## Rician Distribution

### Definition

The Rician distribution has the density function

`${I}_{0}\left(\frac{xs}{{\sigma }^{2}}\right)\frac{x}{{\sigma }^{2}}{e}^{-\left(\frac{{x}^{2}+{s}^{2}}{2{\sigma }^{2}}\right)}$`

with noncentrality parameter s ≥ 0 and scale parameter σ > 0, for x > 0. I0 is the zero-order modified Bessel function of the first kind. If x has a Rician distribution with parameters s and σ, then (x/σ)2 has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter (s/σ)2.

### Background

In communications theory, Nakagami distributions, Rician distributions, and Rayleigh distributions are used to model scattered signals that reach a receiver by multiple paths. Depending on the density of the scatter, the signal will display different fading characteristics. Rayleigh and Nakagami distributions are used to model dense scatters, while Rician distributions model fading with a stronger line-of-sight. Nakagami distributions can be reduced to Rayleigh distributions, but give more control over the extent of the fading.

### Parameters

To estimate distribution parameters, use `mle` or the Distribution Fitter app.

#### Fit Rician Distribution with Known Scale Parameter

Generate sample data of size 1000 from a Rician distribution with noncentrality parameter of 8 and scale parameter of 5. First create the Rician distribution.

`r = makedist('Rician','s',8,'sigma',5);`

Now, generate sample data from the distribution you created above.

```rng default % For reproducibility x = random(r,1000,1);```

Suppose the scale parameter is known, and estimate the noncentrality parameter from sample data. To do this using `mle`, you must custom define the Rician probability density function.

`[phat,pci] = mle(x,'pdf',@(x,s,sigma) pdf('rician',x,s,5),'start',10)`
```phat = 7.8953 ```
```pci = 2×1 7.5405 8.2501 ```

The estimate for the noncentrality parameter is 7.8953, with a 95% confidence interval of 7.5404 and 8.2501. The confidence interval includes the true parameter value of 8.