Rice distribution

In probability theory, the Rice distribution or Rician distribution (or, less commonly, Ricean distribution) is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice (1907–1986). The probability density function is where I0(z) is the modified Bessel function of the first kind with order zero. In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter , defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter , defined as the total power received in all paths. The characteristic function of the Rice distribution is given as: where is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of and . It is given by: where is the rising factorial. The first few raw moments are: and, in general, the raw moments are given by Here Lq(x) denotes a Laguerre polynomial: where is the confluent hypergeometric function of the first kind. When k is even, the raw moments become simple polynomials in σ and ν, as in the examples above. For the case q = 1/2: The second central moment, the variance, is Note that indicates the square of the Laguerre polynomial , not the generalized Laguerre polynomial if where and are statistically independent normal random variables and is any real number. Another case where comes from the following steps: Generate having a Poisson distribution with parameter (also mean, for a Poisson) Generate having a chi-squared distribution with 2P + 2 degrees of freedom. Set If then has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter . If then has a noncentral chi distribution with two degrees of freedom and noncentrality parameter . If then , i.e., for the special case of the Rice distribution given by , the distribution becomes the Rayleigh distribution, for which the variance is .