Abstract

In this paper we present a straightforward, efficient, and computationally fast method for creating a large number of discrete samples with an arbitrary given probability density function and a specified spectral content. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In contrast to previous work, where the analyses were limited to auto regressive and or iterative techniques to obtain satisfactory results, we find that a single application of the inverse transform method yields satisfactory results for a wide class of arbitrary probability distributions. Although a single application of the inverse transform technique does not conserve the power spectra exactly, it yields highly accurate numerical results for a wide range of probability distributions and target power spectra that are sufficient for system simulation purposes and can thus be regarded as an accurate engineering approximation, which can be used for wide range of practical applications. A sufficiency condition is presented regarding the range of parameter values where a single application of the inverse transform method yields satisfactory agreement between the simulated and target power spectra, and a series of examples relevant for the optics community are presented and discussed. Outside this parameter range the agreement gracefully degrades but does not distort in shape. Although we demonstrate the method here focusing on stationary random processes, we see no reason why the method could not be extended to simulate non-stationary random processes.

© 2011 Optical Society of America

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References

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  1. F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
    [CrossRef]
  2. J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
    [CrossRef]
  3. Wolfram Mathematica, Version 7 (Cambridge University, 2008).
  4. P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. (Springer-Verlag, 1987).
    [CrossRef]
  5. R. W. Rubinstein, Simulation and the Monte Carlo Method(Wiley, 1981).
    [CrossRef]
  6. S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes (Dover, 1954).
  7. M. Shinozuka and C. M. Jan, “Digital simulation of random processes and its application,” J. Sound Vib. 25, 111–128 (1972).
    [CrossRef]
  8. M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
    [CrossRef]
  9. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967), Section 6.6.
  10. S. Minfen, F. H. Y. Chan, and P. J. Beadle, “A method for generating non-Gaussian noise series with specified probability distribution and power spectrum,” in ISCAS ’03. Proceedings of the 2003 International Symposium on Circuits and Systems (IEEE, 2003), Vol. 2, II-1–II-4.
    [CrossRef]
  11. J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
    [CrossRef]
  12. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  13. R. Barakat, “Level-crossing statistics of aperture averaged-integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247(1988).
    [CrossRef]
  14. U. Schnell, J. Piot, and R. Dändliker, “Detection of movement with laser speckle patterns: statistical properties,” J. Opt. Soc. Am. A 15, 207–216 (1998).
    [CrossRef]
  15. Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Ser. Opt. Sci. (Springer-Verlag, 2006).
  16. In this regard we note, for ν=1/2, Eq. becomes an exponential, whose first and second derivative evaluated at the origin are −1 and 1, respectively, and thus cannot be representative of a real physical process.
  17. S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
    [CrossRef]
  18. H. T. Yura, “LADAR detection statistics in the presence of pointing errors,” Appl. Opt. 33, 6482–6498 (1994).
    [CrossRef] [PubMed]
  19. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
    [CrossRef]
  20. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).
  21. The CDF of the Rice–Nakagami distribution can be expressed in terms of a Marcum Q function, which are tabulated, but not supported, to the best of our knowledge, by any commercial commuter programs such as Mathematica and MATLAB.
  22. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts, 2006), Section 3.2.2.
  23. D. V. Kiesewetter, “Numerical simulation of a speckle pattern formed by radiation of optical vortices in a multimode optical fibre,” Quantum Electron. 38, 172–180 (2008).
    [CrossRef]
  24. A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
    [CrossRef]
  25. J. W. Goodman, “Speckle with a finite number of steps,” Appl. Opt. 47, A111–A118 (2008).
    [CrossRef] [PubMed]
  26. H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.
  27. P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).
  28. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Section 3.6.
  29. M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
    [CrossRef]

2010 (1)

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

2008 (2)

D. V. Kiesewetter, “Numerical simulation of a speckle pattern formed by radiation of optical vortices in a multimode optical fibre,” Quantum Electron. 38, 172–180 (2008).
[CrossRef]

J. W. Goodman, “Speckle with a finite number of steps,” Appl. Opt. 47, A111–A118 (2008).
[CrossRef] [PubMed]

2007 (1)

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

2004 (1)

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

2000 (1)

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

1998 (1)

1994 (1)

1993 (1)

P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).

