  In the data analysis of oscillatory systems, methods based on phase
reconstruction are widely used to characterize phase-locking properties and
inferring the phase dynamics. The main component in these studies is an
extraction of the phase from a time series of an oscillating scalar observable.
We discuss a practical procedure of phase reconstruction by virtue of a
recently proposed method termed \textit{iterated Hilbert transform embeddings}.
We exemplify the potential benefits and limitations of the approach by applying
it to a generic observable of a forced Stuart-Landau oscillator. Although in
many cases, unavoidable amplitude modulation of the observed signal does not
allow for perfect phase reconstruction, in cases of strong stability of
oscillations and a high frequency of the forcing, iterated Hilbert transform
embeddings significantly improve the quality of the reconstructed phase. We
also demonstrate that for significant amplitude modulation, iterated embeddings
do not provide any improvement.
