Transfer function of delayed noise

Question

Imagine I have a random variable whose fequency spectrum looks like \begin{equation} W(f)=w_0*(2*\pi*f^2) \end{equation} where $w_0$ is some constant. Clearly this frequency spectrum corresponds to some time spectrum, and because the frequency spectrum is not white noise the autocorrelation is not a delta function. Hence there is some correlation between time data, $W(t)$. My question is if I obtain $W(t)$ from the inverse Fourier transform of $W(f)$, what does the frequency spectrum of a new quantity: \begin{equation} Y(t)=W(t)-W(t-\tau) \end{equation} looks like? i.e. is there an analytic form for $Y(f)$?