# Distribution of the square of the sum of independent rayleigh variables

Suppose $$\alpha_i$$ is the $$i$$th independent Rayleigh distribution random variable following a Rayleigh probability density function (PDF) as

$$$$f_{\alpha_i}(r) = \frac{r}{\sigma_i^2} \exp\left(\frac{-r^2}{2\sigma_i^2}\right),r>0,i=1,2,...N$$$$

Define the sum as $$$$S =\Big( \sum_i^N \alpha_i \Big)^2$$$$

Then how to calculate the PDF of $$S$$ ? Is there any close-form expression?