# Nakagami distribution

Parameters Probability density function Cumulative distribution function $m\ or\ \mu >= 0.5$ shape (real) $\Omega\ or\ \omega > 0$ spread (real) $x > 0\!$ $\frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2 \right)$ $\frac{\gamma \left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}$ $\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}$ $\sqrt{\Omega}\!$ $\frac{\sqrt{2}}{2} \left(\frac{(2m-1)\Omega}{m}\right)^{1/2}$ $\Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)$

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter $m$ and a second parameter controlling spread, $\Omega$.

## Characterization

Its probability density function (pdf) is[1]

$f(x;\,m,\Omega) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right).$
$F(x;\,m,\Omega) = P\left(m, \frac{m}{\Omega}x^2\right)$

where P is the incomplete gamma function (regularized).

$\left\{x \Omega f'(x)+f(x) \left(2 m x^2-2 m \Omega +\Omega \right)=0,f(1)=\frac{2 m^m e^{-\frac{m}{\Omega }} \Omega ^{-m}}{\Gamma (m)}\right\}$

## Parameter estimation

The parameters $m$ and $\Omega$ are[2]

$m = \frac{\operatorname{E}^2 \left[X^2 \right]} {\operatorname{Var} \left[X^2 \right]},$

and

$\Omega = \operatorname{E} \left[X^2 \right].$

An alternative way of fitting the distribution is to re-parametrize $\Omega$ and m as σ = Ω/m and m.[3] Then, by taking the derivative of log likelihood with respect to each of the new parameters, the following equations are obtained and these can be solved using the Newton-Raphson method:

$\Gamma(m)= \frac{x^{2m}}{\sigma^m},$

and

$\sigma= \frac{x^2}{m}$

It is reported by authors[who?] that modelling data with Nakagami distribution and estimating parameters by above mention method results in better performance for low data regime compared to moments based methods.

## Generation

The Nakagami distribution is related to the gamma distribution. In particular, given a random variable $Y \, \sim \textrm{Gamma}(k, \theta)$, it is possible to obtain a random variable $X \, \sim \textrm{Nakagami} (m, \Omega)$, by setting $k=m$, $\theta=\Omega / m$, and taking the square root of $Y$:

$X = \sqrt{Y} \,$.

The Nakagami distribution $f(y; \,m,\Omega)$ can be generated from the chi distribution with parameter $k$ set to $2m$ and then following it by a scaling transformation of random variables. That is, a Nakagami random variable $X$ is generated by a simple scaling transformation on a Chi-distributed random variable $Y \sim \chi(2m)$ as below:

$X = \sqrt{(\Omega / 2 m)}\, Y.$

## History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.[4] It has been used to model attenuation of wireless signals traversing multiple paths.[5]

## References

1. ^ a b Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
2. ^ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
3. ^ Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9-12.
4. ^ Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18-20, 1958, pp 3-36. Pergamon Press.
5. ^ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.