@article{GengelPikovskij2022,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings},
  series = {Physica : D, Nonlinear phenomena},
  volume    = {429},
  journal   = {Physica : D, Nonlinear phenomena},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2021.133070},
  pages     = {9},
  year      = {2022},
  abstract  = {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 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.},
  language  = {en}
}
@article{GengelPikovskij2019,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase demodulation with iterative Hilbert transform embeddings},
  series = {Signal processing},
  volume    = {165},
  journal   = {Signal processing},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0165-1684},
  doi       = {10.1016/j.sigpro.2019.07.005},
  pages     = {115 -- 127},
  year      = {2019},
  abstract  = {We propose an efficient method for demodulation of phase modulated signals via iterated Hilbert transform embeddings. We show that while a usual approach based on one application of the Hilbert transform provides only an approximation to a proper phase, with iterations the accuracy is essentially improved, up to precision limited mainly by discretization effects. We demonstrate that the method is applicable to arbitrarily complex waveforms, and to modulations fast compared to the basic frequency. Furthermore, we develop a perturbative theory applicable to a simple cosine waveform, showing convergence of the technique.},
  language  = {en}
}