Abstract

The K distribution is used in a number of areas of scientific endeavor. In optics, it provides a useful statistical description for fluctuations of the irradiance (and the electric field) of light that has been scattered or transmitted through random media (e.g., the turbulent atmosphere). The Poisson transform of the K distribution describes the photon-counting statistics of light whose irradiance is K distributed. The K-distribution family can be represented in a multiply stochastic (compound) form whereby the mean of a gamma distribution is itself stochastic and is described by a member of the gamma family of distributions. Similarly, the family of Poisson transforms of the K distributions can be represented as a family of negative-binomial transforms of the gamma distributions or as Whittaker distributions. The K distributions have heretofore had their origins in random-walk models; the multiply stochastic representations provide an alternative interpretation of the genesis of these distributions and their Poisson transforms. By multiple compounding, we have developed a new transform pair as a possibly useful addition to the K-distribution family. All these distributions decay slowly and are difficult to calculate accurately by conventional formulas. A recursion relation, together with a generalized method of steepest descent, has been developed to evaluate numerically the photon-counting distributions and their factorial moments with excellent accuracy.

© 1989 Optical Society of America

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  1. S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953).
    [CrossRef]
  2. E. J. Williams, “The distribution of larvae of randomly moving insects,” Austral. J. Biol. Sci. 14, 598–604 (1961). Williams’s calculation is virtually identical to that of Broadbent and Kendall.1
  3. N. Yasuda, “The random walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
    [CrossRef] [PubMed]
  4. G. Malécot, “Identical loci and relationship,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Biology and Problems of Health, L. LeCam, J. Neyman, eds. (U. California Press, Berkeley, Calif., 1967), Vol. 4, pp. 317–332.
  5. M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
    [CrossRef]
  6. P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1976), pp. 45–141.
  7. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
    [CrossRef]
  8. E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [CrossRef]
  9. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  10. E. Jakeman, “On the statistics of K-distributed noise,”J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  11. E. Jakeman, “Speckle statistics with a small number of scatterers,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 9–19 (1980).
    [CrossRef]
  12. K. A. O’Donnell, “Speckle statistics of doubly scattered light,”J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [CrossRef]
  13. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
    [CrossRef]
  14. D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
    [CrossRef]
  15. S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).
  16. L. C. Andrews, R. L. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  17. L. C. Andrews, R. L. Phillips, “Mathematical genesis of the I–K distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
    [CrossRef]
  18. R. Barakat, “Weak-scatterer generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986).
    [CrossRef]
  19. E. Jakeman, R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
    [CrossRef]
  20. E. B. Rockower, “Quantum derivation of K-distributed noise for finite 〈N〉,” J. Opt. Soc. Am. A 5, 730–734 (1988).
    [CrossRef]
  21. G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  22. A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and the log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987).
    [CrossRef] [PubMed]
  23. K. D. Ward, “Compound representation of high resolution sea clutter,” Electron. Lett. 17, 561–563 (1981).
    [CrossRef]
  24. R. J. A. Tough, “Fokker–Planck description of K-distributed noise,”J. Phys. A 20, 551–567 (1987).
    [CrossRef]
  25. C. J. Oliver, R. J. A. Tough, “On the simulation of correlated K-distributed random clutter,” Opt. Acta 33, 223–250 (1986).
    [CrossRef]
  26. E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).
  27. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  28. E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 75–116.
    [CrossRef]
  29. Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).
  30. M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920).
    [CrossRef]
  31. J. Gurland, “Some interrelations among compound and generalized distributions,” Biometrika 44, 265–268 (1957).
  32. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964).
  33. R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc. 46, 59–72 (1954).
  34. J. Peřina, “Superposition of coherent and incoherent fields,” Phys. Lett. A 24, 333–334 (1967); “Superposition of thermal and coherent fields,” Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 27, 227–234 (1968); G. Lachs, “Quantum statistics of multiple-mode, superposed coherent and chaotic radiation,” J. Appl. Phys. 38, 3439–3448 (1967).
    [CrossRef]
  35. W. J. McGill, “Neural counting mechanisms and energy detection in audition,”J. Math. Psychol. 4, 351–376 (1967).
    [CrossRef]
  36. M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).
    [CrossRef]
  37. S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979).
    [CrossRef]
  38. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1962).
  39. M. S. Bartlett, An Introduction to Stochastic Processes, 3rd ed. (Cambridge U. Press, Cambridge, 1978).
  40. W. J. McGill, M. C. Teich, “Signal discrimination in an amplifying auditory transmission system,” in Quantitative Analyses of Behavior, Vol. 10 of Signal Detection, M. L. Commons, J. A. Nevins, eds. (Erlbaum, Hillsdale, N.J., to be published).
  41. J. Gurland, “A generalized class of contagious distributions,” Biometrics 14, 229–249 (1958).
    [CrossRef]
  42. M. T. Boswell, G. P. Patil, “Chance mechanisms generating the negative binomial distributions,” in Random Counts in Models and Structures, Vol. 1 of Random Counts in Scientific Work, G. P. Patil, ed., Penn State Statistics Series (Pennsylvania State U. Press, University Park, Pa., 1970), pp. 3–27.

