Abstract

It has been recently reported that reflective semiconductor optical amplifiers (RSOAs) can be used as phase modulators for the cost-effective implementation of the high-speed wavelength-division-multiplexed passive optical networks (WDM PONs). For the detection of the phase-shift-keying (PSK) signal generated by using an RSOA, we should be able to estimate its carrier phase accurately at the digital coherent receiver. However, when the baud rate of this PSK signal is set to be much higher than the RSOA’s modulation bandwidth, the conventional M-th power algorithm cannot estimate its carrier phase accurately. To solve this problem, we develop a simple carrier-phase estimation technique for the high-speed (>10 Gb/s) PSK signals generated by using bandwidth-limited RSOAs. This technique estimates the carrier phase of the PSK signal by measuring the angular direction of the opening occurred in the trajectory of the phasor diagram. By using the proposed technique, we demonstrate the upstream transmission of the 25.78-Gb/s quadrature phase-shift-keying (QPSK) signal generated by using an RSOA (3-dB bandwidth: 3.2 GHz) in a 60-km reach coherent WDM PON.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Carrier-phase-estimation algorithm featuring fast trackability for high-speed coherent WDM PON based on RSOA

Daeho Kim, Byung Gon Kim, Sung Hyun Bae, and Hoon Kim
Opt. Express 25(13) 14282-14289 (2017)

Long-reach 10-Gb/s RSOA-based WDM PON employing QPSK signal and coherent receiver

K. Y. Cho, U. H. Hong, S. P. Jung, Y. Takushima, A. Agata, T. Sano, Y. Horiuchi, M. Suzuki, and Y. C. Chung
Opt. Express 20(14) 15353-15358 (2012)

Transmission of 1.25-Gb/s PSK signal generated by using RSOA in 110-km coherent WDM PON

S. P. Jung, Y. Takushima, and Y. C. Chung
Opt. Express 18(14) 14871-14877 (2010)

References

  • View by:
  • |
  • |
  • |

  1. Y. C. Chung, “Recent advancement in WDM PON technology,” in Proc. of European Conference on Optical Communications 2011, paper Th.11.C.4.
  2. W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
    [Crossref]
  3. K. Y. Cho, Y. Takushima, and Y. C. Chung, “10-Gb/s operation of RSOA for WDM PON,” IEEE Photon. Technol. Lett. 20(18), 1533–1535 (2008).
    [Crossref]
  4. I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
    [Crossref]
  5. H. Kim, “10-Gb/s operation of RSOA using a delay interferometer,” IEEE Photon. Technol. Lett. 22(18), 1379–1381 (2010).
    [Crossref]
  6. K. Y. Cho, Y. Takushima, and Y. C. Chung, “Demonstration of 11-Gb/s, 20-km reach WDM PON using directly-modulated RSOA with 4-ary PAM signal,” in Proc. of Optical Fiber Communication 2010, paper OWG1.
  7. C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
    [Crossref]
  8. J. L. Wei, E. Hugues-Salas, R. P. Giddings, X. Q. Jin, X. Zheng, S. Mansoor, and J. M. Tang, “Wavelength reused bidirectional transmission of adaptively modulated optical OFDM signals in WDM-PONs incorporating SOA and RSOA intensity modulators,” Opt. Express 18(10), 9791–9808 (2010).
    [Crossref] [PubMed]
  9. S. P. Jung, Y. Takushima, and Y. C. Chung, “Transmission of 1.25-Gb/s PSK signal generated by using RSOA in 110-km coherent WDM PON,” Opt. Express 18(14), 14871–14877 (2010).
    [Crossref] [PubMed]
  10. S. P. Jung, Y. Takushima, and Y. C. Chung, “Generation of 5-Gbps QPSK signal using directly modulated RSOA for 100-km coherent WDM PON,” in Proc. of Optical Fiber Communications 2011, paper OTuB3.
  11. K. Y. Cho, U. H. Hong, S. P. Jung, Y. Takushima, A. Agata, T. Sano, Y. Horiuchi, M. Suzuki, and Y. C. Chung, “Long-reach 10-Gb/s RSOA-based WDM PON employing QPSK signal and coherent receiver,” Opt. Express 20(14), 15353–15358 (2012).
    [Crossref] [PubMed]
  12. U. H. Hong, K. Y. Cho, H. G. Choi, and Y. C. Chung, “A simple carrier-phase estimation technique for high-speed RSOA-based coherent WDM PON,” in Proc. of Optical Fiber Communication 2013, paper OM2A.2.
  13. K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12(4), 563–570 (2006).
    [Crossref]
  14. V. K. Prabhu, “PSK performance with imperfect carrier phase recovery,” IEEE Trans. Aerosp. Electron. Syst. 12(2), 275–285 (1976).
    [Crossref]
  15. K. Y. Cho, K. Tanaka, T. Sano, S. P. Jung, J. H. Chang, Y. Takushima, Y. Horiuchi, M. Suzuki, and Y. C. Chung, “Long-reach coherent WDM PON employing self-polarization-stabilization technique,” J. Lightwave Technol. 29(4), 456–462 (2011).
    [Crossref]
  16. S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
    [Crossref]
  17. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001).
  18. M. Fujiwara, J. Kani, H. Suzuki, and K. Iwatsuki, “Impact of backreflection on upstream transmission in WDM single-fiber loopback access networks,” J. Lightwave Technol. 24(2), 740–746 (2006).
    [Crossref]

