Let's say you have a continuous signal $\sigma: \mathbb R \to \mathbb R$. You want to do the Fourier transform at time $t$, so you window it with some function $\omega : \mathbb R \to \mathbb R$, and you look at their product defined by $p(t) = \sigma(t)\omega(t)$.
Let's assume the windowing function is symmetric around 0 and real. Denote the Fourier transforms of the functions with hats: $\hat{\sigma}$, etc...
We are interested in the power spectral density (PSD) of $\sigma$ (at a certain frequency $f$)
It can be defined as $$ PSD(\sigma, f) = \lim_{T \to \infty} \lvert \hat \sigma(f)\rvert^2 $$ But to avoid doing the computation for the whole range of $\sigma$, we instead do it for $\sigma \cdot \omega$.
What is the relationship between $PSD(\sigma, f)$ and $PSD(\sigma \cdot \omega, f$)?
From the literature it looks like they are just scalar multiples of each other, scaled by a factor independent of $f$, but I cannot find any sources. (nor prove it directly...)
Does anyone have any sources/proofs on the relationships between these two? (the discrete case is fine)
Thanks!