), 120603 (2018).
    [Crossref]
  17. J. Siuzdak, “Modulation selection for visible light communications using lighting LEDs,” Photonics Appl. Astron. Commun. Ind. High-Energy Phys. Exp. 2015 9662(September 2015), 966204 (2015).
  18. H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
    [Crossref]
  19. H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
    [Crossref]
  20. M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
    [Crossref]
  21. B. L. Kalman and S. C. Kwasny, “Why tanh: choosing a sigmoidal function,” in IJCNN International Joint Conference on Neural Networks (Proceedings, 1992), pp. 578–581.
    [Crossref]
  22. Y. K. Jain and S. K. Bhandare, “Min Max Normalization Based Data Perturbation Method for Privacy Protection,” Int. J. Comput. Commun. Technol. 2(8), 45–50 (2011).
  23. A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
    [Crossref]

2019 (1)

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

2018 (7)

A. Huang, L. Tao, C. Wang, and L. Zhang, “Error performance of underwater wireless optical communications with spatial diversity under turbulence channels,” Appl. Opt. 57(26), 7600–7608 (2018).
[Crossref] [PubMed]

N. Chi and M. Shi, “Advanced modulation formats for underwater visible light communications [Invited],” Chin. Opt. Lett. 16(12), 120603 (2018).
[Crossref]

H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
[Crossref]

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

N. Chi, Y. Zhou, J. Shi, Y. Wang, and X. Huang, “Enabling technologies for high-speed visible light communication employing CAP modulation,” J. Lightwave Technol. 36(2), 510–518 (2018).
[Crossref]

X. Ma, F. Yang, S. Liu, and J. Song, “Channel estimation for wideband underwater visible light communication: a compressive sensing perspective,” Opt. Express 26(1), 311–321 (2018).
[Crossref] [PubMed]

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

2017 (2)

2016 (2)

2015 (3)

Y. Wang, X. Huang, L. Tao, J. Shi, and N. Chi, “4.5-Gb/s RGB-LED based WDM visible light communication system employing CAP modulation and RLS based adaptive equalization,” Opt. Express 23(10), 13626–13633 (2015).
[Crossref] [PubMed]

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “8-Gb/s RGBY LED-Based WDM VLC System Employing High-Order CAP Modulation and Hybrid Post Equalizer,” IEEE Photonics J. 7(6), 17904507 (2015).
[Crossref]

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

2011 (1)

Y. K. Jain and S. K. Bhandare, “Min Max Normalization Based Data Perturbation Method for Privacy Protection,” Int. J. Comput. Commun. Technol. 2(8), 45–50 (2011).

2008 (1)

S. Jaruwatanadilok, “Channel Modeling and Performance Evaluation using Vector Radiative Transfer Theory,” IEEE J. Sel. Areas Comm. 26(9), 1620–1627 (2008).
[Crossref]

2005 (1)

A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
[Crossref]

Abdallah, M.

Alouini, M.-S.

Bhandare, S. K.

Y. K. Jain and S. K. Bhandare, “Min Max Normalization Based Data Perturbation Method for Privacy Protection,” Int. J. Comput. Commun. Technol. 2(8), 45–50 (2011).

Chi, N.

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

N. Chi, Y. Zhou, J. Shi, Y. Wang, and X. Huang, “Enabling technologies for high-speed visible light communication employing CAP modulation,” J. Lightwave Technol. 36(2), 510–518 (2018).
[Crossref]

N. Chi and M. Shi, “Advanced modulation formats for underwater visible light communications [Invited],” Chin. Opt. Lett. 16(12), 120603 (2018).
[Crossref]

N. Chi, M. Zhang, Y. Zhou, and J. Zhao, “3.375-Gb/s RGB-LED based WDM visible light communication system employing PAM-8 modulation with phase shifted Manchester coding,” Opt. Express 24(19), 21663–21673 (2016).
[Crossref] [PubMed]

Y. Wang, X. Huang, L. Tao, J. Shi, and N. Chi, “4.5-Gb/s RGB-LED based WDM visible light communication system employing CAP modulation and RLS based adaptive equalization,” Opt. Express 23(10), 13626–13633 (2015).
[Crossref] [PubMed]

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “8-Gb/s RGBY LED-Based WDM VLC System Employing High-Order CAP Modulation and Hybrid Post Equalizer,” IEEE Photonics J. 7(6), 17904507 (2015).
[Crossref]

ElAfandy, R. T.

Guo, Y.

Haas, H.

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

Hamdi, M.

He, H.

