Abstract

In this paper, we experimentally demonstrate a 200-G (4×56-Gbit/s) optical 4-level pulse-amplitude modulation (PAM-4) system using 10G-class optics over 10-km standard single-mode fiber and propose a joint Hartley-domain equalizer (HDE) and time-domain equalizer (TDE) algorithm for efficiently compensating the serious high-frequency distortions caused by the bandwidth-limited devices. To the best of our knowledge, the first HDE based on Hartley transform is designed for an optical PAM-4 system. Owing to the real-valued and self-inverse properties of the Hartley transform, the HDE has advantages in processing the real-valued PAM-4 signal. The experimental results show that the joint HDE and TDE algorithm has a better performance than only the HDE or only the TDE. Meanwhile, for obtaining a desired bit error rate, the computational complexity of the joint HDE and TDE algorithm is approximately 6% that of only the TDE with larger tap number. In conclusion, the joint HDE and TDE algorithm shows great potential for high-speed and cost-sensitive optical PAM-4 systems.

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

Full Article  |  PDF Article
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References

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2018 (1)

2017 (2)

2015 (3)

2014 (1)

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

2012 (1)

2010 (2)

1984 (1)

R. N. Bracewell, “The fast Hartley transform,” Proc. IEEE 72, 1010–1018 (1984).
[Crossref]

Bo, T.

T. Bo, B. Kim, and H. Kim,“28-Gbps VCSEL-based Optical Access Network with >14-dB Power Budget Using 10G-Class Optical Components,” in Opto-Electronics and Communications Conference (IEEE, 2017), pp.1–3.

Bracewell, R. N.

R. N. Bracewell, “The fast Hartley transform,” Proc. IEEE 72, 1010–1018 (1984).
[Crossref]

Cai, Z.

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

Cao, Z.

Chen, H.

Chen, J.

Chen, M.

Chen, W.

Cheng, Q.

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

Chi, N.

Cunningham, D. G.

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

Drenski, T.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Fàbrega, J. M.

Gao, Y.

Giacoumidis, E.

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

Grant, A. J.

Griesser, H.

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

Grobe, K.

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

Gui, T.

Guo, Y.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Haykin, S.

S. Haykin, Adaptive Filter Theory (Prentice Hall, 1993), Chap. 9.

Ho, S.

Hsu, D.

Huang, X.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Huo, J.

K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital Signal Processing for Short-Reach Optical Communications: A Review of Current Technologies and Future Trends,” J. Lightwave Technol. 36, 377–400 (2018).
[Crossref]

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-Less Transmission of 56Gbit/s PAM4 over 60km Using 25Gbps EML and APD,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu2D.1.

Ji, Y.

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

Junyent, G.

Kanno, A.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Kim, B.

T. Bo, B. Kim, and H. Kim,“28-Gbps VCSEL-based Optical Access Network with >14-dB Power Budget Using 10G-Class Optical Components,” in Opto-Electronics and Communications Conference (IEEE, 2017), pp.1–3.

Kim, H.

T. Bo, B. Kim, and H. Kim,“28-Gbps VCSEL-based Optical Access Network with >14-dB Power Budget Using 10G-Class Optical Components,” in Opto-Electronics and Communications Conference (IEEE, 2017), pp.1–3.

Kitayama, K.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Lau, A. P. T.

Li, F.

Li, L.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Li, W.

Li, X.

Lin, S.

Ling, W. A.

I. Lyubomirsky and W. A. Ling, “Advanced Modulation for Datacenter Interconnect,” in Optical Fiber Communication Conferenc (Optical Society of America, 2016), paper W4J.3.

Liu, B.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Lu, C.

Luo, L.

Lyubomirsky, I.

I. Lyubomirsky and W. A. Ling, “Advanced Modulation for Datacenter Interconnect,” in Optical Fiber Communication Conferenc (Optical Society of America, 2016), paper W4J.3.

Ma, Z.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Man, J.

Maruta, A.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Matsumoto, K.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Moreolo, M. S.

