  We study the power spectrum which is estimated from a nonstationary signal.
In particular we examine the case when the signal is observed in a measurement
time window $[t_w,t_w+t_m]$, namely the observation started after a waiting
time $t_w$, and $t_m$ is the measurement duration. We introduce a generalized
aging Wiener-Khinchin theorem which relates between the spectrum and the time-
and ensemble-averaged correlation function for arbitrary $t_m$ and $t_w$.
Furthermore we provide a general relation between the non-analytical behavior
of the scale-invariant correlation function and the aging $1/f^{\beta}$ noise.
We illustrate our general results with two-state renewal models with sojourn
times' distributions having a broad tail.
