Abstract

We present a new hardware simulator (HS) for characterization, testing and benchmarking of digital correlators used in various optical correlation spectroscopy experiments where the photon statistics is Gaussian and the corresponding time correlation function can have any arbitrary shape. Starting from the HS developed in [Rev. Sci. Instrum. 74, 4273 (2003)], and using the same I/O board (PCI-6534 National Instrument) mounted on a modern PC (Intel Core i7-CPU, 3.07GHz, 12GB RAM), we have realized an instrument capable of delivering continuous streams of TTL pulses over two channels, with a time resolution of Δt = 50ns, up to a maximum count rate of 〈I〉 ∼ 5MHz. Pulse streams, typically detected in dynamic light scattering and diffuse correlation spectroscopy experiments were generated and measured with a commercial hardware correlator obtaining measured correlation functions that match accurately the expected ones.

© 2014 Optical Society of America

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References

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  1. B.J. Berne and R. Pecora, Dynamic Light Scattering, (Wiley, NY, 1976).
  2. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
    [Crossref] [PubMed]
  3. D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
    [Crossref] [PubMed]
  4. T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
    [Crossref]
  5. K. Schatzel, “Single-photon correlation techniques”, in Dynamic Light Scattering, edited by R.G.W. Brown, ed. (Clarendon, Oxford, 1993). Chapt. 2.
  6. D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering”, Appl. Opt. 40, 4011–4021 (2001).
    [Crossref]
  7. D. Magatti and F. Ferri, “25 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 74, 1135–1144 (2003).
    [Crossref]
  8. M. Wahl, I. Gregor, M. Patting, and J. Enderlein, “Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting”, Opt. Express 11, 3583–3591 (2003).
    [Crossref] [PubMed]
  9. L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
    [Crossref]
  10. J. Buchholz, J. Wolfgang Krieger, G. Mocsar, B. Kreith, E. Charbon, G. Vamosi, U. Kebschull, and J. Langowski, “FPGA implementation of a 32×32 autocorrelator array for analysis of fast image series”, Opt. Express 20, 17767–17781 (2012).
    [Crossref] [PubMed]
  11. E. Schaub, “F2Cor: fast 2-stage correlation algorithm for FCS and DLS”, Opt. Express 20, 2184–2195 (2012).
    [Crossref] [PubMed]
  12. E. Schaub, “High countrate real-time FCS using F2Cor”, Opt. Express 21, 2184–2195 (2012).
    [Crossref]
  13. F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
    [Crossref]
  14. J.W. Goodman, Statistical Optics, (Wiley & Sons Inc., New York, NY, 2000).
  15. J.A. Williamson, “Statistical evaluation of dead time effects and pulse pileup in fast photon counting. Introduction of the sequential model”, Anal. Chem. 60, 2198–2203 (1988).
    [Crossref]
  16. C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain”, Opt. Express,  141125–1144 (2006).
    [Crossref] [PubMed]
  17. K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function”, Quantum Opt.,  2, 287 (1990).
    [Crossref]
  18. R. Peters, “Noise on photon correlation and its effects on data reduction algorithms”, in Dynamic Light Scattering edited by W. Brown, ed. (Clarendon Press, Oxford, 1993) Chapt. 3.
  19. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

2012 (3)

2010 (1)

T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

2009 (1)

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

2006 (1)

2003 (3)

M. Wahl, I. Gregor, M. Patting, and J. Enderlein, “Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting”, Opt. Express 11, 3583–3591 (2003).
[Crossref] [PubMed]

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “25 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 74, 1135–1144 (2003).
[Crossref]

2001 (1)

1995 (1)

D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

1990 (1)

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function”, Quantum Opt.,  2, 287 (1990).
[Crossref]

1988 (2)

J.A. Williamson, “Statistical evaluation of dead time effects and pulse pileup in fast photon counting. Introduction of the sequential model”, Anal. Chem. 60, 2198–2203 (1988).
[Crossref]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

Baker, W.B.

T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Berne, B.J.

B.J. Berne and R. Pecora, Dynamic Light Scattering, (Wiley, NY, 1976).

Boas, D.A.

D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

Buchholz, J.

