Let $x$ and $\hat{x}$ be a random vector and its Fourier transform, respectively. For any practical purpose we can assume it's a finite vector and the Fourier transform is given by the DFT, i.e. there exists an orthogonal matrix $A$ s.t. $\hat{x}=Ax$ and $x=A^T\hat{x}$.

Say I want to find $f(x)$, the density of $x$. Is there a convenient way to express $f(x)$ by $\hat{x}$ somehow, other than the explicit transformation rule?

Also, as the DFT of a real signal may be complex, is there a meaning to the density of a complex vector? (it's not what I'm looking for, but it is interesting)


EDIT: my title was wrong. I fixed it. To clarify - I want to express $f(x)$ via the vector's DFT. I'm not looking for the DFT of the density.


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