Abstract

Binary random numbers with an occurrence ratio of p:(1-p) (0 < p < 1) are needed in some applications. They are obtainable if the phase noise of a laser diode (LD) is used as a continuous-variable random-number generator. The problem is the existence of extra noise in the measuring system. However, most of this noise can be removed by taking the difference between consecutively measured values. This report confirms this by evaluating the randomness of the phase noise of an LD as a continuous quantity. Our findings show that we can easily obtain spontaneous-emission-based continuous-variable random numbers that allow one to set any p.

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

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References

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    [Crossref]
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    [Crossref]

2018 (3)

2017 (1)

2015 (1)

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (2015).
[Crossref]

2014 (2)

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. W. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22(2), 1645–1654 (2014).
[Crossref]

2012 (2)

2011 (2)

2010 (4)

C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18(23), 23584–23597 (2010).
[Crossref]

Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A 81(6), 063814 (2010).
[Crossref]

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35(3), 312–314 (2010).
[Crossref]

2009 (1)

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

2005 (1)

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

2001 (1)

L. Trevisan, “Extractors and pseudorandom generators,” J. Assoc. Comput. Mach. 48(4), 860–879 (2001).
[Crossref]

2000 (1)

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

1999 (1)

N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comp. Sys. Sci. 58(1), 148–173 (1999).
[Crossref]

1984 (1)

K. Kikuchi, T. Okoshi, and R. Arata, “Measurement of linewidth and FM-noise spectrum of 1.52 µm InGaAsP lasers,” Electron. Lett. 20(13), 535–536 (1984).
[Crossref]

1983 (1)

K. Vahala, “Occupation fluctuation noise: A fundamental source of linewidth broadening in semiconductor lasers,” Appl. Phys. Lett. 43(2), 140–142 (1983).
[Crossref]

1982 (2)

D. Welford and A. Mooradian, “Observation of linewidth broadening in (GaAl)As diode lasers due to electron number fluctuations,” Appl. Phys. Lett. 40(7), 560–562 (1982).
[Crossref]

C. Henry, “Theory of the Linewidth of Semiconductor Lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Abellan, C.

M. W. Mitchell, C. Abellan, and W. Amaya, “Strong experimental guarantees in ultrafast quantum random number generation,” Phys. Rev. A 91(1), 012314 (2015).
[Crossref]

Abellán, C.

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

Acín, A.

Amaya, W.

Andersen, U. L.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Arata, R.

K. Kikuchi, T. Okoshi, and R. Arata, “Measurement of linewidth and FM-noise spectrum of 1.52 µm InGaAsP lasers,” Electron. Lett. 20(13), 535–536 (1984).
[Crossref]

Aviad, Y.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

Capmany, J.

Chi, Y.-M.

Cohen, A. B.

Curty, M.

Dong, R.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Dubrova, E.

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Dynes, J. F.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Fan, Y.

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. Li, K. A. Shore, Y. Wang, and A. Wang, “Ultrafast Fully Photonic Random Bit Generator,” J. Lightwave Technol. 36(12), 2531–2540 (2018).
[Crossref]

Fröhlich, B.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Gabriel, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Garcés, I.

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

Guo, X.

Guo, Y.

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. Li, K. A. Shore, Y. Wang, and A. Wang, “Ultrafast Fully Photonic Random Bit Generator,” J. Lightwave Technol. 36(12), 2531–2540 (2018).
[Crossref]

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. Li, K. A. Shore, Y. Wang, and A. Wang, “Ultrafast Fully Photonic Random Bit Generator,” J. Lightwave Technol. 36(12), 2531–2540 (2018).
[Crossref]

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

P. Li, J. Zhang, L. Sang, X. Liu, Y. Guo, X. Guo, A. Wang, K. A. Shore, and Y. Wang, “Real-time online photonic random number generation,” Opt. Lett. 42(14), 2699–2702 (2017).
[Crossref]

Henry, C.

C. Henry, “Theory of the Linewidth of Semiconductor Lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Jennewein, T.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

Jofre, M.

Kanter, I.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

Kennard, J. E.

Kikuchi, K.