Campbell, L.E.

D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

Chang, Y.-R.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Charbon, E.

Choe, R.

T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Daisuke, F.

Durduran, T.

Enderlein, J.

Fann, W.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Ferri, F.

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “25 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 74, 1135–1144 (2003).
[Crossref]

D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering”, Appl. Opt. 40, 4011–4021 (2001).
[Crossref]

Flannery, B.P.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

Goodman, J.W.

J.W. Goodman, Statistical Optics, (Wiley & Sons Inc., New York, NY, 2000).

Greenberg, J. H.

Gregor, I.

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

Hsu, K.-H.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Kebschull, U.

Kreith, B.

Langowski, J.

Lee, H.-Y.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Lin, X.-Y.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Magatti, D.

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “25 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 74, 1135–1144 (2003).
[Crossref]

D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering”, Appl. Opt. 40, 4011–4021 (2001).
[Crossref]

Mocsar, G.

Patting, M.

Pecora, R.

B.J. Berne and R. Pecora, Dynamic Light Scattering, (Wiley, NY, 1976).

Peters, R.

R. Peters, “Noise on photon correlation and its effects on data reduction algorithms”, in Dynamic Light Scattering edited by W. Brown, ed. (Clarendon Press, Oxford, 1993) Chapt. 3.

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

Press, W.H.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

Schatzel, K.

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function”, Quantum Opt.,  2, 287 (1990).
[Crossref]

K. Schatzel, “Single-photon correlation techniques”, in Dynamic Light Scattering, edited by R.G.W. Brown, ed. (Clarendon, Oxford, 1993). Chapt. 2.

Schaub, E.

E. Schaub, “High countrate real-time FCS using F2Cor”, Opt. Express 21, 2184–2195 (2012).
[Crossref]

E. Schaub, “F2Cor: fast 2-stage correlation algorithm for FCS and DLS”, Opt. Express 20, 2184–2195 (2012).
[Crossref] [PubMed]

Teukolsky, S.A.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

Vamosi, G.

Vetterling, W.T.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

Wahl, M.

Wang, M.-K.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

White, J.D.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Williamson, J.A.

J.A. Williamson, “Statistical evaluation of dead time effects and pulse pileup in fast photon counting. Introduction of the sequential model”, Anal. Chem. 60, 2198–2203 (1988).
[Crossref]

Wolfgang Krieger, J.

Yang, L.-L.

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Yodh, A. G.

Yodh, A.G.

T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

Yu, G.

Zhou, C.

Anal. Chem. (1)

J.A. Williamson, “Statistical evaluation of dead time effects and pulse pileup in fast photon counting. Introduction of the sequential model”, Anal. Chem. 60, 2198–2203 (1988).
[Crossref]

Appl. Opt. (1)

J. Microscopy (1)

L.-L. Yang, H.-Y. Lee, M.-K. Wang, X.-Y. Lin, K.-H. Hsu, Y.-R. Chang, W. Fann, and J.D. White, “Real-time data acquisition incorporating high-speed software correlator for single-molecule spectroscopy”, J. Microscopy,  234, 302–310 (2009).
[Crossref]

Opt. Express (5)

Phys. Rev. Lett. (2)

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy”, Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

D.A. Boas, L.E. Campbell, and A.G. Yodh, Scattering and imaging with diffusing temporal field correlations, Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

Quantum Opt. (1)

K. Schatzel, “Noise on photon correlation data: I. Autocorrelation function”, Quantum Opt.,  2, 287 (1990).
[Crossref]

Rep. Prog. Phys. (1)

T. Durduran, R. Choe, W.B. Baker, and A.G. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Rev. Sci. Instrum. (2)

F. Ferri and D. Magatti, “Hardware simulator for photon correlation spectroscopy”, Rev. Sci. Instrum. 74, 4273–4279 (2003).
[Crossref]

D. Magatti and F. Ferri, “25 ns software correlator for photon and fluorescence correlation spectroscopy”, Rev. Sci. Instrum. 74, 1135–1144 (2003).
[Crossref]

Other (5)

B.J. Berne and R. Pecora, Dynamic Light Scattering, (Wiley, NY, 1976).

