Rice distribution

The probability density function is

${\displaystyle f(x\mid \nu ,\sigma )={\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)H(x),}$

where I₀(z) is the modified Bessel function of the first kind with order zero, and H(x) is the Heaviside unit step ^[1].

In the context of Rician fading, the distribution is often also rewritten using the shape parameter ${\displaystyle K={\frac {\nu ^{2}}{2\sigma ^{2}}}}$, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the scale parameter ${\displaystyle \Omega =\nu ^{2}+2\sigma ^{2}}$, defined as the total power received in all paths.^[2]

The characteristic function of the Rice distribution is given as:^[3][4]

${\displaystyle {\begin{aligned}\chi _{X}(t\mid \nu ,\sigma )=\exp \left(-{\frac {\nu ^{2}}{2\sigma ^{2}}}\right)&\left[\Psi _{2}\left(1;1,{\frac {1}{2}};{\frac {\nu ^{2}}{2\sigma ^{2}}},-{\frac {1}{2}}\sigma ^{2}t^{2}\right)\right.\\[8pt]&\left.{}+i{\sqrt {2}}\sigma t\Psi _{2}\left({\frac {3}{2}};1,{\frac {3}{2}};{\frac {\nu ^{2}}{2\sigma ^{2}}},-{\frac {1}{2}}\sigma ^{2}t^{2}\right)\right],\end{aligned}}}$

where ${\displaystyle \Psi _{2}\left(\alpha$ ;\gamma ,\gamma ';x,y\right)} is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of ${\displaystyle x}$ and ⁠${\displaystyle y}$⁠. It is given by:^[5][6]

${\displaystyle \Psi _{2}\left(\alpha$ ;\gamma ,\gamma ';x,y\right)=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{\frac {(\alpha )_{m+n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n!}},}

where

${\displaystyle (x)_{n}=x(x+1)\cdots (x+n-1)={\frac {\Gamma (x+n)}{\Gamma (x)}}}$

is the rising factorial.

• ${\displaystyle R\sim \mathrm {Rice} \left(|\nu |,\sigma \right)}$ if ${\displaystyle R={\sqrt {X^{2}+Y^{2}}}}$ where ${\displaystyle X\sim N\left(\nu \cos \theta ,\sigma ^{2}\right)}$ and ${\displaystyle Y\sim N\left(\nu \sin \theta ,\sigma ^{2}\right)}$ are statistically independent normal random variables and ${\displaystyle \theta }$ is any real number.
• Another case where ${\displaystyle R\sim \mathrm {Rice} \left(\nu ,\sigma \right)}$ comes from the following steps:
  1. Generate ${\displaystyle P}$ having a Poisson distribution with parameter (also mean, for a Poisson) ⁠${\displaystyle \textstyle \lambda ={\frac {\nu ^{2}}{2\sigma ^{2}}}}$⁠.
  2. Generate ${\displaystyle X}$ having a chi-squared distribution with ⁠${\displaystyle 2P+2}$⁠ degrees of freedom.
  3. Set ${\displaystyle R=\sigma {\sqrt {X}}.}$
• If ${\displaystyle R\sim \operatorname {Rice} (\nu ,1)}$ then ${\displaystyle R^{2}}$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter ⁠${\displaystyle \nu ^{2}}$⁠.
• If ${\displaystyle R\sim \operatorname {Rice} (\nu ,1)}$ then ${\displaystyle R}$ has a noncentral chi distribution with two degrees of freedom and noncentrality parameter ⁠${\displaystyle \nu }$⁠.
• If ${\displaystyle R\sim \operatorname {Rice} (0,\sigma )}$ then ⁠${\displaystyle R\sim \operatorname {Rayleigh} (\sigma )}$⁠, i.e., for the special case of the Rice distribution given by ${\displaystyle \nu =0}$, the distribution becomes the Rayleigh distribution, for which the variance is ⁠${\displaystyle \textstyle \mu _{2}={\frac {4-\pi }{2}}\sigma ^{2}}$⁠.
• If ${\displaystyle R\sim \operatorname {Rice} (0,\sigma )}$ then ${\displaystyle R^{2}}$ has an exponential distribution.^[7]
• If ${\displaystyle R\sim \operatorname {Rice} \left(\nu ,\sigma \right)}$ then ${\displaystyle 1/R}$ has an Inverse Rician distribution.^[8]
• The folded normal distribution is the univariate restriction of the Rice distribution.

For large values of the argument, the Laguerre polynomial becomes^[9]

${\displaystyle \lim _{x\to -\infty }L_{\nu }(x)={\frac {|x|^{\nu }}{\Gamma (1+\nu )}}.}$

It is seen that as ⁠${\displaystyle \nu }$⁠ becomes large or ⁠${\displaystyle \sigma }$⁠ becomes small, the mean becomes ⁠${\displaystyle \nu }$⁠ and the variance becomes ⁠${\displaystyle \sigma ^{2}}$⁠.