Muñoz, R.

Nadal, L.

Nishihara, M.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Nuwanpriya, A.

Penty, R. V.

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

Qiao, Y.

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

Rasmussen, J. C.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Shi, J.

Takahara, T.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Tanaka, T.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Tao, L.

Tao, Z.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Tu, Z.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Vílchez, F. J.

Wagner, C.

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

Wang, Y.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-Less Transmission of 56Gbit/s PAM4 over 60km Using 25Gbps EML and APD,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu2D.1.

Wei, C.

Wei, J.

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

Wey, J. S.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

White, I. H.

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

Xie, C.

C. Xie, “Datacenter Optical Interconnects: Requirements and Challenges,” in IEEE Optical Interconnects Conference (OI) (IEEE, 2017), pp. 37–38.

Yamamoto, N.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Yan, W.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

Yan, Y.

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

Yang, B.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Yang, W.

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Yoshida, Y.

K. Matsumoto, Y. Yoshida, A. Maruta, A. Kanno, N. Yamamoto, and K. Kitayama, “On the Impact of Tomlinson-Harashima Precoding in Optical PAM Transmissions for Intra-DCN Communication,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th3D.7.

Yu, C.

K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital Signal Processing for Short-Reach Optical Communications: A Review of Current Technologies and Future Trends,” J. Lightwave Technol. 36, 377–400 (2018).
[Crossref]

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-Less Transmission of 56Gbit/s PAM4 over 60km Using 25Gbps EML and APD,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu2D.1.

Yu, J.

J. Shi, J. Zhang, N. Chi, and J. Yu, “Comparison of 100G PAM-8, CAP-64 and DFT-S OFDM with a bandwidth-limited direct-detection receiver,” Opt. Express 25, 32254–32262 (2017).
[Crossref]

F. Li, J. Yu, Z. Cao, J. Zhang, M. Chen, and X. Li, “Experimental Demonstration of Four-Channel WDM 560 Gbit/s 128QAM-DMT Using IM/DD for 2-km Optical Interconnect,” J. Lightwave Technol. 35, 941–948 (2017).
[Crossref]

J. Zhang, J. S. Wey, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Symmetrical 50-Gb/s/λPAM-4 TDM-PON in O-band with DSP and Semiconductor Optical Amplifier Supporting PR-30 Link Loss Budget,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M1B.4.

Yuang, M. C.

Zeng, L.

K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. T. Lau, and C. Lu, “Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s short reach optical transmission systems,” Opt. Express 23, 1176–1189 (2015).
[Crossref]

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-Less Transmission of 56Gbit/s PAM4 over 60km Using 25Gbps EML and APD,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu2D.1.

Zhang, H.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-Less Transmission of 56Gbit/s PAM4 over 60km Using 25Gbps EML and APD,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu2D.1.

Zhang, J.

Zhong, K.

Zhou, J.

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

Zhou, X.

IEEE Commun. Mag. (1)

J. Wei, Q. Cheng, R. V. Penty, I. H. White, and D. G. Cunningham, “400 Gigabit Ethernet Using Advanced Modulation Formats: Performance, Complexity, and Power Dissipation,” IEEE Commun. Mag. 53, 182–189 (2015).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Zhou, Y. Yan, Z. Cai, Y. Qiao, and Y. Ji, “A Cost-Effective and Efficient Scheme for Optical OFDM in Short-Range IM/DD Systems,” IEEE Photon. Technol. Lett. 26, 1372–1374 (2014).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (4)

Proc. IEEE (1)

R. N. Bracewell, “The fast Hartley transform,” Proc. IEEE 72, 1010–1018 (1984).
[Crossref]

Other (10)

S. Haykin, Adaptive Filter Theory (Prentice Hall, 1993), Chap. 9.

C. Xie, “Datacenter Optical Interconnects: Requirements and Challenges,” in IEEE Optical Interconnects Conference (OI) (IEEE, 2017), pp. 37–38.

