Abstract

The level-crossing theory for Gaussian stochastic processes is used to predict the behavior of an optical channel in the turbulent atmosphere. The log-irradiance is assumed to be Gaussian with a known correlation function and the conditional probability that it be below a “fade” at time τ in the future given that it currently exceeds a threshold level is derived. The probability density, two-point conditional density, and the correlation function of the log-irradiance as determined from experimental data collected under weak, moderate, and strong turbulence conditions are presented. The Gaussian assumption is justified on the basis on the measured probability densities but the correlation functions show that the process is not Markov. The measured correlation functions are used to generate the probability of fading as a function of the time delay τ and a measure of the effectiveness of a channel-monitoring system is defined and calculated.

© 1978 Optical Society of America

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References

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  1. S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).
  2. I. F. Blake and W. C. Lindsay, “Level Crossing Problems for Random Processes,” IEEE Trans. Inf. Theory IT-19, 219 (1973).
  3. J. R. Kerr, “Experiments on turbulence characteristics and multi-wavelength scintillation phenomena,” J. Opt. Soc. Am. 62, 1040 (1972).
    [Crossref]
  4. G. R. Ochs and R. S. Lawrence, “Saturation of laser beam scintillation under conditions of strong atmospheric turbulence,” J. Opt. Soc. Am. 59, 226 (1969).
    [Crossref]
  5. M. Abramowitz and I. Stegun, Handbook of Mathematical Function (Dover, New York, 1965), pp. 931–939.
  6. J. L. Doob, Stochastic Processes (Wiley, Canada, 1953).
  7. W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).
  8. A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
    [Crossref]
  9. R. A. Elliott, J. R. Dunphy, and J. R. Kerr, “Statistical Tests of Distributional Hypotheses Applied to Irradiance Fluctuations,” presented at OSA Topical Meeting on Optical Propagation Through Turbulence Rain and Fog, Boulder, Colorado (August 1977).

1973 (1)

I. F. Blake and W. C. Lindsay, “Level Crossing Problems for Random Processes,” IEEE Trans. Inf. Theory IT-19, 219 (1973).

1972 (1)

1969 (1)

1968 (1)

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Function (Dover, New York, 1965), pp. 931–939.

Blake, I. F.

I. F. Blake and W. C. Lindsay, “Level Crossing Problems for Random Processes,” IEEE Trans. Inf. Theory IT-19, 219 (1973).

Doob, J. L.

J. L. Doob, Stochastic Processes (Wiley, Canada, 1953).

Drijard, D.

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

Dunphy, J. R.

R. A. Elliott, J. R. Dunphy, and J. R. Kerr, “Statistical Tests of Distributional Hypotheses Applied to Irradiance Fluctuations,” presented at OSA Topical Meeting on Optical Propagation Through Turbulence Rain and Fog, Boulder, Colorado (August 1977).

Eadie, W. T.

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

Elliott, R. A.

R. A. Elliott, J. R. Dunphy, and J. R. Kerr, “Statistical Tests of Distributional Hypotheses Applied to Irradiance Fluctuations,” presented at OSA Topical Meeting on Optical Propagation Through Turbulence Rain and Fog, Boulder, Colorado (August 1977).

Gurvich, A. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
[Crossref]

James, F. E.

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

Kallistratova, M. A.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
[Crossref]

Kerr, J. R.

J. R. Kerr, “Experiments on turbulence characteristics and multi-wavelength scintillation phenomena,” J. Opt. Soc. Am. 62, 1040 (1972).
[Crossref]

R. A. Elliott, J. R. Dunphy, and J. R. Kerr, “Statistical Tests of Distributional Hypotheses Applied to Irradiance Fluctuations,” presented at OSA Topical Meeting on Optical Propagation Through Turbulence Rain and Fog, Boulder, Colorado (August 1977).

Lawrence, R. S.

Lindsay, W. C.

I. F. Blake and W. C. Lindsay, “Level Crossing Problems for Random Processes,” IEEE Trans. Inf. Theory IT-19, 219 (1973).

Ochs, G. R.

Rice, S. O.

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

Roos, M.

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

Sadoulet, B.

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Function (Dover, New York, 1965), pp. 931–939.

Time, N. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
[Crossref]

IEEE Trans. Inf. Theory (1)

I. F. Blake and W. C. Lindsay, “Level Crossing Problems for Random Processes,” IEEE Trans. Inf. Theory IT-19, 219 (1973).

J. Opt. Soc. Am. (2)

Radiophys. Quantum Electron. (USSR) (1)

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, “Fluctuations in the Parameters of a Light Wave from a Laser During Propagation in the Atmosphere,” Radiophys. Quantum Electron. (USSR) 11, 771 (1968).
[Crossref]

Other (5)

R. A. Elliott, J. R. Dunphy, and J. R. Kerr, “Statistical Tests of Distributional Hypotheses Applied to Irradiance Fluctuations,” presented at OSA Topical Meeting on Optical Propagation Through Turbulence Rain and Fog, Boulder, Colorado (August 1977).

