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**Fourier transform of "noise"**

Hello,

when we want to get the magnitude of the Fourier frequency spectrum of a function

*f*we typically calculate [tex]F(\omega)=\int_{\mathbb{R}}f(x)e^{-i\omega x}dx[/tex]

and then consider [itex]|F(\omega)|[/itex].

We can do this as long the signal (=function) is deterministic, that is, only one single known value

*f(x)*is associated to every

*x*.

What happens when f(x) is

*not*deterministic anymore? In other words, we don't know what is the exact value of

*f(x)*, but we can say only that

*f(x)*follows a certain probability density function. For example I could say that [tex]f(x) \sim \mathcal{U}(-1 , 1)[/tex] which means that for a given

*x*,

*f(x)*is now a random variable having uniform probability distribution between -1 and 1.

If we plotted such a "function" against

*x*we would see a noisy plot with amplitudes between -1 and 1.

I would like to calculate the magnitude of the Fourier spectrum of such a function, but I don't know from where to start. what can we say about [itex]|F(\omega)|[/itex]? Any hint?

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