J.W. Goodman, Statistical Optics, (Wiley & Sons Inc., New York, NY, 2000).

K. Schatzel, “Single-photon correlation techniques”, in Dynamic Light Scattering, edited by R.G.W. Brown, ed. (Clarendon, Oxford, 1993). Chapt. 2.

R. Peters, “Noise on photon correlation and its effects on data reduction algorithms”, in Dynamic Light Scattering edited by W. Brown, ed. (Clarendon Press, Oxford, 1993) Chapt. 3.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).

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Figures (9)

Fig. 1
Fig. 1

(Adapted from Fig. 1 of [13]). Schematic diagram of the algorithm adopted for generating synthetic correlation data. Ex(t) and Ey(t) are the two components of the scattered electric field, I(t) is the intensity, Ni the sequence of counts (0 or 1) after the Poisson filter and T (t) the integrated photon arrival times that are stored on the hard disk (HD). Optionally, the presence of the detector defects can also be introduced.

Fig. 2
Fig. 2

(a): Normalized Intensity correlation functions (symbols) obtained by measuring with the Flex2k 12×2 correlator (by Correlator.com) a pulse stream characterized by the same decay time τc = 10−5s and different stretched/compressed exponentials with α = 0.5, 1.0, 1.5 (see Eq. (20). The solid lines represent the expected behaviors. The pulses were delivered with a Δt = 50ns clock, at a count rate 〈I〉 = 105Hz, for a measuring time T = 100s. (b): Relative deviations between data and expected curves.

Fig. 3
Fig. 3

(a): Normalized Intensity correlation functions (symbols) obtained by measuring with the Flex2k 12×2 correlator two pulse streams characterized by a field correlation function given by Eq. (21) (see text for parameters values). The solid lines represent the expected behaviors. The pulses were delivered with a Δt = 50ns clock, at a count rate 〈I〉 = 105Hz, for a time T = 100s. (b): Relative deviations between data and expected curves.

Fig. 4
Fig. 4

(a): Normalized Intensity correlation functions (symbols) obtained by measuring with the Flex2k-12×2 correlator a pulse stream characterized by two single exponential decay times [see Eq. (22)] with τ1 = 10−6s, τ2 = 10−4s and A1/(A1 + A2) = 0.30. Different data sets refer to data being acquired within different coherence areas, from Nca = 1 to 5. The solid lines represent the expected behaviors. The pulses were delivered with a Δt = 50ns clock, at a count rate 〈I〉 = 105Hz, for a time T = 100s. (b): Relative deviations between data and expected curves.

Fig. 5
Fig. 5

(a): Normalized intensity correlation functions (symbols) obtained by measuring with the Flex2k 12×2 correlator pulse streams characterized by the same single exponential decay time τc = 10−4s and different count rates from 100kHz to 10MHz. The solid lines represent the fitting to single exponential decay functions. The pulses were delivered with a Δt = 50ns clock, for a time time of T = 100s (blue and green symbols) and T = 10s (red symbols). (b): Relative deviations between data and fitting curves.

Fig. 6
Fig. 6

(a): normalized field correlation function gz(τ) corresponding to a single exponential decay with decay time τc = 10−5s (solid black line) and gz(τ) windowed between ±τmax = ±5 × 10−4s, corresponding to an accuracy ε = 5 × 10−3. (b) normalized impulse response function h(t) (solid blue line) and q resampled at the discrete times q (red crosses).

Fig. 7
Fig. 7

(a): Sketch of the resampling procedure for the function h(t). The original values hj (black dots) sampled at the times tj are replaced by the values q (red crosses) obtained by averaging hj over increasingly wider intervals Δ̃tq [Eq.(B.1)]. An example of the explicit expressions for the first few q values are reported on the right side of the figure.

Fig. 8
Fig. 8

Comparison between the convolution carried out by using Eq. (13) [panel(a)] and Eq.(B.3) [panel (b)]. Equation (13) uses a linear time grid and the generation of each new point zi requires the computation of a very large number of terms, namely the the sum of Np = 2 jmax + 1 products. Conversely, Eq.(B.3) uses a pseudo-geometrical progression time-grid where the last Np yi points of panel (a) are resampled on the same time grid used for h (crosses), with y i 0 (red cross) being the central point with index q = 0. In this case the number of terms to be multiplied and summed is Ñp = 2qmax + 1. Since typically Ñp = 100, while Np might be very large (Np ∼ 104 − 105), the use of Eq.(B.3) with respect to Eq. (13) is very convenient.

