Abstract

In the analysis of the interferometric spectrum, an accurate knowledge of instrument line shape is required. The instrument line shape of the interferometric spectrum can be distorted owing to errors in the measured interferogram, which are generally categorized into the intensity error and the phase error. The intensity error that is due to the effect of the field of view, optics misalignment, instrument noise, and smearing effect causes a symmetric distortion of the instrument line shape, and the phase error causes an asymmetric distortion. This paper describes a fast Fourier transform technique for simulating the distortion effects, and the technique is applied to the analyses of atmospheric absorption spectra and laboratory spectra. It is shown that correct information can be retrieved from the distorted spectrum using the nonlinear least-squares method. From analyses of HF absorption spectra obtained in a laboratory and solar CO absorption spectra obtained from a balloon-borne interferometer, it is found that the retrieved amount of absorbing gas is less than the correct value in most cases if the interferogram distortion effects are not included in the analysis.

© 1983 Optical Society of America

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References

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  1. Y. S. Chang, J. H. Shaw, Appl. Spectrosc. 31, 213 (1977).
    [CrossRef]
  2. W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
    [CrossRef]
  3. D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
    [CrossRef]
  4. C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
    [CrossRef]
  5. H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
    [CrossRef]
  6. J. H. Park, Appl. Opt. 21, 1356 (1982).
    [CrossRef] [PubMed]
  7. J. Connes, “Spectroscopic Studies Using Fourier Transformations,” U.S. Naval Ordnance Test Station NAVWEPS report 8099 (Jan.1963).
  8. J. B. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).
  9. P. R. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley-Interscience, New York, 1975).
  10. G. Guelachvili, “Distortions in Fourier Spectra and Diagnosis,” in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, New York, 1981).
  11. R. H. Norton, R. Beer, J. Opt. Soc. Am. 66, 259 (1976).
    [CrossRef]
  12. J. E. Bertie, “Apodization and Phase Correction,” in Analytical Applications of FT-IR to Molecular and Biological Systems, J. R. Durig, Ed. (Reidel, Dordrecht, 1980).
    [CrossRef]
  13. E. M. Sullivan, NASA Langley Research Center, R. E. Thompson, Systems Applied Science, G. Harvey, J. H. Park, D. Richardson, NASA Langley Research Center, “Halogen Occultation Experiment (HALOE) Gas Cell Life Test Program”; private communication (1982).
  14. L. S. Rothman, Appl. Opt. 20, 1323 (1981).
    [CrossRef] [PubMed]
  15. R. E. Thompson, Systems Applied Science, J. H. Park, M. A. H. Smith, G. Harvey, NASA Langley Research Center, “Line Parameters for HF”; private communication (1982).
  16. C. P. Rinsland, College of William & Mary; private communication (1982).
  17. J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).
  18. M. Minnaert, Z. Astrophys. 10, 40 (1935).
  19. S. Kilston, Pub. Astron. Soc. Pac. 87, 189 (1975).
    [CrossRef]
  20. S. R. Drayson, J. Quant. Spectrosc. Radiat. Transfer 16, 611 (1976).
    [CrossRef]

1982 (1)

1981 (1)

1980 (2)

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

1979 (2)

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

1977 (1)

1976 (2)

R. H. Norton, R. Beer, J. Opt. Soc. Am. 66, 259 (1976).
[CrossRef]

S. R. Drayson, J. Quant. Spectrosc. Radiat. Transfer 16, 611 (1976).
[CrossRef]

1975 (1)

S. Kilston, Pub. Astron. Soc. Pac. 87, 189 (1975).
[CrossRef]

1935 (1)

M. Minnaert, Z. Astrophys. 10, 40 (1935).

Beer, R.

Bell, J. B.

J. B. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

Bertie, J. E.

J. E. Bertie, “Apodization and Phase Correction,” in Analytical Applications of FT-IR to Molecular and Biological Systems, J. R. Durig, Ed. (Reidel, Dordrecht, 1980).
[CrossRef]

Buijs, H. L.

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

Chang, Y. S.

Coffey, M. T.

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

Connes, J.