1988 (1)

1987 (6)

A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and the log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987).
[CrossRef] [PubMed]

E. Jakeman, R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).

R. J. A. Tough, “Fokker–Planck description of K-distributed noise,”J. Phys. A 20, 551–567 (1987).
[CrossRef]

E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).

Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).

1986 (3)

1985 (1)

1983 (1)

D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
[CrossRef]

1982 (1)

1981 (2)

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

K. D. Ward, “Compound representation of high resolution sea clutter,” Electron. Lett. 17, 561–563 (1981).
[CrossRef]

1980 (1)

E. Jakeman, “On the statistics of K-distributed noise,”J. Phys. A 13, 31–48 (1980).
[CrossRef]

1979 (2)

S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979).
[CrossRef]

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

1978 (1)

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

1977 (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

1976 (2)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).
[CrossRef]

1975 (1)

N. Yasuda, “The random walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
[CrossRef] [PubMed]

1970 (1)

M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
[CrossRef]

1967 (2)

J. Peřina, “Superposition of coherent and incoherent fields,” Phys. Lett. A 24, 333–334 (1967); “Superposition of thermal and coherent fields,” Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 27, 227–234 (1968); G. Lachs, “Quantum statistics of multiple-mode, superposed coherent and chaotic radiation,” J. Appl. Phys. 38, 3439–3448 (1967).
[CrossRef]

W. J. McGill, “Neural counting mechanisms and energy detection in audition,”J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

1961 (1)

E. J. Williams, “The distribution of larvae of randomly moving insects,” Austral. J. Biol. Sci. 14, 598–604 (1961). Williams’s calculation is virtually identical to that of Broadbent and Kendall.1

1958 (1)

J. Gurland, “A generalized class of contagious distributions,” Biometrics 14, 229–249 (1958).
[CrossRef]

1957 (1)

J. Gurland, “Some interrelations among compound and generalized distributions,” Biometrika 44, 265–268 (1957).

1954 (1)

R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc. 46, 59–72 (1954).

1953 (1)

S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953).
[CrossRef]

1920 (1)

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920).
[CrossRef]

Andrews, L. C.

Barakat, R.

Bartlett, M. S.

M. S. Bartlett, An Introduction to Stochastic Processes, 3rd ed. (Cambridge U. Press, Cambridge, 1978).

Bertolotti, M.

M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
[CrossRef]

Boswell, M. T.

M. T. Boswell, G. P. Patil, “Chance mechanisms generating the negative binomial distributions,” in Random Counts in Models and Structures, Vol. 1 of Random Counts in Scientific Work, G. P. Patil, ed., Penn State Statistics Series (Pennsylvania State U. Press, University Park, Pa., 1970), pp. 3–27.

Broadbent, S. R.

S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953).
[CrossRef]

Consortini, A.

Conte, E.

E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).

Crosignani, B.

M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
[CrossRef]

DiPorto, P.

M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
[CrossRef]

Gao, M.

Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).

Greenwood, M.

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920).
[CrossRef]

Gurland, J.

J. Gurland, “A generalized class of contagious distributions,” Biometrics 14, 229–249 (1958).
[CrossRef]

J. Gurland, “Some interrelations among compound and generalized distributions,” Biometrika 44, 265–268 (1957).

Hill, R. J.

Jakeman, E.

E. Jakeman, R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

E. Jakeman, “On the statistics of K-distributed noise,”J. Phys. A 13, 31–48 (1980).
[CrossRef]

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

E. Jakeman, “Speckle statistics with a small number of scatterers,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 9–19 (1980).
[CrossRef]

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 75–116.
[CrossRef]

Kendall, D. G.

S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953).
[CrossRef]

Laha, R. G.

R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc. 46, 59–72 (1954).

Lee, P. A.

S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979).
[CrossRef]

Lewinski, D. J.

D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
[CrossRef]

Longo, M.

E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).

Lure, Y. M.

Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).

Malécot, G.

G. Malécot, “Identical loci and relationship,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Biology and Problems of Health, L. LeCam, J. Neyman, eds. (U. California Press, Berkeley, Calif., 1967), Vol. 4, pp. 317–332.

McGill, W. J.

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).
[CrossRef]

W. J. McGill, “Neural counting mechanisms and energy detection in audition,”J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

W. J. McGill, M. C. Teich, “Signal discrimination in an amplifying auditory transmission system,” in Quantitative Analyses of Behavior, Vol. 10 of Signal Detection, M. L. Commons, J. A. Nevins, eds. (Erlbaum, Hillsdale, N.J., to be published).

McWhirter, J. G.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

O’Donnell, K. A.

Oliver, C. J.

C. J. Oliver, R. J. A. Tough, “On the simulation of correlated K-distributed random clutter,” Opt. Acta 33, 223–250 (1986).
[CrossRef]

Ong, S. H.

S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979).
[CrossRef]

Parry, G.

G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Patil, G. P.

M. T. Boswell, G. P. Patil, “Chance mechanisms generating the negative binomial distributions,” in Random Counts in Models and Structures, Vol. 1 of Random Counts in Scientific Work, G. P. Patil, ed., Penn State Statistics Series (Pennsylvania State U. Press, University Park, Pa., 1970), pp. 3–27.

Perina, J.

J. Peřina, “Superposition of coherent and incoherent fields,” Phys. Lett. A 24, 333–334 (1967); “Superposition of thermal and coherent fields,” Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 27, 227–234 (1968); G. Lachs, “Quantum statistics of multiple-mode, superposed coherent and chaotic radiation,” J. Appl. Phys. 38, 3439–3448 (1967).
[CrossRef]

Phillips, R. L.

Pusey, P. N.

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1976), pp. 45–141.

E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 75–116.
[CrossRef]

Rockower, E. B.

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

Teich, M. C.

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).
[CrossRef]

W. J. McGill, M. C. Teich, “Signal discrimination in an amplifying auditory transmission system,” in Quantitative Analyses of Behavior, Vol. 10 of Signal Detection, M. L. Commons, J. A. Nevins, eds. (Erlbaum, Hillsdale, N.J., to be published).

Tough, R. J. A.

E. Jakeman, R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987).
[CrossRef]

R. J. A. Tough, “Fokker–Planck description of K-distributed noise,”J. Phys. A 20, 551–567 (1987).
[CrossRef]

C. J. Oliver, R. J. A. Tough, “On the simulation of correlated K-distributed random clutter,” Opt. Acta 33, 223–250 (1986).
[CrossRef]

Ward, K. D.

S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).

K. D. Ward, “Compound representation of high resolution sea clutter,” Electron. Lett. 17, 561–563 (1981).
[CrossRef]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1962).

Watts, S.

S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1962).

Williams, E. J.

E. J. Williams, “The distribution of larvae of randomly moving insects,” Austral. J. Biol. Sci. 14, 598–604 (1961). Williams’s calculation is virtually identical to that of Broadbent and Kendall.1

Yang, C. C.

Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).

Yasuda, N.

N. Yasuda, “The random walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
[CrossRef] [PubMed]

Yule, G. U.

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920).
[CrossRef]

Austral. J. Biol. Sci. (1)

E. J. Williams, “The distribution of larvae of randomly moving insects,” Austral. J. Biol. Sci. 14, 598–604 (1961). Williams’s calculation is virtually identical to that of Broadbent and Kendall.1

Biom. J. (1)

S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979).
[CrossRef]

Biometrics (2)

J. Gurland, “A generalized class of contagious distributions,” Biometrics 14, 229–249 (1958).
[CrossRef]

S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953).
[CrossRef]

Biometrika (1)

J. Gurland, “Some interrelations among compound and generalized distributions,” Biometrika 44, 265–268 (1957).

Bull. Calcutta Math. Soc. (1)

R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc. 46, 59–72 (1954).

Electron. Lett. (1)

K. D. Ward, “Compound representation of high resolution sea clutter,” Electron. Lett. 17, 561–563 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976).
[CrossRef]

D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983).
[CrossRef]

J. Math. Psychol. (1)

W. J. McGill, “Neural counting mechanisms and energy detection in audition,”J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Phys. A (3)

E. Jakeman, “On the statistics of K-distributed noise,”J. Phys. A 13, 31–48 (1980).
[CrossRef]

R. J. A. Tough, “Fokker–Planck description of K-distributed noise,”J. Phys. A 20, 551–567 (1987).
[CrossRef]

M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970).
[CrossRef]

J. R. Stat. Soc. A (1)

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920).
[CrossRef]

Opt. Acta (2)

C. J. Oliver, R. J. A. Tough, “On the simulation of correlated K-distributed random clutter,” Opt. Acta 33, 223–250 (1986).
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G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
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Opt. Commun. (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
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Opt. Lett. (1)

Phys. Lett. A (1)

J. Peřina, “Superposition of coherent and incoherent fields,” Phys. Lett. A 24, 333–334 (1967); “Superposition of thermal and coherent fields,” Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 27, 227–234 (1968); G. Lachs, “Quantum statistics of multiple-mode, superposed coherent and chaotic radiation,” J. Appl. Phys. 38, 3439–3448 (1967).
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Phys. Rev. Lett. (2)

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).
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E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
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Proc. Inst. Electr. Eng. Part F (2)

S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).

E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).

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N. Yasuda, “The random walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975).
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P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1976), pp. 45–141.

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E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 75–116.
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E. Jakeman, “Speckle statistics with a small number of scatterers,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 9–19 (1980).
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M. T. Boswell, G. P. Patil, “Chance mechanisms generating the negative binomial distributions,” in Random Counts in Models and Structures, Vol. 1 of Random Counts in Scientific Work, G. P. Patil, ed., Penn State Statistics Series (Pennsylvania State U. Press, University Park, Pa., 1970), pp. 3–27.

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W. J. McGill, M. C. Teich, “Signal discrimination in an amplifying auditory transmission system,” in Quantitative Analyses of Behavior, Vol. 10 of Signal Detection, M. L. Commons, J. A. Nevins, eds. (Erlbaum, Hillsdale, N.J., to be published).

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Figures (6)

Fig. 1
Fig. 1

Models representing the effects of a medium with fluctuating transmission (or a scattering medium) on the statistical properties of a light source observed at the output of a detector. (a) The medium (with characteristic fluctuation time τa) stochastically modulates the mean irradiance of the source, the source exhibits intrinsic irradiance fluctuations (with coherence time τc), and the detector records the overall (continuous) integrated irradiance (in the time T). (b) Same as (a) but now the detector records the overall (discrete) photon count (in the time T).

Fig. 2
Fig. 2

Plot of the distribution A(n, 3, 2, 20) = P(n, u) ^G(u, 3, v) ^G(v, 2, 20) versus n.

Fig. 3
Fig. 3

Plot of the distribution A(n, 20) = P(n, u)^E(u, V)^E(v, 20) versus n.

