I have a set of numbers drawn from iid gaussian random variables:

$P(d_0, ..., d_{N-1}) = (\sigma \sqrt{2 \pi})^{-N} exp\left(\frac{-1}{2\sigma^2} (d_0^2 + ... + d_{N-1}^2)\right)$

What is the pdf for the discrete fourier transform $f_0, ... f_{N-1}$ of the $d_k$?

It seems like this should be a fairly strightforward calculation but I'm getting bogged down. One thing that seems to make it tricky is the periodicity property of the $f_k$ means that you need to choose N particular real numbers to work with. I'd actually be happy with an answer that marginalizes over the phase angles, but a full answer would be great.

Is there a good text that addresses this type of question?