J. Connes, “Spectroscopic Studies Using Fourier Transformations,” U.S. Naval Ordnance Test Station NAVWEPS report 8099 (Jan.1963).

Drayson, S. R.

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

S. R. Drayson, J. Quant. Spectrosc. Radiat. Transfer 16, 611 (1976).
[CrossRef]

Farmer, C. B.

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

Goldman, A.

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

Griffith, D. W. T.

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

Griffiths, P. R.

P. R. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley-Interscience, New York, 1975).

Guelachvili, G.

G. Guelachvili, “Distortions in Fourier Spectra and Diagnosis,” in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, New York, 1981).

Kendall, D. J. W.

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

Kilston, S.

S. Kilston, Pub. Astron. Soc. Pac. 87, 189 (1975).
[CrossRef]

Larsen, J. C.

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

Mankin, W. G.

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

Minnaert, M.

M. Minnaert, Z. Astrophys. 10, 40 (1935).

Muller, C.

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

Murcray, D. G.

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

Murcray, F. H.

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

Murcray, F. J.

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

Norton, R. H.

Park, J. H.

J. H. Park, Appl. Opt. 21, 1356 (1982).
[CrossRef] [PubMed]

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

Raper, O. F.

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

Richardson, D.

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

Rinsland, C. P.

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

C. P. Rinsland, College of William & Mary; private communication (1982).

Robbins, B. D.

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

Rothman, L. S.

L. S. Rothman, Appl. Opt. 20, 1323 (1981).
[CrossRef] [PubMed]

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

Shaw, J. H.

Smith, M. A. H.

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

Sullivan, E. M.

E. M. Sullivan, NASA Langley Research Center, R. E. Thompson, Systems Applied Science, G. Harvey, J. H. Park, D. Richardson, NASA Langley Research Center, “Halogen Occultation Experiment (HALOE) Gas Cell Life Test Program”; private communication (1982).

Thompson, R. E.

R. E. Thompson, Systems Applied Science, J. H. Park, M. A. H. Smith, G. Harvey, NASA Langley Research Center, “Line Parameters for HF”; private communication (1982).

Toth, R. A.

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

Tremblay, G.

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

Vail, G. L.

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

William, W. J.

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

Appl. Opt. (2)

Appl. Spectrosc. (1)

Geo-phys. Res. Lett. (1)

H. L. Buijs, G. L. Vail, G. Tremblay, D. J. W. Kendall, Geo-phys. Res. Lett. 7, 205 (1980).
[CrossRef]

Geophys. Res. Lett. (2)

W. G. Mankin, M. T. Coffey, S. R. Drayson, D. W. T. Griffith, Geophys. Res. Lett. 6, 853 (1979).
[CrossRef]

D. G. Murcray, A. Goldman, F. H. Murcray, F. J. Murcray, W. J. William, Geophys. Res. Lett. 6, 857 (1979).
[CrossRef]

J. Geophys. Res. (1)

C. B. Farmer, O. F. Raper, B. D. Robbins, R. A. Toth, C. Muller, J. Geophys. Res. 85, 1621 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

S. R. Drayson, J. Quant. Spectrosc. Radiat. Transfer 16, 611 (1976).
[CrossRef]

Pub. Astron. Soc. Pac. (1)

S. Kilston, Pub. Astron. Soc. Pac. 87, 189 (1975).
[CrossRef]

Z. Astrophys. (1)

M. Minnaert, Z. Astrophys. 10, 40 (1935).

Other (9)

J. E. Bertie, “Apodization and Phase Correction,” in Analytical Applications of FT-IR to Molecular and Biological Systems, J. R. Durig, Ed. (Reidel, Dordrecht, 1980).
[CrossRef]

E. M. Sullivan, NASA Langley Research Center, R. E. Thompson, Systems Applied Science, G. Harvey, J. H. Park, D. Richardson, NASA Langley Research Center, “Halogen Occultation Experiment (HALOE) Gas Cell Life Test Program”; private communication (1982).

J. Connes, “Spectroscopic Studies Using Fourier Transformations,” U.S. Naval Ordnance Test Station NAVWEPS report 8099 (Jan.1963).

