Abstract

A recursive Kalman time-series filter was applied to absorbance measurements obtained with a tunable diode laser spectrometer. The spectrometer uses frequency modulation spectroscopy and a near-infrared diode laser operating at 1.604 μm to monitor the CO2-vapor concentration in a 30-cm absorption cell. The Kalman filter enhanced the signal-to-noise ratio of the spectrometer by an order of magnitude when an absorbance of 6 × 10−5 was monitored.

© 1994 Optical Society of America

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References

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  1. G. C. Bjorklund, “Frequency modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
    [CrossRef] [PubMed]
  2. J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
    [CrossRef]
  3. G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  4. G. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1986).
    [CrossRef]
  5. D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with a lead salt diode laser: theory of line shapes and experimental results,” J. Opt. Soc. Am. B 4, 470–480 (1987).
    [CrossRef]
  6. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–42 (1960).
    [CrossRef]
  7. R. E. Warren, “Adaptive Kalman–Bucy filter for differential absorption lidar time series data,” Appl. Opt. 28, 4755–4760 (1989).
  8. S. M. Bozic, Digital and Kalman Filtering (Arnold, London, 1979), Chap. 7, pp. 100–108.
  9. F. L. Lewis, Optimal Estimation (Wiley, New York, 1986), Chap. 2, p. 67.
  10. A. Gelb, ed., Applied Optimal Estimation (MIT, Cambridge, Mass., 1974), Chap. 4, pp. 107–155.
  11. P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
    [CrossRef]

1993 (1)

P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[CrossRef]

1989 (1)

R. E. Warren, “Adaptive Kalman–Bucy filter for differential absorption lidar time series data,” Appl. Opt. 28, 4755–4760 (1989).

1987 (1)

1986 (1)

1983 (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

1981 (1)

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

1980 (1)

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–42 (1960).
[CrossRef]

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Bjorklund, G. C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

G. C. Bjorklund, “Frequency modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
[CrossRef] [PubMed]

Bozic, S. M.

S. M. Bozic, Digital and Kalman Filtering (Arnold, London, 1979), Chap. 7, pp. 100–108.

Carlisle, C. B.

Cooper, D. E.

Gallagher, T. F.

Hall, J. L.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Janik, G.

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–42 (1960).
[CrossRef]

Lenth, W.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Lewis, F. L.

F. L. Lewis, Optimal Estimation (Wiley, New York, 1986), Chap. 2, p. 67.

Mucke, R.

P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Slemr, F.

P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[CrossRef]

Warren, R. E.

R. E. Warren, “Adaptive Kalman–Bucy filter for differential absorption lidar time series data,” Appl. Opt. 28, 4755–4760 (1989).

D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with a lead salt diode laser: theory of line shapes and experimental results,” J. Opt. Soc. Am. B 4, 470–480 (1987).
[CrossRef]

Werle, P.

P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[CrossRef]

Appl. Opt. (1)

R. E. Warren, “Adaptive Kalman–Bucy filter for differential absorption lidar time series data,” Appl. Opt. 28, 4755–4760 (1989).

Appl. Phys. B (2)

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation spectroscopy: theory of line shapes and signal to noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

P. Werle, R. Mucke, F. Slemr, “The limits of signal averaging in atmospheric trace gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Precision spectroscopy and laser frequency control using FM sideband optical heterodyne techniques,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

J. Basic Eng. (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82, 35–42 (1960).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Other (3)

S. M. Bozic, Digital and Kalman Filtering (Arnold, London, 1979), Chap. 7, pp. 100–108.

F. L. Lewis, Optimal Estimation (Wiley, New York, 1986), Chap. 2, p. 67.

A. Gelb, ed., Applied Optimal Estimation (MIT, Cambridge, Mass., 1974), Chap. 4, pp. 107–155.

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Figures (2)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Application of the Kalman filter to CO2-vapor-concentration measurements.

Equations (10)

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x ( i ) = x ( i - 1 ) + w ( i ) ,
E [ W ( i ) ] = 0 ,             E [ w ( i ) w ( j ) ] = σ w 2 δ i j ,
y ( i ) = c x ( i ) + v ( i ) ,
x ^ ( i ) = x ^ ( i - 1 ) + K ( i ) [ y ( i ) - c x ^ ( i - 1 ) ] ,
p ( i ) = E [ x ^ ( i ) - x ( i ) ] 2 .
K ( i ) = c [ p ( i - 1 ) + σ w 2 ] [ σ v 2 + c 2 σ w 2 + c 2 p ( i - 1 ) ] - 1 ,
p ( i ) = σ v 2 K ( i ) / c .
K ( i ) = c p 1 ( i ) [ c 2 p 1 ( i ) + σ v 2 ] - 1 ,
p ( i ) = p 1 ( i ) [ 1 - c K ( i ) ] ,
p 1 ( i ) = p ( i - 1 ) + σ w 2 .

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