Abstract

The frequency-sampling method is widely used to accommodate nonlinear laser tuning in swept-wavelength interferometric techniques such as optical frequency domain reflectometry (OFDR) and swept-wavelength optical coherence tomography (OCT). In this paper we analyze the frequency-sampling method and identify two sources of sampling errors. One source of error is the limit of an underlying approximation for long interferometer path mismatches and fast laser tuning rates. A second source of error is transmission delays in data acquisition hardware. We show that the measurement system can be configured such that the two error sources cancel to second order. We present experimental verification of sampling error correction using a general swept-wavelength interferometer with a significantly nonlinear laser sweep.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," J. Lightwave Technol. LT-3, 971-977 (1985).
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  7. J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
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  8. L.-T. Wang, K. Iiyama, F. Tsukada, N. Yoshida, and K.-I. Hayashi, "Loss measurement in optical waveguide devices by coherent frequency-modulated continuous-wave reflectometry," Opt. Lett. 18, 1095-1097 (1993).
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  12. T.-J. Ahn, Y. Jung, K. Oh, and D. Y. Kim, "Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber," Opt. Express 13, 10,040-10,047 (2005).
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    [CrossRef] [PubMed]
  28. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, "High-speed optical frequency-domain imaging," Opt. Express 11, 2953-2963 (2003).
    [CrossRef] [PubMed]
  29. K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
    [CrossRef]
  30. K. Iiyama, L.-T. Wang, and K. ichi Hayashi, "Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry," J. Lightwave Technol. 14, 173-178 (1996).
    [CrossRef]
  31. K.-Y. Huang and G. M. Carter, "Coherent optical frequency domain reflectometry (OFDR) using a fiber grating external cavity laser," IEEE Photon. Technol. Lett. 6, 1466-1468 (1994).
    [CrossRef]
  32. M. Kobayashi, K. Takada, and J. Noda, "Optical-frequency encoder using polarization-maintaining fiber," J. Lightwave Technol. 8, 1697-1702 (1990).
    [CrossRef]
  33. K. Takada, "High-resolution OFDR with incorporated fiber-optic frequency encoder," 4, 1069-1072 (1992).
  34. T.-J. Ahn and D. Y. Kim, "Analysis of nonlinear frequency sweep in high-speed tunable laser sources using a self-homodyne measurement and Hilbert transformation," Appl. Opt. 46, 2394-2400 (2007).
    [CrossRef] [PubMed]

2007 (1)

2005 (4)

2003 (2)

G. D. VanWiggeren, A. R. Motamedi, and D. M. Baney, "Single-scan interferometric component analyzer," IEEE Photon. Technol. Lett. 15, 263-265 (2003).
[CrossRef]

S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, "High-speed optical frequency-domain imaging," Opt. Express 11, 2953-2963 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

M. Yoshida, T. Miyamoto, N. Zou, K. Nakamura, and H. Ito, "Novel PMD measurement method based on OFDR using a frequency-shifted feedback fiber laser," Opt. Express 9, 207-211 (2001).
[CrossRef] [PubMed]

M. Yoshida, K. Nakamura, and H. Ito, "A new method for measurement of group velocity dispersion of optical fibers by using a frequency-shifted feedback fiber laser," IEEE Photon. Technol. Lett. 13, 227-229 (2001).
[CrossRef]

1998 (2)

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

M. Froggatt and J. Moore, "High-spatial-resolution distributed strain measurement in optical fiber with Rayleigh backscatter," Appl. Opt. 37, 1735-1740 (1998).
[CrossRef]

1997 (3)

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, "Optical coherence tomography using a frequency-tunable optical source," Opt. Lett. 22, 340-342 (1997).
[CrossRef] [PubMed]

K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
[CrossRef]

J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
[CrossRef]

1996 (2)

K. Iiyama, L.-T. Wang, and K. ichi Hayashi, "Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry," J. Lightwave Technol. 14, 173-178 (1996).
[CrossRef]

M. Froggatt, "Distributed measurement of the complex modulation of a photoinduced Bragg grating in an optical fiber," Appl. Opt. 35, 5162-5164 (1996).
[CrossRef] [PubMed]

1995 (1)

R. Passy, N. Gisin, and J. P. von der Weid, "High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components," IEEE Photon. Technol. Lett. 7, 667-669 (1995).
[CrossRef]

1994 (1)

K.-Y. Huang and G. M. Carter, "Coherent optical frequency domain reflectometry (OFDR) using a fiber grating external cavity laser," IEEE Photon. Technol. Lett. 6, 1466-1468 (1994).
[CrossRef]

1993 (2)

L.-T. Wang, K. Iiyama, F. Tsukada, N. Yoshida, and K.-I. Hayashi, "Loss measurement in optical waveguide devices by coherent frequency-modulated continuous-wave reflectometry," Opt. Lett. 18, 1095-1097 (1993).
[CrossRef] [PubMed]