1991 (2)

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

1988 (2)

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

R. Barakat, “Level-crossing statistics of aperture averaged-integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247(1988).
[CrossRef]

1972 (1)

M. Shinozuka and C. M. Jan, “Digital simulation of random processes and its application,” J. Sound Vib. 25, 111–128 (1972).
[CrossRef]

Aizu, Y.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Ser. Opt. Sci. (Springer-Verlag, 2006).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
[CrossRef]

Asakura, T.

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Ser. Opt. Sci. (Springer-Verlag, 2006).

Barakat, R.

Barthelmie, R. J.

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

Beadle, P. J.

S. Minfen, F. H. Y. Chan, and P. J. Beadle, “A method for generating non-Gaussian noise series with specified probability distribution and power spectrum,” in ISCAS ’03. Proceedings of the 2003 International Symposium on Circuits and Systems (IEEE, 2003), Vol. 2, II-1–II-4.
[CrossRef]

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967), Section 6.6.

Bratley, P.

P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. (Springer-Verlag, 1987).
[CrossRef]

Bucholtz, F.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

Castanedo, S.

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

Chan, F. H. Y.

S. Minfen, F. H. Y. Chan, and P. J. Beadle, “A method for generating non-Gaussian noise series with specified probability distribution and power spectrum,” in ISCAS ’03. Proceedings of the 2003 International Symposium on Circuits and Systems (IEEE, 2003), Vol. 2, II-1–II-4.
[CrossRef]

Dändliker, R.

Deodatis, G.

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

Fox, B. L.

P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. (Springer-Verlag, 1987).
[CrossRef]

Garcia-Martin, A.

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

Gomez-Medina, R.

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Speckle with a finite number of steps,” Appl. Opt. 47, A111–A118 (2008).
[CrossRef] [PubMed]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Section 3.6.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts, 2006), Section 3.2.2.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
[CrossRef]

Jan, C. M.

M. Shinozuka and C. M. Jan, “Digital simulation of random processes and its application,” J. Sound Vib. 25, 111–128 (1972).
[CrossRef]

Katagiri, B.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Kawakami, T.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Kiesewetter, D. V.

D. V. Kiesewetter, “Numerical simulation of a speckle pattern formed by radiation of optical vortices in a multimode optical fibre,” Quantum Electron. 38, 172–180 (2008).
[CrossRef]

Kuratomi, Y.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Lehmann, P.

P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).

Mann, J.

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

Mendez, J.

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

Michalowitz, J. V.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

Minfen, S.

S. Minfen, F. H. Y. Chan, and P. J. Beadle, “A method for generating non-Gaussian noise series with specified probability distribution and power spectrum,” in ISCAS ’03. Proceedings of the 2003 International Symposium on Circuits and Systems (IEEE, 2003), Vol. 2, II-1–II-4.
[CrossRef]

Nichols, J. M.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

Nielsen, M.

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

Nieto-Vesperinas, M.

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

Olson, C. C.

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

Peters, J.

P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
[CrossRef]

Piot, J.

Pryor, S. C.

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes (Dover, 1954).

Rubinstein, R. W.

R. W. Rubinstein, Simulation and the Monte Carlo Method(Wiley, 1981).
[CrossRef]

Saenz, J. J.

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

Satoh, H.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Schnell, U.

Schone, A.

P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).

Schrage, L. E.

P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. (Springer-Verlag, 1987).
[CrossRef]

Sekiya, K.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Shinozuka, M.

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

M. Shinozuka and C. M. Jan, “Digital simulation of random processes and its application,” J. Sound Vib. 25, 111–128 (1972).
[CrossRef]

Suzuki, Y.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

Uchida, T.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

Yamazaki, F.

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

Yura, H. T.

Appl. Mech. Rev. (2)

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

M. Shinozuka and G. Deodatis, “Simulation of stochastic processes by spectral representation,” Appl. Mech. Rev. 44, 191–204 (1991).
[CrossRef]

Appl. Opt. (2)

Coastal Eng. (1)

J. Mendez and S. Castanedo, “A probability distribution for depth-limited extreme wave heights in a sea state,” Coastal Eng. 54, 878–882 (2007).
[CrossRef]

J. Appl. Meteorol. Climatol. (1)

S. C. Pryor, M. Nielsen, R. J. Barthelmie, and J. Mann, “Can satellite sampling of off shore wind speeds realistically represent wind speed distributions? Part II: Quantifying uncertainties associated with distribution fitting methods,” J. Appl. Meteorol. Climatol. 43, 739–750 (2004).
[CrossRef]