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

Ho, K.-T.

Huang, A.

Huang, X.

Huang, X. L.

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

Jain, A.

A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
[Crossref]

Jain, Y. K.

Y. K. Jain and S. K. Bhandare, “Min Max Normalization Based Data Perturbation Method for Privacy Protection,” Int. J. Comput. Commun. Technol. 2(8), 45–50 (2011).

Jaruwatanadilok, S.

S. Jaruwatanadilok, “Channel Modeling and Performance Evaluation using Vector Radiative Transfer Theory,” IEEE J. Sel. Areas Comm. 26(9), 1620–1627 (2008).
[Crossref]

Jiang, F.

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

Jin, S.

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

Juang, B. H.

H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
[Crossref]

Kammoun, A.

Kavehrad, M.

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

Li, G. Y.

H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
[Crossref]

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

Little, T. D. C.

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

Liu, G.

Liu, S.

Liu, Y.

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

Ma, X.

Miramirkhani, F.

F. Miramirkhani and M. Uysal, “Visible Light Communication Channel Modeling for Underwater Environments with Blocking and Shadowing,” IEEE Access 6, 1082–1090 (2017).

Mirzaei, A.

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

Nandakumar, K.

A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
[Crossref]

Ng, T. K.

Ooi, B. S.

Oubei, H. M.

Park, K.-H.

Pourahmadi, V.

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

Ross, A.

A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
[Crossref]

Sheikhzadeh, H.

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

Shen, C.

Shi, J.

Shi, M.

Soltani, M.

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

Song, J.

Tao, L.

Uysal, M.

F. Miramirkhani and M. Uysal, “Visible Light Communication Channel Modeling for Underwater Environments with Blocking and Shadowing,” IEEE Access 6, 1082–1090 (2017).

Wang, C.

Wang, F.

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

Wang, Y.

Wen, C.

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

Yang, F.

Ye, H.

H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
[Crossref]

Zedini, E.

Zhang, L.

Zhang, M.

Zhao, J.

Zhou, Y.

Appl. Opt. (1)

Chin. Opt. Lett. (1)

IEEE Access (1)

F. Miramirkhani and M. Uysal, “Visible Light Communication Channel Modeling for Underwater Environments with Blocking and Shadowing,” IEEE Access 6, 1082–1090 (2017).

IEEE Commun. Lett. (1)

M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep Learning-Based Channel Estimation,” IEEE Commun. Lett. 23(4), 652–655 (2019).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

S. Jaruwatanadilok, “Channel Modeling and Performance Evaluation using Vector Radiative Transfer Theory,” IEEE J. Sel. Areas Comm. 26(9), 1620–1627 (2008).
[Crossref]

IEEE Photonics J. (1)

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “8-Gb/s RGBY LED-Based WDM VLC System Employing High-Order CAP Modulation and Hybrid Post Equalizer,” IEEE Photonics J. 7(6), 17904507 (2015).
[Crossref]

IEEE Wirel. Commun. (1)

N. Chi, H. Haas, M. Kavehrad, T. D. C. Little, and X. L. Huang, “Visible light communications: Demand factors, benefits and opportunities [Guest Editorial],” IEEE Wirel. Commun. 22(2), 5–7 (2015).
[Crossref]

IEEE Wirel. Commun. Lett. (2)

H. Ye, G. Y. Li, and B. H. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” IEEE Wirel. Commun. Lett. 7(1), 114–117 (2018).
[Crossref]

H. He, C. Wen, S. Jin, and G. Y. Li, “Deep Learning-Based Channel Estimation for Beamspace mmWave Massive MIMO Systems,” IEEE Wirel. Commun. Lett. 7(5), 852–855 (2018).
[Crossref]

Int. J. Comput. Commun. Technol. (1)

Y. K. Jain and S. K. Bhandare, “Min Max Normalization Based Data Perturbation Method for Privacy Protection,” Int. J. Comput. Commun. Technol. 2(8), 45–50 (2011).

J. Lightwave Technol. (1)

Opt. Commun. (1)

F. Wang, Y. Liu, F. Jiang, and N. Chi, “High speed underwater visible light communication system based on LED employing maximum ratio combination with multi-PIN reception,” Opt. Commun. 425(April), 106–112 (2018).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Pattern Recognit. (1)

A. Jain, K. Nandakumar, and A. Ross, “Score normalization in multimodal biometric systems,” Pattern Recognit. 38(12), 2270–2285 (2005).
[Crossref]

Other (5)

B. L. Kalman and S. C. Kwasny, “Why tanh: choosing a sigmoidal function,” in IJCNN International Joint Conference on Neural Networks (Proceedings, 1992), pp. 578–581.
[Crossref]

J. Siuzdak, “Modulation selection for visible light communications using lighting LEDs,” Photonics Appl. Astron. Commun. Ind. High-Energy Phys. Exp. 2015 9662(September 2015), 966204 (2015).

