Abstract

In this paper, we demonstrate a novel Gaussian kernel-aided deep neural network (GK-DNN) equalizer that can effectively compensate for the high nonlinear distortion of underwater PAM8 visible light communication (VLC) channels. The application of a Gaussian kernel can reduce the necessary training iterations to 47.06%, enabling it to outperform the traditional DNN equalizer. At the same time, a novel design strategy with respect to the structure of the GK-DNN equalizer is proposed, which can effectively save computing resources and reduce the data volume of the necessary training data set. By using the GK-DNN equalizer, a 1.5 Gbps PAM8 VLC system over 1.2-m underwater transmission is successfully demonstrated.

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

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References

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2018 (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]

2017 (1)

F. Miramirkhani and M. Uysal, “Channel modeling and characterization for visible light communications channel modeling and characterization for visible light communications,” IEEE Photonics J. 7(6), 1–4 (2017).
[Crossref]

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, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (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]

2014 (1)

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

2012 (1)

2010 (1)

K. Burse, R. N. Yadav, and S. C. Shrivastava, “Channel equalization using neural networks: a review,” IEEE Trans. Syst. Man Cybern. C 40(3), 352–357 (2010).
[Crossref]

2009 (1)

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

1989 (1)

G. E. Hinton, “Connectionist learning procedures,” Artif. Intell. 40(1–3), 185–234 (1989).
[Crossref]

1972 (1)

A. A. Bergh and P. J. Dean, “Light-emitting diodes,” Proc. IEEE 60(2), 156–223 (1972).
[Crossref]

Alouini, M.-S.

Arlunno, V.

Bergh, A. A.

A. A. Bergh and P. J. Dean, “Light-emitting diodes,” Proc. IEEE 60(2), 156–223 (1972).
[Crossref]

Borkowski, R.

Burse, K.

K. Burse, R. N. Yadav, and S. C. Shrivastava, “Channel equalization using neural networks: a review,” IEEE Trans. Syst. Man Cybern. C 40(3), 352–357 (2010).
[Crossref]

Caballero, A.

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, 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]

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, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

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]

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

Dean, P. J.

A. A. Bergh and P. J. Dean, “Light-emitting diodes,” Proc. IEEE 60(2), 156–223 (1972).
[Crossref]

Faulkner, G.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Franceschi, N.

Ghassemlooy, Z.

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

Gonzales, N. G.

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]

Haigh, P. A.

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

Hinton, G. E.

G. E. Hinton, “Connectionist learning procedures,” Artif. Intell. 40(1–3), 185–234 (1989).
[Crossref]

Ho, C. L. C.

C. L. C. Ho, “A 10m/10Gbps Underwater wireless laser transmission system,” in IEEE Optical Fiber Communications Conference and Exhibition (2017), paper Th3C.3.

Ho, K.-T.

Huang, X.

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, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

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]

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]

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

Jung, D.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

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]

Krishnapura, N.

N. Krishnapura, S. Pavan, and C. Mathiazhagan, “A baseband pulse shaping filter for Gaussian minimum shift keying,” in IEEE International Symposium on Circuits and Systems. 1,249–252 (1998).

Larsen, K. J.

Lau, A.

T. Shen and A. Lau, “Fiber nonlinearity compensation using extreme learning machine for DSP-based coherent communication systems,” in 16th Opto-Electronics and Communications Conference (2011), pp. 816–817.

Le Minh, H.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Lee, K.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Li, J.

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

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, J.

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

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]

Mao, B.

Mathiazhagan, C.

N. Krishnapura, S. Pavan, and C. Mathiazhagan, “A baseband pulse shaping filter for Gaussian minimum shift keying,” in IEEE International Symposium on Circuits and Systems. 1,249–252 (1998).

Miramirkhani, F.

F. Miramirkhani and M. Uysal, “Channel modeling and characterization for visible light communications channel modeling and characterization for visible light communications,” IEEE Photonics J. 7(6), 1–4 (2017).
[Crossref]

Monroy, I. T.

Ng, T. K.

O’Brien, D.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Oh, Y.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Ooi, B. S.

Oubei, H. M.

Papakonstantinou, I.

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

Park, K.-H.

Pavan, S.

N. Krishnapura, S. Pavan, and C. Mathiazhagan, “A baseband pulse shaping filter for Gaussian minimum shift keying,” in IEEE International Symposium on Circuits and Systems. 1,249–252 (1998).

Popoola, W.

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

Rajbhandari, S.

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

Schmidt, M. N.

Shen, C.

Shen, T.

T. Shen and A. Lau, “Fiber nonlinearity compensation using extreme learning machine for DSP-based coherent communication systems,” in 16th Opto-Electronics and Communications Conference (2011), pp. 816–817.

Shi, J.

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

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]

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

Shi, M.

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

Shrivastava, S. C.

K. Burse, R. N. Yadav, and S. C. Shrivastava, “Channel equalization using neural networks: a review,” IEEE Trans. Syst. Man Cybern. C 40(3), 352–357 (2010).
[Crossref]

Tao, L.

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

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]

Uysal, M.

F. Miramirkhani and M. Uysal, “Channel modeling and characterization for visible light communications channel modeling and characterization for visible light communications,” IEEE Photonics J. 7(6), 1–4 (2017).
[Crossref]

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]

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

Wang, Y.

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, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

X. Huang, J. Shi, J. Li, Y. Wang, Y. Wang, and N. Chi, “750Mbit/s visible light communications employing 64QAM-OFDM based on amplitude equalization circuit,” in Optical Fiber Communications Conference and Exhibition (2015), pp. 1–3.

Winther, O.

Won, E. T.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Yadav, R. N.