K. Kikuchi, T. Okoshi, and R. Arata, “Measurement of linewidth and FM-noise spectrum of 1.52 µm InGaAsP lasers,” Electron. Lett. 20(13), 535–536 (1984).
[Crossref]

Lázaro, J. A.

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

Leuchs, G.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Li, K.

Li, P.

Li, X.

Liu, M.

Liu, X.

Lo, H.-K.

Lucamarini, M.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Ma, X.

Mahler, D. H.

Marquardt, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Matthews, J. C. F.

Mauerer, W.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Mitchell, M. W.

Mooradian, A.

D. Welford and A. Mooradian, “Observation of linewidth broadening in (GaAl)As diode lasers due to electron number fluctuations,” Appl. Phys. Lett. 40(7), 560–562 (1982).
[Crossref]

Murphy, T. E.

Nisan, N.

N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comp. Sys. Sci. 58(1), 148–173 (1999).
[Crossref]

Okoshi, T.

K. Kikuchi, T. Okoshi, and R. Arata, “Measurement of linewidth and FM-noise spectrum of 1.52 µm InGaAsP lasers,” Electron. Lett. 20(13), 535–536 (1984).
[Crossref]

Plews, A.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Pruneri, V.

Qi, B.

Qian, L.

Raffaelli, F.

Reidler, I.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

Rosenbluh, M.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

Roy, R.

Salevan, J. C.

Salinas, I.

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

Sang, L.

Shen, Y.

Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A 81(6), 063814 (2010).
[Crossref]

Shields, A. J.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Shore, K. A.

Sibson, P.

Sych, D.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Ta-Shma, A.

N. Nisan and A. Ta-Shma, “Extracting Randomness: A Survey and New Constructions,” J. Comp. Sys. Sci. 58(1), 148–173 (1999).
[Crossref]

Thompson, M. G.

Tian, L.

Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A 81(6), 063814 (2010).
[Crossref]

Tomaru, T.

Trevisan, L.

L. Trevisan, “Extractors and pseudorandom generators,” J. Assoc. Comput. Mach. 48(4), 860–879 (2001).
[Crossref]

Vahala, K.

K. Vahala, “Occupation fluctuation noise: A fundamental source of linewidth broadening in semiconductor lasers,” Appl. Phys. Lett. 43(2), 140–142 (1983).
[Crossref]

Villafranca, A.

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

Wang, A.

Wang, Y.

Weihs, G.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

Weinfurter, H.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

Welford, D.

D. Welford and A. Mooradian, “Observation of linewidth broadening in (GaAl)As diode lasers due to electron number fluctuations,” Appl. Phys. Lett. 40(7), 560–562 (1982).
[Crossref]

Williams, C. R. S.

Wittmann, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Xu, B.

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Xu, F.

Xu, H.

Xue, L.

Yuan, Z. L.

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Zeilinger, A.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000).
[Crossref]

Zhang, J.

Zhang, M.

Zheng, H.

Zou, H.

Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A 81(6), 063814 (2010).
[Crossref]

APL Photonics (1)

P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. Wang, and A. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Appl. Phys. Lett. (3)

K. Vahala, “Occupation fluctuation noise: A fundamental source of linewidth broadening in semiconductor lasers,” Appl. Phys. Lett. 43(2), 140–142 (1983).
[Crossref]

D. Welford and A. Mooradian, “Observation of linewidth broadening in (GaAl)As diode lasers due to electron number fluctuations,” Appl. Phys. Lett. 40(7), 560–562 (1982).
[Crossref]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014).
[Crossref]

Electron. Lett. (1)

K. Kikuchi, T. Okoshi, and R. Arata, “Measurement of linewidth and FM-noise spectrum of 1.52 µm InGaAsP lasers,” Electron. Lett. 20(13), 535–536 (1984).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Henry, “Theory of the Linewidth of Semiconductor Lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Villafranca, J. A. Lázaro, I. Salinas, and I. Garcés, “Measurement of the Linewidth Enhancement Factor in DFB Lasers Using a High-Resolution Optical Spectrum Analyzer,” IEEE Photon. Technol. Lett. 17(11), 2268–2270 (2005).
[Crossref]