H. Wang, Y. Huang, W. Wang, C. Tsai, C. Cheng, Y. Chi, and G. Lin, “Seawater Communication with Blue Laser Carried 16-QAM OFDM at 3.7 GBaud,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), paper Tu2I.1.
[Crossref]

C. Gabriel, M. Khalighi, S. Bourennane, P. Leon, V. Rigaud, I. Fresnel, and U. M. R. Cnrs, “Channel Modeling for Underwater Optical Communication,” in Globecom Workshops, (2011), pp. 833–837.

H. Wang, Y. Huang, W. Wang, C. Tsai, C. Cheng, Y. Chi, and G. Lin, “Seawater Communication with Blue Laser Carried 16-QAM OFDM at 3.7 GBaud,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2018), paper Tu2I.1.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 The forward propagation process of TTHnet.
Fig. 2
Fig. 2 The hyperparameters and structures of TTHnet, CNN and MLP.
Fig. 3
Fig. 3 The experimental setup.
Fig. 4
Fig. 4 (a) Comparison among of received spectrum, Max-Abs based emulated spectrum, Max-Min normalization based emulated spectrum and Z-Score normalization based emulated spectrum. (b) Corresponding spectrum mismatch between emulated spectrums and received spectrum.
Fig. 5
Fig. 5 (a) Comparison among received spectrum, spectrum emulated by 1st tributary based channel emulator, spectrum emulated by TTHnet based channel emulator without hollow layer and spectrum emulated by TTHnet based channel emulator. (b) The corresponding mismatch between emulated spectrums and received spectrum.
Fig. 6
Fig. 6 BER mismatch and Spectrum mismatch of different neural network channel emulators on the validation set after 30-epochs training. (a) BER mismatch on CAP modulated signals. (b) Spectrum mismatch on CAP modulated signals. (c) BER mismatch on DMT modulated signals. (d) Spectrum mismatch on DMT modulated signals.
Fig. 7
Fig. 7 (a) Comparison of BER between CAP64 transmit signal and DMT64 transmit signal after real UVLC channel and TTHnet emulated UVLC channel under different bias currents. (b) Comparison of BER between CAP64 transmit signal and DMT64 transmit signal after real UVLC channel and TTHnet estimation UVLC channel under different Vpp.
Fig. 8
Fig. 8 Comparison of the BER of the actual received signal with the bit error rate of the TTHnet emulated signal at different bitrates.

Equations (11)

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

N Max_Abs (x)=x/max(abs(x)),
Y out = W ( 3 ) [ W 1 ( 2 ) ( conv( X, W 1 ( 1 ) )+ b 1 ( 1 ) )+ b 1 ( 2 ) tanh( W 2 ( 2 ) tanh( W 2 ( 1 ) hollow( X )+ b 2 ( 1 ) )+ b 2 ( 2 ) ) ]+ b ( 3 ) +AWGN,
tanh( x )= e x e x e x + e x .
hollow( [ X (i l1 2 ) ,... X (i1) , X (i) , X (i+1) ,..., X (i+ l1 2 ) ] )=[ X (i l1 2 ) ,... X (i1) , X (i+1) ,..., X (i+ l1 2 ) ],
W,b= argmin W,b 1 m i=1 m Y ^ (i) Y out ( i ) 2 ,
N Min_Max (x)=[ xmin(x) ]/[ max(x)min(x) ],
N zscore (x)=[ xmean(x) ]/std(x),
spectrummismatch=abs(fft( Y ^ )-fft( Y out )),
Y out = W 1 ( 2 ) ( conv( X, W 1 ( 1 ) )+ b 1 ( 1 ) )+ b 1 ( 2 ) +AWGN,
Y out = W ( 3 ) [ W 1 ( 2 ) ( conv( X, W 1 ( 1 ) )+ b 1 ( 1 ) )+ b 1 ( 2 ) tanh( W 2 ( 2 ) tanh( W 2 ( 1 ) X+ b 2 ( 1 ) )+ b 2 ( 2 ) ) ]+ b ( 3 ) +AWGN.
BERmismatch=abs(BER( Y ^ )-BER( Y out )).

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