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

\begin{equation} 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 \end{equation}

Define the sum as \begin{equation} S =\Big( \sum_i^N \alpha_i \Big)^2 \end{equation}

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

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