Forgive me if any of my terminology isn't right - I come from a physics/stats background, not pure maths.
I have a randomly generated time series, which is normally distributed with a mean of $0$ and standard deviation of $1$. I'm interested in the fourier transform of this time series.
The histogram of the data looks how I'd expect; centred at $0$ and decaying away at each side.
The fourier transform of a gaussian distribution is another gaussian distribution. However, what I'm interested in is the fourier transform of a normally distributed random variable. This has real and imaginary components, which are distributed as
and
Both histograms have a mean close to zero, and a standard deviation of about $223 (\pm 0.5$).
I don't know how I'd go about calculating the fourier transform analytically, or even if it can be done. Are there any resources for me to look at, or a technique I should know before trying?
The above plots were generated in Matlab, using the fft command.
Edit: I now have 10 reputation, so can replace the links to images with the images themselves.
Edit 2: I have some more information from playing further with Matlab. The standard deviation of the FFT of the signal depends on the absolute value of the standard deviation of the signal, like so:
At larger values of $|\sigma_{in}|$, the $\sigma$ for the real part approaches the $\sigma$ for the imaginary part. At a zero input standard deviation, the imaginary part of the FT of the signal has zero standard deviation, but interestingly, the real part has a standard deviation equal to $\sqrt{N}\mu$, where $N$ is the number of data points created, and $\mu$ is the mean of the input.
Hopefully this means something to someone! I have a book on numerical methods with a chapter on FFTs, and have done lots of googling, but I'm yet to find anything that helps me understand what's going on.
Edit 3: I've found this solution on DSP Stack Exchange. It is an excellent answer, which I nearly agree with. When I plot the standard deviation of the imaginary part minus $\sigma \sqrt{N/2}$, the results are consistent with zero:
but for the real part, they aren't quite, when $\mu$ is non-zero. For $\mu=5$, I get
which only approaches zero for large $\sigma$.