Abstract

A theoretical analysis of long-term drift noise in Fourier transform spectroscopy is presented. Theoretical predictions are confirmed by experiment. Fractional Brownian motion is employed as a stochastic process model for drift noise. A formulation of minimum detectable signal is given that properly accounts for drift noise. The spectral exponent of the low-frequency drift noise is calculated from experimental data. A frequency-dependent optimal spectrum averaging time is found to exist beyond which the minimum detectable signal increases indefinitely. It is also shown that the minimum detectable signal in an absorbance or transmission measurement degrades indefinitely with the time elapsed since background spectrum acquisition.

© 1997 Optical Society of America

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  1. E. Voigtman, J. D. Winefordner, “The multiplex disadvantage and excess low-frequency noise,” Appl. Spectrosc. 41(7) , 1182–1184 (1987).
    [CrossRef]
  2. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
    [CrossRef]
  3. J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).
  4. S. Marra, G. Horlick, “Signal-to-noise ratio characteristics of an inductively coupled plasma/Fourier transform spectrometer,” Appl. Spectrosc. 40, 804–813 (1986).
    [CrossRef]
  5. B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10(4) , 422–437 (1968).
    [CrossRef]
  6. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. (Wiley, Chichester, England, 1990).
  7. V. Solo, “Intrinsic random functions and the paradox of 1/f noise,” SIAM J. Appl. Math. 52(1), 270–291 (1992).
    [CrossRef]
  8. J. A. Barnes, “Atomic timekeeping and the statistics of precision signal generators,” Proc. IEEE 54, 207–220 (1966).
    [CrossRef]
  9. D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54(2), 221–220 (1966).
    [CrossRef]
  10. C. M. Van Vliet, P. H. Handel, “A new transform theorem for stochastic processes with special application to counting statistics,” Physica 113A, 261–276 (1982).
    [CrossRef]
  11. J. Beran, “Statistical methods for data with long-range dependence,” Stat. Sci. 7(4), 404–427 (1992).
    [CrossRef]
  12. J-S. Leu, A. Papamarcou, “On estimating the spectral exponent of fractional Brownian motion,” IEEE Trans. Inf. Theory 41(1), 233–244 (1995).
    [CrossRef]
  13. H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
    [CrossRef]
  14. M. A. Sharaf, D. L. Illman, B. R. Kowalski, Chemometrics (Wiley, New York, 1986).

1995 (1)

J-S. Leu, A. Papamarcou, “On estimating the spectral exponent of fractional Brownian motion,” IEEE Trans. Inf. Theory 41(1), 233–244 (1995).
[CrossRef]

1992 (2)

J. Beran, “Statistical methods for data with long-range dependence,” Stat. Sci. 7(4), 404–427 (1992).
[CrossRef]

V. Solo, “Intrinsic random functions and the paradox of 1/f noise,” SIAM J. Appl. Math. 52(1), 270–291 (1992).
[CrossRef]

1987 (1)

1986 (1)

1983 (1)

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

1982 (1)

C. M. Van Vliet, P. H. Handel, “A new transform theorem for stochastic processes with special application to counting statistics,” Physica 113A, 261–276 (1982).
[CrossRef]

1971 (1)

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

1968 (1)

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10(4) , 422–437 (1968).
[CrossRef]

1966 (2)

J. A. Barnes, “Atomic timekeeping and the statistics of precision signal generators,” Proc. IEEE 54, 207–220 (1966).
[CrossRef]

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54(2), 221–220 (1966).
[CrossRef]

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54(2), 221–220 (1966).
[CrossRef]

Barnes, J. A.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

J. A. Barnes, “Atomic timekeeping and the statistics of precision signal generators,” Proc. IEEE 54, 207–220 (1966).
[CrossRef]

Benali, H.

H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
[CrossRef]

Beran, J.

J. Beran, “Statistical methods for data with long-range dependence,” Stat. Sci. 7(4), 404–427 (1992).
[CrossRef]

Bosman, G.

