I am interested seeing the difference between two power spectral densities (PSD) as a noise reduction exercise
The blue line is the psd of my signal, and the the orange line is the psd of the signal without the part I am interested in. So to isolate the part I am interested in I would like to 'subtract' these to see the difference.
However it occurs to me that this would be dangerous since if we consider the DFT's of the 'blue and orange signals' for a particular frequency $k$, to be $F_k$ and $G_k$ respectively. Then the PSD of one minus the other is not the same as the PSD of one minus the PSD of the other since by taking the PSD you lose the phase information. Or less confusingly the problem boils down to this:
$|F_k|^2 - |G_k|^2 \neq |F_k - G_k|^2$
The obvious solution to this is to simply subtract the output of the DFT step, then PSD. However in python this will be difficult since I am using scipys version of welches method to calculate the PSDs which means I would have to rewrite a lot of my code.
So my question is this:
Is there a way to get a good approximation to the difference between the two PSDs using the PSDs rather than doing the earlier subtraction? Or again less confusingly:
Can I get an approximation to $|F_k - G_k|^2$ from $|F_k|^2$ and $|G_k|^2$?