J. Eng. Mech. Div., Am. Soc. Civ. Eng. (1)

F. Yamazaki and M. Shinozuka, “Digital generation of non-Gaussian stochastic fields,” J. Eng. Mech. Div., Am. Soc. Civ. Eng. 114, 1183–1197 (1988).
[CrossRef]

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

J. Sound Vib. (1)

M. Shinozuka and C. M. Jan, “Digital simulation of random processes and its application,” J. Sound Vib. 25, 111–128 (1972).
[CrossRef]

Optik (1)

P. Lehmann, A. Schone, and J. Peters, “Non-Gaussian far field fluctuations in laser light scattered by a random phase screen,” Optik 95, 63–72 (1993).

Phys. Rev. B (1)

A. Garcia-Martin, R. Gomez-Medina, J. J. Saenz, and M. Nieto-Vesperinas, “Finite-size effects in the spatial distribution of the intensity reflected from disordered media,” Phys. Rev. B 62, 9386–9389 (2000).
[CrossRef]

Prob. Eng. Mech. (1)

J. M. Nichols, C. C. Olson, J. V. Michalowitz, and F. Bucholtz, “A simple algorithm for generating spectrally colored non-Gaussian signals,” Prob. Eng. Mech. 25, 315–322 (2010).
[CrossRef]

Quantum Electron. (1)

D. V. Kiesewetter, “Numerical simulation of a speckle pattern formed by radiation of optical vortices in a multimode optical fibre,” Quantum Electron. 38, 172–180 (2008).
[CrossRef]

Other (15)

P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967), Section 6.6.

S. Minfen, F. H. Y. Chan, and P. J. Beadle, “A method for generating non-Gaussian noise series with specified probability distribution and power spectrum,” in ISCAS ’03. Proceedings of the 2003 International Symposium on Circuits and Systems (IEEE, 2003), Vol. 2, II-1–II-4.
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001), Chap. 2.
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

The CDF of the Rice–Nakagami distribution can be expressed in terms of a Marcum Q function, which are tabulated, but not supported, to the best of our knowledge, by any commercial commuter programs such as Mathematica and MATLAB.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts, 2006), Section 3.2.2.

Wolfram Mathematica, Version 7 (Cambridge University, 2008).

P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation, 2nd ed. (Springer-Verlag, 1987).
[CrossRef]

R. W. Rubinstein, Simulation and the Monte Carlo Method(Wiley, 1981).
[CrossRef]

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes (Dover, 1954).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications, Vol. 116 of Springer Ser. Opt. Sci. (Springer-Verlag, 2006).

In this regard we note, for ν=1/2, Eq. becomes an exponential, whose first and second derivative evaluated at the origin are −1 and 1, respectively, and thus cannot be representative of a real physical process.

H. Satoh, K. Sekiya, T. Kawakami, Y. Kuratomi, B. Katagiri, Y. Suzuki, and T. Uchida, “On the effect of small moving diffusers to the speckle reduction in laser projection displays,” in IDW ’09—Proceedings of the 16th International Display Workshops (2009), pp. 1361–1364.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Section 3.6.

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Figures (13)

Fig. 1
Fig. 1

(a) PDF of the beta distribution as a function of the random variable x. (b) Comparison of the CDF of the beta distribution. The simulation [for N = 10 5 sample points and a single application of Eq. (2.1)] and exact results are given by the dotted points, and solid curves, respectively.

Fig. 2
Fig. 2

Simulated and target correlation coefficient as a function of the time lag multiplied by the sample rate for the beta distribution and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively.

Fig. 3
Fig. 3

(a) Simulation of the random process conforming to the PDF given by Eq. (2.7) and the Pierson–Moskowitz PSD for N = 10 4 and κ = 0.5 , a time series segment. (b) Same as (a), except here showing the colored PDF. The dotted points and solid curve are the simulation and target values, respectively. (c) Same as (a), except here showing the colored CDF. The dotted points and solid curve are the simulation and target values, respectively. (d) Same as (a), except here showing the Pierson–Moskowitz correlation coefficient. The dotted points and solid curve are the simulation and target values, respectively. (e) Bimodal PSD. (f) Comparison between the simulated and target correlation coefficient of the bimodal PSD shown in (e).