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Chi, A. R.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Cutler, L. S.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Dipaola, R.

H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
[CrossRef]

Ellis, W. H.

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Falconer, K.

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. (Wiley, Chichester, England, 1990).

Gong, J.

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Guillemet, H.

H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
[CrossRef]

Handel, P.

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Handel, P. H.

C. M. Van Vliet, P. H. Handel, “A new transform theorem for stochastic processes with special application to counting statistics,” Physica 113A, 261–276 (1982).
[CrossRef]

Healey, D. J.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Horlick, G.

Illman, D. L.

M. A. Sharaf, D. L. Illman, B. R. Kowalski, Chemometrics (Wiley, New York, 1986).

Kowalski, B. R.

M. A. Sharaf, D. L. Illman, B. R. Kowalski, Chemometrics (Wiley, New York, 1986).

Leeson, D. B.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Leu, J-S.

J-S. Leu, A. Papamarcou, “On estimating the spectral exponent of fractional Brownian motion,” IEEE Trans. Inf. Theory 41(1), 233–244 (1995).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10(4) , 422–437 (1968).
[CrossRef]

Marra, S.

McGunigal, T. E.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Mullen, J. A.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Papamarcou, A.

J-S. Leu, A. Papamarcou, “On estimating the spectral exponent of fractional Brownian motion,” IEEE Trans. Inf. Theory 41(1), 233–244 (1995).
[CrossRef]

Preteux, F.

H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
[CrossRef]

Sharaf, M. A.

M. A. Sharaf, D. L. Illman, B. R. Kowalski, Chemometrics (Wiley, New York, 1986).

Smith, W. L.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Solo, V.

V. Solo, “Intrinsic random functions and the paradox of 1/f noise,” SIAM J. Appl. Math. 52(1), 270–291 (1992).
[CrossRef]

Sydnor, R. L.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Van Ness, J. W.

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10(4) , 422–437 (1968).
[CrossRef]

Van Vliet, C. M.

C. M. Van Vliet, P. H. Handel, “A new transform theorem for stochastic processes with special application to counting statistics,” Physica 113A, 261–276 (1982).
[CrossRef]

VanVliet, C. M.

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Vessot, R. F. C.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Voigtman, E.

Winefordner, J. D.

Winkler, G. M. R.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Appl. Spectrosc. (2)

IEEE Trans. Inf. Theory (1)

J-S. Leu, A. Papamarcou, “On estimating the spectral exponent of fractional Brownian motion,” IEEE Trans. Inf. Theory 41(1), 233–244 (1995).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. IM-20(2) , 105–120 (1971).
[CrossRef]

Physica (1)

C. M. Van Vliet, P. H. Handel, “A new transform theorem for stochastic processes with special application to counting statistics,” Physica 113A, 261–276 (1982).
[CrossRef]

Proc. IEEE (2)

J. A. Barnes, “Atomic timekeeping and the statistics of precision signal generators,” Proc. IEEE 54, 207–220 (1966).
[CrossRef]

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54(2), 221–220 (1966).
[CrossRef]

SIAM J. Appl. Math. (1)

V. Solo, “Intrinsic random functions and the paradox of 1/f noise,” SIAM J. Appl. Math. 52(1), 270–291 (1992).
[CrossRef]

SIAM Rev. (1)

B. B. Mandelbrot, J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10(4) , 422–437 (1968).
[CrossRef]

Stat. Sci. (1)

J. Beran, “Statistical methods for data with long-range dependence,” Stat. Sci. 7(4), 404–427 (1992).
[CrossRef]

Trans. Am. Nucl. Soc. (1)

J. Gong, W. H. Ellis, C. M. VanVliet, G. Bosman, P. Handel, “Observation of 1/f noise fluctuations in radioactive decay rates,” Trans. Am. Nucl. Soc. 45, 221–222 (1983).

Other (3)

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. (Wiley, Chichester, England, 1990).

