Suppose there is a process $x(t)$ with power spectrum $$S_x(\omega)=\lim_{T\to\infty}\frac{1}{T}\left|\int_{-T/2}^{T/2}\mathrm{d}t\,e^{j\omega t}x(t)\right|^2,$$ ignoring the expectation value for simplicity. $x(t)$ is then modified by $$y(t)=\int_{-\infty}^{\infty}\frac{\mathrm{d\omega}}{2\pi}\,e^{-j\omega t}f(\omega)\int_{-\infty}^{\infty}\mathrm{d}t\,e^{j\omega t}x(t)$$ where $f(\omega)$ is an arbitrary function. I am wondering how to find the power spectrum $S_{y}(\omega)$ of $y(t)$. Intuitively it should be $S_{y}(\omega)=\left|f(\omega)\right|^2S_x(\omega)$ but I just can't seem to prove it using the definition of power spectrum I have given; I'm not sure how to include the $T$ from the time average.
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$\begingroup$ See my answer to Wiener Khinchin theorem : struggle in the derivation. Your question is not identical, but it is very closely related. Your question does not involve the probabilistic expectation, and your question involves a "window" ($f(\omega)$) in the frequency domain. $\endgroup$ – Joe Mack Sep 16 '20 at 20:43
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