Fig. 4
Fig. 4

Plot of A(n, 100, 3, 20) ≈ B(n, 3, 20), the negative-binomial distribution, versus n.

Fig. 5
Fig. 5

Plot of A(n, 100, 1, 20) ≈ B(n, 20), the Bose–Einstein distribution, versus n.

Fig. 6
Fig. 6

Plot of A(n, 100, 100, 20) ≈ P(n, 20), the Poisson distribution, versus n.

Tables (2)

Tables Icon

Table 1 Pdf’s, Moments, and Equivalent Density Functions for Some Continuous Random Variables

Tables Icon

Table 2 Pdf’s, Factorial Moments, and Equivalent Density Functions for Some Discrete Random Variables

Equations (87)

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G ( x , α , N ) = [ ( α / N ) α / Γ ( α ) ] x α - 1 exp ( - α x / N ) .
p ( n , α , N ) = P ( n , W ) G ( W , α , N ) ,
p ( n , α , N ) = 0 P ( n , W ) G ( W , α , N ) d W = 0 ( W n / n ! ) e - W [ ( α / N ) α / Γ ( α ) ] W α - 1 × exp ( - α W / N ) d W .
p ( n , α , N ) = B ( n , α , N ) = [ α α / Γ ( α ) ] [ Γ ( n + α ) / n ! ] N n / ( N + α ) n + α .
n = n p ( n , α , N ) = n P ( n , W ) G ( W , α , N ) d W = n P ( n , W ) G ( W , α , N ) d W = W G ( W , α , N ) d W = N .
p ( n , α , a N ) = P ( n , a W ) G ( W , α , N ) ,
h ( x , a 3 N + b 3 ) = f ( x , a 1 u + b 1 ) g ( u , a 2 N + b 2 ) ,
a 3 = a 1 a 2 ,             b 3 = a 1 b 2 + b 1 .
n = n p ( n , α , N ) = n P ( n , W ) G ( W , α , n ) = W G ( W , α , N ) = N .
x n = x n p ( x , N ) = x n p ( x , N ) d x ,
f ( m , n ) = n ! / ( n - m ) ! = n ( n - 1 ) ( n - 2 ) ( n - m + 1 ) ,
f ( m , n ) = n ! / ( n - m ) ! = f ( m , n ) p ( n , N ) .
f ( m , n ) P ( n , N ) = N m .
f ( m , N ) F ( n , M ) = f ( m , n ) P ( n , x ) p ( x , M ) = x m p ( x , M ) = x m .
h 1 ( x , N ) = f ( x , u ) g ( u , N )
h 2 ( x , N ) = g ( x , u ) f ( u , N ) .
x m = F m N m             for f ( x , N )
x m = G m N m             for g ( x , N ) ;
x m = F m u m g ( u , N ) = F m u m = F m G m N m
x m = G m u m f ( u , N ) = G m u m = G m F m N m .
g ( x , n + 1 ) = P ( n , x ) .
R ( x ) = ( 2 / S ) x exp ( - x 2 / S ) ,
α 1             if T τ c             and            α T / τ c             if T τ c ,
β 1             if T τ a             and            β T / τ a             if T τ a .
K G ( x , α , β , α β + α ( α + 1 ) z ) = [ 2 / Γ ( α ) ] f β - 1 ( x , z ) × g β - α ( x , ( 1 + z ) 1 / 2 ) ,
f ν ( x , u ) = ( x / u ) ν I ν ( 2 u x ) ,             g ν ( x , u ) = ( x / u ) ν K ν ( 2 u x )