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11, 1377-1384 (1993).
[CrossRef]

1990 (2)

E. C. Burrows and K.-Y. Liou, "High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source," Electron. Lett. 26, 577-579 (1990).
[CrossRef]

M. Kobayashi, K. Takada, and J. Noda, "Optical-frequency encoder using polarization-maintaining fiber," J. Lightwave Technol. 8, 1697-1702 (1990).
[CrossRef]

1985 (2)

S. A. Kingsley and D. E. N. Davies, "OFDR diagnostics for fibre and integrated-optic systems," Electron. Lett. 21, 434-435 (1985).
[CrossRef]

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," J. Lightwave Technol. LT-3, 971-977 (1985).
[CrossRef]

1981 (1)

W. Eickhoff and R. Ulrich, "Optical frequency domain reflectometry in single-mode fiber," Appl. Phys. Lett. 39, 693-695 (1981).
[CrossRef]

1960 (1)

A. Hymans and J. Lait, "Analysis of a frequency-modulated continuous-wave ranging system," Proc. IEEE 107, 365-372 (1960).

Ahn, T.-J.

T.-J. Ahn and D. Y. Kim, "Analysis of nonlinear frequency sweep in high-speed tunable laser sources using a self-homodyne measurement and Hilbert transformation," Appl. Opt. 46, 2394-2400 (2007).
[CrossRef] [PubMed]

T.-J. Ahn, Y. Jung, K. Oh, and D. Y. Kim, "Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber," Opt. Express 13, 10,040-10,047 (2005).
[CrossRef]

Baney, D. M.

G. D. VanWiggeren, A. R. Motamedi, and D. M. Baney, "Single-scan interferometric component analyzer," IEEE Photon. Technol. Lett. 15, 263-265 (2003).
[CrossRef]

Bouma, B. E.

Brinkmeyer, E.

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11, 1377-1384 (1993).
[CrossRef]

Burrows, E. C.

E. C. Burrows and K.-Y. Liou, "High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source," Electron. Lett. 26, 577-579 (1990).
[CrossRef]

Carter, G. M.

K.-Y. Huang and G. M. Carter, "Coherent optical frequency domain reflectometry (OFDR) using a fiber grating external cavity laser," IEEE Photon. Technol. Lett. 6, 1466-1468 (1994).
[CrossRef]

Chinn, S. R.

Culshaw, B.

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," J. Lightwave Technol. LT-3, 971-977 (1985).
[CrossRef]

Davies, D. E. N.

S. A. Kingsley and D. E. N. Davies, "OFDR diagnostics for fibre and integrated-optic systems," Electron. Lett. 21, 434-435 (1985).
[CrossRef]

de Boer, J. F.

Eickhoff, W.

W. Eickhoff and R. Ulrich, "Optical frequency domain reflectometry in single-mode fiber," Appl. Phys. Lett. 39, 693-695 (1981).
[CrossRef]

Froggatt, M.

Froggatt, M. E.

Fujimoto, J. G.

Gifford, D. K.

Gisin, N.

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
[CrossRef]

R. Passy, N. Gisin, and J. P. von der Weid, "High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components," IEEE Photon. Technol. Lett. 7, 667-669 (1995).
[CrossRef]

Glombitza, U.

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11, 1377-1384 (1993).
[CrossRef]

Hayashi, K. ichi

K. Iiyama, L.-T. Wang, and K. ichi Hayashi, "Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry," J. Lightwave Technol. 14, 173-178 (1996).
[CrossRef]

Hayashi, K.-I.

Horiguchi, T.

K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
[CrossRef]

Huang, K.-Y.

K.-Y. Huang and G. M. Carter, "Coherent optical frequency domain reflectometry (OFDR) using a fiber grating external cavity laser," IEEE Photon. Technol. Lett. 6, 1466-1468 (1994).
[CrossRef]

Huttner, B.

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

Hymans, A.

A. Hymans and J. Lait, "Analysis of a frequency-modulated continuous-wave ranging system," Proc. IEEE 107, 365-372 (1960).

Iftimia, N.

Iiyama, K.

K. Iiyama, L.-T. Wang, and K. ichi Hayashi, "Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry," J. Lightwave Technol. 14, 173-178 (1996).
[CrossRef]

L.-T. Wang, K. Iiyama, F. Tsukada, N. Yoshida, and K.-I. Hayashi, "Loss measurement in optical waveguide devices by coherent frequency-modulated continuous-wave reflectometry," Opt. Lett. 18, 1095-1097 (1993).
[CrossRef] [PubMed]

Ito, H.

