Abstract

By exploiting a causality property of the nonlinear Fourier transform, a novel decision-feedback detection strategy for nonlinear frequency-division multiplexing (NFDM) systems is introduced. The performance of the proposed strategy is investigated both by simulations and by theoretical bounds and approximations, showing that it achieves a considerable performance improvement compared to previously adopted techniques in terms of Q-factor. The obtained improvement demonstrates that, by tailoring the detection strategy to the peculiar properties of the nonlinear Fourier transform, it is possible to boost the performance of NFDM systems and overcome current limitations imposed by the use of more conventional detection techniques suitable for the linear regime.

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

Full Article  |  PDF Article
OSA Recommended Articles
Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation

Zibo Zheng, Xulun Zhang, Ruihua Yu, Lixia Xi, and Xiaoguang Zhang
Opt. Express 27(20) 28223-28238 (2019)

Polarization-division multiplexing based on the nonlinear Fourier transform

Jan-Willem Goossens, Mansoor I. Yousefi, Yves Jaouën, and Hartmut Hafermann
Opt. Express 25(22) 26437-26452 (2017)

Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing

S. Civelli, S. K. Turitsyn, M. Secondini, and J. E. Prilepsky
Opt. Express 26(13) 17360-17377 (2018)

References

  • View by:
  • |
  • |
  • |

  1. S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
    [Crossref]
  2. S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).
    [Crossref] [PubMed]
  3. S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
    [Crossref]
  4. I. Tavakkolnia and M. Safari, “Dispersion pre-compensation for NFT-based optical fiber communication systems,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO) (IEEE, 2016), pp. 1–2.
  5. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inform. Theory 60, 4312–4369 (2014).
    [Crossref]
  6. M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” in Proceedings of European Conference on Optical Communication (ECOC) (VDE, 2016), pp. 1–3.
  7. H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
    [Crossref]
  8. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, vol. 4 (SIAM, 1981).
    [Crossref]
  9. S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
    [Crossref]
  10. I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
    [Crossref]
  11. J. G. Proakis and M. Salehi, Digital Communications (5th ed.) (McGraw-Hill, 2008).
  12. S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Proceedings of Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.
  13. S. Civelli, E. Forestieri, and M. Secondini, “Impact of discretizations and boundary conditions in nonlinear frequency-division multiplexing,” in Proceedings of Fotonica 2016, (IET, 2016), pp. 1–4.
  14. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19, 12781–12788 (2011).
    [Crossref] [PubMed]
  15. S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
    [Crossref] [PubMed]
  16. R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proceedings of International Conference of Electrical and Computer Engineering (ICECE) (IEEE, 2006), pp. 408–411.
  17. S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
    [Crossref] [PubMed]
  18. S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
    [Crossref]

2017 (4)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
[Crossref]

2016 (2)

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
[Crossref] [PubMed]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
[Crossref]

2015 (2)

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
[Crossref] [PubMed]

2014 (2)

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).
[Crossref] [PubMed]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inform. Theory 60, 4312–4369 (2014).
[Crossref]

2011 (1)

Ablowitz, M. J.

M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, vol. 4 (SIAM, 1981).
[Crossref]

Aref, V.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

Barletti, L.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Proceedings of Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

Bononi, A.

Buelow, H.

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

Bülow, H.

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
[Crossref]

Civelli, S.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Proceedings of Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

S. Civelli, E. Forestieri, and M. Secondini, “Impact of discretizations and boundary conditions in nonlinear frequency-division multiplexing,” in Proceedings of Fotonica 2016, (IET, 2016), pp. 1–4.

DeMenezes, T. D.

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

Derevyanko, S. A.

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4, 307–322 (2017).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
[Crossref] [PubMed]

Ellis, A. D.

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
[Crossref]

Forestieri, E.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

S. Civelli, E. Forestieri, and M. Secondini, “Impact of discretizations and boundary conditions in nonlinear frequency-division multiplexing,” in Proceedings of Fotonica 2016, (IET, 2016), pp. 1–4.

Frumin, L. L.

Grellier, E.

Grigoryan, V.

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

Harper, P.

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
[Crossref]

Islam, A. R.

R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proceedings of International Conference of Electrical and Computer Engineering (ICECE) (IEEE, 2006), pp. 408–411.

Kamalian, M.