J. B. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

P. R. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley-Interscience, New York, 1975).

G. Guelachvili, “Distortions in Fourier Spectra and Diagnosis,” in Spectrometric Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic, New York, 1981).

R. E. Thompson, Systems Applied Science, J. H. Park, M. A. H. Smith, G. Harvey, NASA Langley Research Center, “Line Parameters for HF”; private communication (1982).

C. P. Rinsland, College of William & Mary; private communication (1982).

J. H. Park, L. S. Rothman, C. P. Rinsland, M. A. H. Smith, D. Richardson, J. C. Larsen, “Atlas of Absorption Lines from 0 to 17900 cm−1,” NASA RP1084 (1981).

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Figures (11)

Fig. 1
Fig. 1

Instrument line shape (ILS): (a) symmetric ILS (zero phase error), sinx/x; (b) asymmetric ILS (π/2 phase error), (cos x − 1)/x; (c) phase distorted combination ILS (π/4 phase error), cos(π/4) sinx/x + sin(π/4) (cosx − 1)/x.

Fig. 2
Fig. 2

Absorption spectrum for a Lorentzian line convolved with various ILS for apodization function I3 and L = 4 cm: (a) infinite resolution ILS; (b) symmetric ILS; (c) asymmetric ILS corresponding to (+bj,−aj) of Eq. (B14); (d) asymmetric ILS corresponding to (−bj, +aj); (e) phase distorted combination ILS (π/4 phase error); (f) symmetric ILS with If(0) = 1/(1 − a0); (g) symmetric ILS with If(0) = (1 − 2a0)/(1 − a0).

Fig. 3
Fig. 3

Absorption spectrum for two Lorentzian lines convolved with various ILS for apodization function I3 and L = 4 cm: (a) infinite resolution ILS; (b) symmetric ILS; (c) asymmetric ILS; (d) phase distorted combination ILS (π/4 phase error).

Fig. 4
Fig. 4

ILS distortion due to interferogram smearing for L = 4 cm: (a) smeared apodization function fsm (solid line) and another apodization function fap(δ) = I1 (circles); (b) ILS corresponding to fsm (solid line) and I1 (dotted line).

Fig. 5
Fig. 5

Measured absorption spectra (dotted line) of HF (1–0) band for resolution 0.06 cm−1. The amount of HF and total pressure are retrieved simultaneously with the effective apodization. The simulated spectrum (solid line) is compared with the measurement obtained 11 Nov. 1980: (a) P10 line; (b) P9 line.

Fig. 6
Fig. 6

Simulated spectra (solid line) compared with the measurements (dotted line) obtained 27 May 1981: (a) P10 line; (b) P9 line.

Fig. 7
Fig. 7

Simulated spectra (solid line) compared with the measurements (dotted line) obtained 24 June 1981: (a) P10 line; (b) P9 line; (c) P10 line for HF amount retrieved without the effective apodization.

Fig. 8
Fig. 8

Solar CO absorption spectrum obtained by a balloon-borne interferometer at 39 km (dotted line) for L = 4 cm (scan no. 2305). The solar CO amount is retrieved with and without the effective apodization. (a) Simulated spectrum (solid line) with effective apodization c0 = −0.65 and mCO = 16.5. (b) Simulated spectrum without effective apodization (c0 = 0) and mCO = 10.8.

Fig. 9
Fig. 9

Measured spectrum (dotted line) of R8 line of the HF (1–0) band for resolution 0.06 cm−1 obtained 11 Feb. 1981. The simulated spectrum (solid line) is with the retrieved values of HF and c0. (a) Simulated spectrum with ν0ɛ0 = 0.015 and b0 = 0.055. (b) Simulated spectrum with ν0ɛ0 = 0 and b0 = 0.055. (c) Simulated spectrum with ν0ɛ0 = 0.015 and b0 = 0. (d) Simulated spectrum with ν0ɛ0 = 0 and b0 = 0.

