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Which are the techniques used to approximate a distribution into another? I know that I can model a Gaussian Distribution through the parameters of mean and variance. However how can I approximate Rice Distribution through the Gaussian Distribution? There is some good reference covering that?

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2 Answers

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Center $(\mu_x, \mu_y) =(1,2)$ and standard deviation $\sigma=2.$ If you can generate univariate normal random variables $X \sim \mathsf{Norm}(1,2)$ and $Y \sim \mathsf{Norm}(1,2),$ then you have an uncorrelated bi-variate normal distribution. Similarly, according to the Wikipedia article referenced by @JeanMarie, you can make various kinds of Rice distributions. Here is a plot using R.

x = rnorm(50000, 1, 2); y = rnorm(50000, 2, 2)
plot(x, y, pch=".") abline(v=-10:10, col="lightblue") abline(h=-10:10, col="lightblue") abline(v=0, lwd=2, col="green2") abline(h=0, lwd=2, col="green2")

In order to get normal distributions from randomly generated points in $(0,1)$ you can use the Box-Muller transform. (See Wikipedia.)

Below a histogram suggests the shape of the density function of the distance of points from the origin, and a plot of the corresponding empirical distribution function (ECDF) which approximates the CDF. (Note: The ECDF sorts the $n$ points on the horizontal axis and increments by $1/n$ at corresponding points on the vertical axis)

d = sqrt(s^2 + y^2)
par(mfrow=c(1,2)) hist(d, prob=T, col="skyblue2") plot(ecdf(d))
par(mfrow=c(1,1))

With this introduction in R and reference to the Wikipedia pages, you should be able to figure out how to do this assignment using most any statistical or mathematical software.

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You can generate a Rayleigh distribution simply as:

 x_rayleigh = ( randn(1, 1e6) + 1i*randn(1, 1e6) ) / sqrt(2);

This is a complex normal distribution with zero mean and variance 1/2 per dimension. So the amplitude of x_rayleigh follows a Rayleigh distribution. The phase of x_rayleigh will be uniform.

Now if you add a line-of-sight component in the above random variable, the amplitude will be Rice distributed.

 x_rice = 1 + ( randn(1, 1e6) + 1i*randn(1, 1e6) ) / sqrt(2);

The histogram of the absolute value of the generated random variable is shown in the figure.

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