Standard deviation of the spectrum of white noise I have a temporal signal which looks like $f(t) = t\eta(t)$, where $\eta(t)$ is a white noise with the mean $\eta_m$ and STD $\sigma$. I want to calculate the corresponding spectrum, $F(\omega)$, take its magnitude, $|F(\omega)|$, and finally compute the STD of $|F(\omega)|$ in terms of the given parameters. I have searched some literatures but I couldn't find anything which helps me directly tackle the problem. So, anyone know how to do this, or know which resources you would refer me to?

The following pictures display $f(t)$ and $|F(\omega)|$.


The one below is the histogram of $|F(\omega)|$ excluding some values around the central peak.


I want to derive a mathematical analysis which can compute the standard deviation of the above histogram.
true white noise has infinite power (because the power spectrum with height $ \frac{\eta}{2}$  has infinite area) which means infinite variance $\sigma^2$ which means infinite standard deviation.  white noise is a conceptual instrument.

to do any analysis of a system with white noise in it, you must somehow determine the (single-sided) bandwidth of the system, $B$, and then the power or variance is $\eta B$.