Fig. 10
Fig. 10

Solar CO absorption spectrum (dotted line) (scan no. 2312). The simulated spectrum (solid line) is with effective apodization c0 = −0.24 and mCO = 16.5. (a) Simulated spectrum with phase error ν0ɛ0 = 0.06. (b) Simulated spectrum with zero phase error (symmetric). (c) Simulated spectrum with π/2 phase error (asymmetric).

Fig. 11
Fig. 11

Analysis of solar CO spectrum between 4037 and 4041 cm−1 of scan no. 2312. (a) Effective apodization functions retrieved for the spectral interval: a straight line approximation is shown as a solid line and a polynomial approximation is shown as a dashed line. (b) Observed solar CO absorption spectrum (scan no. 2312) (dotted line) and the simulated spectrum (solid line) using the polynomial form of fin(δ), mCO = 15.6, and ν0ɛ0 = 0.06.

Tables (1)

Tables Icon

Table I Retrieved Values of HF Volume Mixing Ratio (χ), Total Pressure (p, atm), and the Effective Apodization Function (c0) from Analyses of Two HF Absorption Lines, P9 and P10

Equations (54)

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I f ( δ ) = B ( ν ) exp ( i 2 π ν δ ) d ν ,
B ¯ ( ν ) = 2 0 L I f ( δ ) exp ( i 2 π ν δ ) d δ ,
B ¯ ( ν ) = 2 0 L ( B ( ν ) exp { i 2 π ν [ δ + ( ν ) ] } d ν ) f eff ( δ ) × exp ( i 2 π ν δ ) d δ ,
B ¯ ( ν ) = B ( ν ) Φ ( ν ν ) δ ν ,
Φ ( ν ν ) = 2 0 L f eff ( δ ) exp [ i 2 π ν ( ν ) ] exp [ i 2 π δ ( ν ν ) ] d δ .
B ¯ ( ν ) = 2 0 L I f ( δ ) f eff ( δ ) exp ( i 2 π ν δ ) d δ .
f eff ( δ ) = f ap ( δ ) · f sm ( δ ) · f in ( δ ) .
B ¯ ( ν ) = 2 0 L ( B ( ν ) exp { i 2 π ν [ δ + ( ν ) ] } d ν ) × exp ( i 2 π ν δ ) d δ .
B ¯ ( ν ) = B ( ν ) ϕ ( ν ν ) d ν ,
ϕ ( ν ν ) = 2 0 L exp [ i 2 π ν ( ν ) ] exp [ i 2 π δ ( ν ν ) ] d δ ,
ϕ ( ν ν ) = 2 0 L cos [ 2 π δ ( ν ν ) ] d δ = 2 L sin x x ,
ϕ ( ν ν ) = 2 0 L sin [ 2 π δ ( ν ν ) ] d δ = 2 L cos x 1 x .
ϕ ( ν ν ) = 2 0 L { cos ( 2 π ν 0 0 ) cos [ 2 π δ ( ν ν ) ] sin ( 2 π ν 0 0 ) sin [ 2 π δ ( ν ν ) ] } d δ = 2 L cos ( 2 π ν 0 0 ) sin x x + 2 L sin ( 2 π ν 0 0 ) cos x 1 x .
Φ ( ν ν ) = 2 0 L f eff ( δ ) { cos [ 2 π ν ( ν ) ] cos ( 2 π ν δ ) cos ( 2 π ν δ ) sin [ 2 π ν ( ν ) ] sin ( 2 π ν δ ) cos ( 2 π ν δ ) } d δ ,
Φ ( ν ν ) = cos [ 2 π ν ( ν ) ] Φ sym ( ν ν ) + sin [ 2 π ν ( ν ) ] Φ asym ( ν ν ) ,
