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).

Characterization

The probability density function is
:$f(x\mid\nu,\sigma) = \frac\exp\left(\frac \right)I_0\left(\frac\right),$
where ''I''₀(''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'' $K = \frac$, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the ''Scale parameter'' $\Omega = \nu^2+2\sigma^2$, defined as the total power received in all paths.

The characteristic function of the Rice distribution is given as:

:
\begin
\chi_X(t\mid\nu,\sigma)
= \exp \left( -\frac \right) & \left \Psi_2 \left( 1; 1, \frac; \frac, -\frac \sigma^2 t^2 \right) \right. \\[8pt& \left. + i \sqrt \sigma t
\Psi_2 \left( \frac; 1, \frac; \frac, -\frac \sigma^2 t^2 \right) \right],
\end

where $\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right)$ is one of Horn function, Horn's confluent hypergeometric functions with two variables and convergent for all finite values of $x$ and $y$. It is given by:

:$\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right) = \sum_^\sum_^\infty \frac \frac,$

where

: $(x)_n = x(x+1)\cdots(x+n-1) = \frac$

is the rising factorial.

Properties

Moments

The first few raw moments are:

:$\begin \mu_1^&= \sigma \sqrt\,\,L_(-\nu^2/2\sigma^2) \\ \mu_2^&= 2\sigma^2+\nu^2\, \\ \mu_3^&= 3\sigma^3\sqrt\,\,L_(-\nu^2/2\sigma^2) \\ \mu_4^&= 8\sigma^4+8\sigma^2\nu^2+\nu^4\, \\ \mu_5^&=15\sigma^5\sqrt\,\,L_(-\nu^2/2\sigma^2) \\ \mu_6^&=48\sigma^6+72\sigma^4\nu^2+18\sigma^2\nu^4+\nu^6 \end$

and, in general, the raw moments are given by

:$\mu_k^=\sigma^k2^\,\Gamma(1\!+\!k/2)\,L_(-\nu^2/2\sigma^2). \,$

Here ''L''_''q''(''x'') denotes a Laguerre polynomial:

:$L_q(x)=L_q^(x)=M(-q,1,x)=\,_1F_1(-q;1;x)$
where $M(a,b,z) = _1F_1(a;b;z)$ 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:

:
\begin
L_(x) &=\,_1F_1\left( -\frac;1;x\right) \\
&= e^ \left left(1-x\right)I_0\left(-\frac\right) -xI_1\left(-\frac\right) \right
\end

The second central moment, the variance, is
:$\mu_2= 2\sigma^2+\nu^2-(\pi\sigma^2/2)\,L^2_(-\nu^2/2\sigma^2) .$

Note that $L^2_(\cdot)$ indicates the square of the Laguerre polynomial $L_(\cdot)$, not the generalized Laguerre polynomial $L^_(\cdot).$

Related distributions

*$R \sim \mathrm\left(, \nu, ,\sigma\right)$ if $R = \sqrt$ where $X \sim N\left(\nu\cos\theta,\sigma^2\right)$ and $Y \sim N\left(\nu \sin\theta,\sigma^2\right)$ are statistically independent normal random variables and $\theta$ is any real number.
*Another case where $R \sim \mathrm\left(\nu,\sigma\right)$ comes from the following steps:
*# Generate $P$ having a Poisson distribution with parameter (also mean, for a Poisson) $\lambda = \frac.$
*# Generate $X$ having a chi-squared distribution with degrees of freedom.
*# Set $R = \sigma\sqrt.$
*If $R \sim \operatorname(\nu,1)$ then $R^2$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter $\nu^2$.
*If $R \sim \operatorname(\nu,1)$ then $R$ has a noncentral chi distribution with two degrees of freedom and noncentrality parameter $\nu$.
*If $R \sim \operatorname(0,\sigma)$ then $R \sim \operatorname(\sigma)$, i.e., for the special case of the Rice distribution given by $\nu = 0$, the distribution becomes the Rayleigh distribution, for which the variance is $\mu_2= \frac\sigma^2$.
*If $R \sim \operatorname(0,\sigma)$ then $R^2$ has an exponential distribution.
*If $R \sim \operatorname\left(\nu,\sigma\right)$ then $1/R$ has an Inverse Rician distribution.
* The folded normal distribution is the univariate special case of the Rice distribution.