x m = [ Γ ( m + β ) / β m Γ ( β ) ] [ Γ ( m + α ) / α m Γ ( α ) ] × ( α β ) m F ( - m , α + m ; β ; - z ) ,
K G ( x , α , β , α β + α ( α + 1 ) z ) = G ( x , β , u ) G ( u , α , v ) Q ( v , α , β , α β + α ( α + 1 ) z ) = K ( x , β , α , v ) Q ( v , α , β , α β + α ( α + 1 ) z ) ,
x m = ( α β ) m F ( - m , α + m ; β ; - [ ( N - α β ) / α ( α + 1 ) ] ) .
A ( n , N ) = A ( n , 1 , 1 , N ) ,
A ( n , α , N ) = A ( n , 1 , α , N ) .
A ( n , β , α , N ) = G n exp ( α β / 2 N ) ( α β / N ) γ W κ , δ ( α β / N ) ,
G n = Γ ( n + α ) Γ ( n + β ) / Γ ( α ) Γ ( β ) n !
γ = ( α + β - 1 ) / 2 ,             κ = - ( n + γ ) ,             δ = ( α - β ) / 2 ,
A ( n , β , α , N ) = G n ( α β / N ) α U ( n + α , 1 + α - β , α β / N ) = G n ( α β / N ) β U ( n + β , 1 + β - α , α β / N ) ;
A ( n , β , α , N ) = 0 0 P ( n , u ) G ( u , β , v ) G ( v , α , N ) d u d v
x = α v / N ,             y = β u / v ,
u = ( N / α β ) x y ,             v = ( N / α ) x .
d u d v = J ( x , y ) d x d y = ( u , v ) / ( x , y ) d x d y
J ( x , y ) = ( N 2 / α 2 β ) x ,
λ = N / α β ,
A ( n , β , α , N ) = 0 0 P ( n , λ x y ) G ( λ x y , β , β λ x ) G ( β λ x , α , N ) × β λ 2 x d x d y = [ λ n / n ! Γ ( α ) Γ ( β ) ] 0 0 x n + α - 1 x n + β - 1 × exp [ - ( x + y + λ x y ) ] d x d y .
A ( n , β , α , N ) = [ λ n / n ! Γ ( α ) Γ ( β ) ] f ( n + α - 1 , n + β - 1 , λ ) ,
f ( p , q , λ ) = 0 0 x p y q exp [ - ( x + y + λ x y ) ] d x d y
f ( p , q , λ ) / λ = - f ( p + 1 , q + 1 , λ )
f ( p , q , 0 ) = Γ ( p + 1 ) Γ ( q + 1 ) .
· [ x ^ x φ ( x , y ) ] d x d y = x ^ x φ ( x , y ) · d s ,
· [ y ^ y φ ( x , y ) ] d x d y = y ^ y φ ( x , y ) · d s ,
φ ( x , y ) = x p y q exp [ - ( x + y + λ x y ) ] .
· [ x ^ x φ ( x , y ) ] = φ + x φ / x = ( p + 1 - x - λ x y ) φ ( x , y )
· [ y ^ y φ ( x , y ) ] = φ + y φ / y = ( q + 1 - y - λ x y ) φ ( x , y ) .
( p + 1 - x - λ x y ) φ ( x , y ) d x d y = 0
( p + 1 ) f ( p , q , λ ) - f ( p + 1 , q , λ ) - λ f ( p + 1 , q + 1 , λ ) = 0
( q + 1 - y - λ x y ) φ ( x , y ) d x d y = 0
( q + 1 ) f ( p , q , λ ) - f ( p , q + 1 , λ ) - λ f ( p + 1 , q + 1 , λ ) = 0 ,
p q f ( p - 1 , q - 1 , λ ) = [ 1 + λ ( p + q + 1 ) ] f ( p , q , λ ) - λ 2 f ( p + 1 , q + 1 , λ )
( n + α - 1 ) ( n + β - 1 ) A n - 1 = n [ 2 n + α + β - 1 + ( α β / N ) ] A n - n ( n + 1 ) A n + 1 ,
