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Wikipedia / 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 (19071986).

Characterization

The probability density function is

where I0(z) is the modified Bessel function of the first kind with order zero, and H(x) is the Heaviside unit step.

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:

is one of Horn's confluent hypergeometric functions and . It is given by:

where

is the rising factorial.

Properties

Moments

The first few raw moments are:

and, in general, the raw moments are given by

Here denotes a Laguerre polynomial:

is the confluent hypergeometric function of the first kind. When is even, the raw moments become simple polynomials in and , as in the examples above.

For the case :

The second central moment, the variance, is

indicates the square of the Laguerre polynomial , not the generalized Laguerre polynomial .

Related distributions

  • is any real number.

comes from the following steps:

  1. having a Poisson distribution with parameter (also mean, for a Poisson) .

  2. having a chi-squared distribution with degrees of freedom.

  3. has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter .

  4. has a noncentral chi distribution with two degrees of freedom and noncentrality parameter .

  5. then , the distribution becomes the Rayleigh distribution, for which the variance is .

  6. has an exponential distribution.

  7. has an Inverse Rician distribution.

  8. The folded normal distribution is the univariate restriction of the Rice distribution.

Limiting cases

For large values of the argument, the Laguerre polynomial becomes

It is seen that as becomes large or becomes small, the mean becomes and the variance becomes .

The transition to a Gaussian approximation proceeds as follows. From Bessel function theory we have

region, an asymptotic expansion of the Rician distribution:

and finally get the Normal approximation

The approximation becomes usable for .

Parameter estimation (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, and , 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 and the sample standard deviation is an estimate of .

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 , i.e., . The fixed point formula of SNR is expressed as

is the ratio of the parameters, i.e., is given by:

are modified Bessel functions of the first kind.

by:

To find the fixed point, , of , an initial solution is selected, of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition, denotes the composition of the same function, as the fixed point, , i.e., .

are found through the scaling function, , as follows:

and

To speed up the iteration even more, one can use the Newton's method of root-finding. This particular approach is highly efficient.

Applications

See also

References

Further reading

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