# Power spectrum of modified process

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.