  The Wiener-Khinchin theorem shows how the power spectrum of a stationary
random signal $I(t)$ is related to its correlation function $\left\langle
I(t)I(t+\tau)\right\rangle$. We consider non-stationary processes with the
widely observed aging correlation function $\langle I(t) I(t+\tau) \rangle \sim
t^\gamma \phi_{\rm EN}(\tau/t)$ and relate it to the sample spectrum. We
formulate two aging Wiener-Khinchin theorems relating the power spectrum to the
time and ensemble averaged correlation functions, discussing briefly the
advantages of each. When the scaling function $\phi_{\rm EN}(x)$ exhibits a
non-analytical behavior in the vicinity of its small argument we obtain aging
$1/f$ type of spectrum. We demonstrate our results with three examples:
blinking quantum dots, single file diffusion and Brownian motion in a
logarithmic potential, showing that our approach is valid for a wide range of
physical mechanisms.