2012 (1)

2011 (2)

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

K. Y. Cho, K. Tanaka, T. Sano, S. P. Jung, J. H. Chang, Y. Takushima, Y. Horiuchi, M. Suzuki, and Y. C. Chung, “Long-reach coherent WDM PON employing self-polarization-stabilization technique,” J. Lightwave Technol. 29(4), 456–462 (2011).
[Crossref]

2010 (3)

2008 (2)

K. Y. Cho, Y. Takushima, and Y. C. Chung, “10-Gb/s operation of RSOA for WDM PON,” IEEE Photon. Technol. Lett. 20(18), 1533–1535 (2008).
[Crossref]

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

2006 (3)

K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12(4), 563–570 (2006).
[Crossref]

S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
[Crossref]

M. Fujiwara, J. Kani, H. Suzuki, and K. Iwatsuki, “Impact of backreflection on upstream transmission in WDM single-fiber loopback access networks,” J. Lightwave Technol. 24(2), 740–746 (2006).
[Crossref]

2005 (1)

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

1976 (1)

V. K. Prabhu, “PSK performance with imperfect carrier phase recovery,” IEEE Trans. Aerosp. Electron. Syst. 12(2), 275–285 (1976).
[Crossref]

Agata, A.

Birbas, A. N.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Chang, J. H.

Chen, H. Y.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Cho, K. Y.

Cho, S. H.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Chow, C. W.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Chung, Y. C.

Fujiwara, M.

Giddings, R. P.

Hong, U. H.

Horiuchi, Y.

Hugues-Salas, E.

Iwatsuki, K.

Jeong, G.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Jin, X. Q.

Jung, S. P.

Kani, J.

Katoh, K.

S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
[Crossref]

Kikidis, J.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Kikuchi, K.

S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
[Crossref]

K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12(4), 563–570 (2006).
[Crossref]

Kim, B. W.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Kim, C.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Kim, H.

H. Kim, “10-Gb/s operation of RSOA using a delay interferometer,” IEEE Photon. Technol. Lett. 22(18), 1379–1381 (2010).
[Crossref]

Klonidis, D.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Lee, J.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Lee, W.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Lin, Y. H.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Liu, Y.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Mansoor, S.

Omella, M.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Pan, C. L.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Papagiannakis, I.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Park, M. Y.

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

Prabhu, V. K.

V. K. Prabhu, “PSK performance with imperfect carrier phase recovery,” IEEE Trans. Aerosp. Electron. Syst. 12(2), 275–285 (1976).
[Crossref]

Prat, J.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Sano, T.

Sung, J. Y.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Suzuki, H.

Suzuki, M.

Takushima, Y.

Tanaka, K.

Tang, J. M.

Tomkos, I.

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

Tsukamoto, S.

S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
[Crossref]

Wei, J. L.

Wu, Y. F.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Yeh, C. H.

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

Zheng, X.

