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I considered a statistically independent continuous random processf(x)such that Cov(f(x),f(y))=0 for x[itex]\neq[/itex]y and Cov(f(x),f(x))=σ^{2}.

Then I would like to compute the correlation function of the Fourier transform off, that is [itex]Cov\left( F(u),F(v)\right)[/itex].

The result I got from my calculations is that [itex]Cov\left( F(u),F(v)\right)=0[/itex] when u[itex]\neq[/itex]v, and [itex]Cov\left( F(u),F(u)\right)=\sigma^2[/itex].

So again, also the Fourier transform offis supposed to be statistically independent.

At first I thought that this makes sense, but then I started to wonder why F(u) and F(-u) are supposed to be uncorrelated? We know that F(u)=F(-u)^{*}, so the negative frequencies theoretically depend fully on the positive frequencies (they are actually redundant as one is the complex conjugate of the other).

How should I interpret this result? Is it wrong?

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# Integral transforms of random processes

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