K. Burse, R. N. Yadav, and S. C. Shrivastava, “Channel equalization using neural networks: a review,” IEEE Trans. Syst. Man Cybern. C 40(3), 352–357 (2010).
[Crossref]

Ye, Y.

Zeng, L.

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Zhang, M.

Zhao, J.

Zhou, Y.

Zhu, X.

X. Zhu, F. Wang, M. Shi, N. Chi, J. Liu, and F. Jiang, “10.72Gb / s visible light communication system based on single packaged RGBYC LED utilizing QAM-DMT modulation with hardware pre-equalization,” in Optical Fiber Communication Conference, pp. M3K.3 (2018).

Zibar, D.

Artif. Intell. (1)

G. E. Hinton, “Connectionist learning procedures,” Artif. Intell. 40(1–3), 185–234 (1989).
[Crossref]

IEEE Photonics J. (2)

Y. Wang, L. Tao, X. Huang, J. Shi, and N. Chi, “Enhanced performance of a high-Speed WDM CAP64 VLC system employing volterra series-based nonlinear equalizer,” IEEE Photonics J. 7(3), 1–7 (2015).
[Crossref]

F. Miramirkhani and M. Uysal, “Channel modeling and characterization for visible light communications channel modeling and characterization for visible light communications,” IEEE Photonics J. 7(6), 1–4 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (1)

H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photonics Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

IEEE Trans. Syst. Man Cybern. C (1)

K. Burse, R. N. Yadav, and S. C. Shrivastava, “Channel equalization using neural networks: a review,” IEEE Trans. Syst. Man Cybern. C 40(3), 352–357 (2010).
[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]

J. Lightwave Technol. (1)

P. A. Haigh, Z. Ghassemlooy, S. Rajbhandari, I. Papakonstantinou, and W. Popoola, “Visible Light Communications : 170 Mb / s using an artificial neural network equalizer in a low bandwidth white light configuration,” J. Lightwave Technol. 32(c), 1–7 (2014).

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)

Proc. IEEE (1)

A. A. Bergh and P. J. Dean, “Light-emitting diodes,” Proc. IEEE 60(2), 156–223 (1972).
[Crossref]

Other (8)

N. Krishnapura, S. Pavan, and C. Mathiazhagan, “A baseband pulse shaping filter for Gaussian minimum shift keying,” in IEEE International Symposium on Circuits and Systems. 1,249–252 (1998).

J. S. Bridle, “Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters,” Adv. Neural Inf. Process. Syst. (Ml), 211–217 (1990).

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Figures (9)

Fig. 1
Fig. 1 Structure of the GK-DNN.
Fig. 2
Fig. 2 The amplitude coefficients k versus time serial number.
Fig. 3
Fig. 3 Block diagrams and experimental setup of PAM8 underwater VLC system utilizing GK-DNN equalization.
Fig. 4
Fig. 4 (a) The P-I curve of the silicon substrate LED lamp; (b) the I-V curve of the silicon substrate LED lamp.
Fig. 5
Fig. 5 The relationship between the structure of different hidden layer networks and BER performance of a DNN equalizer-based underwater PAM 8 VLC system; (1) the constellation without equalization; (2) the constellation with S-MCMMA equalization; (3) the constellation with S-MCMMA and DNN equalization.
Fig. 6
Fig. 6 (a) DNN equalizer training process under different numbers of features; (b) the effect of the number of features on the BER performance of the underwater PAM8 VLC system.
Fig. 7
Fig. 7 (a) GK-DNN equalizer training process for high nonlinearity channel under various β values and corresponding constellation; (b) GK-DNN equalizer training process for low nonlinearity channel under various β values and corresponding constellation.
Fig. 8
Fig. 8 (a) Relationship between the number of iterations required to reach HD-FEC and β in high and low nonlinearity distortion channels; (b) the efficiency of computing resources.
Fig. 9
Fig. 9 (a) Comparison of BER between S-MCMMA system and S-MCMMA + GK-DNN system under various Vpp; (b) Measured BER performance versus different bitrates; the inset figure.

Equations (15)

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C ( ω ) 10 log 10 ( ( D R θ 1 / e d ) 2 e ( a + b ) d ( D R θ 1 / e d ) S )
H 0 ( ω ) = G T ( ω ) C ( ω ) G R ( ω )
G T ( ω ) = e ω / ω f
s ( t ) = n = a n g T ( t n T s )
r ( k T s + t 0 ) = a n g R ( t 0 ) + n k a n g R ( k T s + t 0 n T s ) + n R ( k T s + t 0 )
i L E D = { i s ( e q V / K T 1 ) , V > V F 0 , V V F
w i , j l W l = ( w 1 , 1 l w 1 , n l w m , 1 l w m , n l ) , l = 1 , 2 , ... , L , L + 1
k ( t , t ' ) i = e - ( π ( t t ' ) a ) 2 = e - ( π ( ( i ) ( i + 1 ) / 2 ) a ) 2 = e - ( π ( i 1 ) 2 a ) 2 , i = 1 , 2 , ... , n f 1 , n f a = 1 β log 2 2
g ( x ) = x k = [ I 1 k 1 , I 2 k 2 , ... , x n f 1 k n f 1 , x n f k n f ]
H j 1 = i = 1 n w i , j 1 k i + b i 1
H j l = i = 1 n w i , j l f ( H j l 1 ) + b i l , 1 < l < L + 1
N h 1 = i = 1 n f C n f i
P ( y = L j | x ) j = e O j / i = 1 n e O i , L j = 7 , 5 , 3 , 1 , 1 , 3 , 5 , 7
O i = i = 1 n w i , j L + 1 f ( H j L + 1 ) + b i L + 1
C ( q , p ) = x q ( x ) log p ( x )

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