# Can realizations of a filtered Gaussian white noise process be represented as a Fourier transform?

Suppose we have a noise process $V(t)$ which is the result of passing Gaussian white noise through a filter with frequency response function $H(\omega)$. Can we represent realizations of this process as a Fourier transform $$V(t) = \int \frac{d\omega}{2\pi} M(\omega) e^{i \phi(\omega)}$$ where $M(\omega)$ and $\phi(\omega)$ are random variables? If so, what are the statistics of $M(\omega)$ and $\phi(\omega)$?

• @DanielSank: What I meant is that for the realization of a random process, especially if it's white, i.e. it has infinite power, there is no reason to believe that the conditions for the existence of the Fourier transform are met. In general they aren't (e.g. the function is not $L_1$, etc.). Jun 14, 2015 at 20:00
• @DanielSank: I think that even in that case my original answer still holds, because also realizations of a filtered white noise process will generally not be $L_1$ or $L_2$, so their Fourier transform will not exist. I added a link to my answer where this problem is discussed. Jun 14, 2015 at 20:23
• Awesome. I actually discovered the construction shown in that book myself when trying to understand this issue. I suppose as you say it doesn't make sense to define a Fourier transform for a noise process. This leaves a funny feeling though, because I thought I'd heard that it's possible to think of white noise as a Fourier integral where the phases $\phi(\omega)$ are all independent and uniformly distributed. Jun 14, 2015 at 20:35