Φ sym ( ν ν ) = 2 0 L f eff ( δ ) cos [ 2 π δ ( ν ν ) ] d δ ,
Φ asym ( ν ν ) = 2 0 L f eff ( δ ) sin [ 2 π δ ( ν ν ) ] d δ .
B ¯ ( ν ) = cos ( 2 π ν 0 0 ) B ¯ sym ( ν ) + sin ( 2 π ν 0 0 ) B ¯ asym ( ν ) ,
B ¯ sym ( ν ) = B ( ν ) Φ sym ( ν ν ) d ν ,
B ¯ asym ( ν ) = B ( ν ) Φ asym ( ν ν ) d ν .
τ ¯ ( ν ) = 1 a ¯ ( ν ) = B ¯ ( ν ) / B ¯ 0 ( ν ) ,
τ ¯ ( ν ) = 1 [ cos ( 2 π ν 0 0 ) a ¯ sym ( ν ) + sin ( 2 π ν 0 0 ) a ¯ asym ( ν ) ] .
f in ( δ ) = 1 + i = 1 N 1 c i 1 ( δ L ) i ,
f in ( δ ) = 1 + c 0 ( δ / L ) .
f 0 ( δ ) f in ( δ ) = 1 + c 0 ( δ L ) ( 0 δ L ) ,
f 0 ( δ ) = 1 ( δ = δ 0 = 0 ) f 1 ( δ ) = f 0 ( δ 0 ) + c 1 ( δ δ 0 ) ( 0 < δ δ 1 ) ,
f 8 ( δ ) = f 0 ( δ 7 ) + c 8 ( δ δ 7 ) ( δ 7 < δ δ 8 ) .
a ¯ ( ν ) = cos ( 2 π ν 0 0 ) a ¯ sym ( ν ) + sin ( 2 π ν 0 0 ) a ¯ asym ( ν ) .
I f ( δ ) = B ( ν ) exp ( i 2 π ν δ ) d ν ,
B ¯ ( ν ) = 2 0 L I f ( δ ) exp ( i 2 π ν δ ) d δ ,
F ( x ) F ¯ ( x ) k = 0 N 1 [ a k cos ( k π x L ) + b k sin ( k π x L ) ] .
Z k = cmplx [ F ( x k ) , 0 ] ( k = 0,1,2 , N 1 ) .
FFT ( Z k , M , I w k ) = Z k exp ( i 2 π ν δ ) d ν ( I w k = work array ) .
Z j = cmplx ( a j , b j ) ( j = 0,1,2 , N 1 ) .
FFT ( Z j , M , I w k ) = Z j exp ( i 2 π ν δ ) d δ .
Z l = cmplx [ F ¯ s ( x l ) , G ¯ s ( x l ) ] ( l = 0,1,2 , N 1 ) ,
Z k = cmplx [ 0 , ± F ( x k ) ] .
FFT ( Z k , M , I w k ) = Z k exp ( i 2 π ν δ ) d ν .
Z j = complx ( c j , d j ) .
Z k = ± i Z k ( i = 1 ) ,
Z j = cmplx ( c j , d j ) = ± i cmplx ( a j , b j ) .
Z j = cmplx ( b j , ± a j ) .
FFT ( Z j , M , I w k ) = Z j exp ( i 2 π ν δ ) d δ ,
Z l = cmplx [ F ¯ a ( x l ) , G ¯ a ( x l ) ] .
F ¯ ( x l ) = cos ( 2 π ν 0 0 ) F ¯ s ( x l ) + sin ( 2 π ν 0 0 ) F ¯ a ( x l ) ,
f i = F i ( data ) F i ( theory ) ( i = 1,2 , N ) ,
I 0 ( ν ) = A 1 + ( A C 1 ) sin 2 [ π L ( ν ν 0 ) ] ,
E = B ¯ ( ν ) d ν = B ¯ 0 ( ν ) [ 1 a ¯ ( ν ) ] d ν = I f ( 0 ) = E 0 ( 1 a 0 ) ,
E 0 = B ¯ 0 ( ν ) d ν ,
a 0 = B ¯ 0 a ( ν ) d ν E 0 .
E = E 0 [ 1 a 0 cos ( 2 π ν 0 0 ) ] · f eff ( 0 )
Δ E = E 0 ( 1 a 0 ) f eff ( 0 ) E 0 ( 1 a 0 ) ,
b 0 = Δ E / E 0 = ( 1 a 0 ) f eff ( 0 ) ( 1 a 0 ) .
τ ¯ ( ν ) = B ¯ ( ν ) / B ¯ 0 ( ν ) b 0 .

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