Certain Statistical Distributions Involving Special Functions and their Applications

Summary

The objective of the paper is to show that a certain generalized Bessel distribution is a useful theoretical tool because, through mixing procedures, specialization of parameter values and integral representation considerations, it unifies the theory of a broad class of special distributions; and that it is a useful tool because it has applications in radio communication. The random sine wave problem and distributions for fluctuating radar targets are studied in terms of this distribution with particular emphasis on amplitude, phase and component distributions. The marginal pdf and characteristic function corresponding to Bennett’s (Rice’s) distribution of a random sine wave plus stationary Gaussian noise are obtained when the sine wave amplitude is assigned a generalized Bessel prior distribution. The Bennett problem is also reduced to a randomly phased sine wave without noise and the corresponding marginal pdf and characteristic function are obtained from the noise corrupted case. Non-uniform phase distributions are also treated in terms of generalized distributions and the corresponding amplitude and component pdf^s are provided. The distribution of a useful quadrat if form in which the random variates are the squares of a generalized component corrupted by Gaussian noise is also provided. The quadratic form distribution is then used as a basic model for fluctuating radar cross section (RCS) and includes the Swerling RCS models and the Nakagami amplitude distributions as special cases. The authors also provide the corresponding pulse-train probability distributions for two pulse integration schemes using either scan-to-scan or pulse-to-pulse amplitude independence. Many of the pdf^s and characteristic functions provided in the paper are expressed in terms of both closed form expressions and mixture representations.