Kschischang, F. R.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inform. Theory 60, 4312–4369 (2014).
[Crossref]

Le, S. T.

Lima, I. T.

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

Menyuk, C. R.

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

O’sullivan, M.

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

Philips, I. D.

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
[Crossref]

Prilepsky, J. E.

Proakis, J. G.

J. G. Proakis and M. Salehi, Digital Communications (5th ed.) (McGraw-Hill, 2008).

Rahman, M. S.

R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proceedings of International Conference of Electrical and Computer Engineering (ICECE) (IEEE, 2006), pp. 408–411.

Safari, M.

I. Tavakkolnia and M. Safari, “Dispersion pre-compensation for NFT-based optical fiber communication systems,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO) (IEEE, 2016), pp. 1–2.

Salehi, M.

J. G. Proakis and M. Salehi, Digital Communications (5th ed.) (McGraw-Hill, 2008).

Secondini, M.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

S. Civelli, E. Forestieri, and M. Secondini, “Impact of discretizations and boundary conditions in nonlinear frequency-division multiplexing,” in Proceedings of Fotonica 2016, (IET, 2016), pp. 1–4.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Proceedings of Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, vol. 4 (SIAM, 1981).
[Crossref]

Shafik, R. A.

R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proceedings of International Conference of Electrical and Computer Engineering (ICECE) (IEEE, 2006), pp. 408–411.

Tavakkolnia, I.

I. Tavakkolnia and M. Safari, “Dispersion pre-compensation for NFT-based optical fiber communication systems,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO) (IEEE, 2016), pp. 1–2.

Turitsyn, S. K.

Wahls, S.

Yangzhang, X.

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” in Proceedings of European Conference on Optical Communication (ECOC) (VDE, 2016), pp. 1–3.

Yousefi, M. I.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inform. Theory 60, 4312–4369 (2014).
[Crossref]

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” in Proceedings of European Conference on Optical Communication (ECOC) (VDE, 2016), pp. 1–3.

IEEE Photon. Technol. Lett. (1)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett. 29, 1332–1335 (2017).
[Crossref]

IEEE Trans. Inform. Theory (1)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Parts I–III,” IEEE Trans. Inform. Theory 60, 4312–4369 (2014).
[Crossref]

J. Lightw. Technol. (3)

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightw. Technol. 33, 1433–1439 (2015).
[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightw. Technol. 34, 2459–2466 (2016).
[Crossref]

I. T. Lima, T. D. DeMenezes, V. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol. 35, 5056 (2017).
[Crossref]

Nat. Commun. (1)

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
[Crossref] [PubMed]

Nat. Photon. (1)

S. T. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nat. Photon. 11, 570 (2017).
[Crossref]

Opt. Express (3)

Optica (1)

Other (7)

I. Tavakkolnia and M. Safari, “Dispersion pre-compensation for NFT-based optical fiber communication systems,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO) (IEEE, 2016), pp. 1–2.

M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, vol. 4 (SIAM, 1981).
[Crossref]

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” in Proceedings of European Conference on Optical Communication (ECOC) (VDE, 2016), pp. 1–3.

R. A. Shafik, M. S. Rahman, and A. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proceedings of International Conference of Electrical and Computer Engineering (ICECE) (IEEE, 2006), pp. 408–411.

J. G. Proakis and M. Salehi, Digital Communications (5th ed.) (McGraw-Hill, 2008).

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Proceedings of Tyrrhenian International Workshop on Digital Communications (TIWDC) (IEEE, 2015), pp. 13–16.

S. Civelli, E. Forestieri, and M. Secondini, “Impact of discretizations and boundary conditions in nonlinear frequency-division multiplexing,” in Proceedings of Fotonica 2016, (IET, 2016), pp. 1–4.

