  A standard approach to study time-dependent stochastic processes is the power
spectral density (PSD), an ensemble-averaged property defined as the Fourier
transform of the autocorrelation function of the process in the asymptotic
limit of long observation times, $T\to\infty$. In many experimental situations
one is able to garner only relatively few stochastic time series of finite $T$,
such that practically neither an ensemble average nor the asymptotic limit
$T\to\infty$ can be achieved. To accommodate for a meaningful analysis of such
finite-length data we here develop the framework of single-trajectory spectral
analysis for one of the standard models of anomalous diffusion, scaled Brownian
motion. We demonstrate that the frequency dependence of the single-trajectory
PSD is exactly the same as for standard Brownian motion, which may lead one to
the erroneous conclusion that the observed motion is normal-diffusive. However,
a distinctive feature is shown to be provided by the explicit dependence on the
measurement time $T$, and this ageing phenomenon can be used to deduce the
anomalous diffusion exponent. We also compare our results to the
single-trajectory PSD behaviour of another standard anomalous diffusion
process, fractional Brownian motion, and work out the commonalities and
differences. Our results represent an important step in establishing
single-trajectory PSDs as an alternative (or complement) to analyses based on
the time-averaged mean squared displacement.