H. Guillemet, H. Benali, F. Preteux, R. Dipaola “Noisy fractional Brownian motion for detection of perturbations in regular textures,” in Statistical and stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 40–51 (1996).
[CrossRef]

M. A. Sharaf, D. L. Illman, B. R. Kowalski, Chemometrics (Wiley, New York, 1986).

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Figures (12)

Fig. 1
Fig. 1

Plots of theoretical measurement variance (σd2) for noisy fractional Brownian motion with α = 2.5 and cγ2 = 0.02. Dashed curve, averaging time T is fixed and measurement period P varies; in solid curve, averaging time and measurement period vary together with zero measurement delay (T = P).

Fig. 2
Fig. 2

Experimental Setup. The current source was a Kepco Model BOP 36-6M; the spectrometer was a Midac Model 2406-C portable emission spectrometer.

Fig. 3
Fig. 3

Typical spectrum from mid-IR FT background data sets. The InSb photodetector cutoff frequency is approximately 1800 cm-1.

Fig. 4
Fig. 4

Time evolution of FTIR spectra at several spectral frequencies from the short-term data set. Data have been mean centered.

Fig. 5
Fig. 5

Time evolution of FTIR spectra at several spectral frequencies from the long-term data set. Data have been mean centered. Note the self-similarity with respect to Fig. 4.

Fig. 6
Fig. 6

Allan variance normalized to mean versus averaging time for the short-term data set at three points in the IR spectrum.

Fig. 7
Fig. 7

Allan variance normalized to mean versus averaging time for the long-term data set at three points in the IR spectrum.

Fig. 8
Fig. 8

Averaging time with minimum Allan variance versus spectral frequency for the short-term data set.

Fig. 9
Fig. 9

Signal-to-noise ratio (sample mean divided by square root of Allan variance). For the optimal averaging trace, data were averaged differently at each spectral frequency with the averaging times recommended by Fig. 8. For the uniform averaging trace, data were averaged for 293.6 s at each spectral frequency. Note that, since such averaging is impossible in practice, the optimal signal-to-noise ratio indicated here cannot be achieved at all spectral frequencies simultaneously.

Fig. 10
Fig. 10

Signal-to-noise ratio (sample mean divided by square root of Allan variance) for two different averaging times.

Fig. 11
Fig. 11

Allan variance normalized to mean versus averaging time with a fixed measurement period of P = 293.6 s.

Fig. 12
Fig. 12

Low-frequency spectral exponent (α) of drift noise at each IR frequency calculated from the log–log plot of Allan variance versus averaging time.

Equations (16)

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γθ=Ev*tvt+θ,
Sω=-γθexp-jωθdθ,
γˆt, θ=1Wt-W/2t+W/2v*τvτ+θdτ.
γˆt, θ=1Wt-W/2t+W/2s*τ; nτsτ+θ; nτ+θdτ1Wt-W/2t+W/2s*τ; nτsτ+θ; nt+θdτlimW1Wt-W/2t+W/2s*τ; ntsτ+θ; nt+θdτ=Es*τ; nτnτ=nt, τ×sτ+θ; nτ+θnτ+θ=nt+θ, τ,
γˆt, θEs*τ; nτnτ=nt, τ×sτ+θ; nτnτ=nt, τγs|ntθ,
Sˆω=θγˆt, θSs|ntω,
γft1, t2=cγ2t1α-1+t2α-1+t1-t2α-1,
Sfω=cSωα,
Pft+T-ftx=12πcγTα-11/2-xexp-u22cγTα-1du
VT=12T2Et-Ttfτ-fτ-Tdτ2.
cV=4cγ2α-1-1αα+1.
VnfBmT=σ2T+cVTα-1.
T0=σ2cVα-11/α.
σd2=12T2Ett+Tfτdτ-tt+Tfτ-Pdτ2.
σd2=cV42α-1-1T2P+Tα+1+P-Tα+1-2Tα+1-2Pα+1,
σd2=σ2T+cV42α-1-1T2P+Tα+1+P-Tα+1-2Tα+1-2Pα+1.

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