Rice distribution explained

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) = (x)/(\exp\left( \sigma ^{2}) (- (x ^{2+\nu} ^{2)})/(2 \sigma ^{2}) \right)I _{0 \left( (x\nu)/(\sigma ^{2})} \right), 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

K = (\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 \Omega = \nu ^{2+2\sigma} ^{2} , defined as the total power received in all paths.^[1]

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

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

where

\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 x and y . It is given by:^{[4] [5]} \Psi _{2} \left( \alpha; \gamma, \gamma'; x, y \right) = \sum ^{inf ty} _{n =0} \sum ^{inf ty} _{m =0} ((\alpha) _{m+n})/((\gamma) _{m(\gamma')} _{n}) (x ^{m} y ^{n})/(m!n!) ,

where

(x) _{n} = x(x +1 ) … (x + n -1 ) = (\Gamma(x + n))/(\Gamma(x))

is the rising factorial.

Properties

Moments

The first few raw moments are:

\begin{align} \mu ^{'} _{1} & = \sigma \sqrt{\pi/ 2 }L _{1/2} ( - \nu ^{2/2\sigma} ^{2)} \\ \mu ^{'} _{2} & = 2 \sigma ^{2+\nu} ^{2} \\ \mu ^{'} _{3} & = 3 \sigma ^{3 \sqrt{\pi/ 2 }L} _{3 / 2} ( - \nu ^{2/2\sigma} ^{2)} \\ \mu ^{'} _{4} & = 8 \sigma ^{4+8\sigma} ^{2\nu} ^{2+\nu} ^{4} \\ \mu ^{'} _{5} & =15 \sigma ^{5 \sqrt{\pi/ 2 }L} _{5 / 2} ( - \nu ^{2/2\sigma} ^{2)} \\ \mu ^{'} _{6} & =48 \sigma ^{6+72\sigma} ^{4\nu} ^{2+18\sigma} ^{2\nu} ^{4+\nu} ^{6 \end{align}}

and, in general, the raw moments are given by

\mu ^{'} _{k} = \sigma ^{k2} ^{k/2} \Gamma( 1+ k/ 2 )L _{k/2} ( - \nu ^{2/2\sigma} ^{2).}

Here L_q(x) denotes a Laguerre polynomial:

L _{q(x)=L} ^{( 0 )} ^{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{align} L _{1/2} (x) & = _{1F} _{1\left(} - (1)/(2) ; 1 ;x\right) \\ & = e ^{x/2} \left[\left( 1- x\right)I _{0 \left( - (x)/(2)} \right) - xI _{1 \left( - (x)/(2)} \right) \right]. \end{align}

The second central moment, the variance, is

\mu _{2=} 2 \sigma ^{2+\nu} ^{2-(\pi\sigma} ^{2/2)L} ^{2} ^{1 / 2} ( - \nu ^{2/2\sigma} ^{2)} .

Note that

L ^{2} _{1 / 2} ( ⋅ ) indicates the square of the Laguerre polynomial L _{1/2} ( ⋅ ) , not the generalized Laguerre polynomial L ^{( 2 )} _{1 / 2} ( ⋅ ).

Related distributions

R \sim Rice \left(|\nu|,\sigma\right) if R = \sqrt{X ^{2} + Y ^{2}} 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 Rice \left(\nu,\sigma\right) comes from the following steps:

• 1. Generate

P having a Poisson distribution with parameter (also mean, for a Poisson) λ = (\nu ^{2})/(2 \sigma ^{2}) .

• 1. Generate

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

• 1. Set

R = \sigma\sqrt{X}.

• If

R \sim \operatorname{Rice}(\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{Rice}(\nu, 1 ) then R has a noncentral chi distribution with two degrees of freedom and noncentrality parameter \nu .

• If

R \sim \operatorname{Rice}( 0 ,\sigma) then R \sim \operatorname{Rayleigh}(\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=} (4- \pi)/(2) \sigma ^{2} .

• If

R \sim \operatorname{Rice}( 0 ,\sigma) then R ^{2} has an exponential distribution.^[6]

• If

R \sim \operatorname{Rice}\left(\nu,\sigma\right) then 1 /R has an Inverse Rician distribution.^[7]

• 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^[8]

\lim _{x} L _{\nu(x) = (|x| ^{\nu})/(\Gamma( 1+ \nu))} .

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 (e ^{z})/(\sqrt{ 2 \pi z) } \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{align} f(x,\nu,\sigma) = {} & (x)/(\exp\left( \sigma ^{2}) (- (x ^{2+\nu} ^{2)})/(2 \sigma ^{2}) \right)I _{0 \left( (x\nu)/(\sigma ^{2})} \right) \\ is \\ & (x)/(\exp\left( \sigma ^{2}) (- (x ^{2} + \nu ^{2)})/(2 \sigma ^{2}) \right) \sqrt{ (\sigma ^{2})/(2 \pi x \nu) } \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 $\backslash nu$ and $|x\; -\; \backslash nu|\; \backslash ll\; \backslash sigma$ because of the Gaussian exponent, we can also write $\backslash sqrt\; \backslash approx\; 1$ and finally get the Normal approximation

f(x,\nu,\sigma) ≈ (1)/(\sigma \sqrt{ 2 \pi) } \exp\left(- \frac\right), \;\;\; \frac \gg 1The approximation becomes usable for (\nu)/(\sigma) > 3