Fig. 4
Fig. 4

(a) Simulated and target correlation coefficient as a function of the time lag for the gamma distribution for m = 1 , a Gaussian-shaped correlation coefficient, and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively. (b) Same as (a), except that m = 6 .

Fig. 5
Fig. 5

(a) Simulated and target correlation coefficient as a function of the time lag for the gamma distribution for m = 1 , a narrowband filtered correlation coefficient, and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively. (b) Same as (a), except that m = 2 .

Fig. 6
Fig. 6

(a) Simulated and target correlation coefficient as a function of time lag for the Weibull distribution for k = 1.8 , and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively.

Fig. 7
Fig. 7

Example of a simulated time series corresponding to the Weibull distribution for a shape parameter of 1.8 and the correlation coefficient of Eq. (3.6) for ν = 3 / 5 . For this example, we have assumed a sample rate of 1 kHz and a correlation time τ C = 10 ms .

Fig. 8
Fig. 8

(a) Simulated and target correlation coefficient as a function of time lag for the power law distribution for m = 1 / 3 , and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively. (b) Same as (a), except that m = 2 .

Fig. 9
Fig. 9

(a) Simulated and target correlation coefficient of Eq. (3.12) as a function of time lag for the lognormal distribution for σ ln I 2 = 0.1 , and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively. (b) Same as (a), except that σ ln I 2 = 0.35 .

Fig. 10
Fig. 10

Comparison of the Rice–Nakagami inverse CDF for C = 1 / 2 obtained via numerical integration (dotted points) to the analytic model obtained by a least square fit (solid curve).

Fig. 11
Fig. 11

Simulated and target Gaussian correlation coefficient as a function of time lag for the Rice–Nakagami distribution for C = 1 / 2 , and a single application of Eq. (2.1). The dotted points and solid curve are the simulation and target values, respectively.

Fig. 12
Fig. 12

Correction factor ψ as a function of the parameter m. Note that, for brevity in notation, we have here denoted the shape parameter k for the Weibull distribution by m.

Fig. 13
Fig. 13

Correction factor ψ for the lognormal distribution as a function of the parameter σ ln I .

Equations (41)