Limiting cases

For large values of the argument, the Laguerre polynomial becomes

:$\lim_L_\nu(x)=\frac.$

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
:$I_\alpha(z) \to \frac \left(1 - \frac + \cdots \right) \text z \rightarrow \infty$
so, in the large $x\nu/\sigma^2$ region, an asymptotic expansion of the Rician distribution:

:$\begin f(x,\nu,\sigma) = & \frac\exp\left(\frac \right)I_0\left(\frac\right) \\ \text \\ & \frac\exp\left(\frac \right) \sqrt \exp \left( \frac \right) \left(1 + \frac + \cdots \right)\\ \rightarrow & \frac\exp\left(-\frac \right) \sqrt , \;\;\; \text \frac \rightarrow \infty \end$

Moreover, when the density is concentrated around $\nu$ and $, x - \nu, \ll \sigma$ because of the Gaussian exponent, we can also write $\sqrt \approx 1$ and finally get the Normal approximation
: $f(x,\nu,\sigma) \approx \frac \exp\left(- \frac\right) , \;\;\; \frac \gg 1$
The approximation becomes usable for $\frac > 3$

Parameter estimation (the Koay inversion technique)

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments, Talukdar et al. 1991 Bonny et al. 1996 Sijbers et al. 1998 den Dekker and Sijbers 2014 (2) method of maximum likelihood, Varadarajan and Haldar 2015 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 μ₂^1/2.

The following is an efficient method, known as the "Koay inversion technique". Koay et al. 2006 (known as the SNR fixed point formula). 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., $r=\mu^_1/\mu^_2$. The fixed point formula of SNR is expressed as

:$g(\theta) = \sqrt,$

where $\theta$ is the ratio of the parameters, i.e., $\theta = /$, and $\xi$ is given by:

: \xi = 2 + \theta^2 - \frac \exp\left (2+\theta^2) I_0 (\theta^2/4) + \theta^2 I_1(\theta^/4)\right2,

where $I_0$ and $I_1$ are modified Bessel functions of the first kind.

Note that $\xi$ is a scaling factor of $\sigma$ and is related to $\mu_$ by:

:$\mu_2 = \xi \sigma^2.$

To find the fixed point, $\theta^$, of $g$, an initial solution is selected, $_$, that is greater than the lower bound, which is $_ = 0$ and occurs when $r = \sqrt$ (Notice that this is the $r=\mu^_1/\mu^_2$ of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition, and this continues until $\left, g^\left(\theta_\right)-\theta_\$ is less than some small positive value. Here, $g^$ denotes the composition of the same function, $g$, $i$ times. In practice, we associate the final $\theta_$ for some integer $n$ as the fixed point, $\theta^$, i.e., $\theta^ = g\left(\theta^\right)$.

Once the fixed point is found, the estimates $\nu$ and $\sigma$ are found through the scaling function, $\xi$, as follows:

:$\sigma = \frac,$

and

:$\nu = \sqrt.$

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

Applications

*The Euclidean norm of a bivariate circularly-symmetric normally distributed random vector.
*Rician fading (for multipath interference))
*Effect of sighting error on target shooting.
*Analysis of diversity receivers in radio communications.
*Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.

See also

* Hoyt distribution
*Rayleigh distribution

References

Further reading

*Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965.
* Rice, S. O., Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945) 46–156.
*
*
*Liu, X. and Hanzo, L.
A Unified Exact BER Performance Analysis of Asynchronous DS-CDMA Systems Using BPSK Modulation over Fading Channels
IEEE Transactions on Wireless Communications, Volume 6, Issue 10, October 2007, pp. 3504–3509.
*Annamalai, A., Tellambura, C. and Bhargava, V. K.
Equal-Gain Diversity Receiver Performance in Wireless Channels
IEEE Transactions on Communications, Volume 48, October 2000, pp. 1732–1745.
*Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.
Higher Transcendental Functions, Volume 1.
McGraw-Hill Book Company Inc., 1953.
*Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series. Ellis Horwood Ltd., 1985.
*Sijbers J., den Dekker A. J., Scheunders P. and Van Dyck D.
"Maximum Likelihood estimation of Rician distribution parameters"
, IEEE Transactions on Medical Imaging, Vol. 17, Nr. 3, pp. 357–361, (1998)
*Varadarajan D. and Haldar J. P.
"A Majorize-Minimize Framework for Rician and Non-Central Chi MR Images"
IEEE Transactions on Medical Imaging, Vol. 34, no. 10, pp. 2191–2202, (2015)
*
* Koay, C.G. and Basser, P. J.
Analytically exact correction scheme for signal extraction from noisy magnitude MR signals
Journal of Magnetic Resonance, Volume 179, Issue = 2, p. 317–322, (2006)
*Abdi, A., Tepedelenlioglu, C., Kaveh, M., and Giannakis, G.br>On the estimation of the K parameter for the Rice fading distribution
IEEE Communications Letters, Volume 5, Number 3, March 2001, pp. 92–94.
*
*

External links

MATLAB code for Rice/Rician distribution
(PDF, mean and variance, and generating random samples)

{{DEFAULTSORT:Rice Distribution
Continuous distributions

he:דעיכות מסוג רייס