Electron. Lett. (1)

C. W. Chow, C. H. Yeh, Y. F. Wu, H. Y. Chen, Y. H. Lin, J. Y. Sung, Y. Liu, and C. L. Pan, “13 Gbit/s WDM-OFDM PON using RSOA-based colourless ONU with seeding light source in local exchange,” Electron. Lett. 47(22), 1235–1236 (2011).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12(4), 563–570 (2006).
[Crossref]

IEEE Photon. Technol. Lett. (5)

S. Tsukamoto, K. Katoh, and K. Kikuchi, “Unrepeated transmission of 20-Gb/s optical quadrature phase-shift-keying signal over 200-km standard single-mode fiber based on digital processing of homodyne-detected signal for group-velocity dispersion compensation,” IEEE Photon. Technol. Lett. 18(9), 1016–1018 (2006).
[Crossref]

W. Lee, M. Y. Park, S. H. Cho, J. Lee, C. Kim, G. Jeong, and B. W. Kim, “Bidirectional WDM-PON based on gain-saturated reflective semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 17(11), 2460–2462 (2005).
[Crossref]

K. Y. Cho, Y. Takushima, and Y. C. Chung, “10-Gb/s operation of RSOA for WDM PON,” IEEE Photon. Technol. Lett. 20(18), 1533–1535 (2008).
[Crossref]

I. Papagiannakis, M. Omella, D. Klonidis, A. N. Birbas, J. Kikidis, I. Tomkos, and J. Prat, “Investigation of 10-Gb/s RSOA-based upstream transmission in WDM-PONs utilizing optical filtering and electronic equalization,” IEEE Photon. Technol. Lett. 20(24), 2168–2170 (2008).
[Crossref]

H. Kim, “10-Gb/s operation of RSOA using a delay interferometer,” IEEE Photon. Technol. Lett. 22(18), 1379–1381 (2010).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

V. K. Prabhu, “PSK performance with imperfect carrier phase recovery,” IEEE Trans. Aerosp. Electron. Syst. 12(2), 275–285 (1976).
[Crossref]

J. Lightwave Technol. (2)

Opt. Express (3)

Other (5)

S. P. Jung, Y. Takushima, and Y. C. Chung, “Generation of 5-Gbps QPSK signal using directly modulated RSOA for 100-km coherent WDM PON,” in Proc. of Optical Fiber Communications 2011, paper OTuB3.

K. Y. Cho, Y. Takushima, and Y. C. Chung, “Demonstration of 11-Gb/s, 20-km reach WDM PON using directly-modulated RSOA with 4-ary PAM signal,” in Proc. of Optical Fiber Communication 2010, paper OWG1.

U. H. Hong, K. Y. Cho, H. G. Choi, and Y. C. Chung, “A simple carrier-phase estimation technique for high-speed RSOA-based coherent WDM PON,” in Proc. of Optical Fiber Communication 2013, paper OM2A.2.

Y. C. Chung, “Recent advancement in WDM PON technology,” in Proc. of European Conference on Optical Communications 2011, paper Th.11.C.4.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 (a) Operating principle of the proposed CPE technique, (b) phasor diagram of the QPSK signal generated by directly modulating an RSOA, and (c) angular histogram obtained from the phasor trajectory in (b).
Fig. 2
Fig. 2 (a) Standard deviations of the phase errors plotted as a function of the data rate of the QSPK signal generated by using the bandwidth-limited modulator (3-dB bandwidth: 3.2 GHz). The black and red curves represent the standard deviations of the phase errors achievable by using the conventional M-th power algorithm and the proposed CPE technique, respectively. Exemplary phase errors obtained by using the M-th power algorithm for the (b) 2-Gb/s QPSK signal, (c) 10-Gb/s QPSK signal, and (d) 14-Gb/s QPSK signal.
Fig. 3
Fig. 3 Experimental setup.
Fig. 4
Fig. 4 Receiver sensitivity (BER = 10−3) of the QPSK signal versus the sample size.
Fig. 5
Fig. 5 (a) Measured BER curves of the QPSK signal in back-to-back operation with different modulation amplitudes. Also shown are the phasor diagrams of the signal when the peak-to-peak modulation currents are (b) 48 mA, (c) 55 mA, (d) 60 mA, and (e) 65 mA.
Fig. 6
Fig. 6 Measured BER curves at various transmission distances. The inset is the constellation diagram of the 25.78-Gb/s QPSK signal measured after the transmission over 60 km of SSMF.

Equations (1)

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

1 4 arg[ k=1 N r k 4 ]

Metrics