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.

In the 2D plane, pick a fixed point at distance ν from the origin. Generate a distribution of 2D points centered around that point, where the x and y coordinates are chosen independently from a Gaussian distribution with standard deviation σ (blue region). If R is the distance from these points to the origin, then R has a Rice distribution.
Probability density function
Rice probability density functions σ = 1.0
Cumulative distribution function
Rice cumulative distribution functions σ = 1.0
Parameters , distance between the reference point and the center of the bivariate distribution,
, spread
Support
PDF
CDF

where Q1 is the Marcum Q-function
Mean
Variance
Skewness (complicated)
Ex. kurtosis (complicated)

CharacterizationEdit

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.[1]

The characteristic function of the Rice distribution is given as:[2][3]

 

where   is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of   and  . It is given by:[4][5]

 

where

 

is the rising factorial.

PropertiesEdit

MomentsEdit

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  

Related distributionsEdit

  •   if   where   and   are statistically independent normal random variables and   is any real number.
  • Another case where   comes from the following steps:
1. Generate   having a Poisson distribution with parameter (also mean, for a Poisson)  
2. Generate   having a chi-squared distribution with 2P + 2 degrees of freedom.
3. 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  .
  • If   then   has an exponential distribution.[6]
  • If   then   has an Inverse Rician distribution.[7]
  • The folded normal distribution is the univariate special case of the Rice distribution.

Limiting casesEdit

For large values of the argument, the Laguerre polynomial becomes[8]

 

It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ2.

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

 

so, in the large   region, an asymptotic expansion of the Rician distribution:

 

Moreover, when the density is concentrated around   and   because of the Gaussian exponent, we can also write   and finally get the Normal approximation

 

The approximation becomes usable for  

Parameter estimation (the Koay inversion technique)Edit

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,[9][10][11][12] (2) method of maximum likelihood,[9][10][11][13] and (3) method of least squares.[citation needed] 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".[14] 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[9][15] 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.,  . The fixed point formula of SNR is expressed as

 

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

 

where   and   are modified Bessel functions of the first kind.

Note that   is a scaling factor of   and is related to   by:

 

To find the fixed point,  , of  , an initial solution is selected,  , that is greater than the lower bound, which is   and occurs when  [14] (Notice that this is the   of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition,[clarification needed] and this continues until   is less than some small positive value. Here,   denotes the composition of the same function,  ,   times. In practice, we associate the final   for some integer   as the fixed point,  , i.e.,  .

Once the fixed point is found, the estimates   and   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.[14] This particular approach is highly efficient.

ApplicationsEdit

See alsoEdit

The multivariate Rician model is used in the analysis of diversity receivers in radio communications.[17][18]

NotesEdit

  1. ^ Abdi, A. and Tepedelenlioglu, C. and Kaveh, M. and Giannakis, G., "On the estimation of the K parameter for the Rice fading distribution", IEEE Communications Letters, March 2001, p. 92–94
  2. ^ Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).
  3. ^ Annamalai 2000 (in a sum of infinite series).
  4. ^ Erdelyi 1953.
  5. ^ Srivastava 1985.
  6. ^ Richards, M.A., Rice Distribution for RCS, Georgia Institute of Technology (Sep 2006)
  7. ^ Jones, Jessica L., Joyce McLaughlin, and Daniel Renzi. "The noise distribution in a shear wave speed image computed using arrival times at fixed spatial positions.", Inverse Problems 33.5 (2017): 055012.
  8. ^ Abramowitz and Stegun (1968) §13.5.1
  9. ^ a b c Talukdar et al. 1991
  10. ^ a b Bonny et al. 1996
  11. ^ a b Sijbers et al. 1998
  12. ^ den Dekker and Sijbers 2014
  13. ^ Varadarajan and Haldar 2015
  14. ^ a b c Koay et al. 2006 (known as the SNR fixed point formula).
  15. ^ Abdi 2001
  16. ^ "Ballistipedia". Retrieved 4 May 2014.
  17. ^ Beaulieu, Norman C; Hemachandra, Kasun (September 2011). "Novel Representations for the Bivariate Rician Distribution". IEEE Transactions on Communications. 59 (11): 2951–2954. doi:10.1109/TCOMM.2011.092011.090171.
  18. ^ Dharmawansa, Prathapasinghe; Rajatheva, Nandana; Tellambura, Chinthananda (March 2009). "New Series Representation for the Trivariate Non-Central Chi-Squared Distribution" (PDF). IEEE Transactions on Communications. 57 (3): 665–675. CiteSeerX 10.1.1.582.533. doi:10.1109/TCOMM.2009.03.070083.

ReferencesEdit

External linksEdit