Fig. 9
Fig. 9

Comparison between the theoretical intensity correlation g2(τ) = 1 + |gz(τ)|2 (solid line) and the two correlations computed by correlating the intensity signal Ii and the corresponding pulse stream Ni. The theoretical gz(τ) is the same as the one of Fig. 6(a). The averaging times were T = 20s and T = 100s for Ii and Ni signals, respectively. Both curves match excellently the theoretical one, with non systematic residuals that, because of the shot noise introduced by the Poisson filter, are higher in the case of the Ni-correlation.

Equations (30)

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z ( t ) = y ( t t ) h ( t ) d t = [ y h ] ( t ) ,
G z ( τ ) = [ G y G h ] ( τ )
G h ( τ ) = h ( t ) h ( t + τ ) d t .
G z ( τ ) = k y 2 G h ( τ )
g z ( τ ) = h ( t ) h ( t + τ ) d t ,
g ^ z ( ω ) = | h ^ ( ω ) | 2
g ^ z ( ω ) e j ϕ ( ω ) = h ^ ( ω )
h ( t ) = g ^ z ( ω ) ^
{ g z ( τ k ) = Δ t j = + h j h j + k | τ k | τ max g z ( τ k ) = 0 | τ k | > τ max .
y ( t ) = i = 0 y i R ( t t i Δ t )
G y ( τ ) = y 2 Λ ( τ / Δ t )
G z ( τ ) = y 2 G h ( τ ) Δ t
z i = Δ t j = j max j max y i j h j
σ z 2 = G z ( 0 ) = σ y 2 G h ( 0 ) Δ t
σ z 2 = σ y 2 h j 2 [ h j ] 2
σ z 2 = σ y 2 h 2 h 2 1 N p ~ σ y 2 Δ t δ h
G 2 ( τ ) = I 2 [ 1 + β | G 1 ( τ ) | 2 I 2 ]
I i = ( E x 2 ) i + ( E y 2 ) i
G 2 ( τ k ) = I 2 [ 1 + 1 N ca ( G z ( τ k ) G z ( 0 ) ) 2 ] .
G 1 ( τ ) = exp [ ( τ τ c ) α ] .
G 1 ( ρ , z , τ ) = v 4 π D [ exp [ K ( τ ) r 1 ] r 1 exp [ K ( τ ) r b ] r b ]
G 1 ( τ ) = A 1 exp ( τ / τ 1 ) + A 2 exp ( τ / τ 2 ) .
G y ( τ ) = lim T 1 T 0 T y ( t ) y ( t + τ ) d t
G y ( τ ) = lim T 1 T m , k = 0 y m y k R ( t t m Δ t ) R ( t + τ t k Δ t ) d t .
G y ( τ ) = lim T y 2 1 T 0 T m = 0 R ( t t m Δ t ) R ( t + τ t m Δ t ) d t .
G y ( τ ) = lim N y 2 1 N Δ t m = 0 N 0 T R ( t t m Δ t ) R ( t + τ t m Δ t ) d t .
G y ( τ ) = y 2 Λ ( τ Δ t )
{ h ˜ q = 1 Δ n ( q ) j = n ( q 1 ) + 1 n ( q ) h j q < 0 h ˜ q = h 0 q = 0 h ˜ q = 1 Δ n ( q ) j = n ( q ) n ( q + 1 ) 1 h j q > 0
{ y ˜ i 0 , q = 1 Δ n ( q ) j = i 0 + n ( q 1 ) + 1 i 0 + n ( q ) y j q < 0 y ˜ i 0 , q = y i 0 q = 0 y ˜ i 0 , q = 1 Δ n ( q ) j = i 0 + n ( q ) i 0 + n ( q + 1 ) 1 y j q > 0 .
z i = q = q max q max y ˜ i 0 , q h ˜ q Δ ˜ t q

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