M. Yoshida, K. Nakamura, and H. Ito, "A new method for measurement of group velocity dispersion of optical fibers by using a frequency-shifted feedback fiber laser," IEEE Photon. Technol. Lett. 13, 227-229 (2001).
[CrossRef]

M. Yoshida, T. Miyamoto, N. Zou, K. Nakamura, and H. Ito, "Novel PMD measurement method based on OFDR using a frequency-shifted feedback fiber laser," Opt. Express 9, 207-211 (2001).
[CrossRef] [PubMed]

Jung, Y.

T.-J. Ahn, Y. Jung, K. Oh, and D. Y. Kim, "Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber," Opt. Express 13, 10,040-10,047 (2005).
[CrossRef]

Kim, D. Y.

T.-J. Ahn and D. Y. Kim, "Analysis of nonlinear frequency sweep in high-speed tunable laser sources using a self-homodyne measurement and Hilbert transformation," Appl. Opt. 46, 2394-2400 (2007).
[CrossRef] [PubMed]

T.-J. Ahn, Y. Jung, K. Oh, and D. Y. Kim, "Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber," Opt. Express 13, 10,040-10,047 (2005).
[CrossRef]

Kingsley, S. A.

S. A. Kingsley and D. E. N. Davies, "OFDR diagnostics for fibre and integrated-optic systems," Electron. Lett. 21, 434-435 (1985).
[CrossRef]

Kobayashi, M.

M. Kobayashi, K. Takada, and J. Noda, "Optical-frequency encoder using polarization-maintaining fiber," J. Lightwave Technol. 8, 1697-1702 (1990).
[CrossRef]

Koyamada, Y.

K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
[CrossRef]

Lait, J.

A. Hymans and J. Lait, "Analysis of a frequency-modulated continuous-wave ranging system," Proc. IEEE 107, 365-372 (1960).

Liou, K.-Y.

E. C. Burrows and K.-Y. Liou, "High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source," Electron. Lett. 26, 577-579 (1990).
[CrossRef]

Miyamoto, T.

Moore, J.

Motamedi, A. R.

G. D. VanWiggeren, A. R. Motamedi, and D. M. Baney, "Single-scan interferometric component analyzer," IEEE Photon. Technol. Lett. 15, 263-265 (2003).
[CrossRef]

Mussi, G.

J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
[CrossRef]

Nakamura, K.

M. Yoshida, T. Miyamoto, N. Zou, K. Nakamura, and H. Ito, "Novel PMD measurement method based on OFDR using a frequency-shifted feedback fiber laser," Opt. Express 9, 207-211 (2001).
[CrossRef] [PubMed]

M. Yoshida, K. Nakamura, and H. Ito, "A new method for measurement of group velocity dispersion of optical fibers by using a frequency-shifted feedback fiber laser," IEEE Photon. Technol. Lett. 13, 227-229 (2001).
[CrossRef]

Noda, J.

M. Kobayashi, K. Takada, and J. Noda, "Optical-frequency encoder using polarization-maintaining fiber," J. Lightwave Technol. 8, 1697-1702 (1990).
[CrossRef]

Oh, K.

T.-J. Ahn, Y. Jung, K. Oh, and D. Y. Kim, "Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber," Opt. Express 13, 10,040-10,047 (2005).
[CrossRef]

Passy, R.

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
[CrossRef]

R. Passy, N. Gisin, and J. P. von der Weid, "High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components," IEEE Photon. Technol. Lett. 7, 667-669 (1995).
[CrossRef]

Reecht, J.

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

Shimizu, K.

K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
[CrossRef]

Soller, B. J.

Swanson, E. A.

Takada, K.

M. Kobayashi, K. Takada, and J. Noda, "Optical-frequency encoder using polarization-maintaining fiber," J. Lightwave Technol. 8, 1697-1702 (1990).
[CrossRef]

Tearney, G. J.

Tsuji, K.

K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, "Spatial-resolution improvement in long-range coherent optical frequency domain reflectometry by frequency-sweep linearisation," Electron. Lett. 33, 408-410 (1997).
[CrossRef]

Tsukada, F.

Ulrich, R.

W. Eickhoff and R. Ulrich, "Optical frequency domain reflectometry in single-mode fiber," Appl. Phys. Lett. 39, 693-695 (1981).
[CrossRef]

Uttam, D.

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," J. Lightwave Technol. LT-3, 971-977 (1985).
[CrossRef]

VanWiggeren, G. D.

G. D. VanWiggeren, A. R. Motamedi, and D. M. Baney, "Single-scan interferometric component analyzer," IEEE Photon. Technol. Lett. 15, 263-265 (2003).
[CrossRef]

von der Weid, J. P.

B. Huttner, J. Reecht, N. Gisin, R. Passy, and J. P. von der Weid, "Local birefringence measurements in singlemode fibers with coherent optical frequency-domain reflectometry," IEEE Photon. Technol. Lett. 10, 1458-1460 (1998).
[CrossRef]

R. Passy, N. Gisin, and J. P. von der Weid, "High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components," IEEE Photon. Technol. Lett. 7, 667-669 (1995).
[CrossRef]

von derWeid, J. P.