f ( p , q , λ ) = 0 0 x p y q exp [ - ( x + y + λ x y ) ] d x d y = 0 0 exp [ - ψ ( x , y ) ] d x d y ,
ψ ( x , y ) = x + y + λ x y - p ln x - q ln y .
ψ = ( 1 + λ y - p / x 1 + λ x - q / y ) = 0.
x + λ x y = p ,             y + λ x y = q .
ψ ( x , y ) = ψ ( x 0 , y 0 ) + r · ψ ( x 0 , y 0 ) + ½ r · ψ ( x 0 , y 0 ) · r + ,
ψ = [ p / x 2 λ λ q / y 2 ] ,
φ ( x , y ) = exp [ - ψ ( x , y ) ] exp [ - ( ψ 0 + ½ r · ψ 0 · r ) = exp ( - ψ 0 ) exp ( - ½ r · ψ 0 · r ) = φ 0 exp ( - ½ r · ψ 0 · r ) ,
f ( p , q , λ ) φ 0 0 0 exp ( - ½ r · ψ 0 · r ) d x d y x 0 p y 0 q exp [ - ( x 0 + y 0 + λ x 0 y 0 ) ] × A exp ( - ½ r · ψ 0 · r ) d A .
A exp ( - ½ r · ψ 0 · r ) d A = 2 π / [ det ψ 0 ] 1 / 2 = 2 π x 0 y 0 / [ p q - ( λ x 0 y 0 ) 2 ] 1 / 2 .
f ( p , q , λ ) 2 π x 0 p + 1 y 0 q + 1 exp [ - ( x 0 + y 0 + λ x 0 y 0 ) ] × [ p q - ( λ x 0 y 0 ) 2 ] - 1 / 2 ,
( λ x y ) 2 - ( p + q + 1 / λ ) ( λ x y ) + p q = 0 ,
λ x 0 y 0 = ( p + q + 1 / λ ) / 2 - { [ ( p + q + 1 / λ ) / 2 ] 2 - p q } 1 / 2 ,
x 0 = p - λ x 0 y 0 ,             y 0 = q - λ x 0 y 0 .
A ( n , β , α , n ) x 0 α y 0 β ( λ x 0 y 0 ) n 2 π exp [ - ( x 0 + y 0 + λ x 0 y 0 ) ] Γ ( α ) Γ ( β ) n ! [ ( n + α - 1 ) ( n + β - 1 ) - ( λ x 0 y 0 ) 2 ] 1 / 2 ,
f ( m , n ) = N m Γ ( m + β ) Γ ( m + α ) β m Γ ( β ) α m Γ ( α ) .
Γ ( m + α ) α m Γ ( α ) = k = 1 m = 1 ( 1 + k / α ) .
σ 2 = n 2 - n 2 = N + N 2 [ ( 1 + 1 / α ) ( 1 + 1 / β ) - 1 ] .
B { Φ } = B ( n , α , x ) Φ ( x , N ) = φ ( n , α , N ) = 0 B ( n , α , x ) Φ ( x , N ) d x .
B { Φ } = [ α α Γ ( n + α ) / n ! Σ ( α ) ] 0 { x n / ( x + α ) n + α } Φ ( x , N ) d x .
B { Φ } = φ ( n , α , N ) = B ( n , α , x ) Φ ( x , N ) = P ( n , u ) G ( u , α , x ) Φ ( x , N ) .
n ! / ( n - m ) ! = [ Γ ( m + α ) / α m Γ ( α ) ] x m .
for m = 1 ,             n = x ;
for m = 2 ,             n ( n - 1 ) = ( 1 + 1 / α ) x 2 ;
for m = 3 ,             n ( n - 1 ) ( n - 2 ) = ( 1 + 1 / α ) ( 1 + 2 / α ) x 3 ;
for m = 4 ,             n ( n - 1 ) ( n - 2 ) ( n - 3 ) = ( 1 + 1 / α ) ( 1 + 2 / α ) ( 1 + 3 / α ) x 4 .
G ( Z ) = Z n = Z n φ ( n , α , N ) ,
φ ( n , α , N ) = ( 1 / n ! ) d n G ( 0 ) / d Z n .
G ( Z ) = Z n = [ 1 + ( Z - 1 ) ] n φ ( n , α , N ) = [ ( Z - 1 ) m / m ! ] [ n ! / ( n - m ) ! ] φ ( n , α , N ) ,
n ! / ( n - m ) ! = d m G ( 1 ) / d Z m .
G ( Z ) = Z n = [ 1 - ( Z - 1 ) ( x / α ) ] - α .

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