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Z = F X 1 ( F Y [ y ] ) ,
F Z ( z ) = Prob ( Z z ) = Prob [ F X 1 ( F Y [ y ] ) z ] = Prob [ F ( y ) Y F X ( z ) ] = Prob [ y F Y 1 ( F X [ z ] ) ] = F Y [ F Y 1 ( F X [ z ] ) ] = F X ( x ) ,
F ( x ) = BetaRegularized [ x , α , β ] , for 0 x 1 ,
F 1 ( x ) = InverseBetaRegularized [ x , α , β ] , for 0 x 1 .
S ( ω ) = σ 2 τ C π exp [ ω 2 τ C 2 / 4 ] ,
R ( τ ) = exp [ τ 2 / τ C 2 ] .
p ( x ) = 2 ϕ 2 ( κ ) ( 1 κ x ) 3 exp [ ϕ 2 ( κ ) ( x 1 κ x ) 2 ] , for 0 x < 1 / κ ,
F ( x ) = 1 exp [ ϕ 2 ( κ ) ( x 1 κ x ) 2 ] , for 0 x < 1 / κ ,
F 1 ( x ) = ϕ ( κ ) log ( 1 x ) + κ log ( 1 x ) ϕ 2 ( κ ) + κ 2 log ( 1 x ) , for 0 x 1 .
S PM ( ω ) = 4 σ 2 ω N 5 exp ( ω N 4 ) ,
R PM ( τ ) = 2 π G 0 , 5 3 , 0 ( ( τ / τ C ) 4 256 | 0 , 1 2 , 1 , 1 4 , 3 4 ) ,
F ( x ) = 1 Γ ( m , m x ) Γ ( m ) , for x 0 ,
F 1 ( x ) = InverseGammaRegularized [ m , 0 , m x ] , for 0 x 1 ,
R NB ( τ ) = cos ( ω 0 τ ) exp [ τ 2 / τ C 2 ] ,
F ( x ) = 1 exp [ x k ] , for x 0 ,
F 1 ( x ) = ( log [ 1 x ] ) 1 / k , for 0 x 1 ,
R ( τ ) = 1 2 ν 1 Γ ( ν ) ( τ / τ C ) ν K ν ( τ / τ C ) , for ν > 1 / 2 ,
S ( ω ) = π Γ ( ν + 1 / 2 ) Γ ( ν ) ( 1 + ω 2 τ C 2 ) ν + 1 / 2 .
F ( x ) = x m , for 0 x 1 ,
F 1 ( x ) = x 1 / m , for 0 x 1 ,
F ( x ) = 1 2 ( 1 + erf [ log x + σ ln I 2 / 2 2 σ ln I ] ) , for x 0 ,
F 1 ( x ) = exp [ x σ ln I σ ln I 2 / 2 ] , for 0 x 1 ,
R ( τ ) = 3.864 Im [ e 11 π i / 12 F 1 1 ( 11 6 ; 1 ; i τ 2 4 ) ] 2.3724 τ 5 / 3 ,
S ( ω ) = 2.577 ω 8 / 3 ( 1 8 Γ ( 17 / 6 ) 11 Γ ( 7 / 3 ) ω 2 Im [ e i ω 2 U ( ω ) ] ) ,
U ( ω ) = π sin ( π b ) ( F 1 1 ( a ; b ; i ω 2 ) Γ ( b ) Γ ( a b + 1 ) ( i ω 2 ) 1 b F 1 1 ( a b + 1 ; 2 b ; i ω 2 ) Γ ( a ) Γ ( 2 b ) ) .
p ( x ) = 2 x exp [ ( x 2 + C 2 ) ] I 0 ( 2 x C ) , for x 0 ,
F ( x ) = 0 x 2 y exp [ ( y 2 + C 2 ) ] I 0 ( 2 y C ) d y , for 0 x 1 ,
Y ˜ ( ω ) = d t Y ( t ) exp ( i ω t ) .
G ˜ ( ω ) = d t S ( ω ) Y ( t ) exp ( i ω t ) ,
G ( t ) = d ω G ˜ ( ω ) exp ( i ω t ) / 2 π .
B G ( τ ) = G ( t ) G * ( t + τ ) = d t 1 d t 2 d ω 1 d ω 2 S ( ω 1 ) S ( ω 2 ) Y ( t 1 ) Y ( t 2 ) × exp [ i ω 1 ( t 1 t ) + i ω 2 ( t 2 t τ ) ] / ( 2 π ) 2 .
B G ( τ ) = d t 1 d ω 1 d ω 2 S ( ω 1 ) S ( ω 2 ) × exp [ i ω 1 ( t 1 t ) + i ω 2 ( t 1 t τ ) ] / ( 2 π ) 2 = d ω 1 d ω 2 2 π δ ( ω 1 ω 2 ) S ( ω 1 ) S ( ω 2 ) × exp [ i ω 1 t i ω 2 ( t + τ ) ] / ( 2 π ) 2 = d ω S ( ω ) exp [ i ω τ ] / 2 π .
Z ( t ) = F 1 [ F G ( t ) ] ,
F G ( t ) = 1 2 [ 1 + erf ( x ( t ) 2 ) ] .
Z ( t ) = n = 0 x n ( t ) n ! d n d x n [ F 1 [ F G ( x ) ] ] x = 0 .
B ( τ ) = Z ( t + τ ) Z ( t ) = n = 0 m = 0 x n ( t + τ ) x m ( t ) n ! m ! ( d n d x n [ F 1 [ F G ( x ) ] ] x = 0 ) ( d m d x m [ F 1 [ F G ( x ) ] ] x = 0 ) ,
B G ( τ ) = x n ( t + τ ) x m ( t ) = i ( n + m ) n ω 1 n m ω 2 m χ ( ω 1 , ω 2 ) | ω 1 = 0 ω 2 = 0 ,
χ ( ω 1 , ω 2 ) = exp [ ω 1 2 + ω 2 2 2 ω 1 ω 2 R G ( τ ) ] ,
R ( τ ) = Z ( t + τ ) Z ( t ) Z 2 Z 2 Z 2 .
R ( τ ) = R G ( τ ) ψ ,
ψ = 4 π [ F 1 ( 1 / 2 ) ] + R G [ F 1 ( 1 / 2 ) ] 4 π [ F 1 ( 1 / 2 ) ] + [ F 1 ( 1 / 2 ) ] .

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