Parameter estimation (the Koay inversion technique)

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. 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".^[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.,

r = \mu ^{1 / 2} _{2} . The fixed point formula of SNR is expressed asg(\theta) = \sqrt{ \xi{(\theta)} \left[ 1+ r ^{2\right]} - 2 },

where

\theta is the ratio of the parameters, i.e., \theta = {\nu}/{\sigma} , and \xi{\left(\theta\right)} is given by:\xi{\left(\theta\right)} = 2 + \theta ^{2} - (\pi)/(8) \exp{( - \theta ^{2/2)}\left[} ( 2+ \theta ^{2)} I _{0} (\theta ^{2/4)} + \theta ^{2} I ^{2} _{1 (\theta} / 4 )\right] ^{2,}

where

I _{0} and I _{1} are modified Bessel functions of the first kind.

Note that

\xi{\left(\theta\right)} is a scaling factor of \sigma and is related to \mu _{2} by:\mu _{2} = \xi{\left(\theta\right)} \sigma ^{2.}

To find the fixed point,

\theta ^{*} , of g , an initial solution is selected, {\theta} _{0} , that is greater than the lower bound, which is {\theta} _{lowerbound} = 0 and occurs when $r\; =\; \backslash sqrt$^[14] (Notice that this is the r = \mu ^{1 / 2} _{2} of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition, and this continues until \left|g ^{i} \left(\theta _{0} \right) - \theta _{i-1} \right| is less than some small positive value. Here, g ^{i} denotes the composition of the same function, g , i times. In practice, we associate the final \theta _{n} 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{\left(\theta\right)} , as follows:\sigma = (\mu ^{1 / 2} _{2})/(\sqrt{\xi\left(\theta ^{*} \right)) },

and

\nu = \sqrt{\left( \mu ^{'~ 2} _{1} + \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.^[14] 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.^[16]
• Analysis of diversity receivers in radio communications.^{[17] [18]}
• Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.^[19]

See also

• Hoyt distribution
• Rayleigh distribution

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.
• I. Soltani Bozchalooi . Ming Liang . 10.1016/j.jsv.2007.07.038 . A smoothness index-guided approach to wavelet parameter selection in signal de-noising and fault detection . Journal of Sound and Vibration . 308 . 1–2 . 20 November 2007 . 253–254 . refBozchalooi2007. 2007JSV...308..246B .
• 10.1016/j.jsv.2017.02.013 . On the distribution of the modulus of Gabor wavelet coefficients and the upper bound of the dimensionless smoothness index in the case of additive Gaussian noises: Revisited . Journal of Sound and Vibration . 395 . 393–400 . 2017 . Wang . Dong . Zhou . Qiang . Tsui . Kwok-Leung .
• 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)
• den Dekker, A.J. . Sijbers, J . 10.1016/j.ejmp.2014.05.002 . 25059432 . Physica Medica . Data distributions in magnetic resonance images: a review . 30 . 7 . December 2014 . 725–741. RefDekker2014.
• 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. On the estimation of the K parameter for the Rice fading distribution, IEEE Communications Letters, Volume 5, Number 3, March 2001, pp. 92 - 94.
• Talukdar, K.K. . Lawing, William D. . 10.1121/1.400532 . Estimation of the parameters of the Rice distribution . Journal of the Acoustical Society of America . 89 . 3 . March 1991 . 1193–1197 . RefTalukdar. 1991ASAJ...89.1193T .
• Bonny, J.M. . Renou, J.P. . Zanca, M. . 10.1006/jmrb.1996.0166 . 8954899 . Optimal Measurement of Magnitude and Phase from MR Data . Journal of Magnetic Resonance, Series B . 113 . 2 . November 1996 . 136–144 . RefBonny. 1996JMRB..113..136B .

External links

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

Notes and References

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. [#refLiu2007|Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).]
3. [#refAnnamalai2000|Annamalai 2000 (in a sum of infinite series).]
4. [#refErdelyi1953|Erdelyi 1953.]
5. [#refSrivastava1985|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. [#RefTalukdar|Talukdar et al. 1991]
10. [#RefBonny|Bonny et al. 1996]
11. [#RefSijbers|Sijbers et al. 1998]
12. [#RefDekker2014|den Dekker and Sijbers 2014]
13. [#RefVaradarajan|Varadarajan and Haldar 2015]
14. [#refKoay2006|Koay et al. 2006 (known as the SNR fixed point formula).]
15. [#RefAbdi|Abdi 2001]
16. Web site: Ballistipedia. 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 . 10.1109/TCOMM.2011.092011.090171. 1221747 .
18. Dharmawansa. Prathapasinghe. Rajatheva. Nandana. Tellambura. Chinthananda. March 2009. New Series Representation for the Trivariate Non-Central Chi-Squared Distribution. IEEE Transactions on Communications. 57 . 3. 665–675. 10.1109/TCOMM.2009.03.070083. 10.1.1.582.533. 15706035 .
19. Laskar . J. . 2008-07-01 . Chaotic diffusion in the Solar System . Icarus . en . 196 . 1 . 1–15 . 10.1016/j.icarus.2008.02.017 . 0802.3371 . 2008Icar..196....1L . 11586168 . 0019-1035.