Weak-scatterer generalization of the K-density function II Probability density of total phase

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Abstract
The probability-density function of the total phase of the random sinusoid model that describes the scattered radiation at a point is studied in the weak-scatterer case in which the nonuniform phases of the individual sinusoids are taken to be a von Mises probability-density function. The number of multipaths (or correlation areas) is taken to be governed by the negative binomial distribution with parameter α. In the limiting case in which the average number of multipaths tends to infinity, it was shown in the first paper in this series [J. Opt. Soc. Am. A 3, 401 (1986)] that the probability-density function of the intensity was a generalization of the K-density function. The probability-density function of the total phase is expressed as a Fourier series in the fundamental interval (-π, π). The special case in which the number of multipaths is governed by a Poisson distribution (i.e., α → ∞) is also evaluated. Numerical results are shown for some typical parameter situations. Finally, the conditional probability-density function of the total phase, given the intensity, is shown to be a von Mises probability-density function that is independent of α. The physical reasoning behind the use of the von Mises density function is discussed in an appendix.