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

${\displaystyle I_{\alpha }(z)\to {\frac {e^{z}}{\sqrt {2\pi z}}}\left(1-{\frac {4\alpha ^{2}-1}{8z}}+\cdots \right){\text{ as }}z\rightarrow \infty }$

so, in the large ${\displaystyle x\nu /\sigma ^{2}}$ region, an asymptotic expansion of the Rician distribution:

${\displaystyle {\begin{aligned}f(x,\nu ,\sigma )={}&{\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)\\{\text{ is }}\\&{\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right){\sqrt {\frac {\sigma ^{2}}{2\pi x\nu }}}\exp \left({\frac {2x\nu }{2\sigma ^{2}}}\right)\left(1+{\frac {\sigma ^{2}}{8x\nu }}+\cdots \right)\\\rightarrow {}&{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\nu )^{2}}{2\sigma ^{2}}}\right){\sqrt {\frac {x}{\nu }}},\;\;\;{\text{ as }}{\frac {x\nu }{\sigma ^{2}}}\rightarrow \infty \end{aligned}}}$

Moreover, when the density is concentrated around ${\textstyle \nu }$ and ${\textstyle |x-\nu |\ll \sigma }$ because of the Gaussian exponent, we can also write ${\textstyle {\sqrt {{x}/{\nu }}}\approx 1}$ and finally get the Normal approximation

${\displaystyle f(x,\nu ,\sigma )\approx {\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\nu )^{2}}{2\sigma ^{2}}}\right),\;\;\;{\frac {\nu }{\sigma }}\gg 1}$

The approximation becomes usable for ⁠${\displaystyle {\nu }/{\sigma }>3}$⁠.

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,^{[10][11][12][13]} (2) method of maximum likelihood,^{[10][11][12][14]} and (3) method of least squares.^{[citation needed]} In the first two methods the interest is in estimating the parameters of the distribution, ⁠${\displaystyle \nu }$⁠ and ⁠${\displaystyle \sigma }$⁠, 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 ⁠${\displaystyle \mu _{1}^{'}}$⁠ and the sample standard deviation is an estimate of ⁠${\displaystyle \mu _{2}^{1/2}}$⁠.

The following is an efficient method, known as the "Koay inversion technique".^[15] 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^[10][16] 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 ⁠${\displaystyle r}$⁠, i.e., ⁠${\displaystyle r=\mu _{1}^{'}/\mu _{2}^{1/2}}$⁠. The fixed point formula of SNR is expressed as

${\displaystyle g(\theta )={\sqrt {\xi {(\theta )}\left[1+r^{2}\right]-2}},}$

where ${\displaystyle \theta }$ is the ratio of the parameters, i.e., ⁠${\displaystyle \theta ={\nu }/{\sigma }}$⁠, and ${\displaystyle \xi {\left(\theta \right)}}$ is given by:

${\displaystyle \xi {\left(\theta \right)}=2+\theta ^{2}-{\frac {\pi }{8}}\exp {(-\theta ^{2}/2)}\left[(2+\theta ^{2})I_{0}(\theta ^{2}/4)+\theta ^{2}I_{1}(\theta ^{2}/4)\right]^{2},}$

where ${\displaystyle I_{0}}$ and ${\displaystyle I_{1}}$ are modified Bessel functions of the first kind.

Note that ${\displaystyle \xi {\left(\theta \right)}}$ is a scaling factor of ${\displaystyle \sigma }$ and is related to ${\displaystyle \mu _{2}}$ by:

${\displaystyle \mu _{2}=\xi {\left(\theta \right)}\sigma ^{2}.}$

To find the fixed point, ⁠${\displaystyle \theta ^{*}}$⁠, of ⁠${\displaystyle g}$⁠, an initial solution is selected, ⁠${\displaystyle {\theta }_{0}}$⁠, that is greater than the lower bound, which is ${\displaystyle {\theta }_{\text{lower bound}}=0}$ and occurs when ${\textstyle r={\sqrt {\pi /(4-\pi )}}}$^[15] (Notice that this is the ${\displaystyle r=\mu _{1}^{'}/\mu _{2}^{1/2}}$ of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition,^{[clarification needed]} and this continues until ${\displaystyle \left|g^{i}\left(\theta _{0}\right)-\theta _{i-1}\right|}$ is less than some small positive value. Here, ${\displaystyle g^{i}}$ denotes the composition of the same function, ⁠${\displaystyle g}$⁠, ${\displaystyle i}$ times. In practice, we associate the final ${\displaystyle \theta _{n}}$ for some integer ${\displaystyle n}$ as the fixed point, ⁠${\displaystyle \theta ^{*}}$⁠, i.e., ⁠${\displaystyle \theta ^{*}=g\left(\theta ^{*}\right)}$⁠.

Once the fixed point is found, the estimates ${\displaystyle \nu }$ and ${\displaystyle \sigma }$ are found through the scaling function, ⁠${\displaystyle \xi {\left(\theta \right)}}$⁠, as follows:

${\displaystyle \sigma ={\frac {\mu _{2}^{1/2}}{\sqrt {\xi \left(\theta ^{*}\right)}}},}$

and

${\displaystyle \nu ={\sqrt {\left(\mu _{1}^{'~2}+\left(\xi \left(\theta ^{*}\right)-2\right)\sigma ^{2}\right)}}.}$

To speed up the iteration even more, one can use the Newton's method of root-finding.^[15] This particular approach is highly efficient.

• The Euclidean norm of a bivariate circularly symmetric normally distributed random vector.
• Rician fading (for multipath interference))
• Effect of sighting error on target shooting.^[17]
• Analysis of diversity receivers in radio communications.^[18][19]
• Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.^[20]
• Distribution of noise in magnetic resonance imaging images is rician^[21]

• Hoyt distribution
• Rayleigh distribution