Abstract

The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer. For emission line data, the MEM process is directly used with the interferogram data in place of the regular Fourier transformation process required in Fourier transform spectroscopy. It produces a spectral estimate with an enhanced resolution. For absorption data with a broad background spectrum, the method is applied to a modified interferogram which corresponds to the Fourier transform of the absorptance spectrum. Two results are presented to demonstrate the power of the technique: for the visible emission spectrum of a spectral, calibration lamp and for the infrared chloroform absorption spectrum. Included in the paper is a discussion of the problems associated with practical use of the MEM.

© 1983 Optical Society of America

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References

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  1. G. A. Vanasse, H. Sakai, “Fourier Spectroscopy,” in Progress in Optics, Vol. VI, E. Wolf Ed. (North Holland, Amsterdam, 1967).
  2. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [CrossRef]
  3. D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).
  4. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef]
  5. A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [CrossRef]
  6. P. A. Jansson, R. H. Hunt, E. K. Plyler, J. Opt. Soc. Am. 60, 596 (1970).
    [CrossRef]
  7. S. Kawata, Y. Ichioka, J. Opt. Soc. Am. 70, 762 (1980).
    [CrossRef]
  8. K. C. Tarn, V. Perez-Mendez, J. Opt. Soc. Am. 71, 582 (1981).
    [CrossRef]
  9. S. Haykin, S. Kesler, “Prediction-Error Filtering and Maximum Entropy Spectral Estimation,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
    [CrossRef]
  10. Special issue on spectral estimation, Proc. IEEE 70, No. 9 (1982).
  11. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  12. R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
    [CrossRef]
  13. T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” submitted to Appl. Opt.
  14. H. J. Caulfield, “Holographic Spectroscopy,” in Advances in Holography, N. Farhat, Ed. (Marcel Dekker, New York, 1976), Vol. 2.
  15. Ref. 9, Chap. 2.9.
  16. T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
    [CrossRef]

1982

Special issue on spectral estimation, Proc. IEEE 70, No. 9 (1982).

1981

K. C. Tarn, V. Perez-Mendez, J. Opt. Soc. Am. 71, 582 (1981).
[CrossRef]

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

1980

1975

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1974

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

1972

1970

1964

1961

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Barton, J. J.

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, “Holographic Spectroscopy,” in Advances in Holography, N. Farhat, Ed. (Marcel Dekker, New York, 1976), Vol. 2.

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Grunthaner, F. J.

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

Harris, J. L.

Haykin, S.

S. Haykin, S. Kesler, “Prediction-Error Filtering and Maximum Entropy Spectral Estimation,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Hunt, R. H.

Ichioka, Y.

Jansson, P. A.

Kawata, S.

S. Kawata, Y. Ichioka, J. Opt. Soc. Am. 70, 762 (1980).
[CrossRef]

T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” submitted to Appl. Opt.

Kesler, S.

S. Haykin, S. Kesler, “Prediction-Error Filtering and Maximum Entropy Spectral Estimation,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Klein, J. D.

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

Minami, S.

T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” submitted to Appl. Opt.

Okamoto, T.

T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” submitted to Appl. Opt.

Ooe, M.

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

Perez-Mendez, V.

Plyler, E. K.

Pollak, H. O.

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Sakai, H.

G. A. Vanasse, H. Sakai, “Fourier Spectroscopy,” in Progress in Optics, Vol. VI, E. Wolf Ed. (North Holland, Amsterdam, 1967).

Slepian, D.

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Tarn, K. C.

Ulrych, T. J.

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Vanasse, G. A.

G. A. Vanasse, H. Sakai, “Fourier Spectroscopy,” in Progress in Optics, Vol. VI, E. Wolf Ed. (North Holland, Amsterdam, 1967).

Vasquez, R. P.

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

Bell Syst. Tech. J.

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

IEEE Trans. Circuits Syst.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

J. Electron Spectrosc.

R. P. Vasquez, J. D. Klein, J. J. Barton, F. J. Grunthaner, J. Electron Spectrosc. 23, 63 (1981).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Proc. IEEE

Special issue on spectral estimation, Proc. IEEE 70, No. 9 (1982).

Other

T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” submitted to Appl. Opt.

H. J. Caulfield, “Holographic Spectroscopy,” in Advances in Holography, N. Farhat, Ed. (Marcel Dekker, New York, 1976), Vol. 2.

Ref. 9, Chap. 2.9.

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

G. A. Vanasse, H. Sakai, “Fourier Spectroscopy,” in Progress in Optics, Vol. VI, E. Wolf Ed. (North Holland, Amsterdam, 1967).

S. Haykin, S. Kesler, “Prediction-Error Filtering and Maximum Entropy Spectral Estimation,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Interferogram of the low-pressure mercury lamp with 512 sampling points obtained by the Fourier transform spectrophotometer system developed by the authors.13

Fig. 2
Fig. 2

(a) Spectrum of the interferogram shown in Fig. 1 by FFT and (b) same as (a) but by the MEM.

Fig. 3
Fig. 3

Interferogram of a Na–K lamp with 512 sampling points obtained by the Fourier transform spectrophotometer system developed by the authors.13

Fig. 4
Fig. 4

(a) Spectrum of the interferogram shown in Fig. 3 by FFT and (b) same as (a) but by the MEM.

Fig. 5
Fig. 5

Procedure of superresolved spectral estimation of absorption spectra with the aid of the MEM.

Fig. 6
Fig. 6

Interferograms of FT-IR absorption data (512 sample points): (a) chloroform with background; (b) background only; and (c) intermediate result of background-free interferogram computed from (a) and (b).

Fig. 7
Fig. 7

Spectra of chloroform computed by (a) FFT from the interferograms of Figs. 6(a) and (b); (b) MEM from the interferogram of Fig. 6(c); and (c) FFT from the interferogram with 4096 sampling points.

Fig. 8
Fig. 8

Plot of the values of FPE vs the order M.

Equations (11)

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Ŝ ( ν ) = P M Δ t | 1 + m = 1 M a m exp ( 2 π j ν m Δ t ) | 2 ,
[ R ( 0 ) R ( 1 ) R ( M ) R ( 1 ) R ( 0 ) R ( 1 M ) R ( M ) R ( M 1 ) R ( 0 ) ] [ 1 a 1 a 2 a M ] = [ P M 0 0 ] ,
R ( m ) = E [ x k x k + m ] ,
P M = E [ f k f k ] + E [ b k b k ] min ,
f k = x k + m = 1 M a m x k m , b k = m = 0 M 1 a M m x k m + x k M .
c 1 = M 2 ( 8 N 3 M + 1 ) .
c 1 8 k 3 2 k 2 N 2 .
c 2 = ( M + 2 ) B / Δ .
c 2 = ( M + 2 ) N MN .
c 2 > MN .
c 2 < MN .

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