W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s Optical IM-DD Transmission with 10G-Class Devices Enabled by 65 GSamples/s CMOS DAC Core,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OM3H.1.

T. Bo, B. Kim, and H. Kim,“28-Gbps VCSEL-based Optical Access Network with >14-dB Power Budget Using 10G-Class Optical Components,” in Opto-Electronics and Communications Conference (IEEE, 2017), pp.1–3.

J. Wei, K. Grobe, C. Wagner, E. Giacoumidis, and H. Griesser, “40 Gb/s Lane Rate NG-PON using Electrical/Optical Duobinary, PAM-4 and Low Complex Equalizations,” in Optical Fiber Communication Conference (Optical Society of America, 2016), paper Tu3C.5.

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Figures (7)

Fig. 1
Fig. 1 The designed structure of joint HDE and TDE algorithm.
Fig. 2
Fig. 2 (a) Block diagram of optical PAM-4 system with joint HDE and TDE algorithm using 10G-class optics; (b) Frequency response of the system.
Fig. 3
Fig. 3 (a) Optical spectrum of optical carriers (blue lines) and 4×56-Gbit/s optical PAM-4 signal (red lines); (b) Electrical spectrum of 56-Gbit/s PAM-4 signal.
Fig. 4
Fig. 4 (a) BER against tap number of FFE-DFE in joint HDE and TDE algorithm; (b) BER versus the ROP for the joint HDE and (5, 1) TDE algorithm, only HDE, and only (5, 1) TDE, respectively.
Fig. 5
Fig. 5 The eye diagram of the recovered PAM-4 at the ROP of −9 dBm after 10-km SSMF transmission. (a) Without equalization; (b) After only HDE; (c) After only (5, 1) TDE; (d) After joint HDE and (5, 1) TDE algorithm.
Fig. 6
Fig. 6 BER against ROP for 4×56-Gbit/s PAM-4 system with joint HDE and TDE algorithm after 10-km SSMF transmission.
Fig. 7
Fig. 7 (a) The comparison of BER performance between joint HDE and (5, 1) TDE algorithm and only TDE; (b) The comparison of computational complexity between joint HDE and (5, 1) TDE algorithm and only TDE.

Equations (11)

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x n = 1 N k = 0 N 1 X k [ cos ( 2 π n k N ) + sin ( 2 π n k N ) ]
X k = 1 N n = 0 N 1 x n [ cos ( 2 π n k N ) + sin ( 2 π n k N ) ]
r ( t ) = h ( t ) x ( t ) + n ( t )
R ( f ) = + r ( t ) × cas ( 2 π f t ) d t = + + h ( τ ) x ( t τ ) d τ cas ( 2 π f t ) d t + N ( f ) = X ( f ) H e ( f ) + X ( f ) H o ( f ) + N ( f )
X ( f ) = H e ( f ) R ( f ) H o ( f ) R ( f ) H o 2 ( f ) + H e 2 ( f ) H e ( f ) N ( f ) H o ( f ) N ( f ) H o 2 ( f ) + H e 2 ( f )
X ( f ) = H o ( f ) R ( f ) H e ( f ) R ( f ) H o 2 ( f ) + H e 2 ( f ) H o ( f ) N ( f ) H e ( f ) N ( f ) H o 2 ( f ) + H e 2 ( f )
X ( f ) H e ( f ) R ( f ) H o ( f ) R ( f ) H o 2 ( f ) + H e 2 ( f )
X ( f ) H o ( f ) R ( f ) H e ( f ) R ( f ) H o 2 ( f ) + H e 2 ( f )
TDE output ( m ) = l = 0 L 1 a ( l ) y ( m + 1 ) + n = 1 K b ( m ) d ( m n )
C HDE = 2 N log 2 N / ( N log 2 M ) = 2 log 2 N / log 2 M
C TDE = [ 3 ( L + K ) 2 + 5 ( L + K ) ] F / ( F log 2 M ) = [ 3 ( L + k ) 2 + 5 L + 5 K ] / log 2 M

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