J. Assoc. Comput. Mach. (1)

L. Trevisan, “Extractors and pseudorandom generators,” J. Assoc. Comput. Mach. 48(4), 860–879 (2001).
[Crossref]

J. Comp. Sys. Sci. (1)

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Only two tests are responsible for most of the 188 terms in NIST SP800-22. Therefore, SP800-22 does not evaluate binary random numbers equally with respect to the 15 kinds of tests. Although there is room for improvement, it is beyond the scope of our study. Thus, we decided to use SP800-22 as it stands in order to obtain one of evaluation figures.

http://csrc.nist.gov/publications/PubsSPs.html

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Figures (7)

Fig. 1.
Fig. 1. Model of phase-noise measuring system.
Fig. 2.
Fig. 2. Block diagram of experimental setup. The right figure shows an example of phase noise measured at 100 MSps.
Fig. 3.
Fig. 3. How to translate 256-value fluctuations into 16-value random numbers. (a) Simple way. (b) Way of finely equalizing the probability distribution of 4-value random numbers.
Fig. 4.
Fig. 4. Probability distribution of phase noise. Gaussian curves are also drawn. Data were acquired at three different sampling rates. Measured phase noise is ideally distributed with respect to probability distribution.
Fig. 5.
Fig. 5. Power spectrum of phase noise.
Fig. 6.
Fig. 6. (a) Power spectrum of phase noise. The horizontal axis of the inset is a log scale. (b) Power spectrum of intensity noise. The extra noise at 20 MHz comes from the temperature controller.
Fig. 7.
Fig. 7. Autocorrelation. Insets show mutual correlation. The horizontal axis in the insets is the number of data used for calculating the mutual correlation, i.e., 1×, 2×, 3×, 4×, and 5×105. (a) Autocorrelation for raw data. Autocorrelation is large in the small-delay region owing mainly to the low-frequency extra noise in the measuring system. (b) Autocorrelation for differential data. The correlation due to the extra noise in the measuring system is mostly removed by differentiating the consecutively measured values. When the LD is driven at a near-threshold current, i.e., Id = 12 mA, the variance of the autocorrelation is almost constant except for the case of 2.5 GSps sampling. The dotted lines indicate 2σsd (or 2σmd in the inset) for the ideal case.

Tables (2)

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Table 1. Results of NIST SP800-22 test.

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Table 2. Failure counts and rates in NIST SP800-22 test.

Equations (7)

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I^(t)12ξ[a^(t+τ)a^(t)+a^(t)a^(t+τ)]
I(t)(1)nξa0(t+τ)a0(t)sinδφ(t,τ).
jδxj2¯=j(xj+1xj)2¯=j(xj+122xj+1xj+xj2)¯=2jxj2¯,
jδxjδxj+1¯=j(xj+1xj)(xj+2xj+1)¯=j(xj+1xj+2xj+12xjxj+2+xjxj+1)¯=jxj+12¯.
s(d)=jδxjδxj+1/jδxjδxj+1jδxj2jδxj2¯=jδxjδxj+1¯/s(d)=jδxjδxj+1/jδxjδxj+1jδxj2jδxj2¯=jδxjδxj+1¯jδxj2¯jδxj2¯=0.5.
(j=1Nδxjδxj+d)2¯=[j=1N(xj+1xj+1+dxj+1xj+dxjxj+1+d+xjxj+d)]2¯=j=1N(xj+12xj+1+d2¯+xj+12xj+d2¯+xj2xj+1+d2¯+xj2xj+d2¯)+2j=2Nxj2xj+d2¯=4Nxj2¯2+2(N1)xj2¯26Nxj2¯2.
(j=1Nδxjδxj+1)2¯=[j=1N(xj+1xj+2xj+12xjxj+2+xjxj+1)]2¯=j=1N(xj+12xj+22¯+xj2xj+22¯+xj2xj+12¯)+2j=2Nxj2xj+12¯+j=1Nxj+12j=1Nxj+12¯=3Nxj2¯2+2(N1)xj2¯2+N(xj4¯xj2¯2)+N2xj2¯27Nxj2¯2+N2xj2¯2,

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