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
where I0(z) is the modified Bessel function of the first kind with order zero, and H(x) is the Heaviside unit step.Johnson, Norman L., Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous univariate distributions, volume 1. Vol. 1. John wiley & sons, 1994.
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.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, pp. 92–94
The characteristic function of the Rice distribution is given as:Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).Annamalai 2000 (in a sum of infinite series).
where is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of and . It is given by:Erdelyi 1953.Srivastava 1985.
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:
where 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
Note that indicates the square of the Laguerre polynomial , not the generalized Laguerre polynomial .
Related distributions if where and are statistically independent normal random variables and is any real number. Another case where comes from the following steps: Generate having a Poisson distribution with parameter (also mean, for a Poisson) . Generate having a chi-squared distribution with degrees of freedom. 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.Richards, M.A., Rice Distribution for RCS, Georgia Institute of Technology (Sep 2006) If then has an Inverse Rician distribution.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. The folded normal distribution is the univariate restriction of the Rice distribution.
Limiting cases For large values of the argument, the Laguerre polynomial becomesAbramowitz and Stegun (1968) §13.5.1
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
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 (Koay inversion technique) There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,Talukdar et al. 1991Bonny et al. 1996Sijbers et al. 1998den 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, 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".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 worksAbdi 2001 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
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 (Notice that this is the of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition, 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. 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. Distribution of noise in magnetic resonance imaging images is rician
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, pp. 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.
External links MATLAB code for Rice/Rician distribution (PDF, mean and variance, and generating random samples)
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