Signals Cross Correlation from Their Power Spectral Densities?

Question

For two real valued signals $f$ and $g$ with their corresponding power spectral densities $|F(k)|^2$ and $|G(k)|^2$ Is there a way to get an intuition of how correlated $f$ and $g$ are just by looking at the power spectral densities (PSDs)?

For example does the fact that the PSDs of these two signals are so similar imply that they are cross correlated?

So far I have considered that $$|F(k)|^2 = F(k) \overline{F(k)}$$ $$|G(k)|^2 = G(k) \overline{G(k)}$$

and the cross correlation we want to find is

$$ F[f \star g] = F(k)\overline{G(k)} = G(-k)\overline{F(-k)}$$

and so it seems that there could be a way to get an exact result for the cross correlation from the PSDs, however I would also be happy just to get a hand wavy idea of if these two signals are cross correlated

Answer 1

No,

For instance 2 instances of AWGN with the same STD have the same PSD while being totally uncorrelated (Moreover, Statistically Independent).

Also, pay attention that Correlation is not commutative:

$$ f \ast g \neq g \ast f \Leftrightarrow F \left( k \right) \overline{G \left( k \right)} \neq G \left( k \right) \overline{F \left( k \right)} $$