J. P. von derWeid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical fiber network components with optical frequency domain reflectometry," J. Lightwave Technol. 15, 1131-1141 (1997).
[CrossRef]

Waagaard, O. H.

Wang, L.-T.

K. Iiyama, L.-T. Wang, and K. ichi Hayashi, "Linearizing optical frequency-sweep of a laser diode for FMCW reflectometry," J. Lightwave Technol. 14, 173-178 (1996).
[CrossRef]

L.-T. Wang, K. Iiyama, F. Tsukada, N. Yoshida, and K.-I. Hayashi, "Loss measurement in optical waveguide devices by coherent frequency-modulated continuous-wave reflectometry," Opt. Lett. 18, 1095-1097 (1993).
[CrossRef] [PubMed]

Wegmuller, M.

Wolfe, M. S.

Yoshida, M.

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[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

An interferometer layout suitable for producing an analog clock signal for triggering swept-wavelength data acquisition.

Fig. 2.
Fig. 2.

(Color online) Measured data and processing steps for characterizing nonlinear laser frequency sweeps using the Fourier domain filter method. Note that different sections of the 65 ms measurement are shown in the time domain plots. (A) Measured interference fringes for an Agilent 81680A laser swept at a nominal rate of 40 nm/s using a relative path delay of 13.2 ns. (B) Fourier transform of the fringe pattern and the digital filter function. (C) The filtered data is shifted so the selected sideband occupies the DC location in the data array. (D) The phase of the inverse FFT of the shifted filtered data before and after unwrapping. (E) Measured laser tuning rate as a function of time.

Fig. 3.
Fig. 3.

Simplified schematic diagram of the SWI system used for experimental verification of sampling error correction.

Fig. 4.
Fig. 4.

(Color online) Measured linear phase deviation data for a swept-wavelength interferometer using the frequency-sampling method. The dot-dash curve shows significant error due to sampling errors for a system with unmatched system delays. These errors are present both in single-scan and averaged data. The solid red curve shows a corrected measurement of linear phase deviation using a system with an optical delay line for sampling error correction.

Fig. 5.
Fig. 5.

(Color online) Standard deviation of the deviation from linear phase of the measured fringe pattern as a function of added delay. The filled diamonds are measured data and the dot-dash line is a piecewise linear fit.

Equations (21)

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E ( t ) = E 0 e i ϕ ( t ) ,
v ( t ) = 1 2 π d ϕ dt .
U ( t ) = σ E ( t ) + E ( t + τ ) 2
= U 0 { 1 + cos [ ϕ ( t + τ ) ϕ ( t ) ] } ,
ϕ ( t + τ ) = n = 0 τ n n ! ϕ ( n ) ( t ) .
ϕ ( t + τ ) ϕ ( t ) = 2 π v ( t ) τ + 2 π n = 2 τ n n ! v ( n 1 ) ( t ) .
τ 2 dv dt 1 ,
2 π τ [ v ( t i + 1 ) v ( t i ) ] + 2 π n = 2 τ n n ! [ v ( n 1 ) ( t i 1 ) v ( n 1 ) ( t 1 ) ] = 2 π .
Δ v 1 v ( t i + 1 ) v ( t i ) = τ 1 + δ v i I ,
δ v i I = n = 2 τ n 1 n ! [ v ( n 1 ) ( t i + 1 ) v ( n 1 ) ( t i ) ] .
U ~ ( t ) = U ~ 0 { 1 + cos [ 2 π v ( t ) τ ] } .
v ( t i + 1 ) v ( t i ) = τ 1 .
v ( t i + 1 + δ t ) v ( t i + δ t ) = τ 1 + δ v i D .
v ( t + δ t ) = n = 0 δ t n n ! v ( n ) ( t ) .
v ( t i + 1 ) v ( t i ) + n = 1 δ t n n ! v ( n ) ( t i + 1 ) n = 1 δ t n n ! v ( n ) ( t i ) = τ 1 + δ v i D .
δ v i D = n = 1 δ t n n ! [ v ( n ) ( t i + 1 ) v ( n ) ( t i ) ] .
Δ v i = τ 1 + δ v i I + δ v i D
= τ 1 n = 1 τ n ( n + 1 ) ! [ v ( n ) ( t i + 1 ) v ( n ) ( t i ) ]
+ n = 1 δ t n n ! [ v ( n ) ( t i + 1 ) v ( n ) ( t i ) ] ,
Δ v i = τ 1 + [ δ t τ 2 ] [ v ( t i + 1 ) v ( t i ) ] .
δ t e = sin 1 V A 2 π f ,

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