S. O. Rice, “Mathematical Analysis of Random Noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954).

M. Abramowitz and I. Stegun, Handbook of Mathematical Function (Dover, New York, 1965), pp. 931–939.

J. L. Doob, Stochastic Processes (Wiley, Canada, 1953).

W. T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, London, 1971).

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Figures (6)

FIG. 1
FIG. 1

Autocorrelation functions of the log-irradiance fluctuations. □, run A; log-irradiance variance, 〈χ2(t)〉 = σ2 = 0.006; wind speed, 9 m/s. ○, run B; σ2 = 0.04; wind speed, 3.5 m/s. △, run C; σ2 = 1.23; wind speed, 1 m/s variable. λ = 10.6 μm.

FIG. 2
FIG. 2

Autocorrelation functions of the log-irradiance plotted on a logarithmic scale. □, run A; ○, run B; △ run C.

FIG. 3
FIG. 3

Cumulative probability distribution of the log-irradiance fluctuations. □, run A; ○, run B; △ run C. The straight line is the standard normal distribution.

FIG. 4
FIG. 4

(a) Conditional probability distribution for run A. ○, χ0 = 2σ; τ = 2 ms. △, χ0 = σ, τ = 4 ms. □, χ0 = −σ, τ = 16 ms. The straight lines are maximum likelihood fitted normal distributions. The theoretical means χ0C(τ) are indicated by the solid circle, triangle, and square, respectively. (b). Conditional probability distributions for run B. ○, χ0 = 2σ, τ = 8 ms. △, χ0 = 0, τ = 2 ms. □, χ0 = −σ, τ = 2 ms. (c). Conditional probability distributions for run C. ○, χ0 = 1.78σ, τ = 1 ms. △, χ0 = 1.78 σ, τ = 8 ms. □, χ0 = −0.21 σ, τ = 2 ms.

FIG. 5
FIG. 5

Conditional probability of fading as a function of time delay, τ for runs A, B, and C. Fade level, If = 0.85I. Threshold level: (—), Ip = 0.9I; (– · – – –), Ip = 0.95I; (– – –), Ip = I.

FIG. 6
FIG. 6

Effectiveness of path monitoring as a function of time delay, Rf(τ) = 1 − Pf (I(t + τ) ≤ If|I(t) > Ip)/P(If),with If = 0.85I and Ip = I for runs A, B, C.

Tables (1)

Tables Icon

TABLE I Comparison of measured variances with predicted ones.

Equations (16)

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Z ( t ) = ( 1 / 2 π ) exp ( - t 2 / 2 ) ,
P ( t ) = - t Z ( t ) d t ,
Q ( t ) = 1 - P ( t ) = t Z ( t ) d t .
p 1 ( χ ( t ) ) = Z ( χ ( t ) / σ ) ,
σ 2 = χ 2 ( t )
C ( t + τ , t ) = C ( τ ) = χ ( t + τ ) χ ( t ) / σ 2
p 2 ( χ ( t ) , χ ( t + τ ) ) = 1 2 π σ 2 ( 1 - C 2 ( τ ) ) 1 / 2 exp ( - 1 2 σ 2 ( 1 - C 2 ( τ ) ) [ χ 2 ( t ) - 2 χ ( t ) χ ( t + τ ) C ( τ ) + χ 2 ( t + τ ) ] ) .
P f ( χ ( t + τ ) χ f χ ( t ) χ P ) = χ P d χ ( t ) - χ f d χ ( t + τ ) p 2 ( χ ( t ) , χ ( t + τ ) ) χ P d χ ( t ) p 1 ( χ ( t ) ) .
L ( h , k , ρ ) = 1 2 π ( 1 - ρ 2 ) 1 / 2 × h d x k d y exp ( - ( x 2 - 2 x y ρ + y 2 ) 2 ( 1 - ρ 2 ) )
P f ( χ ( t + τ ) χ f χ ( t ) χ p ) = L ( χ p / σ , - χ f / σ , C ( τ ) ) / Q ( χ p / σ ) .
L ( h , k , ρ ) = Q ( h ) Q ( k ) + n = 0 Z ( n ) ( h ) Z ( n ) ( k ) ( n + 1 ) ! ρ n + 1
Z ( n ) ( t ) = d n d t n ( Z ( t ) )
C ( t 0 , t 2 ) = C ( t 0 , t 1 ) · C ( t 1 , t 2 ) ; t 0 < t 1 < t 2 .
C ( τ ) = exp ( - τ / τ 0 ) .
p ( χ ( t + τ ) χ ( t ) = χ 0 ) = p 2 ( χ ( t ) = χ 0 , χ ( t + τ ) ) / Z ( χ 0 / σ ) = 1 / 2 π σ ( 1 - C 2 ( τ ) ) 1 / 2 × exp { - 1 / 2 σ 2 ( 1 - C 2 ( τ ) ) [ χ ( t + τ ) - C ( τ ) χ 0 ] 2 } = Z ( χ ( t + τ ) - C ( τ ) χ 0 σ ( 1 - C 2 ( τ ) ) 1 / 2 )
R f ( τ ) = 1 - P f ( χ ( t + τ ) χ f χ ( t ) χ p ) / P ( χ f )