  Time-dependent processes are often analysed using the power spectral density
(PSD), calculated by taking an appropriate Fourier transform of individual
trajectories and finding the associated ensemble-average. Frequently, the
available experimental data sets are too small for such ensemble averages, and
hence it is of a great conceptual and practical importance to understand to
which extent relevant information can be gained from $S(f,T)$, the PSD of a
single trajectory. Here we focus on the behavior of this random,
realization-dependent variable, parametrized by frequency $f$ and
observation-time $T$, for a broad family of anomalous diffusions---fractional
Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability
density function. We show that $S(f,T)$ is proportional---up to a random
numerical factor whose universal distribution we determine---to the
ensemble-averaged PSD. For subdiffusion ($H<1/2$) we find that $S(f,T)\sim
A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion
$(H>1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for
$H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a
deceptive property that may lead to false conclusions when interpreting
experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on
$T$. Our predictions for both sub- and superdiffusion are confirmed by
experiments in live cells and in agarose hydrogels, and by extensive
simulations.