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 (9)

Fig. 1
Fig. 1 NFDM with DF-BNFT detection.
Fig. 2
Fig. 2 The NFT causality property for NIS with no ISI on s(t). A train of Gaussian pulses, modulated by 16QAM symbols, and almost ISI-free, is shown before (on the left) and after (on the right) the BNFT. The red signal is generated by 8 symbols, while for the blue one only the first 6 are taken into account. The two optical signals are superimposed for tt6, as for Eq. (5) (baudrate Rs = 50 GBd, optical power Ps = 7 dBm).
Fig. 3
Fig. 3 Performance of the NFDM system for DF-BNFT (solid lines) and standard FNFT (dashed lines) detection for different burst length Nb (and rate efficiency η). 16QAM symbols, β2 = −20.39 ps2/km, Nz = 2000, L = 2000 km, and Rs = 50 GBd.
Fig. 4
Fig. 4 Impact of fiber propagation: performance of DF-BNFT on the fiber link without (solid lines) and with average nonlinear phase compensation (dashed lines) and on the AWGN channel (dotted line). Same scenario of Fig. 3.
Fig. 5
Fig. 5 Impact of error propagation due to decision feedback in the proposed DF-BNFT detection strategy: actual performance (solid lines), and error-propagation-free performance (dotted lines). Same scenario of Fig. 3.
Fig. 6
Fig. 6 Best achievable performance vs rate efficiency for NFDM with different detection strategies and for conventional systems with EDC or DBP. Same scenario of Fig. 3.
Fig. 7
Fig. 7 Best achievable performance vs rate efficiency for NFDM with different detection strategies and for conventional systems with EDC or DBP: (a) low-dispersion fiber with 16QAM symbols, β2 = −1.27 ps2/km, Nz = 125, L = 2000 km, and Rs = 50 GBd; (b) QPSK symbols with β2 = −20.39 ps2/km, Nz = 160, L = 4000 km, and Rs = 10 GBd.
Fig. 8
Fig. 8 Validation of the semianalytic approximation and bounds for the performance of DF-BNFT detection. Same scenario of Fig. 3, with (a) Nb = 256 and (b) Nb = 1024.
Fig. 9
Fig. 9 Convergence of the numerical simulations and of the semianalytic approximation and bounds with the number of iterations (transmitted sequences). Same scenario of Fig. 3, with Nb = 256 at Ps = −9dBm (above) and at optimal power Ps = −4dBm (below).

Equations (24)

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

K ( x , y ) σ F * ( x + y ) + σ x x K ( x , r ) F ( r + s ) F * ( s + y ) d r d s = 0 ,
F ( y ) = 1 2 π ρ ( λ ) e j λ y d λ .
s ( t ) = k = 1 N b x k g [ t ( k 1 ) T s ]
ρ ( λ ) = S ( λ / π ) .
r ( t ) | t t k = 𝒢 { x 1 , . . , x k } ,
r ˜ ( t ) = r ( t ) + n ( t ) ,
x ^ = argmax x p ( r ˜ | x ) .
x ^ = argmax x k = 1 N b p ( r ˜ k | x ) = argmax x k = 1 N b ln p ( r ˜ k | x ) ,
x ^ = argmax x k = 1 N b ln p ( r ˜ k | ( x 1 , , x k ) ) .
p ( r ˜ k | ( x 1 , , x k ) = 1 ( π σ 2 ) ν exp ( r ˜ k r k 2 / σ 2 )
x ^ k = argmax X i { X 1 , . . , X M } ln p ( r ˜ k | ( x ^ 1 , , x ^ k 1 , X i ) )
x ^ k = argmin X i { X 1 , . . , X M } r ˜ k r k ( i ) 2
P e = 1 M N b k = 1 N b m = 1 M P k ( m ) .
E m = { X m is not preferred when deciding on x k } = i = 1 i m M E m , i
P k ( m ) = P ( E m | ( x ^ 1 , , x ^ k 1 , X m ) ) = P ( i = 1 i m M E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) i = 1 i m M P ( E m , i | ( x ^ 1 , , x ^ k 1 , X m ) )
P ( E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) = 𝒬 ( d k ( m , i ) 2 σ )
d k ( m , i ) = r k ( m ) r k ( i )
P k ( m ) = P ( i = 1 i m M E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) = 1 P ( i = 1 i m M C m , i | ( x ^ 1 , , x ^ k 1 , X m ) )
P k ( m ) 1 i = 1 i m M ( 1 P ( E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) )
P k ( m ) = P ( i = 1 i m M E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) max i m P ( E m , i | ( x ^ 1 , , x ^ k 1 , X m ) ) .
P k ( m ) i = 1 i m M 𝒬 ( d k ( m , i ) 2 σ )
P k ( m ) 1 i = 1 i m M ( 1 𝒬 ( d k ( m , i ) 2 σ ) )
P k ( m ) 𝒬 ( d k 2 σ )
d k = min i m d k ( m , i )

Metrics