Support x ∈ [0, +∞) | ||
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Parameters ν ≥ 0 — distance between the reference point and the center of the bivariate distribution,σ ≥ 0 — scale PDF x σ 2 exp ( − ( x 2 + ν 2 ) 2 σ 2 ) I 0 ( x ν σ 2 ) {displaystyle {rac {x}{sigma ^{2}}}exp left({rac {-(x^{2}+u ^{2})}{2sigma ^{2}}}ight)I_{0}left({rac {xu }{sigma ^{2}}}ight)} CDF 1 − Q 1 ( ν σ , x σ ) {displaystyle 1-Q_{1}left({rac {u }{sigma }},{rac {x}{sigma }}ight)} where Q1 is the Marcum Q-function Mean σ π / 2 L 1 / 2 ( − ν 2 / 2 σ 2 ) {displaystyle sigma {sqrt {pi /2}},,L_{1/2}(-u ^{2}/2sigma ^{2})} Variance 2 σ 2 + ν 2 − π σ 2 2 L 1 / 2 2 ( − ν 2 2 σ 2 ) {displaystyle 2sigma ^{2}+u ^{2}-{rac {pi sigma ^{2}}{2}}L_{1/2}^{2}left({rac {-u ^{2}}{2sigma ^{2}}}ight)} |
In probability theory, the Rice distribution, Rician distribution or Ricean distribution is the probability distribution of the magnitude of a circular bivariate normal random variable with potentially non-zero mean. It was named after Stephen O. Rice.
Contents
Characterization
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
The characteristic function of the Rice distribution is given as:
where
where
is the rising factorial.
Moments
The first few raw moments are:
and, in general, the raw moments are given by
Here Lq(x) denotes a Laguerre polynomial:
where
For the case q = 1/2:
The second central moment, the variance, is
Note that
Differential equation
The pdf of the Rice distribution is a solution of the following differential equation:
Related distributions
Limiting cases
For large values of the argument, the Laguerre polynomial becomes
It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ2.
Parameter estimation (the Koay inversion technique)
There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments, (2) method of maximum likelihood, and (3) method of least squares. In the first two methods the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ1' and the sample standard deviation is an estimate of μ21/2.
The following is an efficient method, known as the "Koay inversion technique". for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier works on the method of moments usually use a root-finding method to solve the problem, which is not efficient.
First, the ratio of the sample mean to the sample standard deviation is defined as r, i.e.,
where
where
Note that
To find the fixed point,
Once the fixed point is found, the estimates
and
To speed up the iteration even more, one can use the Newton's method of root-finding. This particular approach is highly efficient.