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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 measurementtime window $[t_w,t_w+t_m]$, namely the observation started after a waitingtime $t_w$, and $t_m$ is the measurement duration. We introduce a generalizedaging 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 behaviorof the scale-invariant correlation function and the aging $1/f^{\beta}$ noise.We illustrate our general results with two-state renewal models with sojourntimes' distributions having a broad tail.

$1/f^{\beta}$ noise for scale-invariant processes: How long you wait matters