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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 28, Iss. 4 — Apr. 1, 2011
  • pp: 675–685

Digital simulation of an arbitrary stationary stochastic process by spectral representation

Harold T. Yura and Steen G. Hanson  »View Author Affiliations


JOSA A, Vol. 28, Issue 4, pp. 675-685 (2011)
http://dx.doi.org/10.1364/JOSAA.28.000675


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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

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.1670) Coherence and statistical optics : Coherent optical effects
(030.6140) Coherence and statistical optics : Speckle
(030.6600) Coherence and statistical optics : Statistical optics
(120.7250) Instrumentation, measurement, and metrology : Velocimetry

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: October 12, 2010
Revised Manuscript: January 14, 2011
Manuscript Accepted: January 17, 2011
Published: March 31, 2011

Citation
Harold T. Yura and Steen G. Hanson, "Digital simulation of an arbitrary stationary stochastic process by spectral representation," J. Opt. Soc. Am. A 28, 675-685 (2011)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-28-4-675


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References

  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]

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